DT2H2X---1.9.5.THM-Ref.s

%------------------------------------------------------------------------------
% File     : DT2H2X---1.9.5
% Problem  : ITP236^3 : 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 : n016.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 09:08:38 AM UTC 2026

% Result   : Theorem 20.23s 7.37s
% Output   : Refutation 20.23s
% Verified : 
% SZS Type : Refutation
%            Derivation depth      :    9
%            Number of leaves      :   10
% Syntax   : Number of formulae    :   50 (  19 unt;   0 typ;   5 def)
%            Number of atoms       :  177 (  42 equ;   0 cnn)
%            Maximal formula atoms :    3 (   3 avg)
%            Number of connectives :  215 (  28   ~;  34   |;   0   &; 144   @)
%                                         (   5 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    7 (   3 avg)
%            Number of types       :  174 ( 173 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1228 (1225 usr;  81 con; 0-5 aty)
%            Number of variables   :   29 (  12   ^;  13   !;   4   ?;  29   :)

% 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,
    complex: $tType ).

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

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

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

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

thf(type_def_16,type,
    produc6271795597528267376eger_o: $tType ).

thf(type_def_17,type,
    produc8923325533196201883nteger: $tType ).

thf(type_def_18,type,
    set_fi4554929511873752355omplex: $tType ).

thf(type_def_19,type,
    filter6041513312241820739omplex: $tType ).

thf(type_def_20,type,
    set_fi7789364187291644575l_real: $tType ).

thf(type_def_21,type,
    filter2146258269922977983l_real: $tType ).

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

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

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

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

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

thf(type_def_27,type,
    filter1242075044329608583at_nat: $tType ).

thf(type_def_28,type,
    set_Pr5085853215250843933omplex: $tType ).

thf(type_def_29,type,
    set_Pr6218003697084177305l_real: $tType ).

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

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

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

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

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

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

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

thf(type_def_37,type,
    set_li5450038453877631591at_nat: $tType ).

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

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

thf(type_def_40,type,
    set_Pr4329608150637261639at_nat: $tType ).

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

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

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

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

thf(type_def_45,type,
    set_se7855581050983116737at_nat: $tType ).

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

thf(type_def_47,type,
    set_Pr8693737435421807431at_nat: $tType ).

thf(type_def_48,type,
    produc1319942482725812455at_nat: $tType ).

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

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

thf(type_def_51,type,
    produc3843707927480180839at_nat: $tType ).

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

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

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

thf(type_def_55,type,
    set_Pr958786334691620121nt_int: $tType ).

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

thf(type_def_57,type,
    list_P6011104703257516679at_nat: $tType ).

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

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

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

thf(type_def_61,type,
    list_P7333126701944960589_nat_o: $tType ).

thf(type_def_62,type,
    list_P3521021558325789923at_int: $tType ).

thf(type_def_63,type,
    list_P5647936690300460905T_VEBT: $tType ).

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

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

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

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

thf(type_def_68,type,
    option4927543243414619207at_nat: $tType ).

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

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

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

thf(type_def_72,type,
    list_C878401137130745250e_term: $tType ).

thf(type_def_73,type,
    set_Co9149898834107579976e_term: $tType ).

thf(type_def_74,type,
    list_i8448526496819171953e_term: $tType ).

thf(type_def_75,type,
    set_in3461395444621081367e_term: $tType ).

thf(type_def_76,type,
    list_P1316552470764441098e_term: $tType ).

thf(type_def_77,type,
    set_Pr7604974323444597168e_term: $tType ).

thf(type_def_78,type,
    list_P1743416141875011707e_term: $tType ).

thf(type_def_79,type,
    set_Pr3642885161833720865e_term: $tType ).

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

thf(type_def_81,type,
    list_P5311841565141990158nteger: $tType ).

thf(type_def_82,type,
    set_Pr8056137968301705908nteger: $tType ).

thf(type_def_83,type,
    list_P8915022641806594461nt_int: $tType ).

thf(type_def_84,type,
    set_Pr1872883991513573699nt_int: $tType ).

thf(type_def_85,type,
    list_P7828571989066258726nteger: $tType ).

thf(type_def_86,type,
    set_Pr1281608226676607948nteger: $tType ).

thf(type_def_87,type,
    list_P651320350408439699nt_int: $tType ).

thf(type_def_88,type,
    set_Pr9222295170931077689nt_int: $tType ).

thf(type_def_89,type,
    list_P4002435161011370285od_o_o: $tType ).

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

thf(type_def_91,type,
    list_P3795440434834930179_o_int: $tType ).

thf(type_def_92,type,
    set_Pr8834758594704517033_o_int: $tType ).

thf(type_def_93,type,
    list_P6285523579766656935_o_nat: $tType ).

thf(type_def_94,type,
    set_Pr2101469702781467981_o_nat: $tType ).

thf(type_def_95,type,
    list_P5232166724548748803o_real: $tType ).

thf(type_def_96,type,
    set_Pr6573716822653411497o_real: $tType ).

thf(type_def_97,type,
    list_P7495141550334521929T_VEBT: $tType ).

thf(type_def_98,type,
    set_Pr7543698050874017315T_VEBT: $tType ).

thf(type_def_99,type,
    list_P5578671422887162913nteger: $tType ).

thf(type_def_100,type,
    set_Pr4811707699266497531nteger: $tType ).

thf(type_def_101,type,
    list_P5707943133018811711nt_int: $tType ).

thf(type_def_102,type,
    list_P6863124054624500543t_real: $tType ).

thf(type_def_103,type,
    set_Pr3538720872664544793t_real: $tType ).

thf(type_def_104,type,
    list_P7524865323317820941T_VEBT: $tType ).

thf(type_def_105,type,
    set_Pr8044002425091019955T_VEBT: $tType ).

thf(type_def_106,type,
    list_P8469869581646625389at_nat: $tType ).

thf(type_def_107,type,
    list_P3595434254542482545real_o: $tType ).

thf(type_def_108,type,
    set_Pr4936984352647145239real_o: $tType ).

thf(type_def_109,type,
    list_P4344331454722006975al_int: $tType ).

thf(type_def_110,type,
    set_Pr1019928272762051225al_int: $tType ).

thf(type_def_111,type,
    list_P6834414599653733731al_nat: $tType ).

thf(type_def_112,type,
    set_Pr3510011417693777981al_nat: $tType ).

thf(type_def_113,type,
    list_P8689742595348180415l_real: $tType ).

thf(type_def_114,type,
    list_P877281246627933069T_VEBT: $tType ).

thf(type_def_115,type,
    set_Pr6019664923565264691T_VEBT: $tType ).

thf(type_def_116,type,
    list_P6254988961118846195et_nat: $tType ).

thf(type_def_117,type,
    set_Pr5488025237498180813et_nat: $tType ).

thf(type_def_118,type,
    list_P5464809261938338413at_nat: $tType ).

thf(type_def_119,type,
    list_P3126845725202233233VEBT_o: $tType ).

thf(type_def_120,type,
    set_Pr3175402225741728619VEBT_o: $tType ).

thf(type_def_121,type,
    list_P4547456442757143711BT_int: $tType ).

thf(type_def_122,type,
    set_Pr5066593544530342725BT_int: $tType ).

thf(type_def_123,type,
    list_P7037539587688870467BT_nat: $tType ).

thf(type_def_124,type,
    set_Pr7556676689462069481BT_nat: $tType ).

thf(type_def_125,type,
    list_P2623026923184700063T_real: $tType ).

thf(type_def_126,type,
    set_Pr7765410600122031685T_real: $tType ).

thf(type_def_127,type,
    list_P7413028617227757229T_VEBT: $tType ).

thf(type_def_128,type,
    set_Pr6192946355708809607T_VEBT: $tType ).

thf(type_def_129,type,
    option6357759511663192854e_term: $tType ).

thf(type_def_130,type,
    produc6241069584506657477e_term: $tType ).

thf(type_def_131,type,
    produc8551481072490612790e_term: $tType ).

thf(type_def_132,type,
    produc8763457246119570046nteger: $tType ).

thf(type_def_133,type,
    produc7773217078559923341nt_int: $tType ).

thf(type_def_134,type,
    produc1908205239877642774nteger: $tType ).

thf(type_def_135,type,
    produc2285326912895808259nt_int: $tType ).

thf(type_def_136,type,
    set_Pr6227168374412355847list_o: $tType ).

thf(type_def_137,type,
    set_Pr765067013931698361st_int: $tType ).

thf(type_def_138,type,
    set_Pr3451248702717554689st_nat: $tType ).

thf(type_def_139,type,
    set_Pr1916528119006554503T_VEBT: $tType ).

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

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

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

thf(type_def_143,type,
    produc2504756804600209347T_VEBT: $tType ).

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

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

thf(type_def_146,type,
    produc8025551001238799321T_VEBT: $tType ).

thf(type_def_147,type,
    produc334124729049499915VEBT_o: $tType ).

thf(type_def_148,type,
    produc4894624898956917775BT_int: $tType ).

thf(type_def_149,type,
    produc9072475918466114483BT_nat: $tType ).

thf(type_def_150,type,
    produc8243902056947475879T_VEBT: $tType ).

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

thf(type_def_152,type,
    produc4953844613479565601on_nat: $tType ).

thf(type_def_153,type,
    produc2233624965454879586on_nat: $tType ).

thf(type_def_154,type,
    produc8306885398267862888on_nat: $tType ).

thf(type_def_155,type,
    produc3447558737645232053on_num: $tType ).

thf(type_def_156,type,
    produc7036089656553540234on_num: $tType ).

thf(type_def_157,type,
    produc1193250871479095198on_num: $tType ).

thf(type_def_158,type,
    produc6121120109295599847at_nat: $tType ).

thf(type_def_159,type,
    produc5491161045314408544at_nat: $tType ).

thf(type_def_160,type,
    produc5542196010084753463at_nat: $tType ).

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

thf(type_def_162,type,
    produc679980390762269497t_real: $tType ).

thf(type_def_163,type,
    produc1531783533982839933T_VEBT: $tType ).

thf(type_def_164,type,
    produc7102631898165422375list_o: $tType ).

thf(type_def_165,type,
    produc1186641810826059865st_int: $tType ).

thf(type_def_166,type,
    produc1828647624359046049st_nat: $tType ).

thf(type_def_167,type,
    produc9211091688327510695T_VEBT: $tType ).

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

thf(type_def_169,type,
    product_prod_num_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_list_nat: $tType ).

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

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

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

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

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

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

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

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

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

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

thf(func_def_141,type,
    set_Code_integer: $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,
    binomial: nat > nat > nat ).

thf(func_def_184,type,
    gbinomial_complex: complex > nat > complex ).

thf(func_def_185,type,
    gbinomial_int: int > nat > int ).

thf(func_def_186,type,
    gbinomial_nat: nat > nat > nat ).

thf(func_def_187,type,
    gbinomial_rat: rat > nat > rat ).

thf(func_def_188,type,
    gbinomial_real: real > nat > real ).

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

thf(func_def_190,type,
    bit_and_not_num: num > num > option_num ).

thf(func_def_191,type,
    bit_concat_bit: nat > int > int > int ).

thf(func_def_192,type,
    bit_ri7919022796975470100ot_int: int > int ).

thf(func_def_193,type,
    bit_ri6519982836138164636nteger: nat > code_integer > code_integer ).

thf(func_def_194,type,
    bit_ri631733984087533419it_int: nat > int > int ).

thf(func_def_195,type,
    bit_se725231765392027082nd_int: int > int > int ).

thf(func_def_196,type,
    bit_se727722235901077358nd_nat: nat > nat > nat ).

thf(func_def_197,type,
    bit_se8568078237143864401it_int: nat > int > int ).

thf(func_def_198,type,
    bit_se8570568707652914677it_nat: nat > nat > nat ).

thf(func_def_199,type,
    bit_se1345352211410354436nteger: nat > code_integer > code_integer ).

thf(func_def_200,type,
    bit_se2159334234014336723it_int: nat > int > int ).

thf(func_def_201,type,
    bit_se2161824704523386999it_nat: nat > nat > nat ).

thf(func_def_202,type,
    bit_se2119862282449309892nteger: nat > code_integer ).

thf(func_def_203,type,
    bit_se2000444600071755411sk_int: nat > int ).

thf(func_def_204,type,
    bit_se2002935070580805687sk_nat: nat > nat ).

thf(func_def_205,type,
    bit_se1409905431419307370or_int: int > int > int ).

thf(func_def_206,type,
    bit_se1412395901928357646or_nat: nat > nat > nat ).

thf(func_def_207,type,
    bit_se545348938243370406it_int: nat > int > int ).

thf(func_def_208,type,
    bit_se547839408752420682it_nat: nat > nat > nat ).

thf(func_def_209,type,
    bit_se2793503036327961859nteger: nat > code_integer > code_integer ).

thf(func_def_210,type,
    bit_se7879613467334960850it_int: nat > int > int ).

thf(func_def_211,type,
    bit_se7882103937844011126it_nat: nat > nat > nat ).

thf(func_def_212,type,
    bit_se1745604003318907178nteger: nat > code_integer > code_integer ).

thf(func_def_213,type,
    bit_se2923211474154528505it_int: nat > int > int ).

thf(func_def_214,type,
    bit_se2925701944663578781it_nat: nat > nat > nat ).

thf(func_def_215,type,
    bit_se8260200283734997820nteger: nat > code_integer > code_integer ).

thf(func_def_216,type,
    bit_se4203085406695923979it_int: nat > int > int ).

thf(func_def_217,type,
    bit_se4205575877204974255it_nat: nat > nat > nat ).

thf(func_def_218,type,
    bit_se3222712562003087583nteger: code_integer > code_integer > code_integer ).

thf(func_def_219,type,
    bit_se6526347334894502574or_int: int > int > int ).

thf(func_def_220,type,
    bit_se6528837805403552850or_nat: nat > nat > nat ).

thf(func_def_221,type,
    bit_se9216721137139052372nteger: code_integer > nat > $o ).

thf(func_def_222,type,
    bit_se1146084159140164899it_int: int > nat > $o ).

thf(func_def_223,type,
    bit_se1148574629649215175it_nat: nat > nat > $o ).

thf(func_def_224,type,
    bit_take_bit_num: nat > num > option_num ).

thf(func_def_225,type,
    bit_un7362597486090784418nd_num: num > num > option_num ).

thf(func_def_226,type,
    bit_un2480387367778600638or_num: num > num > option_num ).

thf(func_def_227,type,
    code_bit_cut_integer: code_integer > produc6271795597528267376eger_o ).

thf(func_def_228,type,
    code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_229,type,
    code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_230,type,
    code_int_of_integer: code_integer > int ).

thf(func_def_231,type,
    code_integer_of_int: int > code_integer ).

thf(func_def_232,type,
    code_nat_of_integer: code_integer > nat ).

thf(func_def_233,type,
    comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex ).

thf(func_def_234,type,
    comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real ).

thf(func_def_235,type,
    complete_Inf_Inf_nat: set_nat > nat ).

thf(func_def_236,type,
    comple7806235888213564991et_nat: set_set_nat > set_nat ).

thf(func_def_237,type,
    complete_Sup_Sup_nat: set_nat > nat ).

thf(func_def_238,type,
    arg: complex > real ).

thf(func_def_239,type,
    cis: real > complex ).

thf(func_def_240,type,
    complex2: real > real > complex ).

thf(func_def_241,type,
    condit2214826472909112428ve_nat: set_nat > $o ).

thf(func_def_242,type,
    differ6690327859849518006l_real: ( real > real ) > filter_real > $o ).

thf(func_def_243,type,
    has_de1759254742604945161l_real: ( real > real ) > ( real > real ) > filter_real > $o ).

thf(func_def_244,type,
    has_fi5821293074295781190e_real: ( real > real ) > real > filter_real > $o ).

thf(func_def_245,type,
    adjust_div: product_prod_int_int > int ).

thf(func_def_246,type,
    divmod_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_247,type,
    eucl_rel_int: int > int > product_prod_int_int > $o ).

thf(func_def_248,type,
    unique6319869463603278526ux_int: product_prod_int_int > $o ).

thf(func_def_249,type,
    unique6322359934112328802ux_nat: product_prod_nat_nat > $o ).

thf(func_def_250,type,
    unique3479559517661332726nteger: num > num > produc8923325533196201883nteger ).

thf(func_def_251,type,
    unique5052692396658037445od_int: num > num > product_prod_int_int ).

thf(func_def_252,type,
    unique5055182867167087721od_nat: num > num > product_prod_nat_nat ).

thf(func_def_253,type,
    unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_254,type,
    unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int ).

thf(func_def_255,type,
    unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_256,type,
    comm_s8582702949713902594nteger: code_integer > nat > code_integer ).

thf(func_def_257,type,
    comm_s2602460028002588243omplex: complex > nat > complex ).

thf(func_def_258,type,
    comm_s4660882817536571857er_int: int > nat > int ).

thf(func_def_259,type,
    comm_s4663373288045622133er_nat: nat > nat > nat ).

thf(func_def_260,type,
    comm_s4028243227959126397er_rat: rat > nat > rat ).

thf(func_def_261,type,
    comm_s7457072308508201937r_real: real > nat > real ).

thf(func_def_262,type,
    semiri3624122377584611663nteger: nat > code_integer ).

thf(func_def_263,type,
    semiri5044797733671781792omplex: nat > complex ).

thf(func_def_264,type,
    semiri1406184849735516958ct_int: nat > int ).

thf(func_def_265,type,
    semiri1408675320244567234ct_nat: nat > nat ).

thf(func_def_266,type,
    semiri773545260158071498ct_rat: nat > rat ).

thf(func_def_267,type,
    semiri2265585572941072030t_real: nat > real ).

thf(func_def_268,type,
    inverse_inverse_rat: rat > rat ).

thf(func_def_269,type,
    inverse_inverse_real: real > real ).

thf(func_def_270,type,
    at_bot_real: filter_real ).

thf(func_def_271,type,
    at_top_nat: filter_nat ).

thf(func_def_272,type,
    at_top_real: filter_real ).

thf(func_def_273,type,
    eventually_nat: ( nat > $o ) > filter_nat > $o ).

thf(func_def_274,type,
    eventu1038000079068216329at_nat: ( product_prod_nat_nat > $o ) > filter1242075044329608583at_nat > $o ).

thf(func_def_275,type,
    eventually_real: ( real > $o ) > filter_real > $o ).

thf(func_def_276,type,
    filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).

thf(func_def_277,type,
    filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).

thf(func_def_278,type,
    filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).

thf(func_def_279,type,
    princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex ).

thf(func_def_280,type,
    princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real ).

thf(func_def_281,type,
    prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).

thf(func_def_282,type,
    finite_card_complex: set_complex > nat ).

thf(func_def_283,type,
    finite_card_list_nat: set_list_nat > nat ).

thf(func_def_284,type,
    finite_card_nat: set_nat > nat ).

thf(func_def_285,type,
    finite_card_set_nat: set_set_nat > nat ).

thf(func_def_286,type,
    finite_finite_o: set_o > $o ).

thf(func_def_287,type,
    finite3207457112153483333omplex: set_complex > $o ).

thf(func_def_288,type,
    finite_finite_int: set_int > $o ).

thf(func_def_289,type,
    finite_finite_list_o: set_list_o > $o ).

thf(func_def_290,type,
    finite8712137658972009173omplex: set_list_complex > $o ).

thf(func_def_291,type,
    finite3922522038869484883st_int: set_list_int > $o ).

thf(func_def_292,type,
    finite8100373058378681591st_nat: set_list_nat > $o ).

thf(func_def_293,type,
    finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o ).

thf(func_def_294,type,
    finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o ).

thf(func_def_295,type,
    finite_finite_nat: set_nat > $o ).

thf(func_def_296,type,
    finite_finite_num: set_num > $o ).

thf(func_def_297,type,
    finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_298,type,
    finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o ).

thf(func_def_299,type,
    finite_finite_rat: set_rat > $o ).

thf(func_def_300,type,
    finite_finite_real: set_real > $o ).

thf(func_def_301,type,
    finite6551019134538273531omplex: set_set_complex > $o ).

thf(func_def_302,type,
    finite6197958912794628473et_int: set_set_int > $o ).

thf(func_def_303,type,
    finite1152437895449049373et_nat: set_set_nat > $o ).

thf(func_def_304,type,
    finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o ).

thf(func_def_305,type,
    finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o ).

thf(func_def_306,type,
    bij_be1856998921033663316omplex: ( complex > complex ) > set_complex > set_complex > $o ).

thf(func_def_307,type,
    bij_betw_nat_complex: ( nat > complex ) > set_nat > set_complex > $o ).

thf(func_def_308,type,
    bij_betw_nat_nat: ( nat > nat ) > set_nat > set_nat > $o ).

thf(func_def_309,type,
    comp_C8797469213163452608nteger: ( ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_310,type,
    comp_C1593894019821074884nteger: ( code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ) > ( code_integer > code_integer ) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_311,type,
    comp_int_nat_int: ( int > nat ) > ( int > int ) > int > nat ).

thf(func_def_312,type,
    comp_nat_nat_nat: ( nat > nat ) > ( nat > nat ) > nat > nat ).

thf(func_def_313,type,
    id_o: $o > $o ).

thf(func_def_314,type,
    id_nat: nat > nat ).

thf(func_def_315,type,
    inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_316,type,
    inj_on_real_real: ( real > real ) > set_real > $o ).

thf(func_def_317,type,
    inj_on_set_nat_nat: ( set_nat > nat ) > set_set_nat > $o ).

thf(func_def_318,type,
    map_fu434086159418415080_int_o: ( int > product_prod_nat_nat ) > ( ( product_prod_nat_nat > $o ) > int > $o ) > ( product_prod_nat_nat > product_prod_nat_nat > $o ) > int > int > $o ).

thf(func_def_319,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_320,type,
    map_fu4826362097070443709at_o_o: ( int > product_prod_nat_nat ) > ( $o > $o ) > ( product_prod_nat_nat > $o ) > int > $o ).

thf(func_def_321,type,
    map_fu2345160673673942751at_nat: ( int > product_prod_nat_nat ) > ( nat > nat ) > ( product_prod_nat_nat > nat ) > int > nat ).

thf(func_def_322,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_323,type,
    map_fu898904425404107465nt_o_o: ( rat > product_prod_int_int ) > ( $o > $o ) > ( product_prod_int_int > $o ) > rat > $o ).

thf(func_def_324,type,
    strict1292158309912662752at_nat: ( nat > nat ) > set_nat > $o ).

thf(func_def_325,type,
    the_in5290026491893676941l_real: set_real > ( real > real ) > real > real ).

thf(func_def_326,type,
    fun_is_measure_int: ( int > nat ) > $o ).

thf(func_def_327,type,
    fun_max_strict: set_Pr4329608150637261639at_nat ).

thf(func_def_328,type,
    fun_max_weak: set_Pr4329608150637261639at_nat ).

thf(func_def_329,type,
    fun_min_strict: set_Pr4329608150637261639at_nat ).

thf(func_def_330,type,
    fun_min_weak: set_Pr4329608150637261639at_nat ).

thf(func_def_331,type,
    fun_pair_leq: set_Pr8693737435421807431at_nat ).

thf(func_def_332,type,
    fun_pair_less: set_Pr8693737435421807431at_nat ).

thf(func_def_333,type,
    fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o ).

thf(func_def_334,type,
    gcd_Gcd_int: set_int > int ).

thf(func_def_335,type,
    gcd_Gcd_nat: set_nat > nat ).

thf(func_def_336,type,
    bezw: nat > nat > product_prod_int_int ).

thf(func_def_337,type,
    bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_338,type,
    gcd_gcd_Code_integer: code_integer > code_integer > code_integer ).

thf(func_def_339,type,
    gcd_gcd_int: int > int > int ).

thf(func_def_340,type,
    gcd_gcd_nat: nat > nat > nat ).

thf(func_def_341,type,
    gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_342,type,
    abs_abs_Code_integer: code_integer > code_integer ).

thf(func_def_343,type,
    abs_abs_complex: complex > complex ).

thf(func_def_344,type,
    abs_abs_int: int > int ).

thf(func_def_345,type,
    abs_abs_rat: rat > rat ).

thf(func_def_346,type,
    abs_abs_real: real > real ).

thf(func_def_347,type,
    minus_minus_int_o: ( int > $o ) > ( int > $o ) > int > $o ).

thf(func_def_348,type,
    minus_1139252259498527702_nat_o: ( list_nat > $o ) > ( list_nat > $o ) > list_nat > $o ).

thf(func_def_349,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(func_def_350,type,
    minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_351,type,
    minus_minus_real_o: ( real > $o ) > ( real > $o ) > real > $o ).

thf(func_def_352,type,
    minus_6910147592129066416_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > set_nat > $o ).

thf(func_def_353,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

thf(func_def_354,type,
    minus_minus_complex: complex > complex > complex ).

thf(func_def_355,type,
    minus_minus_int: int > int > int ).

thf(func_def_356,type,
    minus_minus_nat: nat > nat > nat ).

thf(func_def_357,type,
    minus_minus_rat: rat > rat > rat ).

thf(func_def_358,type,
    minus_minus_real: real > real > real ).

thf(func_def_359,type,
    minus_811609699411566653omplex: set_complex > set_complex > set_complex ).

thf(func_def_360,type,
    minus_minus_set_int: set_int > set_int > set_int ).

thf(func_def_361,type,
    minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat ).

thf(func_def_362,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_363,type,
    minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_364,type,
    minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_365,type,
    minus_minus_set_real: set_real > set_real > set_real ).

thf(func_def_366,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(func_def_367,type,
    one_one_Code_integer: code_integer ).

thf(func_def_368,type,
    one_one_complex: complex ).

thf(func_def_369,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(func_def_370,type,
    one_one_int: int ).

thf(func_def_371,type,
    one_one_nat: nat ).

thf(func_def_372,type,
    one_one_rat: rat ).

thf(func_def_373,type,
    one_one_real: real ).

thf(func_def_374,type,
    plus_p5714425477246183910nteger: code_integer > code_integer > code_integer ).

thf(func_def_375,type,
    plus_plus_complex: complex > complex > complex ).

thf(func_def_376,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_377,type,
    plus_plus_int: int > int > int ).

thf(func_def_378,type,
    plus_plus_nat: nat > nat > nat ).

thf(func_def_379,type,
    plus_plus_num: num > num > num ).

thf(func_def_380,type,
    plus_plus_rat: rat > rat > rat ).

thf(func_def_381,type,
    plus_plus_real: real > real > real ).

thf(func_def_382,type,
    sgn_sgn_Code_integer: code_integer > code_integer ).

thf(func_def_383,type,
    sgn_sgn_complex: complex > complex ).

thf(func_def_384,type,
    sgn_sgn_int: int > int ).

thf(func_def_385,type,
    sgn_sgn_rat: rat > rat ).

thf(func_def_386,type,
    sgn_sgn_real: real > real ).

thf(func_def_387,type,
    times_3573771949741848930nteger: code_integer > code_integer > code_integer ).

thf(func_def_388,type,
    times_times_complex: complex > complex > complex ).

thf(func_def_389,type,
    times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat ).

thf(func_def_390,type,
    times_times_int: int > int > int ).

thf(func_def_391,type,
    times_times_nat: nat > nat > nat ).

thf(func_def_392,type,
    times_times_rat: rat > rat > rat ).

thf(func_def_393,type,
    times_times_real: real > real > real ).

thf(func_def_394,type,
    uminus_uminus_int_o: ( int > $o ) > int > $o ).

thf(func_def_395,type,
    uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).

thf(func_def_396,type,
    uminus_uminus_nat_o: ( nat > $o ) > nat > $o ).

thf(func_def_397,type,
    uminus8676089048583255045_nat_o: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_398,type,
    uminus_uminus_real_o: ( real > $o ) > real > $o ).

thf(func_def_399,type,
    uminus6401447641752708672_nat_o: ( set_nat > $o ) > set_nat > $o ).

thf(func_def_400,type,
    uminus1351360451143612070nteger: code_integer > code_integer ).

thf(func_def_401,type,
    uminus1482373934393186551omplex: complex > complex ).

thf(func_def_402,type,
    uminus_uminus_int: int > int ).

thf(func_def_403,type,
    uminus_uminus_rat: rat > rat ).

thf(func_def_404,type,
    uminus_uminus_real: real > real ).

thf(func_def_405,type,
    uminus1532241313380277803et_int: set_int > set_int ).

thf(func_def_406,type,
    uminus3195874150345416415st_nat: set_list_nat > set_list_nat ).

thf(func_def_407,type,
    uminus5710092332889474511et_nat: set_nat > set_nat ).

thf(func_def_408,type,
    uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_409,type,
    uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_410,type,
    uminus612125837232591019t_real: set_real > set_real ).

thf(func_def_411,type,
    uminus613421341184616069et_nat: set_set_nat > set_set_nat ).

thf(func_def_412,type,
    zero_z3403309356797280102nteger: code_integer ).

thf(func_def_413,type,
    zero_zero_complex: complex ).

thf(func_def_414,type,
    zero_z5237406670263579293d_enat: extended_enat ).

thf(func_def_415,type,
    zero_zero_int: int ).

thf(func_def_416,type,
    zero_zero_nat: nat ).

thf(func_def_417,type,
    zero_zero_rat: rat ).

thf(func_def_418,type,
    zero_zero_real: real ).

thf(func_def_419,type,
    groups8507830703676809646_o_nat: ( $o > nat ) > set_o > nat ).

thf(func_def_420,type,
    groups7754918857620584856omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_421,type,
    groups5690904116761175830ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_422,type,
    groups5693394587270226106ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_423,type,
    groups5058264527183730370ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_424,type,
    groups5808333547571424918x_real: ( complex > real ) > set_complex > real ).

thf(func_def_425,type,
    groups7873554091576472773nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_426,type,
    groups3049146728041665814omplex: ( int > complex ) > set_int > complex ).

thf(func_def_427,type,
    groups4538972089207619220nt_int: ( int > int ) > set_int > int ).

thf(func_def_428,type,
    groups4541462559716669496nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_429,type,
    groups3906332499630173760nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_430,type,
    groups8778361861064173332t_real: ( int > real ) > set_int > real ).

thf(func_def_431,type,
    groups7501900531339628137nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_432,type,
    groups2073611262835488442omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_433,type,
    groups7108830773950497114d_enat: ( nat > extended_enat ) > set_nat > extended_enat ).

thf(func_def_434,type,
    groups3539618377306564664at_int: ( nat > int ) > set_nat > int ).

thf(func_def_435,type,
    groups3542108847815614940at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_436,type,
    groups2906978787729119204at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_437,type,
    groups6591440286371151544t_real: ( nat > real ) > set_nat > real ).

thf(func_def_438,type,
    groups977919841031483927at_nat: ( product_prod_nat_nat > nat ) > set_Pr1261947904930325089at_nat > nat ).

thf(func_def_439,type,
    groups3860910324918113789at_nat: ( produc3843707927480180839at_nat > nat ) > set_Pr4329608150637261639at_nat > nat ).

thf(func_def_440,type,
    groups7713935264441627589nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_441,type,
    groups5754745047067104278omplex: ( real > complex ) > set_real > complex ).

thf(func_def_442,type,
    groups1932886352136224148al_int: ( real > int ) > set_real > int ).

thf(func_def_443,type,
    groups1935376822645274424al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_444,type,
    groups1300246762558778688al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_445,type,
    groups8097168146408367636l_real: ( real > real ) > set_real > real ).

thf(func_def_446,type,
    groups8294997508430121362at_nat: ( set_nat > nat ) > set_set_nat > nat ).

thf(func_def_447,type,
    groups771621172384141258BT_nat: ( vEBT_VEBT > nat ) > set_VEBT_VEBT > nat ).

thf(func_def_448,type,
    groups8682486955453173170nteger: ( complex > code_integer ) > set_complex > code_integer ).

thf(func_def_449,type,
    groups3708469109370488835omplex: ( complex > complex ) > set_complex > complex ).

thf(func_def_450,type,
    groups858564598930262913ex_int: ( complex > int ) > set_complex > int ).

thf(func_def_451,type,
    groups861055069439313189ex_nat: ( complex > nat ) > set_complex > nat ).

thf(func_def_452,type,
    groups225925009352817453ex_rat: ( complex > rat ) > set_complex > rat ).

thf(func_def_453,type,
    groups766887009212190081x_real: ( complex > real ) > set_complex > real ).

thf(func_def_454,type,
    groups3827104343326376752nteger: ( int > code_integer ) > set_int > code_integer ).

thf(func_def_455,type,
    groups7440179247065528705omplex: ( int > complex ) > set_int > complex ).

thf(func_def_456,type,
    groups1705073143266064639nt_int: ( int > int ) > set_int > int ).

thf(func_def_457,type,
    groups1707563613775114915nt_nat: ( int > nat ) > set_int > nat ).

thf(func_def_458,type,
    groups1072433553688619179nt_rat: ( int > rat ) > set_int > rat ).

thf(func_def_459,type,
    groups2316167850115554303t_real: ( int > real ) > set_int > real ).

thf(func_def_460,type,
    groups3455450783089532116nteger: ( nat > code_integer ) > set_nat > code_integer ).

thf(func_def_461,type,
    groups6464643781859351333omplex: ( nat > complex ) > set_nat > complex ).

thf(func_def_462,type,
    groups705719431365010083at_int: ( nat > int ) > set_nat > int ).

thf(func_def_463,type,
    groups708209901874060359at_nat: ( nat > nat ) > set_nat > nat ).

thf(func_def_464,type,
    groups73079841787564623at_rat: ( nat > rat ) > set_nat > rat ).

thf(func_def_465,type,
    groups129246275422532515t_real: ( nat > real ) > set_nat > real ).

thf(func_def_466,type,
    groups6225526099057966256nteger: ( real > code_integer ) > set_real > code_integer ).

thf(func_def_467,type,
    groups713298508707869441omplex: ( real > complex ) > set_real > complex ).

thf(func_def_468,type,
    groups4694064378042380927al_int: ( real > int ) > set_real > int ).

thf(func_def_469,type,
    groups4696554848551431203al_nat: ( real > nat ) > set_real > nat ).

thf(func_def_470,type,
    groups4061424788464935467al_rat: ( real > rat ) > set_real > rat ).

thf(func_def_471,type,
    groups1681761925125756287l_real: ( real > real ) > set_real > real ).

thf(func_def_472,type,
    groups9116527308978886569_o_int: ( $o > int ) > int > list_o > int ).

thf(func_def_473,type,
    groups9119017779487936845_o_nat: ( $o > nat ) > nat > list_o > nat ).

thf(func_def_474,type,
    groups4561878855575611511st_nat: list_nat > nat ).

thf(func_def_475,type,
    the_int: ( int > $o ) > int ).

thf(func_def_476,type,
    the_real: ( real > $o ) > real ).

thf(func_def_477,type,
    if_int_int: $o > ( int > int ) > ( int > int ) > int > int ).

thf(func_def_478,type,
    if_Code_integer: $o > code_integer > code_integer > code_integer ).

thf(func_def_479,type,
    if_complex: $o > complex > complex > complex ).

thf(func_def_480,type,
    if_Extended_enat: $o > extended_enat > extended_enat > extended_enat ).

thf(func_def_481,type,
    if_int: $o > int > int > int ).

thf(func_def_482,type,
    if_list_int: $o > list_int > list_int > list_int ).

thf(func_def_483,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(func_def_484,type,
    if_nat: $o > nat > nat > nat ).

thf(func_def_485,type,
    if_option_num: $o > option_num > option_num > option_num ).

thf(func_def_486,type,
    if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o ).

thf(func_def_487,type,
    if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_488,type,
    if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int ).

thf(func_def_489,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_490,type,
    if_rat: $o > rat > rat > rat ).

thf(func_def_491,type,
    if_real: $o > real > real > real ).

thf(func_def_492,type,
    if_set_int: $o > set_int > set_int > set_int ).

thf(func_def_493,type,
    abs_Integ: product_prod_nat_nat > int ).

thf(func_def_494,type,
    rep_Integ: int > product_prod_nat_nat ).

thf(func_def_495,type,
    int_ge_less_than: int > set_Pr958786334691620121nt_int ).

thf(func_def_496,type,
    int_ge_less_than2: int > set_Pr958786334691620121nt_int ).

thf(func_def_497,type,
    nat2: int > nat ).

thf(func_def_498,type,
    power_int_real: real > int > real ).

thf(func_def_499,type,
    ring_18347121197199848620nteger: int > code_integer ).

thf(func_def_500,type,
    ring_17405671764205052669omplex: int > complex ).

thf(func_def_501,type,
    ring_1_of_int_int: int > int ).

thf(func_def_502,type,
    ring_1_of_int_rat: int > rat ).

thf(func_def_503,type,
    ring_1_of_int_real: int > real ).

thf(func_def_504,type,
    inf_inf_int: int > int > int ).

thf(func_def_505,type,
    inf_inf_nat: nat > nat > nat ).

thf(func_def_506,type,
    inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_507,type,
    semila1623282765462674594er_nat: ( nat > nat > nat ) > nat > ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_508,type,
    sup_sup_int: int > int > int ).

thf(func_def_509,type,
    sup_sup_nat: nat > nat > nat ).

thf(func_def_510,type,
    sup_sup_set_nat: set_nat > set_nat > set_nat ).

thf(func_def_511,type,
    sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_512,type,
    lattic8263393255366662781ax_int: set_int > int ).

thf(func_def_513,type,
    lattic8265883725875713057ax_nat: set_nat > nat ).

thf(func_def_514,type,
    lattic5364784637807008409ex_nat: ( complex > nat ) > set_complex > complex ).

thf(func_def_515,type,
    lattic1922116423962787043ex_num: ( complex > num ) > set_complex > complex ).

thf(func_def_516,type,
    lattic4729654577720512673ex_rat: ( complex > rat ) > set_complex > complex ).

thf(func_def_517,type,
    lattic8794016678065449205x_real: ( complex > real ) > set_complex > complex ).

thf(func_def_518,type,
    lattic5003618458639192673nt_num: ( int > num ) > set_int > int ).

thf(func_def_519,type,
    lattic7811156612396918303nt_rat: ( int > rat ) > set_int > int ).

thf(func_def_520,type,
    lattic2675449441010098035t_real: ( int > real ) > set_int > int ).

thf(func_def_521,type,
    lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).

thf(func_def_522,type,
    lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).

thf(func_def_523,type,
    lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).

thf(func_def_524,type,
    lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).

thf(func_def_525,type,
    lattic1613168225601753569al_num: ( real > num ) > set_real > real ).

thf(func_def_526,type,
    lattic4420706379359479199al_rat: ( real > rat ) > set_real > real ).

thf(func_def_527,type,
    lattic8440615504127631091l_real: ( real > real ) > set_real > real ).

thf(func_def_528,type,
    append_int: list_int > list_int > list_int ).

thf(func_def_529,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(func_def_530,type,
    count_list_o: list_o > $o > nat ).

thf(func_def_531,type,
    count_list_complex: list_complex > complex > nat ).

thf(func_def_532,type,
    count_list_int: list_int > int > nat ).

thf(func_def_533,type,
    count_list_nat: list_nat > nat > nat ).

thf(func_def_534,type,
    count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).

thf(func_def_535,type,
    count_list_real: list_real > real > nat ).

thf(func_def_536,type,
    count_list_set_nat: list_set_nat > set_nat > nat ).

thf(func_def_537,type,
    count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).

thf(func_def_538,type,
    distinct_int: list_int > $o ).

thf(func_def_539,type,
    distinct_nat: list_nat > $o ).

thf(func_def_540,type,
    drop_nat: nat > list_nat > list_nat ).

thf(func_def_541,type,
    enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_542,type,
    enumerate_int: nat > list_int > list_P3521021558325789923at_int ).

thf(func_def_543,type,
    enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_544,type,
    enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_545,type,
    find_o: ( $o > $o ) > list_o > option_o ).

thf(func_def_546,type,
    find_int: ( int > $o ) > list_int > option_int ).

thf(func_def_547,type,
    find_nat: ( nat > $o ) > list_nat > option_nat ).

thf(func_def_548,type,
    find_num: ( num > $o ) > list_num > option_num ).

thf(func_def_549,type,
    find_P8199882355184865565at_nat: ( product_prod_nat_nat > $o ) > list_P6011104703257516679at_nat > option4927543243414619207at_nat ).

thf(func_def_550,type,
    find_real: ( real > $o ) > list_real > option_real ).

thf(func_def_551,type,
    find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).

thf(func_def_552,type,
    find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).

thf(func_def_553,type,
    gen_length_o: nat > list_o > nat ).

thf(func_def_554,type,
    gen_length_int: nat > list_int > nat ).

thf(func_def_555,type,
    gen_length_nat: nat > list_nat > nat ).

thf(func_def_556,type,
    gen_length_VEBT_VEBT: nat > list_VEBT_VEBT > nat ).

thf(func_def_557,type,
    linord2614967742042102400et_nat: set_nat > list_nat ).

thf(func_def_558,type,
    cons_int: int > list_int > list_int ).

thf(func_def_559,type,
    cons_nat: nat > list_nat > list_nat ).

thf(func_def_560,type,
    nil_int: list_int ).

thf(func_def_561,type,
    nil_nat: list_nat ).

thf(func_def_562,type,
    hd_nat: list_nat > nat ).

thf(func_def_563,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(func_def_564,type,
    set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term ).

thf(func_def_565,type,
    set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term ).

thf(func_def_566,type,
    set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term ).

thf(func_def_567,type,
    set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term ).

thf(func_def_568,type,
    set_o2: list_o > set_o ).

thf(func_def_569,type,
    set_complex2: list_complex > set_complex ).

thf(func_def_570,type,
    set_int2: list_int > set_int ).

thf(func_def_571,type,
    set_list_nat2: list_list_nat > set_list_nat ).

thf(func_def_572,type,
    set_nat2: list_nat > set_nat ).

thf(func_def_573,type,
    set_num2: list_num > set_num ).

thf(func_def_574,type,
    set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger ).

thf(func_def_575,type,
    set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int ).

thf(func_def_576,type,
    set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger ).

thf(func_def_577,type,
    set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int ).

thf(func_def_578,type,
    set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).

thf(func_def_579,type,
    set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int ).

thf(func_def_580,type,
    set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat ).

thf(func_def_581,type,
    set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real ).

thf(func_def_582,type,
    set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT ).

thf(func_def_583,type,
    set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger ).

thf(func_def_584,type,
    set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int ).

thf(func_def_585,type,
    set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real ).

thf(func_def_586,type,
    set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).

thf(func_def_587,type,
    set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_588,type,
    set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).

thf(func_def_589,type,
    set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).

thf(func_def_590,type,
    set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).

thf(func_def_591,type,
    set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).

thf(func_def_592,type,
    set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).

thf(func_def_593,type,
    set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).

thf(func_def_594,type,
    set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).

thf(func_def_595,type,
    set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_596,type,
    set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).

thf(func_def_597,type,
    set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).

thf(func_def_598,type,
    set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).

thf(func_def_599,type,
    set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).

thf(func_def_600,type,
    set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).

thf(func_def_601,type,
    set_real2: list_real > set_real ).

thf(func_def_602,type,
    set_set_nat2: list_set_nat > set_set_nat ).

thf(func_def_603,type,
    set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_604,type,
    size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).

thf(func_def_605,type,
    tl_nat: list_nat > list_nat ).

thf(func_def_606,type,
    list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > ( code_integer > option6357759511663192854e_term ) > list_C878401137130745250e_term ).

thf(func_def_607,type,
    list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).

thf(func_def_608,type,
    list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > list_P1316552470764441098e_term ).

thf(func_def_609,type,
    list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term ).

thf(func_def_610,type,
    list_update_o: list_o > nat > $o > list_o ).

thf(func_def_611,type,
    list_update_int: list_int > nat > int > list_int ).

thf(func_def_612,type,
    list_update_nat: list_nat > nat > nat > list_nat ).

thf(func_def_613,type,
    list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger ).

thf(func_def_614,type,
    list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int ).

thf(func_def_615,type,
    list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger ).

thf(func_def_616,type,
    list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int ).

thf(func_def_617,type,
    list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger ).

thf(func_def_618,type,
    list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).

thf(func_def_619,type,
    list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(func_def_620,type,
    list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).

thf(func_def_621,type,
    list_update_real: list_real > nat > real > list_real ).

thf(func_def_622,type,
    list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).

thf(func_def_623,type,
    list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_624,type,
    listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).

thf(func_def_625,type,
    listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).

thf(func_def_626,type,
    listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).

thf(func_def_627,type,
    listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).

thf(func_def_628,type,
    listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).

thf(func_def_629,type,
    nth_o: list_o > nat > $o ).

thf(func_def_630,type,
    nth_int: list_int > nat > int ).

thf(func_def_631,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(func_def_632,type,
    nth_nat: list_nat > nat > nat ).

thf(func_def_633,type,
    nth_num: list_num > nat > num ).

thf(func_def_634,type,
    nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).

thf(func_def_635,type,
    nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).

thf(func_def_636,type,
    nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).

thf(func_def_637,type,
    nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).

thf(func_def_638,type,
    nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).

thf(func_def_639,type,
    nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).

thf(func_def_640,type,
    nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).

thf(func_def_641,type,
    nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).

thf(func_def_642,type,
    nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).

thf(func_def_643,type,
    nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).

thf(func_def_644,type,
    nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).

thf(func_def_645,type,
    nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).

thf(func_def_646,type,
    nth_real: list_real > nat > real ).

thf(func_def_647,type,
    nth_set_nat: list_set_nat > nat > set_nat ).

thf(func_def_648,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_649,type,
    product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_650,type,
    product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_651,type,
    product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(func_def_652,type,
    product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_653,type,
    product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_654,type,
    produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_655,type,
    product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_656,type,
    produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_657,type,
    produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_658,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_659,type,
    remdups_nat: list_nat > list_nat ).

thf(func_def_660,type,
    replicate_o: nat > $o > list_o ).

thf(func_def_661,type,
    replicate_int: nat > int > list_int ).

thf(func_def_662,type,
    replicate_nat: nat > nat > list_nat ).

thf(func_def_663,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

thf(func_def_664,type,
    replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).

thf(func_def_665,type,
    replicate_real: nat > real > list_real ).

thf(func_def_666,type,
    replicate_set_nat: nat > set_nat > list_set_nat ).

thf(func_def_667,type,
    replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_668,type,
    sorted_wrt_int: ( int > int > $o ) > list_int > $o ).

thf(func_def_669,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(func_def_670,type,
    take_nat: nat > list_nat > list_nat ).

thf(func_def_671,type,
    upt: nat > nat > list_nat ).

thf(func_def_672,type,
    upto: int > int > list_int ).

thf(func_def_673,type,
    upto_aux: int > int > list_int > list_int ).

thf(func_def_674,type,
    upto_rel: product_prod_int_int > product_prod_int_int > $o ).

thf(func_def_675,type,
    zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger ).

thf(func_def_676,type,
    zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int ).

thf(func_def_677,type,
    zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger ).

thf(func_def_678,type,
    zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int ).

thf(func_def_679,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

thf(func_def_680,type,
    zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).

thf(func_def_681,type,
    zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).

thf(func_def_682,type,
    zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(func_def_683,type,
    zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

thf(func_def_684,type,
    zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).

thf(func_def_685,type,
    zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).

thf(func_def_686,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

thf(func_def_687,type,
    zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).

thf(func_def_688,type,
    zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(func_def_689,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

thf(func_def_690,type,
    zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).

thf(func_def_691,type,
    zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(func_def_692,type,
    zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(func_def_693,type,
    zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(func_def_694,type,
    zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(func_def_695,type,
    zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(func_def_696,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

thf(func_def_697,type,
    zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(func_def_698,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

thf(func_def_699,type,
    zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).

thf(func_def_700,type,
    zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

thf(func_def_701,type,
    zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(func_def_702,type,
    suc: nat > nat ).

thf(func_def_703,type,
    compow_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_704,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(func_def_705,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(func_def_706,type,
    case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).

thf(func_def_707,type,
    pred: nat > nat ).

thf(func_def_708,type,
    semiring_1_Nats_int: set_int ).

thf(func_def_709,type,
    semiri4939895301339042750nteger: nat > code_integer ).

thf(func_def_710,type,
    semiri8010041392384452111omplex: nat > complex ).

thf(func_def_711,type,
    semiri4216267220026989637d_enat: nat > extended_enat ).

thf(func_def_712,type,
    semiri1314217659103216013at_int: nat > int ).

thf(func_def_713,type,
    semiri1316708129612266289at_nat: nat > nat ).

thf(func_def_714,type,
    semiri681578069525770553at_rat: nat > rat ).

thf(func_def_715,type,
    semiri5074537144036343181t_real: nat > real ).

thf(func_def_716,type,
    semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).

thf(func_def_717,type,
    semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).

thf(func_def_718,type,
    semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).

thf(func_def_719,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(func_def_720,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

thf(func_def_721,type,
    size_size_list_o: list_o > nat ).

thf(func_def_722,type,
    size_s3451745648224563538omplex: list_complex > nat ).

thf(func_def_723,type,
    size_size_list_int: list_int > nat ).

thf(func_def_724,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(func_def_725,type,
    size_size_list_nat: list_nat > nat ).

thf(func_def_726,type,
    size_size_list_num: list_num > nat ).

thf(func_def_727,type,
    size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).

thf(func_def_728,type,
    size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).

thf(func_def_729,type,
    size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).

thf(func_def_730,type,
    size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).

thf(func_def_731,type,
    size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).

thf(func_def_732,type,
    size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).

thf(func_def_733,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(func_def_734,type,
    size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).

thf(func_def_735,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

thf(func_def_736,type,
    size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).

thf(func_def_737,type,
    size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).

thf(func_def_738,type,
    size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).

thf(func_def_739,type,
    size_size_list_real: list_real > nat ).

thf(func_def_740,type,
    size_s3254054031482475050et_nat: list_set_nat > nat ).

thf(func_def_741,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

thf(func_def_742,type,
    size_size_num: num > nat ).

thf(func_def_743,type,
    size_size_option_nat: option_nat > nat ).

thf(func_def_744,type,
    size_size_option_num: option_num > nat ).

thf(func_def_745,type,
    size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).

thf(func_def_746,type,
    size_size_VEBT_VEBT: vEBT_VEBT > nat ).

thf(func_def_747,type,
    nat_list_encode: list_nat > nat ).

thf(func_def_748,type,
    nat_list_encode_rel: list_nat > list_nat > $o ).

thf(func_def_749,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

thf(func_def_750,type,
    nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_751,type,
    nat_prod_encode: product_prod_nat_nat > nat ).

thf(func_def_752,type,
    nat_set_decode: nat > set_nat ).

thf(func_def_753,type,
    nat_set_encode: set_nat > nat ).

thf(func_def_754,type,
    nat_triangle: nat > nat ).

thf(func_def_755,type,
    root: nat > real > real ).

thf(func_def_756,type,
    sqrt: real > real ).

thf(func_def_757,type,
    bitM: num > num ).

thf(func_def_758,type,
    inc: num > num ).

thf(func_def_759,type,
    neg_nu7757733837767384882nteger: code_integer > code_integer ).

thf(func_def_760,type,
    neg_nu6511756317524482435omplex: complex > complex ).

thf(func_def_761,type,
    neg_nu3811975205180677377ec_int: int > int ).

thf(func_def_762,type,
    neg_nu3179335615603231917ec_rat: rat > rat ).

thf(func_def_763,type,
    neg_nu6075765906172075777c_real: real > real ).

thf(func_def_764,type,
    neg_numeral_sub_int: num > num > int ).

thf(func_def_765,type,
    bit0: num > num ).

thf(func_def_766,type,
    bit1: num > num ).

thf(func_def_767,type,
    one: num ).

thf(func_def_768,type,
    case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).

thf(func_def_769,type,
    size_num: num > nat ).

thf(func_def_770,type,
    num_of_nat: nat > num ).

thf(func_def_771,type,
    numera6620942414471956472nteger: num > code_integer ).

thf(func_def_772,type,
    numera6690914467698888265omplex: num > complex ).

thf(func_def_773,type,
    numera1916890842035813515d_enat: num > extended_enat ).

thf(func_def_774,type,
    numeral_numeral_int: num > int ).

thf(func_def_775,type,
    numeral_numeral_nat: num > nat ).

thf(func_def_776,type,
    numeral_numeral_rat: num > rat ).

thf(func_def_777,type,
    numeral_numeral_real: num > real ).

thf(func_def_778,type,
    pred_numeral: num > nat ).

thf(func_def_779,type,
    none_int: option_int ).

thf(func_def_780,type,
    none_nat: option_nat ).

thf(func_def_781,type,
    none_num: option_num ).

thf(func_def_782,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(func_def_783,type,
    none_real: option_real ).

thf(func_def_784,type,
    none_set_nat: option_set_nat ).

thf(func_def_785,type,
    none_VEBT_VEBT: option_VEBT_VEBT ).

thf(func_def_786,type,
    some_o: $o > option_o ).

thf(func_def_787,type,
    some_int: int > option_int ).

thf(func_def_788,type,
    some_nat: nat > option_nat ).

thf(func_def_789,type,
    some_num: num > option_num ).

thf(func_def_790,type,
    some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).

thf(func_def_791,type,
    some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).

thf(func_def_792,type,
    case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).

thf(func_def_793,type,
    case_option_num_num: num > ( num > num ) > option_num > num ).

thf(func_def_794,type,
    case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).

thf(func_def_795,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

thf(func_def_796,type,
    size_option_nat: ( nat > nat ) > option_nat > nat ).

thf(func_def_797,type,
    size_option_num: ( num > nat ) > option_num > nat ).

thf(func_def_798,type,
    size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).

thf(func_def_799,type,
    the_nat: option_nat > nat ).

thf(func_def_800,type,
    the_num: option_num > num ).

thf(func_def_801,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(func_def_802,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(func_def_803,type,
    bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_804,type,
    bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_805,type,
    bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(func_def_806,type,
    bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).

thf(func_def_807,type,
    bot_bot_int_int_o: int > int > $o ).

thf(func_def_808,type,
    bot_bot_int_o: int > $o ).

thf(func_def_809,type,
    bot_bot_list_nat_o: list_nat > $o ).

thf(func_def_810,type,
    bot_bot_nat_nat_o: nat > nat > $o ).

thf(func_def_811,type,
    bot_bot_nat_o: nat > $o ).

thf(func_def_812,type,
    bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).

thf(func_def_813,type,
    bot_bot_real_o: real > $o ).

thf(func_def_814,type,
    bot_bot_set_nat_o: set_nat > $o ).

thf(func_def_815,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(func_def_816,type,
    bot_bot_filter_nat: filter_nat ).

thf(func_def_817,type,
    bot_bot_nat: nat ).

thf(func_def_818,type,
    bot_bot_set_complex: set_complex ).

thf(func_def_819,type,
    bot_bot_set_int: set_int ).

thf(func_def_820,type,
    bot_bot_set_list_nat: set_list_nat ).

thf(func_def_821,type,
    bot_bot_set_nat: set_nat ).

thf(func_def_822,type,
    bot_bot_set_num: set_num ).

thf(func_def_823,type,
    bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).

thf(func_def_824,type,
    bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).

thf(func_def_825,type,
    bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).

thf(func_def_826,type,
    bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).

thf(func_def_827,type,
    bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int ).

thf(func_def_828,type,
    bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat ).

thf(func_def_829,type,
    bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).

thf(func_def_830,type,
    bot_bot_set_rat: set_rat ).

thf(func_def_831,type,
    bot_bot_set_real: set_real ).

thf(func_def_832,type,
    bot_bot_set_set_int: set_set_int ).

thf(func_def_833,type,
    bot_bot_set_set_nat: set_set_nat ).

thf(func_def_834,type,
    bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT ).

thf(func_def_835,type,
    ord_less_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_836,type,
    ord_less_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_837,type,
    ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_838,type,
    ord_less_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_839,type,
    ord_less_set_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(func_def_840,type,
    ord_le6747313008572928689nteger: code_integer > code_integer > $o ).

thf(func_def_841,type,
    ord_le72135733267957522d_enat: extended_enat > extended_enat > $o ).

thf(func_def_842,type,
    ord_less_int: int > int > $o ).

thf(func_def_843,type,
    ord_less_nat: nat > nat > $o ).

thf(func_def_844,type,
    ord_less_num: num > num > $o ).

thf(func_def_845,type,
    ord_less_rat: rat > rat > $o ).

thf(func_def_846,type,
    ord_less_real: real > real > $o ).

thf(func_def_847,type,
    ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_848,type,
    ord_less_set_complex: set_complex > set_complex > $o ).

thf(func_def_849,type,
    ord_less_set_int: set_int > set_int > $o ).

thf(func_def_850,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(func_def_851,type,
    ord_less_set_num: set_num > set_num > $o ).

thf(func_def_852,type,
    ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_853,type,
    ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_854,type,
    ord_less_set_rat: set_rat > set_rat > $o ).

thf(func_def_855,type,
    ord_less_set_real: set_real > set_real > $o ).

thf(func_def_856,type,
    ord_less_set_set_int: set_set_int > set_set_int > $o ).

thf(func_def_857,type,
    ord_less_set_set_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_858,type,
    ord_le3636971675376928563eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_859,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_860,type,
    ord_le4340812435750786203eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > $o ).

thf(func_def_861,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_862,type,
    ord_le6741204236512500942_int_o: ( int > int > $o ) > ( int > int > $o ) > $o ).

thf(func_def_863,type,
    ord_less_eq_int_o: ( int > $o ) > ( int > $o ) > $o ).

thf(func_def_864,type,
    ord_le2646555220125990790_nat_o: ( nat > nat > $o ) > ( nat > nat > $o ) > $o ).

thf(func_def_865,type,
    ord_less_eq_nat_o: ( nat > $o ) > ( nat > $o ) > $o ).

thf(func_def_866,type,
    ord_le704812498762024988_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

thf(func_def_867,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

thf(func_def_868,type,
    ord_le3964352015994296041_nat_o: ( set_nat > $o ) > ( set_nat > $o ) > $o ).

thf(func_def_869,type,
    ord_le3102999989581377725nteger: code_integer > code_integer > $o ).

thf(func_def_870,type,
    ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o ).

thf(func_def_871,type,
    ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o ).

thf(func_def_872,type,
    ord_less_eq_int: int > int > $o ).

thf(func_def_873,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(func_def_874,type,
    ord_less_eq_num: num > num > $o ).

thf(func_def_875,type,
    ord_less_eq_rat: rat > rat > $o ).

thf(func_def_876,type,
    ord_less_eq_real: real > real > $o ).

thf(func_def_877,type,
    ord_less_eq_set_o: set_o > set_o > $o ).

thf(func_def_878,type,
    ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o ).

thf(func_def_879,type,
    ord_le211207098394363844omplex: set_complex > set_complex > $o ).

thf(func_def_880,type,
    ord_less_eq_set_int: set_int > set_int > $o ).

thf(func_def_881,type,
    ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o ).

thf(func_def_882,type,
    ord_less_eq_set_nat: set_nat > set_nat > $o ).

thf(func_def_883,type,
    ord_less_eq_set_num: set_num > set_num > $o ).

thf(func_def_884,type,
    ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_885,type,
    ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_886,type,
    ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_887,type,
    ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_888,type,
    ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_889,type,
    ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_890,type,
    ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_891,type,
    ord_less_eq_set_rat: set_rat > set_rat > $o ).

thf(func_def_892,type,
    ord_less_eq_set_real: set_real > set_real > $o ).

thf(func_def_893,type,
    ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).

thf(func_def_894,type,
    ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o ).

thf(func_def_895,type,
    ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_896,type,
    ord_max_int: int > int > int ).

thf(func_def_897,type,
    ord_max_nat: nat > nat > nat ).

thf(func_def_898,type,
    ord_min_int: int > int > int ).

thf(func_def_899,type,
    ord_min_nat: nat > nat > nat ).

thf(func_def_900,type,
    order_Greatest_nat: ( nat > $o ) > nat ).

thf(func_def_901,type,
    order_mono_nat_nat: ( nat > nat ) > $o ).

thf(func_def_902,type,
    order_5726023648592871131at_nat: ( nat > nat ) > $o ).

thf(func_def_903,type,
    order_7092887310737990675l_real: ( real > real ) > $o ).

thf(func_def_904,type,
    top_top_set_int: set_int ).

thf(func_def_905,type,
    top_top_set_nat: set_nat ).

thf(func_def_906,type,
    top_top_set_real: set_real ).

thf(func_def_907,type,
    power_8256067586552552935nteger: code_integer > nat > code_integer ).

thf(func_def_908,type,
    power_power_complex: complex > nat > complex ).

thf(func_def_909,type,
    power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).

thf(func_def_910,type,
    power_power_int: int > nat > int ).

thf(func_def_911,type,
    power_power_nat: nat > nat > nat ).

thf(func_def_912,type,
    power_power_rat: rat > nat > rat ).

thf(func_def_913,type,
    power_power_real: real > nat > real ).

thf(func_def_914,type,
    produc6137756002093451184nteger: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ).

thf(func_def_915,type,
    produc4305682042979456191nt_int: ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ).

thf(func_def_916,type,
    produc4035269172776083154on_nat: ( nat > nat > $o ) > produc4953844613479565601on_nat > produc2233624965454879586on_nat ).

thf(func_def_917,type,
    produc8929957630744042906on_nat: ( nat > nat > nat ) > produc4953844613479565601on_nat > produc8306885398267862888on_nat ).

thf(func_def_918,type,
    produc3576312749637752826on_num: ( num > num > $o ) > produc3447558737645232053on_num > produc7036089656553540234on_num ).

thf(func_def_919,type,
    produc5778274026573060048on_num: ( num > num > num ) > produc3447558737645232053on_num > produc1193250871479095198on_num ).

thf(func_def_920,type,
    produc8603105652947943368nteger: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ).

thf(func_def_921,type,
    produc5700946648718959541nt_int: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ).

thf(func_def_922,type,
    produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).

thf(func_def_923,type,
    produc2899441246263362727at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > produc6121120109295599847at_nat > produc5542196010084753463at_nat ).

thf(func_def_924,type,
    product_Pair_o_o: $o > $o > product_prod_o_o ).

thf(func_def_925,type,
    product_Pair_o_int: $o > int > product_prod_o_int ).

thf(func_def_926,type,
    product_Pair_o_nat: $o > nat > product_prod_o_nat ).

thf(func_def_927,type,
    product_Pair_o_real: $o > real > product_prod_o_real ).

thf(func_def_928,type,
    produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).

thf(func_def_929,type,
    produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o ).

thf(func_def_930,type,
    produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger ).

thf(func_def_931,type,
    product_Pair_int_int: int > int > product_prod_int_int ).

thf(func_def_932,type,
    produc801115645435158769t_real: int > real > produc679980390762269497t_real ).

thf(func_def_933,type,
    produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT ).

thf(func_def_934,type,
    produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o ).

thf(func_def_935,type,
    produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).

thf(func_def_936,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(func_def_937,type,
    produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT ).

thf(func_def_938,type,
    product_Pair_nat_o: nat > $o > product_prod_nat_o ).

thf(func_def_939,type,
    product_Pair_nat_int: nat > int > product_prod_nat_int ).

thf(func_def_940,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(func_def_941,type,
    product_Pair_nat_num: nat > num > product_prod_nat_num ).

thf(func_def_942,type,
    produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT ).

thf(func_def_943,type,
    product_Pair_num_num: num > num > product_prod_num_num ).

thf(func_def_944,type,
    produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).

thf(func_def_945,type,
    produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num ).

thf(func_def_946,type,
    produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat ).

thf(func_def_947,type,
    produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat ).

thf(func_def_948,type,
    product_Pair_real_o: real > $o > product_prod_real_o ).

thf(func_def_949,type,
    produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).

thf(func_def_950,type,
    produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).

thf(func_def_951,type,
    produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).

thf(func_def_952,type,
    produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT ).

thf(func_def_953,type,
    produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).

thf(func_def_954,type,
    produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat ).

thf(func_def_955,type,
    produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat ).

thf(func_def_956,type,
    produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o ).

thf(func_def_957,type,
    produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int ).

thf(func_def_958,type,
    produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat ).

thf(func_def_959,type,
    produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real ).

thf(func_def_960,type,
    produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT ).

thf(func_def_961,type,
    produc457027306803732586at_nat: set_nat > ( nat > set_nat ) > set_Pr1261947904930325089at_nat ).

thf(func_def_962,type,
    produc6499014454317279255nteger: ( code_integer > code_integer ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_963,type,
    produc127349428274296955eger_o: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc8763457246119570046nteger > $o ).

thf(func_def_964,type,
    produc3906647086178084059nteger: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc8763457246119570046nteger ) > produc8763457246119570046nteger > produc8763457246119570046nteger ).

thf(func_def_965,type,
    produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).

thf(func_def_966,type,
    produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).

thf(func_def_967,type,
    produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).

thf(func_def_968,type,
    produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).

thf(func_def_969,type,
    produc5122537100556696953nt_int: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ) > produc7773217078559923341nt_int > produc7773217078559923341nt_int ).

thf(func_def_970,type,
    produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).

thf(func_def_971,type,
    produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).

thf(func_def_972,type,
    produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).

thf(func_def_973,type,
    produc6512950862096126219nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ) > produc1908205239877642774nteger > produc1908205239877642774nteger ).

thf(func_def_974,type,
    produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).

thf(func_def_975,type,
    produc8492565224438309093nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int ).

thf(func_def_976,type,
    produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).

thf(func_def_977,type,
    produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).

thf(func_def_978,type,
    produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).

thf(func_def_979,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

thf(func_def_980,type,
    produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).

thf(func_def_981,type,
    produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).

thf(func_def_982,type,
    produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).

thf(func_def_983,type,
    produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).

thf(func_def_984,type,
    produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).

thf(func_def_985,type,
    produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).

thf(func_def_986,type,
    produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).

thf(func_def_987,type,
    produc5233655623923918146et_nat: ( int > int > set_set_nat ) > product_prod_int_int > set_set_nat ).

thf(func_def_988,type,
    produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(func_def_989,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_990,type,
    produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_991,type,
    produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).

thf(func_def_992,type,
    produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).

thf(func_def_993,type,
    produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).

thf(func_def_994,type,
    produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).

thf(func_def_995,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(func_def_996,type,
    produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).

thf(func_def_997,type,
    produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).

thf(func_def_998,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

thf(func_def_999,type,
    produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).

thf(func_def_1000,type,
    product_fst_int_int: product_prod_int_int > int ).

thf(func_def_1001,type,
    product_fst_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_1002,type,
    product_snd_int_int: product_prod_int_int > int ).

thf(func_def_1003,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

thf(func_def_1004,type,
    fract: int > int > rat ).

thf(func_def_1005,type,
    frct: product_prod_int_int > rat ).

thf(func_def_1006,type,
    rep_Rat: rat > product_prod_int_int ).

thf(func_def_1007,type,
    field_5140801741446780682s_real: set_real ).

thf(func_def_1008,type,
    normalize: product_prod_int_int > product_prod_int_int ).

thf(func_def_1009,type,
    of_int: int > rat ).

thf(func_def_1010,type,
    positive: rat > $o ).

thf(func_def_1011,type,
    quotient_of: rat > product_prod_int_int ).

thf(func_def_1012,type,
    real_V3694042436643373181omplex: complex > complex > real ).

thf(func_def_1013,type,
    real_V975177566351809787t_real: real > real > real ).

thf(func_def_1014,type,
    real_V1022390504157884413omplex: complex > real ).

thf(func_def_1015,type,
    real_V7735802525324610683m_real: real > real ).

thf(func_def_1016,type,
    real_V4546457046886955230omplex: real > complex ).

thf(func_def_1017,type,
    real_V1803761363581548252l_real: real > real ).

thf(func_def_1018,type,
    field_nat: set_Pr1261947904930325089at_nat > set_nat ).

thf(func_def_1019,type,
    divide6298287555418463151nteger: code_integer > code_integer > code_integer ).

thf(func_def_1020,type,
    divide1717551699836669952omplex: complex > complex > complex ).

thf(func_def_1021,type,
    divide_divide_int: int > int > int ).

thf(func_def_1022,type,
    divide_divide_nat: nat > nat > nat ).

thf(func_def_1023,type,
    divide_divide_rat: rat > rat > rat ).

thf(func_def_1024,type,
    divide_divide_real: real > real > real ).

thf(func_def_1025,type,
    dvd_dvd_Code_integer: code_integer > code_integer > $o ).

thf(func_def_1026,type,
    dvd_dvd_complex: complex > complex > $o ).

thf(func_def_1027,type,
    dvd_dvd_int: int > int > $o ).

thf(func_def_1028,type,
    dvd_dvd_nat: nat > nat > $o ).

thf(func_def_1029,type,
    dvd_dvd_rat: rat > rat > $o ).

thf(func_def_1030,type,
    dvd_dvd_real: real > real > $o ).

thf(func_def_1031,type,
    modulo364778990260209775nteger: code_integer > code_integer > code_integer ).

thf(func_def_1032,type,
    modulo_modulo_int: int > int > int ).

thf(func_def_1033,type,
    modulo_modulo_nat: nat > nat > nat ).

thf(func_def_1034,type,
    zero_n356916108424825756nteger: $o > code_integer ).

thf(func_def_1035,type,
    zero_n1201886186963655149omplex: $o > complex ).

thf(func_def_1036,type,
    zero_n2684676970156552555ol_int: $o > int ).

thf(func_def_1037,type,
    zero_n2687167440665602831ol_nat: $o > nat ).

thf(func_def_1038,type,
    zero_n2052037380579107095ol_rat: $o > rat ).

thf(func_def_1039,type,
    zero_n3304061248610475627l_real: $o > real ).

thf(func_def_1040,type,
    suminf_complex: ( nat > complex ) > complex ).

thf(func_def_1041,type,
    suminf_int: ( nat > int ) > int ).

thf(func_def_1042,type,
    suminf_nat: ( nat > nat ) > nat ).

thf(func_def_1043,type,
    suminf_real: ( nat > real ) > real ).

thf(func_def_1044,type,
    summable_complex: ( nat > complex ) > $o ).

thf(func_def_1045,type,
    summable_int: ( nat > int ) > $o ).

thf(func_def_1046,type,
    summable_nat: ( nat > nat ) > $o ).

thf(func_def_1047,type,
    summable_real: ( nat > real ) > $o ).

thf(func_def_1048,type,
    sums_complex: ( nat > complex ) > complex > $o ).

thf(func_def_1049,type,
    sums_int: ( nat > int ) > int > $o ).

thf(func_def_1050,type,
    sums_nat: ( nat > nat ) > nat > $o ).

thf(func_def_1051,type,
    sums_real: ( nat > real ) > real > $o ).

thf(func_def_1052,type,
    collect_o: ( $o > $o ) > set_o ).

thf(func_def_1053,type,
    collect_Code_integer: ( code_integer > $o ) > set_Code_integer ).

thf(func_def_1054,type,
    collect_complex: ( complex > $o ) > set_complex ).

thf(func_def_1055,type,
    collect_int: ( int > $o ) > set_int ).

thf(func_def_1056,type,
    collect_list_o: ( list_o > $o ) > set_list_o ).

thf(func_def_1057,type,
    collect_list_complex: ( list_complex > $o ) > set_list_complex ).

thf(func_def_1058,type,
    collect_list_int: ( list_int > $o ) > set_list_int ).

thf(func_def_1059,type,
    collect_list_nat: ( list_nat > $o ) > set_list_nat ).

thf(func_def_1060,type,
    collec3343600615725829874at_nat: ( list_P6011104703257516679at_nat > $o ) > set_li5450038453877631591at_nat ).

thf(func_def_1061,type,
    collec5608196760682091941T_VEBT: ( list_VEBT_VEBT > $o ) > set_list_VEBT_VEBT ).

thf(func_def_1062,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(func_def_1063,type,
    collect_num: ( num > $o ) > set_num ).

thf(func_def_1064,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(func_def_1065,type,
    collec213857154873943460nt_int: ( product_prod_int_int > $o ) > set_Pr958786334691620121nt_int ).

thf(func_def_1066,type,
    collec3392354462482085612at_nat: ( product_prod_nat_nat > $o ) > set_Pr1261947904930325089at_nat ).

thf(func_def_1067,type,
    collec3799799289383736868l_real: ( produc2422161461964618553l_real > $o ) > set_Pr6218003697084177305l_real ).

thf(func_def_1068,type,
    collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).

thf(func_def_1069,type,
    collect_rat: ( rat > $o ) > set_rat ).

thf(func_def_1070,type,
    collect_real: ( real > $o ) > set_real ).

thf(func_def_1071,type,
    collect_set_complex: ( set_complex > $o ) > set_set_complex ).

thf(func_def_1072,type,
    collect_set_int: ( set_int > $o ) > set_set_int ).

thf(func_def_1073,type,
    collect_set_nat: ( set_nat > $o ) > set_set_nat ).

thf(func_def_1074,type,
    collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).

thf(func_def_1075,type,
    collect_VEBT_VEBT: ( vEBT_VEBT > $o ) > set_VEBT_VEBT ).

thf(func_def_1076,type,
    pow_nat: set_nat > set_set_nat ).

thf(func_def_1077,type,
    image_int_int: ( int > int ) > set_int > set_int ).

thf(func_def_1078,type,
    image_int_nat: ( int > nat ) > set_int > set_nat ).

thf(func_def_1079,type,
    image_nat_int: ( nat > int ) > set_nat > set_int ).

thf(func_def_1080,type,
    image_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_1081,type,
    image_nat_set_nat: ( nat > set_nat ) > set_nat > set_set_nat ).

thf(func_def_1082,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(func_def_1083,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

thf(func_def_1084,type,
    image_real_real: ( real > real ) > set_real > set_real ).

thf(func_def_1085,type,
    insert_complex: complex > set_complex > set_complex ).

thf(func_def_1086,type,
    insert_int: int > set_int > set_int ).

thf(func_def_1087,type,
    insert_list_nat: list_nat > set_list_nat > set_list_nat ).

thf(func_def_1088,type,
    insert_nat: nat > set_nat > set_nat ).

thf(func_def_1089,type,
    insert_num: num > set_num > set_num ).

thf(func_def_1090,type,
    insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1091,type,
    insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1092,type,
    insert_rat: rat > set_rat > set_rat ).

thf(func_def_1093,type,
    insert_real: real > set_real > set_real ).

thf(func_def_1094,type,
    insert_set_nat: set_nat > set_set_nat > set_set_nat ).

thf(func_def_1095,type,
    insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT ).

thf(func_def_1096,type,
    the_elem_int: set_int > int ).

thf(func_def_1097,type,
    the_elem_nat: set_nat > nat ).

thf(func_def_1098,type,
    the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat ).

thf(func_def_1099,type,
    the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).

thf(func_def_1100,type,
    the_elem_real: set_real > real ).

thf(func_def_1101,type,
    vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).

thf(func_def_1102,type,
    set_fo1517530859248394432omplex: ( nat > complex > complex ) > nat > nat > complex > complex ).

thf(func_def_1103,type,
    set_fo2581907887559384638at_int: ( nat > int > int ) > nat > nat > int > int ).

thf(func_def_1104,type,
    set_fo2584398358068434914at_nat: ( nat > nat > nat ) > nat > nat > nat > nat ).

thf(func_def_1105,type,
    set_fo1949268297981939178at_rat: ( nat > rat > rat ) > nat > nat > rat > rat ).

thf(func_def_1106,type,
    set_fo3111899725591712190t_real: ( nat > real > real ) > nat > nat > real > real ).

thf(func_def_1107,type,
    set_or1266510415728281911st_int: int > int > set_int ).

thf(func_def_1108,type,
    set_or1269000886237332187st_nat: nat > nat > set_nat ).

thf(func_def_1109,type,
    set_or7049704709247886629st_num: num > num > set_num ).

thf(func_def_1110,type,
    set_or633870826150836451st_rat: rat > rat > set_rat ).

thf(func_def_1111,type,
    set_or1222579329274155063t_real: real > real > set_real ).

thf(func_def_1112,type,
    set_or370866239135849197et_int: set_int > set_int > set_set_int ).

thf(func_def_1113,type,
    set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat ).

thf(func_def_1114,type,
    set_or4662586982721622107an_int: int > int > set_int ).

thf(func_def_1115,type,
    set_or4665077453230672383an_nat: nat > nat > set_nat ).

thf(func_def_1116,type,
    set_ord_atLeast_nat: nat > set_nat ).

thf(func_def_1117,type,
    set_ord_atMost_int: int > set_int ).

thf(func_def_1118,type,
    set_ord_atMost_nat: nat > set_nat ).

thf(func_def_1119,type,
    set_ord_atMost_num: num > set_num ).

thf(func_def_1120,type,
    set_ord_atMost_rat: rat > set_rat ).

thf(func_def_1121,type,
    set_ord_atMost_real: real > set_real ).

thf(func_def_1122,type,
    set_or58775011639299419et_int: set_int > set_set_int ).

thf(func_def_1123,type,
    set_or4236626031148496127et_nat: set_nat > set_set_nat ).

thf(func_def_1124,type,
    set_or6656581121297822940st_int: int > int > set_int ).

thf(func_def_1125,type,
    set_or6659071591806873216st_nat: nat > nat > set_nat ).

thf(func_def_1126,type,
    set_or5832277885323065728an_int: int > int > set_int ).

thf(func_def_1127,type,
    set_or5834768355832116004an_nat: nat > nat > set_nat ).

thf(func_def_1128,type,
    set_or1633881224788618240n_real: real > real > set_real ).

thf(func_def_1129,type,
    set_or1210151606488870762an_nat: nat > set_nat ).

thf(func_def_1130,type,
    set_or5849166863359141190n_real: real > set_real ).

thf(func_def_1131,type,
    set_ord_lessThan_int: int > set_int ).

thf(func_def_1132,type,
    set_ord_lessThan_nat: nat > set_nat ).

thf(func_def_1133,type,
    set_ord_lessThan_num: num > set_num ).

thf(func_def_1134,type,
    set_ord_lessThan_rat: rat > set_rat ).

thf(func_def_1135,type,
    set_or5984915006950818249n_real: real > set_real ).

thf(func_def_1136,type,
    set_or890127255671739683et_nat: set_nat > set_set_nat ).

thf(func_def_1137,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

thf(func_def_1138,type,
    topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).

thf(func_def_1139,type,
    topolo4899668324122417113eq_int: ( nat > int ) > $o ).

thf(func_def_1140,type,
    topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).

thf(func_def_1141,type,
    topolo1459490580787246023eq_num: ( nat > num ) > $o ).

thf(func_def_1142,type,
    topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).

thf(func_def_1143,type,
    topolo6980174941875973593q_real: ( nat > real ) > $o ).

thf(func_def_1144,type,
    topolo3100542954746470799et_int: ( nat > set_int ) > $o ).

thf(func_def_1145,type,
    topolo2177554685111907308n_real: real > set_real > filter_real ).

thf(func_def_1146,type,
    topolo2815343760600316023s_real: real > filter_real ).

thf(func_def_1147,type,
    topolo4055970368930404560y_real: ( nat > real ) > $o ).

thf(func_def_1148,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(func_def_1149,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

thf(func_def_1150,type,
    arccos: real > real ).

thf(func_def_1151,type,
    arcosh_real: real > real ).

thf(func_def_1152,type,
    arcsin: real > real ).

thf(func_def_1153,type,
    arctan: real > real ).

thf(func_def_1154,type,
    arsinh_real: real > real ).

thf(func_def_1155,type,
    artanh_real: real > real ).

thf(func_def_1156,type,
    cos_complex: complex > complex ).

thf(func_def_1157,type,
    cos_real: real > real ).

thf(func_def_1158,type,
    cos_coeff: nat > real ).

thf(func_def_1159,type,
    cosh_real: real > real ).

thf(func_def_1160,type,
    cot_real: real > real ).

thf(func_def_1161,type,
    diffs_complex: ( nat > complex ) > nat > complex ).

thf(func_def_1162,type,
    diffs_int: ( nat > int ) > nat > int ).

thf(func_def_1163,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

thf(func_def_1164,type,
    diffs_real: ( nat > real ) > nat > real ).

thf(func_def_1165,type,
    exp_complex: complex > complex ).

thf(func_def_1166,type,
    exp_real: real > real ).

thf(func_def_1167,type,
    ln_ln_real: real > real ).

thf(func_def_1168,type,
    log: real > real > real ).

thf(func_def_1169,type,
    pi: real ).

thf(func_def_1170,type,
    powr_real: real > real > real ).

thf(func_def_1171,type,
    sin_complex: complex > complex ).

thf(func_def_1172,type,
    sin_real: real > real ).

thf(func_def_1173,type,
    sin_coeff: nat > real ).

thf(func_def_1174,type,
    sinh_real: real > real ).

thf(func_def_1175,type,
    tan_complex: complex > complex ).

thf(func_def_1176,type,
    tan_real: real > real ).

thf(func_def_1177,type,
    tanh_complex: complex > complex ).

thf(func_def_1178,type,
    tanh_real: real > real ).

thf(func_def_1179,type,
    transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1180,type,
    transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).

thf(func_def_1181,type,
    vEBT_Leaf: $o > $o > vEBT_VEBT ).

thf(func_def_1182,type,
    vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1183,type,
    vEBT_size_VEBT: vEBT_VEBT > nat ).

thf(func_def_1184,type,
    vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).

thf(func_def_1185,type,
    vEBT_VEBT_high: nat > nat > nat ).

thf(func_def_1186,type,
    vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).

thf(func_def_1187,type,
    vEBT_VEBT_low: nat > nat > nat ).

thf(func_def_1188,type,
    vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).

thf(func_def_1189,type,
    vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1190,type,
    vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).

thf(func_def_1191,type,
    vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1192,type,
    vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).

thf(func_def_1193,type,
    vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1194,type,
    vEBT_invar_vebt: vEBT_VEBT > nat > $o ).

thf(func_def_1195,type,
    vEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1196,type,
    vEBT_vebt_buildup: nat > vEBT_VEBT ).

thf(func_def_1197,type,
    vEBT_v4011308405150292612up_rel: nat > nat > $o ).

thf(func_def_1198,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

thf(func_def_1199,type,
    vEBT_VEBT_minNull: vEBT_VEBT > $o ).

thf(func_def_1200,type,
    vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1201,type,
    vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).

thf(func_def_1202,type,
    vEBT_vebt_member: vEBT_VEBT > nat > $o ).

thf(func_def_1203,type,
    vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).

thf(func_def_1204,type,
    vEBT_VEBT_add: option_nat > option_nat > option_nat ).

thf(func_def_1205,type,
    vEBT_VEBT_greater: option_nat > option_nat > $o ).

thf(func_def_1206,type,
    vEBT_VEBT_less: option_nat > option_nat > $o ).

thf(func_def_1207,type,
    vEBT_VEBT_lesseq: option_nat > option_nat > $o ).

thf(func_def_1208,type,
    vEBT_VEBT_max_in_set: set_nat > nat > $o ).

thf(func_def_1209,type,
    vEBT_VEBT_min_in_set: set_nat > nat > $o ).

thf(func_def_1210,type,
    vEBT_VEBT_mul: option_nat > option_nat > option_nat ).

thf(func_def_1211,type,
    vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).

thf(func_def_1212,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(func_def_1213,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_1214,type,
    vEBT_VEBT_power: option_nat > option_nat > option_nat ).

thf(func_def_1215,type,
    vEBT_vebt_maxt: vEBT_VEBT > option_nat ).

thf(func_def_1216,type,
    vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1217,type,
    vEBT_vebt_mint: vEBT_VEBT > option_nat ).

thf(func_def_1218,type,
    vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).

thf(func_def_1219,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

thf(func_def_1220,type,
    accp_nat: ( nat > nat > $o ) > nat > $o ).

thf(func_def_1221,type,
    accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).

thf(func_def_1222,type,
    accp_P4275260045618599050at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > $o ).

thf(func_def_1223,type,
    accp_P2887432264394892906BT_nat: ( produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ) > produc9072475918466114483BT_nat > $o ).

thf(func_def_1224,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

thf(func_def_1225,type,
    max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1226,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(func_def_1227,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(func_def_1228,type,
    wf_int: set_Pr958786334691620121nt_int > $o ).

thf(func_def_1229,type,
    wf_nat: set_Pr1261947904930325089at_nat > $o ).

thf(func_def_1230,type,
    fChoice_real: ( real > $o ) > real ).

thf(func_def_1231,type,
    member1535805642427569193e_term: ( code_integer > option6357759511663192854e_term ) > set_Co9149898834107579976e_term > $o ).

thf(func_def_1232,type,
    member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).

thf(func_def_1233,type,
    member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).

thf(func_def_1234,type,
    member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).

thf(func_def_1235,type,
    member_o: $o > set_o > $o ).

thf(func_def_1236,type,
    member_complex: complex > set_complex > $o ).

thf(func_def_1237,type,
    member_int: int > set_int > $o ).

thf(func_def_1238,type,
    member_list_o: list_o > set_list_o > $o ).

thf(func_def_1239,type,
    member_list_int: list_int > set_list_int > $o ).

thf(func_def_1240,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(func_def_1241,type,
    member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o ).

thf(func_def_1242,type,
    member_nat: nat > set_nat > $o ).

thf(func_def_1243,type,
    member_num: num > set_num > $o ).

thf(func_def_1244,type,
    member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).

thf(func_def_1245,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(func_def_1246,type,
    member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).

thf(func_def_1247,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(func_def_1248,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(func_def_1249,type,
    member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).

thf(func_def_1250,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(func_def_1251,type,
    member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).

thf(func_def_1252,type,
    member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).

thf(func_def_1253,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).

thf(func_def_1254,type,
    member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).

thf(func_def_1255,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(func_def_1256,type,
    member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).

thf(func_def_1257,type,
    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).

thf(func_def_1258,type,
    member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).

thf(func_def_1259,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(func_def_1260,type,
    member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).

thf(func_def_1261,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(func_def_1262,type,
    member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).

thf(func_def_1263,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).

thf(func_def_1264,type,
    member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).

thf(func_def_1265,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(func_def_1266,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(func_def_1267,type,
    member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).

thf(func_def_1268,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(func_def_1269,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(func_def_1270,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(func_def_1271,type,
    member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).

thf(func_def_1272,type,
    member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).

thf(func_def_1273,type,
    member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).

thf(func_def_1274,type,
    member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).

thf(func_def_1275,type,
    member_rat: rat > set_rat > $o ).

thf(func_def_1276,type,
    member_real: real > set_real > $o ).

thf(func_def_1277,type,
    member_set_int: set_int > set_set_int > $o ).

thf(func_def_1278,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(func_def_1279,type,
    member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).

thf(func_def_1280,type,
    deg: nat ).

thf(func_def_1281,type,
    m: nat ).

thf(func_def_1282,type,
    ma: nat ).

thf(func_def_1283,type,
    maxs: nat ).

thf(func_def_1284,type,
    mi: nat ).

thf(func_def_1285,type,
    n: nat ).

thf(func_def_1286,type,
    summary: vEBT_VEBT ).

thf(func_def_1288,type,
    treeList: list_VEBT_VEBT ).

thf(func_def_1307,type,
    sP0: ( nat > vEBT_VEBT > $o ) > nat > $o ).

thf(func_def_1308,type,
    sP1: nat > ( nat > vEBT_VEBT > $o ) > $o ).

thf(func_def_1309,type,
    sP2: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1310,type,
    sP3: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1311,type,
    sP4: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1312,type,
    sP5: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1313,type,
    sP6: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1314,type,
    sP7: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_1315,type,
    sP8: nat > vEBT_VEBT > $o ).

thf(func_def_1316,type,
    sP9: vEBT_VEBT > nat > $o ).

thf(func_def_1317,type,
    sP10: vEBT_VEBT > nat > $o ).

thf(func_def_1318,type,
    sP11: nat > vEBT_VEBT > $o ).

thf(func_def_1319,type,
    sP12: nat > list_VEBT_VEBT > $o ).

thf(func_def_1320,type,
    sP13: nat > $o ).

thf(func_def_1321,type,
    sP14: list_VEBT_VEBT > $o ).

thf(func_def_1322,type,
    sP15: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1323,type,
    sP16: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1324,type,
    sP17: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1325,type,
    sP18: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1326,type,
    sP19: nat > nat > list_VEBT_VEBT > $o ).

thf(func_def_1327,type,
    sP20: nat > vEBT_VEBT > list_VEBT_VEBT > $o ).

thf(func_def_1328,type,
    sP21: nat > list_VEBT_VEBT > $o ).

thf(func_def_1329,type,
    sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o ).

thf(func_def_1330,type,
    sP23: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1331,type,
    sP24: nat > nat > list_VEBT_VEBT > nat > nat > $o ).

thf(func_def_1332,type,
    sP25: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1333,type,
    sP26: list_VEBT_VEBT > $o ).

thf(func_def_1334,type,
    sP27: nat > list_VEBT_VEBT > $o ).

thf(func_def_1335,type,
    sP28: nat > list_VEBT_VEBT > nat > nat > nat > $o ).

thf(func_def_1336,type,
    sP29: vEBT_VEBT > list_VEBT_VEBT > nat > $o ).

thf(func_def_1337,type,
    sP30: list_VEBT_VEBT > nat > $o ).

thf(func_def_1338,type,
    sK31: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1339,type,
    sK32: list_VEBT_VEBT > ( vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1340,type,
    sK33: nat > ( nat > vEBT_VEBT > $o ) > nat ).

thf(func_def_1341,type,
    sK34: nat > ( nat > vEBT_VEBT > $o ) > vEBT_VEBT ).

thf(func_def_1342,type,
    sK35: list_VEBT_VEBT > ( nat > vEBT_VEBT > $o ) > nat > nat ).

thf(func_def_1343,type,
    sK36: ( nat > vEBT_VEBT > $o ) > nat > list_VEBT_VEBT ).

thf(func_def_1344,type,
    sK37: nat > vEBT_VEBT > nat ).

thf(func_def_1345,type,
    sK38: nat > vEBT_VEBT > nat ).

thf(func_def_1346,type,
    sK39: nat > vEBT_VEBT > nat ).

thf(func_def_1347,type,
    sK40: nat > vEBT_VEBT > nat ).

thf(func_def_1348,type,
    sK41: nat > vEBT_VEBT > nat ).

thf(func_def_1349,type,
    sK42: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1350,type,
    sK43: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1351,type,
    sK44: vEBT_VEBT > nat > nat ).

thf(func_def_1352,type,
    sK45: vEBT_VEBT > nat > nat ).

thf(func_def_1353,type,
    sK46: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1354,type,
    sK47: vEBT_VEBT > nat > nat ).

thf(func_def_1355,type,
    sK48: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1356,type,
    sK49: vEBT_VEBT > nat > nat ).

thf(func_def_1357,type,
    sK50: vEBT_VEBT > nat > nat ).

thf(func_def_1358,type,
    sK51: vEBT_VEBT > nat > nat ).

thf(func_def_1359,type,
    sK52: vEBT_VEBT > nat > nat ).

thf(func_def_1360,type,
    sK53: vEBT_VEBT > nat > list_VEBT_VEBT ).

thf(func_def_1361,type,
    sK54: vEBT_VEBT > nat > nat ).

thf(func_def_1362,type,
    sK55: vEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1363,type,
    sK56: nat > vEBT_VEBT > nat ).

thf(func_def_1364,type,
    sK57: nat > vEBT_VEBT > nat ).

thf(func_def_1365,type,
    sK58: nat > vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1366,type,
    sK59: nat > vEBT_VEBT > nat ).

thf(func_def_1367,type,
    sK60: nat > vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1368,type,
    sK61: nat > list_VEBT_VEBT > nat ).

thf(func_def_1369,type,
    sK62: nat > list_VEBT_VEBT > nat ).

thf(func_def_1370,type,
    sK63: vEBT_VEBT > $o ).

thf(func_def_1371,type,
    sK64: vEBT_VEBT > $o ).

thf(func_def_1372,type,
    sK65: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1373,type,
    sK66: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1374,type,
    sK67: list_VEBT_VEBT > nat ).

thf(func_def_1375,type,
    sK68: vEBT_VEBT > nat ).

thf(func_def_1376,type,
    sK69: nat ).

thf(func_def_1377,type,
    sK70: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1378,type,
    sK71: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > nat ).

thf(func_def_1379,type,
    sK72: vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1380,type,
    sK73: list_VEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1381,type,
    sK74: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1382,type,
    sK75: nat > vEBT_VEBT > list_VEBT_VEBT > nat ).

thf(func_def_1383,type,
    sK76: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1384,type,
    sK77: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1385,type,
    sK78: list_VEBT_VEBT > nat > nat ).

thf(func_def_1386,type,
    sK79: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1387,type,
    sK80: list_VEBT_VEBT > nat ).

thf(func_def_1388,type,
    sK81: ( list_VEBT_VEBT > $o ) > list_VEBT_VEBT ).

thf(func_def_1389,type,
    sK82: vEBT_VEBT > $o ).

thf(func_def_1390,type,
    sK83: vEBT_VEBT > $o ).

thf(func_def_1391,type,
    sK84: vEBT_VEBT > nat ).

thf(func_def_1392,type,
    sK85: vEBT_VEBT > list_VEBT_VEBT ).

thf(func_def_1393,type,
    sK86: vEBT_VEBT > option4927543243414619207at_nat ).

thf(func_def_1394,type,
    sK87: vEBT_VEBT > vEBT_VEBT ).

thf(func_def_1395,type,
    sK88: vEBT_VEBT > list_VEBT_VEBT > nat > nat ).

thf(func_def_1396,type,
    sK89: nat > list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1397,type,
    sK90: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1398,type,
    sK91: list_VEBT_VEBT > nat ).

thf(func_def_1399,type,
    sK92: list_VEBT_VEBT > nat > nat ).

thf(func_def_1400,type,
    sK93: nat > nat > list_VEBT_VEBT > nat > nat > nat ).

thf(func_def_1401,type,
    sK94: list_VEBT_VEBT > nat > nat > nat > nat > nat ).

thf(func_def_1402,type,
    sK95: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > nat ).

thf(func_def_1403,type,
    sK96: nat > nat ).

thf(func_def_1404,type,
    sK97: vEBT_VEBT > nat ).

thf(func_def_1405,type,
    sK98: list_VEBT_VEBT > nat > vEBT_VEBT ).

thf(func_def_1406,type,
    sK99: vEBT_VEBT > nat ).

thf(func_def_1407,type,
    sK100: list_VEBT_VEBT > vEBT_VEBT ).

thf(func_def_1408,type,
    sK101: list_VEBT_VEBT > nat ).

thf(func_def_1409,type,
    sK102: nat > list_VEBT_VEBT ).

thf(func_def_1411,type,
    ph104: 
      !>[X0: $tType] : X0 ).

thf(f24734,plain,
    $false,
    inference(avatar_sat_refutation,[status(thm)],[f24679,f24683,f24701,f24718,f24730,f24733]) ).

thf(f24733,plain,
    ( ~ spl103_1
    | ~ spl103_8 ),
    inference(avatar_contradiction_clause,[status(thm)],[f24732]) ).

thf(f24732,plain,
    ( $false
    | ~ spl103_1
    | ~ spl103_8 ),
    inference(trivial_inequality_removal,[status(thm)],[f24731]) ).

thf(f24731,plain,
    ( ( $false = $true )
    | ~ spl103_1
    | ~ spl103_8 ),
    inference(forward_demodulation,[status(thm)],[f24700,f24675]) ).

thf(f24675,plain,
    ( ! [X0: nat] :
        ( $false
        = ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) )
    | ~ spl103_1 ),
    inference(avatar_component_clause,[status(thm)],[f24674]) ).

thf(f24674,definition,
    ( spl103_1
  <=> ! [X0: nat] :
        ( $false
        = ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) ) ),
    introduced(definition,[new_symbols(naming,[spl103_1])],[avatar_definition]) ).

thf(f24700,plain,
    ( ( ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true )
    | ~ spl103_8 ),
    inference(avatar_component_clause,[status(thm)],[f24699]) ).

thf(f24699,definition,
    ( spl103_8
  <=> ( ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true ) ),
    introduced(definition,[new_symbols(naming,[spl103_8])],[avatar_definition]) ).

thf(f24730,plain,
    ( ~ spl103_3
    | ~ spl103_7 ),
    inference(avatar_contradiction_clause,[status(thm)],[f24729]) ).

thf(f24729,plain,
    ( $false
    | ~ spl103_3
    | ~ spl103_7 ),
    inference(trivial_inequality_removal,[status(thm)],[f24728]) ).

thf(f24728,plain,
    ( ( $false = $true )
    | ~ spl103_3
    | ~ spl103_7 ),
    inference(backward_demodulation,[status(thm)],[f24697,f24682]) ).

thf(f24682,plain,
    ( ! [X0: nat] :
        ( $false
        = ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) )
    | ~ spl103_3 ),
    inference(avatar_component_clause,[status(thm)],[f24681]) ).

thf(f24681,definition,
    ( spl103_3
  <=> ! [X0: nat] :
        ( $false
        = ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) ) ),
    introduced(definition,[new_symbols(naming,[spl103_3])],[avatar_definition]) ).

thf(f24697,plain,
    ( ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true )
    | ~ spl103_7 ),
    inference(avatar_component_clause,[status(thm)],[f24696]) ).

thf(f24696,definition,
    ( spl103_7
  <=> ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true ) ),
    introduced(definition,[new_symbols(naming,[spl103_7])],[avatar_definition]) ).

thf(f24718,plain,
    ~ spl103_2,
    inference(avatar_contradiction_clause,[status(thm)],[f24717]) ).

thf(f24717,plain,
    ( $false
    | ~ spl103_2 ),
    inference(subsumption_resolution,[status(thm)],[f24678,f24423]) ).

thf(f24423,plain,
    thesis != $true,
    inference(cnf_transformation,[status(thm)],[f24024]) ).

thf(f24024,plain,
    thesis != $true,
    inference(flattening,[status(thm)],[f21580]) ).

thf(f21580,plain,
    thesis != $true,
    inference(fool_elimination,[status(thm)],[f21579]) ).

thf(f21579,plain,
    ~ thesis,
    inference(rectify,[status(thm)],[f9832]) ).

thf(f9832,negated_conjecture,
    ~ thesis,
    inference(negated_conjecture,[status(cth)],[f9831]) ).

thf(f9831,conjecture,
    thesis,
    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',conj_1) ).

thf(f24678,plain,
    ( ( thesis = $true )
    | ~ spl103_2 ),
    inference(avatar_component_clause,[status(thm)],[f24677]) ).

thf(f24677,definition,
    ( spl103_2
  <=> ( thesis = $true ) ),
    introduced(definition,[new_symbols(naming,[spl103_2])],[avatar_definition]) ).

thf(f24701,plain,
    ( spl103_7
    | spl103_8 ),
    inference(avatar_split_clause,[status(thm)],[f24552,f24699,f24696]) ).

thf(f24552,plain,
    ( ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true )
    | ( ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true ) ),
    inference(binary_proxy_clausification,[status(thm)],[f24551]) ).

thf(f24551,plain,
    ( ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      | ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 ) )
    = $true ),
    inference(beta_eta_normalization,[status(thm)],[f24458]) ).

thf(f24458,plain,
    ( ( ^ [Y0: vEBT_VEBT,Y1: nat] :
          ( ( vEBT_V5719532721284313246member @ Y0 @ Y1 )
          | ( vEBT_VEBT_membermima @ Y0 @ Y1 ) )
      @ ( nth_VEBT_VEBT @ treeList @ maxs )
      @ sK69 )
    = $true ),
    inference(definition_unfolding,[status(thm)],[f24351,f24401]) ).

thf(f24401,plain,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [Y0: vEBT_VEBT,Y1: nat] :
          ( ( vEBT_V5719532721284313246member @ Y0 @ Y1 )
          | ( vEBT_VEBT_membermima @ Y0 @ Y1 ) ) ) ),
    inference(cnf_transformation,[status(thm)],[f9895]) ).

thf(f9895,plain,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [Y0: vEBT_VEBT,Y1: nat] :
          ( ( vEBT_V5719532721284313246member @ Y0 @ Y1 )
          | ( vEBT_VEBT_membermima @ Y0 @ Y1 ) ) ) ),
    inference(fool_elimination,[status(thm)],[f9894]) ).

thf(f9894,plain,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [X0: vEBT_VEBT,X1: nat] :
          ( ( vEBT_VEBT_membermima @ X0 @ X1 )
          | ( vEBT_V5719532721284313246member @ X0 @ X1 ) ) ) ),
    inference(rectify,[status(thm)],[f4]) ).

thf(f4,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [X2: vEBT_VEBT,X3: nat] :
          ( ( vEBT_VEBT_membermima @ X2 @ X3 )
          | ( vEBT_V5719532721284313246member @ X2 @ X3 ) ) ) ),
    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',fact_3_both__member__options__def) ).

thf(f24351,plain,
    ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
    = $true ),
    inference(cnf_transformation,[status(thm)],[f24174]) ).

thf(f24174,plain,
    ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
    = $true ),
    inference(skolemisation,[status(esa),new_symbols(skolem,[sK69])],[f16117,f24173]) ).

thf(f24173,plain,
    ( ? [X0: nat] :
        ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
        = $true )
   => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ sK69 )
      = $true ) ),
    introduced(definition,[],[choice_axiom]) ).

thf(f16117,plain,
    ? [X0: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
      = $true ),
    inference(fool_elimination,[status(thm)],[f16116]) ).

thf(f16116,plain,
    ? [X0: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ),
    inference(rectify,[status(thm)],[f1]) ).

thf(f1,axiom,
    ? [X0: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ),
    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062) ).

thf(f24683,plain,
    ( spl103_3
    | spl103_2 ),
    inference(avatar_split_clause,[status(thm)],[f24637,f24677,f24681]) ).

thf(f24637,plain,
    ! [X0: nat] :
      ( ( $false
        = ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) )
      | ( thesis = $true ) ),
    inference(binary_proxy_clausification,[status(thm)],[f24635]) ).

thf(f24635,plain,
    ! [X0: nat] :
      ( ( ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
          | ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) )
       != $true )
      | ( thesis = $true ) ),
    inference(beta_eta_normalization,[status(thm)],[f24462]) ).

thf(f24462,plain,
    ! [X0: nat] :
      ( ( thesis = $true )
      | ( ( ^ [Y0: vEBT_VEBT,Y1: nat] :
              ( ( vEBT_V5719532721284313246member @ Y0 @ Y1 )
              | ( vEBT_VEBT_membermima @ Y0 @ Y1 ) )
          @ ( nth_VEBT_VEBT @ treeList @ maxs )
          @ X0 )
       != $true ) ),
    inference(definition_unfolding,[status(thm)],[f24370,f24401]) ).

thf(f24370,plain,
    ! [X0: nat] :
      ( ( thesis = $true )
      | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
       != $true ) ),
    inference(cnf_transformation,[status(thm)],[f24047]) ).

thf(f24047,plain,
    ! [X0: nat] :
      ( ( thesis = $true )
      | ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
       != $true ) ),
    inference(ennf_transformation,[status(thm)],[f15842]) ).

thf(f15842,plain,
    ! [X0: nat] :
      ( ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
        = $true )
     => ( thesis = $true ) ),
    inference(fool_elimination,[status(thm)],[f15841]) ).

thf(f15841,plain,
    ! [X0: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 )
     => thesis ),
    inference(rectify,[status(thm)],[f9830]) ).

thf(f9830,axiom,
    ! [X4: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X4 )
     => thesis ),
    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',conj_0) ).

thf(f24679,plain,
    ( spl103_1
    | spl103_2 ),
    inference(avatar_split_clause,[status(thm)],[f24636,f24677,f24674]) ).

thf(f24636,plain,
    ! [X0: nat] :
      ( ( thesis = $true )
      | ( $false
        = ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X0 ) ) ),
    inference(binary_proxy_clausification,[status(thm)],[f24635]) ).

%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11  % Problem  : ITP236^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.12/0.13  % Command  : /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.15/0.34  % Computer : n016.cluster.edu
% 0.15/0.34  % Model    : x86_64 x86_64
% 0.15/0.34  % CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.15/0.34  % Memory   : 8042.1875MB
% 0.15/0.34  % OS       : Linux 3.10.0-693.el7.x86_64
% 0.15/0.34  % CPULimit : 300
% 0.15/0.34  % WCLimit  : 300
% 0.15/0.34  % DateTime : Thu Feb 26 16:00:51 EST 2026
% 0.15/0.34  % CPUTime  : 
% 0.15/0.34  Running /export/starexec/sandbox/solver/bin/run_DT2H2X /export/starexec/sandbox/benchmark/theBenchmark.p 300
% 0.29/0.46  ---- Original DTF file ---
% 0.29/0.46  thf(spec,logic,$$dhol).
% 0.29/0.46  %------------------------------------------------------------------------------
% 0.29/0.46  % File     : ITP236^3 : TPTP v9.2.1. Bugfixed v9.2.1.
% 0.29/0.46  % Domain   : Interactive Theorem Proving
% 0.29/0.46  % Problem  : Sledgehammer problem VEBT_MinMax 00214_010433
% 0.29/0.46  % Version  : [Des22] axioms.
% 0.29/0.46  % English  :
% 0.29/0.46  
% 0.29/0.46  % Refs     : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.29/0.46  %          : [Des22] Desharnais (2022), Email to Geoff Sutcliffe
% 0.29/0.46  % Source   : [Des22]
% 0.29/0.46  % Names    : 0068_VEBT_MinMax_00214_010433 [Des22]
% 0.29/0.46  
% 0.29/0.46  % Status   : Theorem
% 0.29/0.46  % Rating   : ? v9.2.1
% 0.29/0.46  % Syntax   : Number of formulae    : 11120 (5066 unt;1289 typ;   0 def)
% 0.29/0.46  %            Number of atoms       : 28552 (11652 equ;   0 cnn)
% 0.29/0.46  %            Maximal formula atoms :   71 (   2 avg)
% 0.29/0.46  %            Number of connectives : 105320 (3164   ~; 537   |;2070   &;88288   @)
% 0.29/0.46  %                                         (   0 <=>;11261  =>;   0  <=;   0 <~>)
% 0.29/0.46  %            Maximal formula depth :   39 (   6 avg)
% 0.29/0.46  %            Number of types       :  174 ( 173 usr)
% 0.29/0.46  %            Number of type conns  : 4900 (4900   >;   0   *;   0   +;   0  <<)
% 0.29/0.46  %            Number of symbols     : 1119 (1116 usr;  75 con; 0-5 aty)
% 0.29/0.46  %            Number of variables   : 25757 (2081   ^;22827   !; 849   ?;25757   :)
% 0.29/0.46  % SPC      : TH0_THM_EQU_NAR
% 0.29/0.46  
% 0.29/0.46  % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.29/0.46  %            from the van Emde Boas Trees session in the Archive of Formal
% 0.29/0.46  %            proofs - 
% 0.29/0.46  %            www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 0.29/0.46  %            2022-02-17 22:12:27.135
% 0.29/0.46  % Bugfixes : Added ()s around negated sides of equations.
% 0.29/0.46  %------------------------------------------------------------------------------
% 0.29/0.46  % Could-be-implicit typings (173)
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% 0.29/0.46  thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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% 0.29/0.46  thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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% 0.29/0.47  thf(sy_c_List_Ogen__length_001t__Nat__Onat,type,
% 0.29/0.47      gen_length_nat: nat > list_nat > nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Ogen__length_001t__VEBT____Definitions__OVEBT,type,
% 0.29/0.47      gen_length_VEBT_VEBT: nat > list_VEBT_VEBT > nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
% 0.29/0.47      linord2614967742042102400et_nat: set_nat > list_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
% 0.29/0.47      cons_int: int > list_int > list_int ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 0.29/0.47      cons_nat: nat > list_nat > list_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
% 0.29/0.47      nil_int: list_int ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
% 0.29/0.47      nil_nat: list_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
% 0.29/0.47      hd_nat: list_nat > nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 0.29/0.47      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
% 0.29/0.47  
% 0.29/0.47  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,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001_Eo,type,
% 0.29/0.47      set_o2: list_o > set_o ).
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
% 0.29/0.47      set_nat2: list_nat > set_nat ).
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Num__Onum,type,
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% 0.29/0.47  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,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.29/0.47      set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o ).
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47      set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 0.29/0.47      set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 0.29/0.47      set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.29/0.47      set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.29/0.47      set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat ).
% 0.29/0.47  
% 0.29/0.47  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,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.29/0.47      set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 0.29/0.47      set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 0.29/0.47      set_real2: list_real > set_real ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47      set_set_nat2: list_set_nat > set_set_nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 0.29/0.47      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 0.29/0.47      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 0.29/0.47      tl_nat: list_nat > list_nat ).
% 0.29/0.47  
% 0.29/0.47  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,
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% 0.29/0.47  
% 0.29/0.47  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,
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% 0.29/0.47  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,
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% 0.29/0.47  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,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist__update_001_Eo,type,
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% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 0.29/0.47      list_update_int: list_int > nat > int > list_int ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 0.29/0.47      list_update_nat: list_nat > nat > nat > list_nat ).
% 0.29/0.47  
% 0.29/0.47  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,
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% 0.29/0.47  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,
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% 0.29/0.47  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,
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% 0.29/0.47  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.29/0.47  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
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% 0.29/0.47  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
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% 0.29/0.47  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
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% 0.29/0.47  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
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% 0.29/0.47  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
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% 0.29/0.47  thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
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% 0.29/0.47  thf(sy_c_List_Oupt,type,
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% 0.29/0.47  thf(sy_c_List_Oupto__rel,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
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% 0.29/0.47  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
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% 0.29/0.47  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,
% 0.29/0.47      member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 0.29/0.47      member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 0.29/0.47      member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 0.29/0.47      member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
% 0.29/0.47      member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 0.29/0.47      member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 0.29/0.47      member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
% 0.29/0.47      member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
% 0.29/0.47      member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
% 0.29/0.47      member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
% 0.29/0.47      member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 0.29/0.47      member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_M_Eo_J,type,
% 0.29/0.47      member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
% 0.29/0.47      member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
% 0.29/0.47      member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
% 0.29/0.47      member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
% 0.29/0.47      member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  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,
% 0.29/0.47      member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 0.29/0.47      member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 0.29/0.47      member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 0.29/0.47      member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 0.29/0.47      member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 0.29/0.47      member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Rat__Orat,type,
% 0.29/0.47      member_rat: rat > set_rat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Real__Oreal,type,
% 0.29/0.47      member_real: real > set_real > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
% 0.29/0.47      member_set_int: set_int > set_set_int > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
% 0.29/0.47      member_set_nat: set_nat > set_set_nat > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_c_member_001t__VEBT____Definitions__OVEBT,type,
% 0.29/0.47      member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_deg,type,
% 0.29/0.47      deg: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_m____,type,
% 0.29/0.47      m: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_ma,type,
% 0.29/0.47      ma: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_maxs____,type,
% 0.29/0.47      maxs: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_mi,type,
% 0.29/0.47      mi: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_n____,type,
% 0.29/0.47      n: nat ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_summary,type,
% 0.29/0.47      summary: vEBT_VEBT ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_thesis____,type,
% 0.29/0.47      thesis: $o ).
% 0.29/0.47  
% 0.29/0.47  thf(sy_v_treeList,type,
% 0.29/0.47      treeList: list_VEBT_VEBT ).
% 0.29/0.47  
% 0.29/0.47  % Relevant facts (9795)
% 0.29/0.47  thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom,
% 0.29/0.47      ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X_1 ) ).
% 0.29/0.47  
% 0.29/0.47  % \<open>\<exists>x. both_member_options (treeList ! maxs) x\<close>
% 0.29/0.47  thf(fact_1_not__min__Null__member,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT] :
% 0.29/0.47        ( ~ ( vEBT_VEBT_minNull @ T )
% 0.29/0.47       => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).
% 0.29/0.47  
% 0.29/0.47  % not_min_Null_member
% 0.29/0.47  thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom,
% 0.29/0.47      ( ( some_nat @ maxs )
% 0.29/0.47      = ( vEBT_vebt_maxt @ summary ) ) ).
% 0.29/0.47  
% 0.29/0.47  % \<open>Some maxs = vebt_maxt summary\<close>
% 0.29/0.47  thf(fact_3_both__member__options__def,axiom,
% 0.29/0.47      ( vEBT_V8194947554948674370ptions
% 0.29/0.47      = ( ^ [T2: vEBT_VEBT,X: nat] :
% 0.29/0.47            ( ( vEBT_V5719532721284313246member @ T2 @ X )
% 0.29/0.47            | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % both_member_options_def
% 0.29/0.47  thf(fact_4__C5_C_I7_J,axiom,
% 0.29/0.47      ( ( mi = ma )
% 0.29/0.47     => ! [X2: vEBT_VEBT] :
% 0.29/0.47          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.29/0.47         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % "5"(7)
% 0.29/0.47  thf(fact_5__C5_C_I1_J,axiom,
% 0.29/0.47      ! [X2: vEBT_VEBT] :
% 0.29/0.47        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 0.29/0.47       => ( vEBT_invar_vebt @ X2 @ n ) ) ).
% 0.29/0.47  
% 0.29/0.47  % "5"(1)
% 0.29/0.47  thf(fact_6_in__children__def,axiom,
% 0.29/0.47      ( vEBT_V5917875025757280293ildren
% 0.29/0.47      = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % in_children_def
% 0.29/0.47  thf(fact_7_both__member__options__equiv__member,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.47        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.47       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 0.29/0.47          = ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % both_member_options_equiv_member
% 0.29/0.47  thf(fact_8_valid__member__both__member__options,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.47        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.47       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 0.29/0.47         => ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % valid_member_both_member_options
% 0.29/0.47  thf(fact_9_intind,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,P: nat > $o,X3: nat] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( P @ X3 )
% 0.29/0.47         => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % intind
% 0.29/0.47  thf(fact_10_intind,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,P: int > $o,X3: int] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( P @ X3 )
% 0.29/0.47         => ( P @ ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % intind
% 0.29/0.47  thf(fact_11_intind,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( P @ X3 )
% 0.29/0.47         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % intind
% 0.29/0.47  thf(fact_12_assms_I2_J,axiom,
% 0.29/0.47      mi != ma ).
% 0.29/0.47  
% 0.29/0.47  % assms(2)
% 0.29/0.47  thf(fact_13_list__update__id,axiom,
% 0.29/0.47      ! [Xs: list_VEBT_VEBT,I: nat] :
% 0.29/0.47        ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
% 0.29/0.47        = Xs ) ).
% 0.29/0.47  
% 0.29/0.47  % list_update_id
% 0.29/0.47  thf(fact_14_list__update__id,axiom,
% 0.29/0.47      ! [Xs: list_nat,I: nat] :
% 0.29/0.47        ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
% 0.29/0.47        = Xs ) ).
% 0.29/0.47  
% 0.29/0.47  % list_update_id
% 0.29/0.47  thf(fact_15_list__update__id,axiom,
% 0.29/0.47      ! [Xs: list_int,I: nat] :
% 0.29/0.47        ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
% 0.29/0.47        = Xs ) ).
% 0.29/0.47  
% 0.29/0.47  % list_update_id
% 0.29/0.47  thf(fact_16_nth__list__update__neq,axiom,
% 0.29/0.47      ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.47        ( ( I != J )
% 0.29/0.47       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 0.29/0.47          = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_list_update_neq
% 0.29/0.47  thf(fact_17_nth__list__update__neq,axiom,
% 0.29/0.47      ! [I: nat,J: nat,Xs: list_nat,X3: nat] :
% 0.29/0.47        ( ( I != J )
% 0.29/0.47       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 0.29/0.47          = ( nth_nat @ Xs @ J ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_list_update_neq
% 0.29/0.47  thf(fact_18_nth__list__update__neq,axiom,
% 0.29/0.47      ! [I: nat,J: nat,Xs: list_int,X3: int] :
% 0.29/0.47        ( ( I != J )
% 0.29/0.47       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 0.29/0.47          = ( nth_int @ Xs @ J ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_list_update_neq
% 0.29/0.47  thf(fact_19_min__Null__member,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT,X3: nat] :
% 0.29/0.47        ( ( vEBT_VEBT_minNull @ T )
% 0.29/0.47       => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).
% 0.29/0.47  
% 0.29/0.47  % min_Null_member
% 0.29/0.47  thf(fact_20__C5_C_I2_J,axiom,
% 0.29/0.47      vEBT_invar_vebt @ summary @ m ).
% 0.29/0.47  
% 0.29/0.47  % "5"(2)
% 0.29/0.47  thf(fact_21_bit__split__inv,axiom,
% 0.29/0.47      ! [X3: nat,D: nat] :
% 0.29/0.47        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
% 0.29/0.47        = X3 ) ).
% 0.29/0.47  
% 0.29/0.47  % bit_split_inv
% 0.29/0.47  thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.29/0.47      ~ ! [Maxs: nat] :
% 0.29/0.47          ( ( some_nat @ Maxs )
% 0.29/0.47         != ( vEBT_vebt_maxt @ summary ) ) ).
% 0.29/0.47  
% 0.29/0.47  % \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.29/0.47  thf(fact_23_member__valid__both__member__options,axiom,
% 0.29/0.47      ! [Tree: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.47        ( ( vEBT_invar_vebt @ Tree @ N2 )
% 0.29/0.47       => ( ( vEBT_vebt_member @ Tree @ X3 )
% 0.29/0.47         => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
% 0.29/0.47            | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % member_valid_both_member_options
% 0.29/0.47  thf(fact_24_maxt__member,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
% 0.29/0.47        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.47       => ( ( ( vEBT_vebt_maxt @ T )
% 0.29/0.47            = ( some_nat @ Maxi ) )
% 0.29/0.47         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % maxt_member
% 0.29/0.47  thf(fact_25_member__correct,axiom,
% 0.29/0.47      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.47        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.47       => ( ( vEBT_vebt_member @ T @ X3 )
% 0.29/0.47          = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % member_correct
% 0.29/0.47  thf(fact_26__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 0.29/0.47      vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ).
% 0.29/0.47  
% 0.29/0.47  % \<open>both_member_options summary (high ma n)\<close>
% 0.29/0.47  thf(fact_27_nth__replicate,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,X3: vEBT_VEBT] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I )
% 0.29/0.47          = X3 ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_replicate
% 0.29/0.47  thf(fact_28_nth__replicate,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,X3: int] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I )
% 0.29/0.47          = X3 ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_replicate
% 0.29/0.47  thf(fact_29_nth__replicate,axiom,
% 0.29/0.47      ! [I: nat,N2: nat,X3: nat] :
% 0.29/0.47        ( ( ord_less_nat @ I @ N2 )
% 0.29/0.47       => ( ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I )
% 0.29/0.47          = X3 ) ) ).
% 0.29/0.47  
% 0.29/0.47  % nth_replicate
% 0.29/0.47  thf(fact_30__C5_C_I8_J,axiom,
% 0.29/0.47      ord_less_eq_nat @ mi @ ma ).
% 0.29/0.47  
% 0.29/0.47  % "5"(8)
% 0.29/0.47  thf(fact_31_greater__shift,axiom,
% 0.29/0.47      ( ord_less_nat
% 0.29/0.47      = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % greater_shift
% 0.29/0.47  thf(fact_32_less__shift,axiom,
% 0.29/0.47      ( ord_less_nat
% 0.29/0.47      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.29/0.47  
% 0.29/0.47  % less_shift
% 0.29/0.47  thf(fact_33__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
% 0.29/0.48      ord_less_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% 0.29/0.48  
% 0.29/0.48  % \<open>high ma n < the (vebt_maxt summary)\<close>
% 0.29/0.48  thf(fact_34_mint__member,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( ( vEBT_vebt_mint @ T )
% 0.29/0.48            = ( some_nat @ Maxi ) )
% 0.29/0.48         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mint_member
% 0.29/0.48  thf(fact_35_maxt__corr__help,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( ( vEBT_vebt_maxt @ T )
% 0.29/0.48            = ( some_nat @ Maxi ) )
% 0.29/0.48         => ( ( vEBT_vebt_member @ T @ X3 )
% 0.29/0.48           => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % maxt_corr_help
% 0.29/0.48  thf(fact_36_maxt__corr,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( ( vEBT_vebt_maxt @ T )
% 0.29/0.48            = ( some_nat @ X3 ) )
% 0.29/0.48         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % maxt_corr
% 0.29/0.48  thf(fact_37_maxt__sound,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 0.29/0.48         => ( ( vEBT_vebt_maxt @ T )
% 0.29/0.48            = ( some_nat @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % maxt_sound
% 0.29/0.48  thf(fact_38_buildup__nothing__in__leaf,axiom,
% 0.29/0.48      ! [N2: nat,X3: nat] :
% 0.29/0.48        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % buildup_nothing_in_leaf
% 0.29/0.48  thf(fact_39_buildup__nothing__in__min__max,axiom,
% 0.29/0.48      ! [N2: nat,X3: nat] :
% 0.29/0.48        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % buildup_nothing_in_min_max
% 0.29/0.48  thf(fact_40_option_Oinject,axiom,
% 0.29/0.48      ! [X22: nat,Y2: nat] :
% 0.29/0.48        ( ( ( some_nat @ X22 )
% 0.29/0.48          = ( some_nat @ Y2 ) )
% 0.29/0.48        = ( X22 = Y2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % option.inject
% 0.29/0.48  thf(fact_41_option_Oinject,axiom,
% 0.29/0.48      ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 0.29/0.48        ( ( ( some_P7363390416028606310at_nat @ X22 )
% 0.29/0.48          = ( some_P7363390416028606310at_nat @ Y2 ) )
% 0.29/0.48        = ( X22 = Y2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % option.inject
% 0.29/0.48  thf(fact_42_option_Oinject,axiom,
% 0.29/0.48      ! [X22: num,Y2: num] :
% 0.29/0.48        ( ( ( some_num @ X22 )
% 0.29/0.48          = ( some_num @ Y2 ) )
% 0.29/0.48        = ( X22 = Y2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % option.inject
% 0.29/0.48  thf(fact_43_valid__eq,axiom,
% 0.29/0.48      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 0.29/0.48  
% 0.29/0.48  % valid_eq
% 0.29/0.48  thf(fact_44_valid__eq1,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,D: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ D )
% 0.29/0.48       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 0.29/0.48  
% 0.29/0.48  % valid_eq1
% 0.29/0.48  thf(fact_45_valid__eq2,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,D: nat] :
% 0.29/0.48        ( ( vEBT_VEBT_valid @ T @ D )
% 0.29/0.48       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 0.29/0.48  
% 0.29/0.48  % valid_eq2
% 0.29/0.48  thf(fact_46_inthall,axiom,
% 0.29/0.48      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: product_prod_nat_nat] :
% 0.29/0.48            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_47_inthall,axiom,
% 0.29/0.48      ! [Xs: list_real,P: real > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: real] :
% 0.29/0.48            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_48_inthall,axiom,
% 0.29/0.48      ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: set_nat] :
% 0.29/0.48            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_49_inthall,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: vEBT_VEBT] :
% 0.29/0.48            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_50_inthall,axiom,
% 0.29/0.48      ! [Xs: list_o,P: $o > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: $o] :
% 0.29/0.48            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_51_inthall,axiom,
% 0.29/0.48      ! [Xs: list_nat,P: nat > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: nat] :
% 0.29/0.48            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_52_inthall,axiom,
% 0.29/0.48      ! [Xs: list_int,P: int > $o,N2: nat] :
% 0.29/0.48        ( ! [X4: int] :
% 0.29/0.48            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.29/0.48           => ( P @ X4 ) )
% 0.29/0.48       => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48         => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % inthall
% 0.29/0.48  thf(fact_53_deg__not__0,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % deg_not_0
% 0.29/0.48  thf(fact_54__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
% 0.29/0.48      ord_less_eq_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% 0.29/0.48  
% 0.29/0.48  % \<open>high ma n \<le> the (vebt_maxt summary)\<close>
% 0.29/0.48  thf(fact_55_valid__tree__deg__neq__0,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT] :
% 0.29/0.48        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % valid_tree_deg_neq_0
% 0.29/0.48  thf(fact_56_valid__0__not,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT] :
% 0.29/0.48        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % valid_0_not
% 0.29/0.48  thf(fact_57_min__in__set__def,axiom,
% 0.29/0.48      ( vEBT_VEBT_min_in_set
% 0.29/0.48      = ( ^ [Xs2: set_nat,X: nat] :
% 0.29/0.48            ( ( member_nat @ X @ Xs2 )
% 0.29/0.48            & ! [Y: nat] :
% 0.29/0.48                ( ( member_nat @ Y @ Xs2 )
% 0.29/0.48               => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % min_in_set_def
% 0.29/0.48  thf(fact_58_max__in__set__def,axiom,
% 0.29/0.48      ( vEBT_VEBT_max_in_set
% 0.29/0.48      = ( ^ [Xs2: set_nat,X: nat] :
% 0.29/0.48            ( ( member_nat @ X @ Xs2 )
% 0.29/0.48            & ! [Y: nat] :
% 0.29/0.48                ( ( member_nat @ Y @ Xs2 )
% 0.29/0.48               => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % max_in_set_def
% 0.29/0.48  thf(fact_59_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.29/0.48        ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_60_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: real,P: real > $o] :
% 0.29/0.48        ( ( member_real @ A @ ( collect_real @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_61_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: list_nat,P: list_nat > $o] :
% 0.29/0.48        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_62_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: set_nat,P: set_nat > $o] :
% 0.29/0.48        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_63_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: nat,P: nat > $o] :
% 0.29/0.48        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_64_mem__Collect__eq,axiom,
% 0.29/0.48      ! [A: int,P: int > $o] :
% 0.29/0.48        ( ( member_int @ A @ ( collect_int @ P ) )
% 0.29/0.48        = ( P @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mem_Collect_eq
% 0.29/0.48  thf(fact_65_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.48        ( ( collec3392354462482085612at_nat
% 0.29/0.48          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_66_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_real] :
% 0.29/0.48        ( ( collect_real
% 0.29/0.48          @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_67_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_list_nat] :
% 0.29/0.48        ( ( collect_list_nat
% 0.29/0.48          @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_68_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_set_nat] :
% 0.29/0.48        ( ( collect_set_nat
% 0.29/0.48          @ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_69_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_nat] :
% 0.29/0.48        ( ( collect_nat
% 0.29/0.48          @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_70_Collect__mem__eq,axiom,
% 0.29/0.48      ! [A2: set_int] :
% 0.29/0.48        ( ( collect_int
% 0.29/0.48          @ ^ [X: int] : ( member_int @ X @ A2 ) )
% 0.29/0.48        = A2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_mem_eq
% 0.29/0.48  thf(fact_71_Collect__cong,axiom,
% 0.29/0.48      ! [P: real > $o,Q: real > $o] :
% 0.29/0.48        ( ! [X4: real] :
% 0.29/0.48            ( ( P @ X4 )
% 0.29/0.48            = ( Q @ X4 ) )
% 0.29/0.48       => ( ( collect_real @ P )
% 0.29/0.48          = ( collect_real @ Q ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_cong
% 0.29/0.48  thf(fact_72_Collect__cong,axiom,
% 0.29/0.48      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.29/0.48        ( ! [X4: list_nat] :
% 0.29/0.48            ( ( P @ X4 )
% 0.29/0.48            = ( Q @ X4 ) )
% 0.29/0.48       => ( ( collect_list_nat @ P )
% 0.29/0.48          = ( collect_list_nat @ Q ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_cong
% 0.29/0.48  thf(fact_73_Collect__cong,axiom,
% 0.29/0.48      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.29/0.48        ( ! [X4: set_nat] :
% 0.29/0.48            ( ( P @ X4 )
% 0.29/0.48            = ( Q @ X4 ) )
% 0.29/0.48       => ( ( collect_set_nat @ P )
% 0.29/0.48          = ( collect_set_nat @ Q ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_cong
% 0.29/0.48  thf(fact_74_Collect__cong,axiom,
% 0.29/0.48      ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.48        ( ! [X4: nat] :
% 0.29/0.48            ( ( P @ X4 )
% 0.29/0.48            = ( Q @ X4 ) )
% 0.29/0.48       => ( ( collect_nat @ P )
% 0.29/0.48          = ( collect_nat @ Q ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_cong
% 0.29/0.48  thf(fact_75_Collect__cong,axiom,
% 0.29/0.48      ! [P: int > $o,Q: int > $o] :
% 0.29/0.48        ( ! [X4: int] :
% 0.29/0.48            ( ( P @ X4 )
% 0.29/0.48            = ( Q @ X4 ) )
% 0.29/0.48       => ( ( collect_int @ P )
% 0.29/0.48          = ( collect_int @ Q ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Collect_cong
% 0.29/0.48  thf(fact_76_buildup__gives__valid,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.48       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % buildup_gives_valid
% 0.29/0.48  thf(fact_77_set__vebt__set__vebt_H__valid,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( vEBT_set_vebt @ T )
% 0.29/0.48          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_vebt_set_vebt'_valid
% 0.29/0.48  thf(fact_78_mint__corr__help,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( ( vEBT_vebt_mint @ T )
% 0.29/0.48            = ( some_nat @ Mini ) )
% 0.29/0.48         => ( ( vEBT_vebt_member @ T @ X3 )
% 0.29/0.48           => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mint_corr_help
% 0.29/0.48  thf(fact_79_replicate__eq__replicate,axiom,
% 0.29/0.48      ! [M: nat,X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
% 0.29/0.48        ( ( ( replicate_VEBT_VEBT @ M @ X3 )
% 0.29/0.48          = ( replicate_VEBT_VEBT @ N2 @ Y3 ) )
% 0.29/0.48        = ( ( M = N2 )
% 0.29/0.48          & ( ( M != zero_zero_nat )
% 0.29/0.48           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eq_replicate
% 0.29/0.48  thf(fact_80_replicate__eq__replicate,axiom,
% 0.29/0.48      ! [M: nat,X3: int,N2: nat,Y3: int] :
% 0.29/0.48        ( ( ( replicate_int @ M @ X3 )
% 0.29/0.48          = ( replicate_int @ N2 @ Y3 ) )
% 0.29/0.48        = ( ( M = N2 )
% 0.29/0.48          & ( ( M != zero_zero_nat )
% 0.29/0.48           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eq_replicate
% 0.29/0.48  thf(fact_81_replicate__eq__replicate,axiom,
% 0.29/0.48      ! [M: nat,X3: nat,N2: nat,Y3: nat] :
% 0.29/0.48        ( ( ( replicate_nat @ M @ X3 )
% 0.29/0.48          = ( replicate_nat @ N2 @ Y3 ) )
% 0.29/0.48        = ( ( M = N2 )
% 0.29/0.48          & ( ( M != zero_zero_nat )
% 0.29/0.48           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eq_replicate
% 0.29/0.48  thf(fact_82_length__replicate,axiom,
% 0.29/0.48      ! [N2: nat,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % length_replicate
% 0.29/0.48  thf(fact_83_length__replicate,axiom,
% 0.29/0.48      ! [N2: nat,X3: $o] :
% 0.29/0.48        ( ( size_size_list_o @ ( replicate_o @ N2 @ X3 ) )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % length_replicate
% 0.29/0.48  thf(fact_84_length__replicate,axiom,
% 0.29/0.48      ! [N2: nat,X3: nat] :
% 0.29/0.48        ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X3 ) )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % length_replicate
% 0.29/0.48  thf(fact_85_length__replicate,axiom,
% 0.29/0.48      ! [N2: nat,X3: int] :
% 0.29/0.48        ( ( size_size_list_int @ ( replicate_int @ N2 @ X3 ) )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % length_replicate
% 0.29/0.48  thf(fact_86_length__list__update,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) )
% 0.29/0.48        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_list_update
% 0.29/0.48  thf(fact_87_length__list__update,axiom,
% 0.29/0.48      ! [Xs: list_o,I: nat,X3: $o] :
% 0.29/0.48        ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X3 ) )
% 0.29/0.48        = ( size_size_list_o @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_list_update
% 0.29/0.48  thf(fact_88_length__list__update,axiom,
% 0.29/0.48      ! [Xs: list_nat,I: nat,X3: nat] :
% 0.29/0.48        ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X3 ) )
% 0.29/0.48        = ( size_size_list_nat @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_list_update
% 0.29/0.48  thf(fact_89_length__list__update,axiom,
% 0.29/0.48      ! [Xs: list_int,I: nat,X3: int] :
% 0.29/0.48        ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X3 ) )
% 0.29/0.48        = ( size_size_list_int @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_list_update
% 0.29/0.48  thf(fact_90_mint__sound,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 0.29/0.48         => ( ( vEBT_vebt_mint @ T )
% 0.29/0.48            = ( some_nat @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mint_sound
% 0.29/0.48  thf(fact_91_mint__corr,axiom,
% 0.29/0.48      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.48        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.48       => ( ( ( vEBT_vebt_mint @ T )
% 0.29/0.48            = ( some_nat @ X3 ) )
% 0.29/0.48         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % mint_corr
% 0.29/0.48  thf(fact_92_Ball__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.48        ( ( ! [X: vEBT_VEBT] :
% 0.29/0.48              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Ball_set_replicate
% 0.29/0.48  thf(fact_93_Ball__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: int,P: int > $o] :
% 0.29/0.48        ( ( ! [X: int] :
% 0.29/0.48              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Ball_set_replicate
% 0.29/0.48  thf(fact_94_Ball__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: nat,P: nat > $o] :
% 0.29/0.48        ( ( ! [X: nat] :
% 0.29/0.48              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Ball_set_replicate
% 0.29/0.48  thf(fact_95_Bex__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.48        ( ( ? [X: vEBT_VEBT] :
% 0.29/0.48              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 0.29/0.48              & ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Bex_set_replicate
% 0.29/0.48  thf(fact_96_Bex__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: int,P: int > $o] :
% 0.29/0.48        ( ( ? [X: int] :
% 0.29/0.48              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 0.29/0.48              & ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Bex_set_replicate
% 0.29/0.48  thf(fact_97_Bex__set__replicate,axiom,
% 0.29/0.48      ! [N2: nat,A: nat,P: nat > $o] :
% 0.29/0.48        ( ( ? [X: nat] :
% 0.29/0.48              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
% 0.29/0.48              & ( P @ X ) ) )
% 0.29/0.48        = ( ( P @ A )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Bex_set_replicate
% 0.29/0.48  thf(fact_98_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: product_prod_nat_nat,N2: nat,Y3: product_prod_nat_nat] :
% 0.29/0.48        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_99_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: real,N2: nat,Y3: real] :
% 0.29/0.48        ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_100_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: set_nat,N2: nat,Y3: set_nat] :
% 0.29/0.48        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_101_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
% 0.29/0.48        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_102_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: int,N2: nat,Y3: int] :
% 0.29/0.48        ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_103_in__set__replicate,axiom,
% 0.29/0.48      ! [X3: nat,N2: nat,Y3: nat] :
% 0.29/0.48        ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y3 ) ) )
% 0.29/0.48        = ( ( X3 = Y3 )
% 0.29/0.48          & ( N2 != zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_replicate
% 0.29/0.48  thf(fact_104_list__update__beyond,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 0.29/0.48       => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_beyond
% 0.29/0.48  thf(fact_105_list__update__beyond,axiom,
% 0.29/0.48      ! [Xs: list_o,I: nat,X3: $o] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 0.29/0.48       => ( ( list_update_o @ Xs @ I @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_beyond
% 0.29/0.48  thf(fact_106_list__update__beyond,axiom,
% 0.29/0.48      ! [Xs: list_nat,I: nat,X3: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 0.29/0.48       => ( ( list_update_nat @ Xs @ I @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_beyond
% 0.29/0.48  thf(fact_107_list__update__beyond,axiom,
% 0.29/0.48      ! [Xs: list_int,I: nat,X3: int] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
% 0.29/0.48       => ( ( list_update_int @ Xs @ I @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_beyond
% 0.29/0.48  thf(fact_108__C5_C_I4_J,axiom,
% 0.29/0.48      ( m
% 0.29/0.48      = ( suc @ n ) ) ).
% 0.29/0.48  
% 0.29/0.48  % "5"(4)
% 0.29/0.48  thf(fact_109_nth__list__update__eq,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I )
% 0.29/0.48          = X3 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update_eq
% 0.29/0.48  thf(fact_110_nth__list__update__eq,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_o,X3: $o] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ I )
% 0.29/0.48          = X3 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update_eq
% 0.29/0.48  thf(fact_111_nth__list__update__eq,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_nat,X3: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ I )
% 0.29/0.48          = X3 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update_eq
% 0.29/0.48  thf(fact_112_nth__list__update__eq,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_int,X3: int] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ I )
% 0.29/0.48          = X3 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update_eq
% 0.29/0.48  thf(fact_113_lesseq__shift,axiom,
% 0.29/0.48      ( ord_less_eq_nat
% 0.29/0.48      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % lesseq_shift
% 0.29/0.48  thf(fact_114_set__swap,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 0.29/0.48            = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_swap
% 0.29/0.48  thf(fact_115_set__swap,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_o,J: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 0.29/0.48         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 0.29/0.48            = ( set_o2 @ Xs ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_swap
% 0.29/0.48  thf(fact_116_set__swap,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_nat,J: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 0.29/0.48            = ( set_nat2 @ Xs ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_swap
% 0.29/0.48  thf(fact_117_set__swap,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_int,J: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 0.29/0.48         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 0.29/0.48            = ( set_int2 @ Xs ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_swap
% 0.29/0.48  thf(fact_118_Ex__list__of__length,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48      ? [Xs3: list_VEBT_VEBT] :
% 0.29/0.48        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Ex_list_of_length
% 0.29/0.48  thf(fact_119_Ex__list__of__length,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48      ? [Xs3: list_o] :
% 0.29/0.48        ( ( size_size_list_o @ Xs3 )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Ex_list_of_length
% 0.29/0.48  thf(fact_120_Ex__list__of__length,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48      ? [Xs3: list_nat] :
% 0.29/0.48        ( ( size_size_list_nat @ Xs3 )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Ex_list_of_length
% 0.29/0.48  thf(fact_121_Ex__list__of__length,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48      ? [Xs3: list_int] :
% 0.29/0.48        ( ( size_size_list_int @ Xs3 )
% 0.29/0.48        = N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % Ex_list_of_length
% 0.29/0.48  thf(fact_122_neq__if__length__neq,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.29/0.48        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.48         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.48       => ( Xs != Ys ) ) ).
% 0.29/0.48  
% 0.29/0.48  % neq_if_length_neq
% 0.29/0.48  thf(fact_123_neq__if__length__neq,axiom,
% 0.29/0.48      ! [Xs: list_o,Ys: list_o] :
% 0.29/0.48        ( ( ( size_size_list_o @ Xs )
% 0.29/0.48         != ( size_size_list_o @ Ys ) )
% 0.29/0.48       => ( Xs != Ys ) ) ).
% 0.29/0.48  
% 0.29/0.48  % neq_if_length_neq
% 0.29/0.48  thf(fact_124_neq__if__length__neq,axiom,
% 0.29/0.48      ! [Xs: list_nat,Ys: list_nat] :
% 0.29/0.48        ( ( ( size_size_list_nat @ Xs )
% 0.29/0.48         != ( size_size_list_nat @ Ys ) )
% 0.29/0.48       => ( Xs != Ys ) ) ).
% 0.29/0.48  
% 0.29/0.48  % neq_if_length_neq
% 0.29/0.48  thf(fact_125_neq__if__length__neq,axiom,
% 0.29/0.48      ! [Xs: list_int,Ys: list_int] :
% 0.29/0.48        ( ( ( size_size_list_int @ Xs )
% 0.29/0.48         != ( size_size_list_int @ Ys ) )
% 0.29/0.48       => ( Xs != Ys ) ) ).
% 0.29/0.48  
% 0.29/0.48  % neq_if_length_neq
% 0.29/0.48  thf(fact_126_option_Osel,axiom,
% 0.29/0.48      ! [X22: nat] :
% 0.29/0.48        ( ( the_nat @ ( some_nat @ X22 ) )
% 0.29/0.48        = X22 ) ).
% 0.29/0.48  
% 0.29/0.48  % option.sel
% 0.29/0.48  thf(fact_127_option_Osel,axiom,
% 0.29/0.48      ! [X22: product_prod_nat_nat] :
% 0.29/0.48        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.48        = X22 ) ).
% 0.29/0.48  
% 0.29/0.48  % option.sel
% 0.29/0.48  thf(fact_128_option_Osel,axiom,
% 0.29/0.48      ! [X22: num] :
% 0.29/0.48        ( ( the_num @ ( some_num @ X22 ) )
% 0.29/0.48        = X22 ) ).
% 0.29/0.48  
% 0.29/0.48  % option.sel
% 0.29/0.48  thf(fact_129_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.48        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_130_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: real,Xs: list_real] :
% 0.29/0.48        ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_131_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: set_nat,Xs: list_set_nat] :
% 0.29/0.48        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_132_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.29/0.48        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_133_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: $o,Xs: list_o] :
% 0.29/0.48        ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_134_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: nat,Xs: list_nat] :
% 0.29/0.48        ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_135_length__pos__if__in__set,axiom,
% 0.29/0.48      ! [X3: int,Xs: list_int] :
% 0.29/0.48        ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.48       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_pos_if_in_set
% 0.29/0.48  thf(fact_136_length__induct,axiom,
% 0.29/0.48      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.29/0.48        ( ! [Xs3: list_VEBT_VEBT] :
% 0.29/0.48            ( ! [Ys2: list_VEBT_VEBT] :
% 0.29/0.48                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 0.29/0.48               => ( P @ Ys2 ) )
% 0.29/0.48           => ( P @ Xs3 ) )
% 0.29/0.48       => ( P @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_induct
% 0.29/0.48  thf(fact_137_length__induct,axiom,
% 0.29/0.48      ! [P: list_o > $o,Xs: list_o] :
% 0.29/0.48        ( ! [Xs3: list_o] :
% 0.29/0.48            ( ! [Ys2: list_o] :
% 0.29/0.48                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 0.29/0.48               => ( P @ Ys2 ) )
% 0.29/0.48           => ( P @ Xs3 ) )
% 0.29/0.48       => ( P @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_induct
% 0.29/0.48  thf(fact_138_length__induct,axiom,
% 0.29/0.48      ! [P: list_nat > $o,Xs: list_nat] :
% 0.29/0.48        ( ! [Xs3: list_nat] :
% 0.29/0.48            ( ! [Ys2: list_nat] :
% 0.29/0.48                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 0.29/0.48               => ( P @ Ys2 ) )
% 0.29/0.48           => ( P @ Xs3 ) )
% 0.29/0.48       => ( P @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_induct
% 0.29/0.48  thf(fact_139_length__induct,axiom,
% 0.29/0.48      ! [P: list_int > $o,Xs: list_int] :
% 0.29/0.48        ( ! [Xs3: list_int] :
% 0.29/0.48            ( ! [Ys2: list_int] :
% 0.29/0.48                ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 0.29/0.48               => ( P @ Ys2 ) )
% 0.29/0.48           => ( P @ Xs3 ) )
% 0.29/0.48       => ( P @ Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % length_induct
% 0.29/0.48  thf(fact_140_list__eq__iff__nth__eq,axiom,
% 0.29/0.48      ( ( ^ [Y4: list_VEBT_VEBT,Z: list_VEBT_VEBT] : ( Y4 = Z ) )
% 0.29/0.48      = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 0.29/0.48            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.29/0.48              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 0.29/0.48            & ! [I2: nat] :
% 0.29/0.48                ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 0.29/0.48               => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 0.29/0.48                  = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_eq_iff_nth_eq
% 0.29/0.48  thf(fact_141_list__eq__iff__nth__eq,axiom,
% 0.29/0.48      ( ( ^ [Y4: list_o,Z: list_o] : ( Y4 = Z ) )
% 0.29/0.48      = ( ^ [Xs2: list_o,Ys3: list_o] :
% 0.29/0.48            ( ( ( size_size_list_o @ Xs2 )
% 0.29/0.48              = ( size_size_list_o @ Ys3 ) )
% 0.29/0.48            & ! [I2: nat] :
% 0.29/0.48                ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 0.29/0.48               => ( ( nth_o @ Xs2 @ I2 )
% 0.29/0.48                  = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_eq_iff_nth_eq
% 0.29/0.48  thf(fact_142_list__eq__iff__nth__eq,axiom,
% 0.29/0.48      ( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
% 0.29/0.48      = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 0.29/0.48            ( ( ( size_size_list_nat @ Xs2 )
% 0.29/0.48              = ( size_size_list_nat @ Ys3 ) )
% 0.29/0.48            & ! [I2: nat] :
% 0.29/0.48                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 0.29/0.48               => ( ( nth_nat @ Xs2 @ I2 )
% 0.29/0.48                  = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_eq_iff_nth_eq
% 0.29/0.48  thf(fact_143_list__eq__iff__nth__eq,axiom,
% 0.29/0.48      ( ( ^ [Y4: list_int,Z: list_int] : ( Y4 = Z ) )
% 0.29/0.48      = ( ^ [Xs2: list_int,Ys3: list_int] :
% 0.29/0.48            ( ( ( size_size_list_int @ Xs2 )
% 0.29/0.48              = ( size_size_list_int @ Ys3 ) )
% 0.29/0.48            & ! [I2: nat] :
% 0.29/0.48                ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 0.29/0.48               => ( ( nth_int @ Xs2 @ I2 )
% 0.29/0.48                  = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_eq_iff_nth_eq
% 0.29/0.48  thf(fact_144_Skolem__list__nth,axiom,
% 0.29/0.48      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 0.29/0.48        ( ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48             => ? [X5: vEBT_VEBT] : ( P @ I2 @ X5 ) ) )
% 0.29/0.48        = ( ? [Xs2: list_VEBT_VEBT] :
% 0.29/0.48              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.29/0.48                = K )
% 0.29/0.48              & ! [I2: nat] :
% 0.29/0.48                  ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48                 => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Skolem_list_nth
% 0.29/0.48  thf(fact_145_Skolem__list__nth,axiom,
% 0.29/0.48      ! [K: nat,P: nat > $o > $o] :
% 0.29/0.48        ( ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48             => ? [X5: $o] : ( P @ I2 @ X5 ) ) )
% 0.29/0.48        = ( ? [Xs2: list_o] :
% 0.29/0.48              ( ( ( size_size_list_o @ Xs2 )
% 0.29/0.48                = K )
% 0.29/0.48              & ! [I2: nat] :
% 0.29/0.48                  ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48                 => ( P @ I2 @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Skolem_list_nth
% 0.29/0.48  thf(fact_146_Skolem__list__nth,axiom,
% 0.29/0.48      ! [K: nat,P: nat > nat > $o] :
% 0.29/0.48        ( ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48             => ? [X5: nat] : ( P @ I2 @ X5 ) ) )
% 0.29/0.48        = ( ? [Xs2: list_nat] :
% 0.29/0.48              ( ( ( size_size_list_nat @ Xs2 )
% 0.29/0.48                = K )
% 0.29/0.48              & ! [I2: nat] :
% 0.29/0.48                  ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48                 => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Skolem_list_nth
% 0.29/0.48  thf(fact_147_Skolem__list__nth,axiom,
% 0.29/0.48      ! [K: nat,P: nat > int > $o] :
% 0.29/0.48        ( ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48             => ? [X5: int] : ( P @ I2 @ X5 ) ) )
% 0.29/0.48        = ( ? [Xs2: list_int] :
% 0.29/0.48              ( ( ( size_size_list_int @ Xs2 )
% 0.29/0.48                = K )
% 0.29/0.48              & ! [I2: nat] :
% 0.29/0.48                  ( ( ord_less_nat @ I2 @ K )
% 0.29/0.48                 => ( P @ I2 @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Skolem_list_nth
% 0.29/0.48  thf(fact_148_nth__equalityI,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.29/0.48        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.48          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.48       => ( ! [I3: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 0.29/0.48                = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 0.29/0.48         => ( Xs = Ys ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_equalityI
% 0.29/0.48  thf(fact_149_nth__equalityI,axiom,
% 0.29/0.48      ! [Xs: list_o,Ys: list_o] :
% 0.29/0.48        ( ( ( size_size_list_o @ Xs )
% 0.29/0.48          = ( size_size_list_o @ Ys ) )
% 0.29/0.48       => ( ! [I3: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48             => ( ( nth_o @ Xs @ I3 )
% 0.29/0.48                = ( nth_o @ Ys @ I3 ) ) )
% 0.29/0.48         => ( Xs = Ys ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_equalityI
% 0.29/0.48  thf(fact_150_nth__equalityI,axiom,
% 0.29/0.48      ! [Xs: list_nat,Ys: list_nat] :
% 0.29/0.48        ( ( ( size_size_list_nat @ Xs )
% 0.29/0.48          = ( size_size_list_nat @ Ys ) )
% 0.29/0.48       => ( ! [I3: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48             => ( ( nth_nat @ Xs @ I3 )
% 0.29/0.48                = ( nth_nat @ Ys @ I3 ) ) )
% 0.29/0.48         => ( Xs = Ys ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_equalityI
% 0.29/0.48  thf(fact_151_nth__equalityI,axiom,
% 0.29/0.48      ! [Xs: list_int,Ys: list_int] :
% 0.29/0.48        ( ( ( size_size_list_int @ Xs )
% 0.29/0.48          = ( size_size_list_int @ Ys ) )
% 0.29/0.48       => ( ! [I3: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48             => ( ( nth_int @ Xs @ I3 )
% 0.29/0.48                = ( nth_int @ Ys @ I3 ) ) )
% 0.29/0.48         => ( Xs = Ys ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_equalityI
% 0.29/0.48  thf(fact_152_replicate__length__same,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.48        ( ! [X4: vEBT_VEBT] :
% 0.29/0.48            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48           => ( X4 = X3 ) )
% 0.29/0.48       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_length_same
% 0.29/0.48  thf(fact_153_replicate__length__same,axiom,
% 0.29/0.48      ! [Xs: list_o,X3: $o] :
% 0.29/0.48        ( ! [X4: $o] :
% 0.29/0.48            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.29/0.48           => ( X4 = X3 ) )
% 0.29/0.48       => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_length_same
% 0.29/0.48  thf(fact_154_replicate__length__same,axiom,
% 0.29/0.48      ! [Xs: list_nat,X3: nat] :
% 0.29/0.48        ( ! [X4: nat] :
% 0.29/0.48            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.29/0.48           => ( X4 = X3 ) )
% 0.29/0.48       => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_length_same
% 0.29/0.48  thf(fact_155_replicate__length__same,axiom,
% 0.29/0.48      ! [Xs: list_int,X3: int] :
% 0.29/0.48        ( ! [X4: int] :
% 0.29/0.48            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.29/0.48           => ( X4 = X3 ) )
% 0.29/0.48       => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X3 )
% 0.29/0.48          = Xs ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_length_same
% 0.29/0.48  thf(fact_156_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_P6011104703257516679at_nat,N2: nat,X3: product_prod_nat_nat] :
% 0.29/0.48        ( ( ( size_s5460976970255530739at_nat @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: product_prod_nat_nat] :
% 0.29/0.48              ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replic4235873036481779905at_nat @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_157_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_real,N2: nat,X3: real] :
% 0.29/0.48        ( ( ( size_size_list_real @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: real] :
% 0.29/0.48              ( ( member_real @ Y5 @ ( set_real2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_real @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_158_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_set_nat,N2: nat,X3: set_nat] :
% 0.29/0.48        ( ( ( size_s3254054031482475050et_nat @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: set_nat] :
% 0.29/0.48              ( ( member_set_nat @ Y5 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_set_nat @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_159_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,N2: nat,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: vEBT_VEBT] :
% 0.29/0.48              ( ( member_VEBT_VEBT @ Y5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_VEBT_VEBT @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_160_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_o,N2: nat,X3: $o] :
% 0.29/0.48        ( ( ( size_size_list_o @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: $o] :
% 0.29/0.48              ( ( member_o @ Y5 @ ( set_o2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_o @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_161_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_nat,N2: nat,X3: nat] :
% 0.29/0.48        ( ( ( size_size_list_nat @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: nat] :
% 0.29/0.48              ( ( member_nat @ Y5 @ ( set_nat2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_nat @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_162_replicate__eqI,axiom,
% 0.29/0.48      ! [Xs: list_int,N2: nat,X3: int] :
% 0.29/0.48        ( ( ( size_size_list_int @ Xs )
% 0.29/0.48          = N2 )
% 0.29/0.48       => ( ! [Y5: int] :
% 0.29/0.48              ( ( member_int @ Y5 @ ( set_int2 @ Xs ) )
% 0.29/0.48             => ( Y5 = X3 ) )
% 0.29/0.48         => ( Xs
% 0.29/0.48            = ( replicate_int @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % replicate_eqI
% 0.29/0.48  thf(fact_163_all__set__conv__all__nth,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.48        ( ( ! [X: vEBT_VEBT] :
% 0.29/0.48              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48             => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_set_conv_all_nth
% 0.29/0.48  thf(fact_164_all__set__conv__all__nth,axiom,
% 0.29/0.48      ! [Xs: list_o,P: $o > $o] :
% 0.29/0.48        ( ( ! [X: $o] :
% 0.29/0.48              ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48             => ( P @ ( nth_o @ Xs @ I2 ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_set_conv_all_nth
% 0.29/0.48  thf(fact_165_all__set__conv__all__nth,axiom,
% 0.29/0.48      ! [Xs: list_nat,P: nat > $o] :
% 0.29/0.48        ( ( ! [X: nat] :
% 0.29/0.48              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48             => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_set_conv_all_nth
% 0.29/0.48  thf(fact_166_all__set__conv__all__nth,axiom,
% 0.29/0.48      ! [Xs: list_int,P: int > $o] :
% 0.29/0.48        ( ( ! [X: int] :
% 0.29/0.48              ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.48             => ( P @ X ) ) )
% 0.29/0.48        = ( ! [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48             => ( P @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_set_conv_all_nth
% 0.29/0.48  thf(fact_167_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_168_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_real,P: real > $o,X3: real] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_real @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_169_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_set_nat,P: set_nat > $o,X3: set_nat] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_set_nat @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_170_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_171_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_o,P: $o > $o,X3: $o] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_o @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_172_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_nat,P: nat > $o,X3: nat] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_nat @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_173_all__nth__imp__all__set,axiom,
% 0.29/0.48      ! [Xs: list_int,P: int > $o,X3: int] :
% 0.29/0.48        ( ! [I3: nat] :
% 0.29/0.48            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48           => ( P @ ( nth_int @ Xs @ I3 ) ) )
% 0.29/0.48       => ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.48         => ( P @ X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % all_nth_imp_all_set
% 0.29/0.48  thf(fact_174_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.48        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.48              & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_175_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: real,Xs: list_real] :
% 0.29/0.48        ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 0.29/0.48              & ( ( nth_real @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_176_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: set_nat,Xs: list_set_nat] :
% 0.29/0.48        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.48              & ( ( nth_set_nat @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_177_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.29/0.48        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48              & ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_178_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: $o,Xs: list_o] :
% 0.29/0.48        ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48              & ( ( nth_o @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_179_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: nat,Xs: list_nat] :
% 0.29/0.48        ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48              & ( ( nth_nat @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_180_in__set__conv__nth,axiom,
% 0.29/0.48      ! [X3: int,Xs: list_int] :
% 0.29/0.48        ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.48        = ( ? [I2: nat] :
% 0.29/0.48              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48              & ( ( nth_int @ Xs @ I2 )
% 0.29/0.48                = X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % in_set_conv_nth
% 0.29/0.48  thf(fact_181_list__ball__nth,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( ! [X4: vEBT_VEBT] :
% 0.29/0.48              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.48             => ( P @ X4 ) )
% 0.29/0.48         => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_ball_nth
% 0.29/0.48  thf(fact_182_list__ball__nth,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_o,P: $o > $o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( ! [X4: $o] :
% 0.29/0.48              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 0.29/0.48             => ( P @ X4 ) )
% 0.29/0.48         => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_ball_nth
% 0.29/0.48  thf(fact_183_list__ball__nth,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_nat,P: nat > $o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( ! [X4: nat] :
% 0.29/0.48              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 0.29/0.48             => ( P @ X4 ) )
% 0.29/0.48         => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_ball_nth
% 0.29/0.48  thf(fact_184_list__ball__nth,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_int,P: int > $o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( ! [X4: int] :
% 0.29/0.48              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 0.29/0.48             => ( P @ X4 ) )
% 0.29/0.48         => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_ball_nth
% 0.29/0.48  thf(fact_185_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.48       => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_186_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_real] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.29/0.48       => ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_187_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_set_nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.48       => ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_188_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_VEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_189_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_190_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_191_nth__mem,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_int] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_mem
% 0.29/0.48  thf(fact_192_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.48       => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_193_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_real,X3: real] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 0.29/0.48       => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_194_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_set_nat,X3: set_nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 0.29/0.48       => ( member_set_nat @ X3 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_195_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_196_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_o,X3: $o] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_197_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_nat,X3: nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_198_set__update__memI,axiom,
% 0.29/0.48      ! [N2: nat,Xs: list_int,X3: int] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % set_update_memI
% 0.29/0.48  thf(fact_199_list__update__same__conv,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
% 0.29/0.48            = Xs )
% 0.29/0.48          = ( ( nth_VEBT_VEBT @ Xs @ I )
% 0.29/0.48            = X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_same_conv
% 0.29/0.48  thf(fact_200_list__update__same__conv,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_o,X3: $o] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( ( ( list_update_o @ Xs @ I @ X3 )
% 0.29/0.48            = Xs )
% 0.29/0.48          = ( ( nth_o @ Xs @ I )
% 0.29/0.48            = X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_same_conv
% 0.29/0.48  thf(fact_201_list__update__same__conv,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_nat,X3: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( ( ( list_update_nat @ Xs @ I @ X3 )
% 0.29/0.48            = Xs )
% 0.29/0.48          = ( ( nth_nat @ Xs @ I )
% 0.29/0.48            = X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_same_conv
% 0.29/0.48  thf(fact_202_list__update__same__conv,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_int,X3: int] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( ( ( list_update_int @ Xs @ I @ X3 )
% 0.29/0.48            = Xs )
% 0.29/0.48          = ( ( nth_int @ Xs @ I )
% 0.29/0.48            = X3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % list_update_same_conv
% 0.29/0.48  thf(fact_203_nth__list__update,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.48       => ( ( ( I = J )
% 0.29/0.48           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = X3 ) )
% 0.29/0.48          & ( ( I != J )
% 0.29/0.48           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update
% 0.29/0.48  thf(fact_204_nth__list__update,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_o,X3: $o,J: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.48       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ J )
% 0.29/0.48          = ( ( ( I = J )
% 0.29/0.48             => X3 )
% 0.29/0.48            & ( ( I != J )
% 0.29/0.48             => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update
% 0.29/0.48  thf(fact_205_nth__list__update,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_nat,J: nat,X3: nat] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.48       => ( ( ( I = J )
% 0.29/0.48           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = X3 ) )
% 0.29/0.48          & ( ( I != J )
% 0.29/0.48           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update
% 0.29/0.48  thf(fact_206_nth__list__update,axiom,
% 0.29/0.48      ! [I: nat,Xs: list_int,J: nat,X3: int] :
% 0.29/0.48        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 0.29/0.48       => ( ( ( I = J )
% 0.29/0.48           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = X3 ) )
% 0.29/0.48          & ( ( I != J )
% 0.29/0.48           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 0.29/0.48              = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nth_list_update
% 0.29/0.48  thf(fact_207_le0,axiom,
% 0.29/0.48      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.29/0.48  
% 0.29/0.48  % le0
% 0.29/0.48  thf(fact_208_bot__nat__0_Oextremum,axiom,
% 0.29/0.48      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot_nat_0.extremum
% 0.29/0.48  thf(fact_209_less__nat__zero__code,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % less_nat_zero_code
% 0.29/0.48  thf(fact_210_neq0__conv,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( N2 != zero_zero_nat )
% 0.29/0.48        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % neq0_conv
% 0.29/0.48  thf(fact_211_bot__nat__0_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( A != zero_zero_nat )
% 0.29/0.48        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot_nat_0.not_eq_extremum
% 0.29/0.48  thf(fact_212_not__gr__zero,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.48        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % not_gr_zero
% 0.29/0.48  thf(fact_213_le__zero__eq,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.29/0.48        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_zero_eq
% 0.29/0.48  thf(fact_214__C5_C_I5_J,axiom,
% 0.29/0.48      ( deg
% 0.29/0.48      = ( plus_plus_nat @ n @ m ) ) ).
% 0.29/0.48  
% 0.29/0.48  % "5"(5)
% 0.29/0.48  thf(fact_215_ex__least__nat__le,axiom,
% 0.29/0.48      ! [P: nat > $o,N2: nat] :
% 0.29/0.48        ( ( P @ N2 )
% 0.29/0.48       => ( ~ ( P @ zero_zero_nat )
% 0.29/0.48         => ? [K2: nat] :
% 0.29/0.48              ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.29/0.48              & ! [I4: nat] :
% 0.29/0.48                  ( ( ord_less_nat @ I4 @ K2 )
% 0.29/0.48                 => ~ ( P @ I4 ) )
% 0.29/0.48              & ( P @ K2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % ex_least_nat_le
% 0.29/0.48  thf(fact_216_buildup__gives__empty,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
% 0.29/0.48        = bot_bot_set_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % buildup_gives_empty
% 0.29/0.48  thf(fact_217_dual__order_Orefl,axiom,
% 0.29/0.48      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % dual_order.refl
% 0.29/0.48  thf(fact_218_dual__order_Orefl,axiom,
% 0.29/0.48      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % dual_order.refl
% 0.29/0.48  thf(fact_219_dual__order_Orefl,axiom,
% 0.29/0.48      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % dual_order.refl
% 0.29/0.48  thf(fact_220_dual__order_Orefl,axiom,
% 0.29/0.48      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % dual_order.refl
% 0.29/0.48  thf(fact_221_dual__order_Orefl,axiom,
% 0.29/0.48      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % dual_order.refl
% 0.29/0.48  thf(fact_222_even__odd__cases,axiom,
% 0.29/0.48      ! [X3: nat] :
% 0.29/0.48        ( ! [N3: nat] :
% 0.29/0.48            ( X3
% 0.29/0.48           != ( plus_plus_nat @ N3 @ N3 ) )
% 0.29/0.48       => ~ ! [N3: nat] :
% 0.29/0.48              ( X3
% 0.29/0.48             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % even_odd_cases
% 0.29/0.48  thf(fact_223_add__shift,axiom,
% 0.29/0.48      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.29/0.48          = Z2 )
% 0.29/0.48        = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.29/0.48          = ( some_nat @ Z2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_shift
% 0.29/0.48  thf(fact_224_order__refl,axiom,
% 0.29/0.48      ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % order_refl
% 0.29/0.48  thf(fact_225_order__refl,axiom,
% 0.29/0.48      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % order_refl
% 0.29/0.48  thf(fact_226_order__refl,axiom,
% 0.29/0.48      ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % order_refl
% 0.29/0.48  thf(fact_227_order__refl,axiom,
% 0.29/0.48      ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % order_refl
% 0.29/0.48  thf(fact_228_order__refl,axiom,
% 0.29/0.48      ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 0.29/0.48  
% 0.29/0.48  % order_refl
% 0.29/0.48  thf(fact_229_add__left__cancel,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.48          = ( plus_plus_real @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_cancel
% 0.29/0.48  thf(fact_230_add__left__cancel,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.48          = ( plus_plus_rat @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_cancel
% 0.29/0.48  thf(fact_231_add__left__cancel,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.48          = ( plus_plus_nat @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_cancel
% 0.29/0.48  thf(fact_232_add__left__cancel,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.48          = ( plus_plus_int @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_cancel
% 0.29/0.48  thf(fact_233_add__right__cancel,axiom,
% 0.29/0.48      ! [B: real,A: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ B @ A )
% 0.29/0.48          = ( plus_plus_real @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_cancel
% 0.29/0.48  thf(fact_234_add__right__cancel,axiom,
% 0.29/0.48      ! [B: rat,A: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ B @ A )
% 0.29/0.48          = ( plus_plus_rat @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_cancel
% 0.29/0.48  thf(fact_235_add__right__cancel,axiom,
% 0.29/0.48      ! [B: nat,A: nat,C: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.48          = ( plus_plus_nat @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_cancel
% 0.29/0.48  thf(fact_236_add__right__cancel,axiom,
% 0.29/0.48      ! [B: int,A: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ B @ A )
% 0.29/0.48          = ( plus_plus_int @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_cancel
% 0.29/0.48  thf(fact_237_nat_Oinject,axiom,
% 0.29/0.48      ! [X22: nat,Y2: nat] :
% 0.29/0.48        ( ( ( suc @ X22 )
% 0.29/0.48          = ( suc @ Y2 ) )
% 0.29/0.48        = ( X22 = Y2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nat.inject
% 0.29/0.48  thf(fact_238_old_Onat_Oinject,axiom,
% 0.29/0.48      ! [Nat: nat,Nat2: nat] :
% 0.29/0.48        ( ( ( suc @ Nat )
% 0.29/0.48          = ( suc @ Nat2 ) )
% 0.29/0.48        = ( Nat = Nat2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % old.nat.inject
% 0.29/0.48  thf(fact_239_add__le__cancel__left,axiom,
% 0.29/0.48      ! [C: real,A: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.29/0.48        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_left
% 0.29/0.48  thf(fact_240_add__le__cancel__left,axiom,
% 0.29/0.48      ! [C: rat,A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.29/0.48        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_left
% 0.29/0.48  thf(fact_241_add__le__cancel__left,axiom,
% 0.29/0.48      ! [C: nat,A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.48        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_left
% 0.29/0.48  thf(fact_242_add__le__cancel__left,axiom,
% 0.29/0.48      ! [C: int,A: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.29/0.48        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_left
% 0.29/0.48  thf(fact_243_add__le__cancel__right,axiom,
% 0.29/0.48      ! [A: real,C: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.29/0.48        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_right
% 0.29/0.48  thf(fact_244_add__le__cancel__right,axiom,
% 0.29/0.48      ! [A: rat,C: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.48        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_right
% 0.29/0.48  thf(fact_245_add__le__cancel__right,axiom,
% 0.29/0.48      ! [A: nat,C: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.48        = ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_right
% 0.29/0.48  thf(fact_246_add__le__cancel__right,axiom,
% 0.29/0.48      ! [A: int,C: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.29/0.48        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_cancel_right
% 0.29/0.48  thf(fact_247_add_Oright__neutral,axiom,
% 0.29/0.48      ! [A: complex] :
% 0.29/0.48        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_neutral
% 0.29/0.48  thf(fact_248_add_Oright__neutral,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_neutral
% 0.29/0.48  thf(fact_249_add_Oright__neutral,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_neutral
% 0.29/0.48  thf(fact_250_add_Oright__neutral,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_neutral
% 0.29/0.48  thf(fact_251_add_Oright__neutral,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_neutral
% 0.29/0.48  thf(fact_252_double__zero__sym,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( zero_zero_real
% 0.29/0.48          = ( plus_plus_real @ A @ A ) )
% 0.29/0.48        = ( A = zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_zero_sym
% 0.29/0.48  thf(fact_253_double__zero__sym,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( zero_zero_rat
% 0.29/0.48          = ( plus_plus_rat @ A @ A ) )
% 0.29/0.48        = ( A = zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_zero_sym
% 0.29/0.48  thf(fact_254_double__zero__sym,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( zero_zero_int
% 0.29/0.48          = ( plus_plus_int @ A @ A ) )
% 0.29/0.48        = ( A = zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_zero_sym
% 0.29/0.48  thf(fact_255_add__cancel__left__left,axiom,
% 0.29/0.48      ! [B: complex,A: complex] :
% 0.29/0.48        ( ( ( plus_plus_complex @ B @ A )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_complex ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_left
% 0.29/0.48  thf(fact_256_add__cancel__left__left,axiom,
% 0.29/0.48      ! [B: real,A: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ B @ A )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_left
% 0.29/0.48  thf(fact_257_add__cancel__left__left,axiom,
% 0.29/0.48      ! [B: rat,A: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ B @ A )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_left
% 0.29/0.48  thf(fact_258_add__cancel__left__left,axiom,
% 0.29/0.48      ! [B: nat,A: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_left
% 0.29/0.48  thf(fact_259_add__cancel__left__left,axiom,
% 0.29/0.48      ! [B: int,A: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ B @ A )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_left
% 0.29/0.48  thf(fact_260_add__cancel__left__right,axiom,
% 0.29/0.48      ! [A: complex,B: complex] :
% 0.29/0.48        ( ( ( plus_plus_complex @ A @ B )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_complex ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_right
% 0.29/0.48  thf(fact_261_add__cancel__left__right,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_right
% 0.29/0.48  thf(fact_262_add__cancel__left__right,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_right
% 0.29/0.48  thf(fact_263_add__cancel__left__right,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_right
% 0.29/0.48  thf(fact_264_add__cancel__left__right,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.48          = A )
% 0.29/0.48        = ( B = zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_left_right
% 0.29/0.48  thf(fact_265_add__cancel__right__left,axiom,
% 0.29/0.48      ! [A: complex,B: complex] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_complex @ B @ A ) )
% 0.29/0.48        = ( B = zero_zero_complex ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_left
% 0.29/0.48  thf(fact_266_add__cancel__right__left,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_real @ B @ A ) )
% 0.29/0.48        = ( B = zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_left
% 0.29/0.48  thf(fact_267_add__cancel__right__left,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_rat @ B @ A ) )
% 0.29/0.48        = ( B = zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_left
% 0.29/0.48  thf(fact_268_add__cancel__right__left,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_nat @ B @ A ) )
% 0.29/0.48        = ( B = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_left
% 0.29/0.48  thf(fact_269_add__cancel__right__left,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_int @ B @ A ) )
% 0.29/0.48        = ( B = zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_left
% 0.29/0.48  thf(fact_270_add__cancel__right__right,axiom,
% 0.29/0.48      ! [A: complex,B: complex] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_complex @ A @ B ) )
% 0.29/0.48        = ( B = zero_zero_complex ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_right
% 0.29/0.48  thf(fact_271_add__cancel__right__right,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_real @ A @ B ) )
% 0.29/0.48        = ( B = zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_right
% 0.29/0.48  thf(fact_272_add__cancel__right__right,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_rat @ A @ B ) )
% 0.29/0.48        = ( B = zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_right
% 0.29/0.48  thf(fact_273_add__cancel__right__right,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_nat @ A @ B ) )
% 0.29/0.48        = ( B = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_right
% 0.29/0.48  thf(fact_274_add__cancel__right__right,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( A
% 0.29/0.48          = ( plus_plus_int @ A @ B ) )
% 0.29/0.48        = ( B = zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_cancel_right_right
% 0.29/0.48  thf(fact_275_add__eq__0__iff__both__eq__0,axiom,
% 0.29/0.48      ! [X3: nat,Y3: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.29/0.48          = zero_zero_nat )
% 0.29/0.48        = ( ( X3 = zero_zero_nat )
% 0.29/0.48          & ( Y3 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_eq_0_iff_both_eq_0
% 0.29/0.48  thf(fact_276_zero__eq__add__iff__both__eq__0,axiom,
% 0.29/0.48      ! [X3: nat,Y3: nat] :
% 0.29/0.48        ( ( zero_zero_nat
% 0.29/0.48          = ( plus_plus_nat @ X3 @ Y3 ) )
% 0.29/0.48        = ( ( X3 = zero_zero_nat )
% 0.29/0.48          & ( Y3 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_eq_add_iff_both_eq_0
% 0.29/0.48  thf(fact_277_add__0,axiom,
% 0.29/0.48      ! [A: complex] :
% 0.29/0.48        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add_0
% 0.29/0.48  thf(fact_278_add__0,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add_0
% 0.29/0.48  thf(fact_279_add__0,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add_0
% 0.29/0.48  thf(fact_280_add__0,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add_0
% 0.29/0.48  thf(fact_281_add__0,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add_0
% 0.29/0.48  thf(fact_282_add__less__cancel__left,axiom,
% 0.29/0.48      ! [C: real,A: real,B: real] :
% 0.29/0.48        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.29/0.48        = ( ord_less_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_left
% 0.29/0.48  thf(fact_283_add__less__cancel__left,axiom,
% 0.29/0.48      ! [C: rat,A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.29/0.48        = ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_left
% 0.29/0.48  thf(fact_284_add__less__cancel__left,axiom,
% 0.29/0.48      ! [C: nat,A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.48        = ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_left
% 0.29/0.48  thf(fact_285_add__less__cancel__left,axiom,
% 0.29/0.48      ! [C: int,A: int,B: int] :
% 0.29/0.48        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.29/0.48        = ( ord_less_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_left
% 0.29/0.48  thf(fact_286_add__less__cancel__right,axiom,
% 0.29/0.48      ! [A: real,C: real,B: real] :
% 0.29/0.48        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.29/0.48        = ( ord_less_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_right
% 0.29/0.48  thf(fact_287_add__less__cancel__right,axiom,
% 0.29/0.48      ! [A: rat,C: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.48        = ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_right
% 0.29/0.48  thf(fact_288_add__less__cancel__right,axiom,
% 0.29/0.48      ! [A: nat,C: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.48        = ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_right
% 0.29/0.48  thf(fact_289_add__less__cancel__right,axiom,
% 0.29/0.48      ! [A: int,C: int,B: int] :
% 0.29/0.48        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.29/0.48        = ( ord_less_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_cancel_right
% 0.29/0.48  thf(fact_290_lessI,axiom,
% 0.29/0.48      ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % lessI
% 0.29/0.48  thf(fact_291_Suc__mono,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.48       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Suc_mono
% 0.29/0.48  thf(fact_292_Suc__less__eq,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.29/0.48        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Suc_less_eq
% 0.29/0.48  thf(fact_293_Suc__le__mono,axiom,
% 0.29/0.48      ! [N2: nat,M: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
% 0.29/0.48        = ( ord_less_eq_nat @ N2 @ M ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Suc_le_mono
% 0.29/0.48  thf(fact_294_add__Suc__right,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
% 0.29/0.48        = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_Suc_right
% 0.29/0.48  thf(fact_295_add__is__0,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ M @ N2 )
% 0.29/0.48          = zero_zero_nat )
% 0.29/0.48        = ( ( M = zero_zero_nat )
% 0.29/0.48          & ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_is_0
% 0.29/0.48  thf(fact_296_Nat_Oadd__0__right,axiom,
% 0.29/0.48      ! [M: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 0.29/0.48        = M ) ).
% 0.29/0.48  
% 0.29/0.48  % Nat.add_0_right
% 0.29/0.48  thf(fact_297_nat__add__left__cancel__less,axiom,
% 0.29/0.48      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.29/0.48        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nat_add_left_cancel_less
% 0.29/0.48  thf(fact_298_nat__add__left__cancel__le,axiom,
% 0.29/0.48      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.29/0.48        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nat_add_left_cancel_le
% 0.29/0.48  thf(fact_299_add__le__same__cancel1,axiom,
% 0.29/0.48      ! [B: real,A: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel1
% 0.29/0.48  thf(fact_300_add__le__same__cancel1,axiom,
% 0.29/0.48      ! [B: rat,A: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel1
% 0.29/0.48  thf(fact_301_add__le__same__cancel1,axiom,
% 0.29/0.48      ! [B: nat,A: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel1
% 0.29/0.48  thf(fact_302_add__le__same__cancel1,axiom,
% 0.29/0.48      ! [B: int,A: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel1
% 0.29/0.48  thf(fact_303_add__le__same__cancel2,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel2
% 0.29/0.48  thf(fact_304_add__le__same__cancel2,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel2
% 0.29/0.48  thf(fact_305_add__le__same__cancel2,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel2
% 0.29/0.48  thf(fact_306_add__le__same__cancel2,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_same_cancel2
% 0.29/0.48  thf(fact_307_le__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.29/0.48        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel1
% 0.29/0.48  thf(fact_308_le__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.48        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel1
% 0.29/0.48  thf(fact_309_le__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.48        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel1
% 0.29/0.48  thf(fact_310_le__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.29/0.48        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel1
% 0.29/0.48  thf(fact_311_le__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.29/0.48        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel2
% 0.29/0.48  thf(fact_312_le__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.29/0.48        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel2
% 0.29/0.48  thf(fact_313_le__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.29/0.48        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel2
% 0.29/0.48  thf(fact_314_le__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.29/0.48        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_add_same_cancel2
% 0.29/0.48  thf(fact_315_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.29/0.48        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.48  thf(fact_316_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.29/0.48        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.48  thf(fact_317_double__add__le__zero__iff__single__add__le__zero,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.29/0.48        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_le_zero_iff_single_add_le_zero
% 0.29/0.48  thf(fact_318_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.29/0.48        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.48  thf(fact_319_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.29/0.48        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.48  thf(fact_320_zero__le__double__add__iff__zero__le__single__add,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.29/0.48        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_le_double_add_iff_zero_le_single_add
% 0.29/0.48  thf(fact_321_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 0.29/0.48        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.48  thf(fact_322_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 0.29/0.48        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.48  thf(fact_323_zero__less__double__add__iff__zero__less__single__add,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 0.29/0.48        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_less_double_add_iff_zero_less_single_add
% 0.29/0.48  thf(fact_324_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 0.29/0.48        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.48  thf(fact_325_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 0.29/0.48        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.48  thf(fact_326_double__add__less__zero__iff__single__add__less__zero,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 0.29/0.48        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % double_add_less_zero_iff_single_add_less_zero
% 0.29/0.48  thf(fact_327_less__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.29/0.48        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel2
% 0.29/0.48  thf(fact_328_less__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.29/0.48        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel2
% 0.29/0.48  thf(fact_329_less__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.29/0.48        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel2
% 0.29/0.48  thf(fact_330_less__add__same__cancel2,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.29/0.48        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel2
% 0.29/0.48  thf(fact_331_less__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.29/0.48        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel1
% 0.29/0.48  thf(fact_332_less__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.48        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel1
% 0.29/0.48  thf(fact_333_less__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.48        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel1
% 0.29/0.48  thf(fact_334_less__add__same__cancel1,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.29/0.48        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_same_cancel1
% 0.29/0.48  thf(fact_335_add__less__same__cancel2,axiom,
% 0.29/0.48      ! [A: real,B: real] :
% 0.29/0.48        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel2
% 0.29/0.48  thf(fact_336_add__less__same__cancel2,axiom,
% 0.29/0.48      ! [A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel2
% 0.29/0.48  thf(fact_337_add__less__same__cancel2,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel2
% 0.29/0.48  thf(fact_338_add__less__same__cancel2,axiom,
% 0.29/0.48      ! [A: int,B: int] :
% 0.29/0.48        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.29/0.48        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel2
% 0.29/0.48  thf(fact_339_add__less__same__cancel1,axiom,
% 0.29/0.48      ! [B: real,A: real] :
% 0.29/0.48        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel1
% 0.29/0.48  thf(fact_340_add__less__same__cancel1,axiom,
% 0.29/0.48      ! [B: rat,A: rat] :
% 0.29/0.48        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel1
% 0.29/0.48  thf(fact_341_add__less__same__cancel1,axiom,
% 0.29/0.48      ! [B: nat,A: nat] :
% 0.29/0.48        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel1
% 0.29/0.48  thf(fact_342_add__less__same__cancel1,axiom,
% 0.29/0.48      ! [B: int,A: int] :
% 0.29/0.48        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.29/0.48        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_less_same_cancel1
% 0.29/0.48  thf(fact_343_zero__less__Suc,axiom,
% 0.29/0.48      ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % zero_less_Suc
% 0.29/0.48  thf(fact_344_less__Suc0,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.29/0.48        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_Suc0
% 0.29/0.48  thf(fact_345_add__gr__0,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.48        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.48          | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_gr_0
% 0.29/0.48  thf(fact_346_one__is__add,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ( suc @ zero_zero_nat )
% 0.29/0.48          = ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.48        = ( ( ( M
% 0.29/0.48              = ( suc @ zero_zero_nat ) )
% 0.29/0.48            & ( N2 = zero_zero_nat ) )
% 0.29/0.48          | ( ( M = zero_zero_nat )
% 0.29/0.48            & ( N2
% 0.29/0.48              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % one_is_add
% 0.29/0.48  thf(fact_347_add__is__1,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ M @ N2 )
% 0.29/0.48          = ( suc @ zero_zero_nat ) )
% 0.29/0.48        = ( ( ( M
% 0.29/0.48              = ( suc @ zero_zero_nat ) )
% 0.29/0.48            & ( N2 = zero_zero_nat ) )
% 0.29/0.48          | ( ( M = zero_zero_nat )
% 0.29/0.48            & ( N2
% 0.29/0.48              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_is_1
% 0.29/0.48  thf(fact_348_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % ab_semigroup_add_class.add_ac(1)
% 0.29/0.48  thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % ab_semigroup_add_class.add_ac(1)
% 0.29/0.48  thf(fact_350_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % ab_semigroup_add_class.add_ac(1)
% 0.29/0.48  thf(fact_351_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % ab_semigroup_add_class.add_ac(1)
% 0.29/0.48  thf(fact_352_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ( plus_plus_real @ I @ K )
% 0.29/0.48          = ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(4)
% 0.29/0.48  thf(fact_353_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ( plus_plus_rat @ I @ K )
% 0.29/0.48          = ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(4)
% 0.29/0.48  thf(fact_354_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ( plus_plus_nat @ I @ K )
% 0.29/0.48          = ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(4)
% 0.29/0.48  thf(fact_355_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ( plus_plus_int @ I @ K )
% 0.29/0.48          = ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(4)
% 0.29/0.48  thf(fact_356_nat__arith_Osuc1,axiom,
% 0.29/0.48      ! [A2: nat,K: nat,A: nat] :
% 0.29/0.48        ( ( A2
% 0.29/0.48          = ( plus_plus_nat @ K @ A ) )
% 0.29/0.48       => ( ( suc @ A2 )
% 0.29/0.48          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % nat_arith.suc1
% 0.29/0.48  thf(fact_357_add__Suc,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 0.29/0.48        = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_Suc
% 0.29/0.48  thf(fact_358_group__cancel_Oadd1,axiom,
% 0.29/0.48      ! [A2: real,K: real,A: real,B: real] :
% 0.29/0.48        ( ( A2
% 0.29/0.48          = ( plus_plus_real @ K @ A ) )
% 0.29/0.48       => ( ( plus_plus_real @ A2 @ B )
% 0.29/0.48          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add1
% 0.29/0.48  thf(fact_359_group__cancel_Oadd1,axiom,
% 0.29/0.48      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.29/0.48        ( ( A2
% 0.29/0.48          = ( plus_plus_rat @ K @ A ) )
% 0.29/0.48       => ( ( plus_plus_rat @ A2 @ B )
% 0.29/0.48          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add1
% 0.29/0.48  thf(fact_360_group__cancel_Oadd1,axiom,
% 0.29/0.48      ! [A2: nat,K: nat,A: nat,B: nat] :
% 0.29/0.48        ( ( A2
% 0.29/0.48          = ( plus_plus_nat @ K @ A ) )
% 0.29/0.48       => ( ( plus_plus_nat @ A2 @ B )
% 0.29/0.48          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add1
% 0.29/0.48  thf(fact_361_group__cancel_Oadd1,axiom,
% 0.29/0.48      ! [A2: int,K: int,A: int,B: int] :
% 0.29/0.48        ( ( A2
% 0.29/0.48          = ( plus_plus_int @ K @ A ) )
% 0.29/0.48       => ( ( plus_plus_int @ A2 @ B )
% 0.29/0.48          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add1
% 0.29/0.48  thf(fact_362_group__cancel_Oadd2,axiom,
% 0.29/0.48      ! [B2: real,K: real,B: real,A: real] :
% 0.29/0.48        ( ( B2
% 0.29/0.48          = ( plus_plus_real @ K @ B ) )
% 0.29/0.48       => ( ( plus_plus_real @ A @ B2 )
% 0.29/0.48          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add2
% 0.29/0.48  thf(fact_363_group__cancel_Oadd2,axiom,
% 0.29/0.48      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.29/0.48        ( ( B2
% 0.29/0.48          = ( plus_plus_rat @ K @ B ) )
% 0.29/0.48       => ( ( plus_plus_rat @ A @ B2 )
% 0.29/0.48          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add2
% 0.29/0.48  thf(fact_364_group__cancel_Oadd2,axiom,
% 0.29/0.48      ! [B2: nat,K: nat,B: nat,A: nat] :
% 0.29/0.48        ( ( B2
% 0.29/0.48          = ( plus_plus_nat @ K @ B ) )
% 0.29/0.48       => ( ( plus_plus_nat @ A @ B2 )
% 0.29/0.48          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add2
% 0.29/0.48  thf(fact_365_group__cancel_Oadd2,axiom,
% 0.29/0.48      ! [B2: int,K: int,B: int,A: int] :
% 0.29/0.48        ( ( B2
% 0.29/0.48          = ( plus_plus_int @ K @ B ) )
% 0.29/0.48       => ( ( plus_plus_int @ A @ B2 )
% 0.29/0.48          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % group_cancel.add2
% 0.29/0.48  thf(fact_366_add_Oassoc,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.assoc
% 0.29/0.48  thf(fact_367_add_Oassoc,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.assoc
% 0.29/0.48  thf(fact_368_add_Oassoc,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.assoc
% 0.29/0.48  thf(fact_369_add_Oassoc,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.48        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.assoc
% 0.29/0.48  thf(fact_370_add_Oleft__cancel,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.48          = ( plus_plus_real @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_cancel
% 0.29/0.48  thf(fact_371_add_Oleft__cancel,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.48          = ( plus_plus_rat @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_cancel
% 0.29/0.48  thf(fact_372_add_Oleft__cancel,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.48          = ( plus_plus_int @ A @ C ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_cancel
% 0.29/0.48  thf(fact_373_add_Oright__cancel,axiom,
% 0.29/0.48      ! [B: real,A: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ B @ A )
% 0.29/0.48          = ( plus_plus_real @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_cancel
% 0.29/0.48  thf(fact_374_add_Oright__cancel,axiom,
% 0.29/0.48      ! [B: rat,A: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ B @ A )
% 0.29/0.48          = ( plus_plus_rat @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_cancel
% 0.29/0.48  thf(fact_375_add_Oright__cancel,axiom,
% 0.29/0.48      ! [B: int,A: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ B @ A )
% 0.29/0.48          = ( plus_plus_int @ C @ A ) )
% 0.29/0.48        = ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.right_cancel
% 0.29/0.48  thf(fact_376_add_Ocommute,axiom,
% 0.29/0.48      ( plus_plus_real
% 0.29/0.48      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.commute
% 0.29/0.48  thf(fact_377_add_Ocommute,axiom,
% 0.29/0.48      ( plus_plus_rat
% 0.29/0.48      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.commute
% 0.29/0.48  thf(fact_378_add_Ocommute,axiom,
% 0.29/0.48      ( plus_plus_nat
% 0.29/0.48      = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.commute
% 0.29/0.48  thf(fact_379_add_Ocommute,axiom,
% 0.29/0.48      ( plus_plus_int
% 0.29/0.48      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.commute
% 0.29/0.48  thf(fact_380_add_Oleft__commute,axiom,
% 0.29/0.48      ! [B: real,A: real,C: real] :
% 0.29/0.48        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 0.29/0.48        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_commute
% 0.29/0.48  thf(fact_381_add_Oleft__commute,axiom,
% 0.29/0.48      ! [B: rat,A: rat,C: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 0.29/0.48        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_commute
% 0.29/0.48  thf(fact_382_add_Oleft__commute,axiom,
% 0.29/0.48      ! [B: nat,A: nat,C: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 0.29/0.48        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_commute
% 0.29/0.48  thf(fact_383_add_Oleft__commute,axiom,
% 0.29/0.48      ! [B: int,A: int,C: int] :
% 0.29/0.48        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.29/0.48        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add.left_commute
% 0.29/0.48  thf(fact_384_add__left__imp__eq,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ A @ B )
% 0.29/0.48          = ( plus_plus_real @ A @ C ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_imp_eq
% 0.29/0.48  thf(fact_385_add__left__imp__eq,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ A @ B )
% 0.29/0.48          = ( plus_plus_rat @ A @ C ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_imp_eq
% 0.29/0.48  thf(fact_386_add__left__imp__eq,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ A @ B )
% 0.29/0.48          = ( plus_plus_nat @ A @ C ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_imp_eq
% 0.29/0.48  thf(fact_387_add__left__imp__eq,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ A @ B )
% 0.29/0.48          = ( plus_plus_int @ A @ C ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_imp_eq
% 0.29/0.48  thf(fact_388_add__right__imp__eq,axiom,
% 0.29/0.48      ! [B: real,A: real,C: real] :
% 0.29/0.48        ( ( ( plus_plus_real @ B @ A )
% 0.29/0.48          = ( plus_plus_real @ C @ A ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_imp_eq
% 0.29/0.48  thf(fact_389_add__right__imp__eq,axiom,
% 0.29/0.48      ! [B: rat,A: rat,C: rat] :
% 0.29/0.48        ( ( ( plus_plus_rat @ B @ A )
% 0.29/0.48          = ( plus_plus_rat @ C @ A ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_imp_eq
% 0.29/0.48  thf(fact_390_add__right__imp__eq,axiom,
% 0.29/0.48      ! [B: nat,A: nat,C: nat] :
% 0.29/0.48        ( ( ( plus_plus_nat @ B @ A )
% 0.29/0.48          = ( plus_plus_nat @ C @ A ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_imp_eq
% 0.29/0.48  thf(fact_391_add__right__imp__eq,axiom,
% 0.29/0.48      ! [B: int,A: int,C: int] :
% 0.29/0.48        ( ( ( plus_plus_int @ B @ A )
% 0.29/0.48          = ( plus_plus_int @ C @ A ) )
% 0.29/0.48       => ( B = C ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_imp_eq
% 0.29/0.48  thf(fact_392_Suc__inject,axiom,
% 0.29/0.48      ! [X3: nat,Y3: nat] :
% 0.29/0.48        ( ( ( suc @ X3 )
% 0.29/0.48          = ( suc @ Y3 ) )
% 0.29/0.48       => ( X3 = Y3 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % Suc_inject
% 0.29/0.48  thf(fact_393_n__not__Suc__n,axiom,
% 0.29/0.48      ! [N2: nat] :
% 0.29/0.48        ( N2
% 0.29/0.48       != ( suc @ N2 ) ) ).
% 0.29/0.48  
% 0.29/0.48  % n_not_Suc_n
% 0.29/0.48  thf(fact_394_add__Suc__shift,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 0.29/0.48        = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_Suc_shift
% 0.29/0.48  thf(fact_395_less__imp__Suc__add,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.48       => ? [K2: nat] :
% 0.29/0.48            ( N2
% 0.29/0.48            = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_imp_Suc_add
% 0.29/0.48  thf(fact_396_less__iff__Suc__add,axiom,
% 0.29/0.48      ( ord_less_nat
% 0.29/0.48      = ( ^ [M2: nat,N: nat] :
% 0.29/0.48          ? [K3: nat] :
% 0.29/0.48            ( N
% 0.29/0.48            = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_iff_Suc_add
% 0.29/0.48  thf(fact_397_less__add__Suc2,axiom,
% 0.29/0.48      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_Suc2
% 0.29/0.48  thf(fact_398_less__add__Suc1,axiom,
% 0.29/0.48      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_add_Suc1
% 0.29/0.48  thf(fact_399_less__natE,axiom,
% 0.29/0.48      ! [M: nat,N2: nat] :
% 0.29/0.48        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.48       => ~ ! [Q2: nat] :
% 0.29/0.48              ( N2
% 0.29/0.48             != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_natE
% 0.29/0.48  thf(fact_400_bot_Oextremum,axiom,
% 0.29/0.48      ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum
% 0.29/0.48  thf(fact_401_bot_Oextremum,axiom,
% 0.29/0.48      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum
% 0.29/0.48  thf(fact_402_bot_Oextremum,axiom,
% 0.29/0.48      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum
% 0.29/0.48  thf(fact_403_bot_Oextremum,axiom,
% 0.29/0.48      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum
% 0.29/0.48  thf(fact_404_bot_Oextremum,axiom,
% 0.29/0.48      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum
% 0.29/0.48  thf(fact_405_bot_Oextremum__unique,axiom,
% 0.29/0.48      ! [A: set_Pr1261947904930325089at_nat] :
% 0.29/0.48        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.29/0.48        = ( A = bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_unique
% 0.29/0.48  thf(fact_406_bot_Oextremum__unique,axiom,
% 0.29/0.48      ! [A: set_real] :
% 0.29/0.48        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.29/0.48        = ( A = bot_bot_set_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_unique
% 0.29/0.48  thf(fact_407_bot_Oextremum__unique,axiom,
% 0.29/0.48      ! [A: set_nat] :
% 0.29/0.48        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.29/0.48        = ( A = bot_bot_set_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_unique
% 0.29/0.48  thf(fact_408_bot_Oextremum__unique,axiom,
% 0.29/0.48      ! [A: set_int] :
% 0.29/0.48        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.29/0.48        = ( A = bot_bot_set_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_unique
% 0.29/0.48  thf(fact_409_bot_Oextremum__unique,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.48        = ( A = bot_bot_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_unique
% 0.29/0.48  thf(fact_410_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48      ! [A: set_Pr1261947904930325089at_nat] :
% 0.29/0.48        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.29/0.48       => ( A = bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_uniqueI
% 0.29/0.48  thf(fact_411_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48      ! [A: set_real] :
% 0.29/0.48        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 0.29/0.48       => ( A = bot_bot_set_real ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_uniqueI
% 0.29/0.48  thf(fact_412_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48      ! [A: set_nat] :
% 0.29/0.48        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 0.29/0.48       => ( A = bot_bot_set_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_uniqueI
% 0.29/0.48  thf(fact_413_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48      ! [A: set_int] :
% 0.29/0.48        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 0.29/0.48       => ( A = bot_bot_set_int ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_uniqueI
% 0.29/0.48  thf(fact_414_bot_Oextremum__uniqueI,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 0.29/0.48       => ( A = bot_bot_nat ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_uniqueI
% 0.29/0.48  thf(fact_415_bot_Oextremum__strict,axiom,
% 0.29/0.48      ! [A: set_Pr1261947904930325089at_nat] :
% 0.29/0.48        ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_strict
% 0.29/0.48  thf(fact_416_bot_Oextremum__strict,axiom,
% 0.29/0.48      ! [A: set_real] :
% 0.29/0.48        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_strict
% 0.29/0.48  thf(fact_417_bot_Oextremum__strict,axiom,
% 0.29/0.48      ! [A: set_nat] :
% 0.29/0.48        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_strict
% 0.29/0.48  thf(fact_418_bot_Oextremum__strict,axiom,
% 0.29/0.48      ! [A: set_int] :
% 0.29/0.48        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_strict
% 0.29/0.48  thf(fact_419_bot_Oextremum__strict,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.extremum_strict
% 0.29/0.48  thf(fact_420_bot_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: set_Pr1261947904930325089at_nat] :
% 0.29/0.48        ( ( A != bot_bo2099793752762293965at_nat )
% 0.29/0.48        = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.not_eq_extremum
% 0.29/0.48  thf(fact_421_bot_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: set_real] :
% 0.29/0.48        ( ( A != bot_bot_set_real )
% 0.29/0.48        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.not_eq_extremum
% 0.29/0.48  thf(fact_422_bot_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: set_nat] :
% 0.29/0.48        ( ( A != bot_bot_set_nat )
% 0.29/0.48        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.not_eq_extremum
% 0.29/0.48  thf(fact_423_bot_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: set_int] :
% 0.29/0.48        ( ( A != bot_bot_set_int )
% 0.29/0.48        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.not_eq_extremum
% 0.29/0.48  thf(fact_424_bot_Onot__eq__extremum,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( A != bot_bot_nat )
% 0.29/0.48        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 0.29/0.48  
% 0.29/0.48  % bot.not_eq_extremum
% 0.29/0.48  thf(fact_425_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( ord_less_eq_real @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(3)
% 0.29/0.48  thf(fact_426_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( ord_less_eq_rat @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(3)
% 0.29/0.48  thf(fact_427_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(3)
% 0.29/0.48  thf(fact_428_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( ord_less_eq_int @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(3)
% 0.29/0.48  thf(fact_429_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_eq_real @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(2)
% 0.29/0.48  thf(fact_430_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_eq_rat @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(2)
% 0.29/0.48  thf(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_eq_nat @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(2)
% 0.29/0.48  thf(fact_432_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_eq_int @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(2)
% 0.29/0.48  thf(fact_433_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( ord_less_eq_real @ I @ J )
% 0.29/0.48          & ( ord_less_eq_real @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(1)
% 0.29/0.48  thf(fact_434_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( ord_less_eq_rat @ I @ J )
% 0.29/0.48          & ( ord_less_eq_rat @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(1)
% 0.29/0.48  thf(fact_435_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.48          & ( ord_less_eq_nat @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(1)
% 0.29/0.48  thf(fact_436_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( ord_less_eq_int @ I @ J )
% 0.29/0.48          & ( ord_less_eq_int @ K @ L ) )
% 0.29/0.48       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_semiring(1)
% 0.29/0.48  thf(fact_437_add__mono,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.48       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.48         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono
% 0.29/0.48  thf(fact_438_add__mono,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.48       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.48         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono
% 0.29/0.48  thf(fact_439_add__mono,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.48         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono
% 0.29/0.48  thf(fact_440_add__mono,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.48       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.48         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono
% 0.29/0.48  thf(fact_441_add__left__mono,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.48       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_mono
% 0.29/0.48  thf(fact_442_add__left__mono,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.48       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_mono
% 0.29/0.48  thf(fact_443_add__left__mono,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_mono
% 0.29/0.48  thf(fact_444_add__left__mono,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.48       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_left_mono
% 0.29/0.48  thf(fact_445_less__eqE,axiom,
% 0.29/0.48      ! [A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48       => ~ ! [C2: nat] :
% 0.29/0.48              ( B
% 0.29/0.48             != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % less_eqE
% 0.29/0.48  thf(fact_446_add__right__mono,axiom,
% 0.29/0.48      ! [A: real,B: real,C: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.48       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_mono
% 0.29/0.48  thf(fact_447_add__right__mono,axiom,
% 0.29/0.48      ! [A: rat,B: rat,C: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.48       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_mono
% 0.29/0.48  thf(fact_448_add__right__mono,axiom,
% 0.29/0.48      ! [A: nat,B: nat,C: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.48       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_mono
% 0.29/0.48  thf(fact_449_add__right__mono,axiom,
% 0.29/0.48      ! [A: int,B: int,C: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.48       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_right_mono
% 0.29/0.48  thf(fact_450_le__iff__add,axiom,
% 0.29/0.48      ( ord_less_eq_nat
% 0.29/0.48      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.48          ? [C3: nat] :
% 0.29/0.48            ( B3
% 0.29/0.48            = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % le_iff_add
% 0.29/0.48  thf(fact_451_add__le__imp__le__left,axiom,
% 0.29/0.48      ! [C: real,A: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.29/0.48       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_left
% 0.29/0.48  thf(fact_452_add__le__imp__le__left,axiom,
% 0.29/0.48      ! [C: rat,A: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.29/0.48       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_left
% 0.29/0.48  thf(fact_453_add__le__imp__le__left,axiom,
% 0.29/0.48      ! [C: nat,A: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.48       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_left
% 0.29/0.48  thf(fact_454_add__le__imp__le__left,axiom,
% 0.29/0.48      ! [C: int,A: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.29/0.48       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_left
% 0.29/0.48  thf(fact_455_add__le__imp__le__right,axiom,
% 0.29/0.48      ! [A: real,C: real,B: real] :
% 0.29/0.48        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.29/0.48       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_right
% 0.29/0.48  thf(fact_456_add__le__imp__le__right,axiom,
% 0.29/0.48      ! [A: rat,C: rat,B: rat] :
% 0.29/0.48        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.48       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_right
% 0.29/0.48  thf(fact_457_add__le__imp__le__right,axiom,
% 0.29/0.48      ! [A: nat,C: nat,B: nat] :
% 0.29/0.48        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.48       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_right
% 0.29/0.48  thf(fact_458_add__le__imp__le__right,axiom,
% 0.29/0.48      ! [A: int,C: int,B: int] :
% 0.29/0.48        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.29/0.48       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_le_imp_le_right
% 0.29/0.48  thf(fact_459_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.48      ! [A: complex] :
% 0.29/0.48        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % comm_monoid_add_class.add_0
% 0.29/0.48  thf(fact_460_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % comm_monoid_add_class.add_0
% 0.29/0.48  thf(fact_461_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % comm_monoid_add_class.add_0
% 0.29/0.48  thf(fact_462_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % comm_monoid_add_class.add_0
% 0.29/0.48  thf(fact_463_comm__monoid__add__class_Oadd__0,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % comm_monoid_add_class.add_0
% 0.29/0.48  thf(fact_464_add_Ocomm__neutral,axiom,
% 0.29/0.48      ! [A: complex] :
% 0.29/0.48        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.comm_neutral
% 0.29/0.48  thf(fact_465_add_Ocomm__neutral,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.comm_neutral
% 0.29/0.48  thf(fact_466_add_Ocomm__neutral,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.comm_neutral
% 0.29/0.48  thf(fact_467_add_Ocomm__neutral,axiom,
% 0.29/0.48      ! [A: nat] :
% 0.29/0.48        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.comm_neutral
% 0.29/0.48  thf(fact_468_add_Ocomm__neutral,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.comm_neutral
% 0.29/0.48  thf(fact_469_add_Ogroup__left__neutral,axiom,
% 0.29/0.48      ! [A: complex] :
% 0.29/0.48        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.group_left_neutral
% 0.29/0.48  thf(fact_470_add_Ogroup__left__neutral,axiom,
% 0.29/0.48      ! [A: real] :
% 0.29/0.48        ( ( plus_plus_real @ zero_zero_real @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.group_left_neutral
% 0.29/0.48  thf(fact_471_add_Ogroup__left__neutral,axiom,
% 0.29/0.48      ! [A: rat] :
% 0.29/0.48        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.group_left_neutral
% 0.29/0.48  thf(fact_472_add_Ogroup__left__neutral,axiom,
% 0.29/0.48      ! [A: int] :
% 0.29/0.48        ( ( plus_plus_int @ zero_zero_int @ A )
% 0.29/0.48        = A ) ).
% 0.29/0.48  
% 0.29/0.48  % add.group_left_neutral
% 0.29/0.48  thf(fact_473_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( ord_less_real @ I @ J )
% 0.29/0.48          & ( ord_less_real @ K @ L ) )
% 0.29/0.48       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(5)
% 0.29/0.48  thf(fact_474_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( ord_less_rat @ I @ J )
% 0.29/0.48          & ( ord_less_rat @ K @ L ) )
% 0.29/0.48       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(5)
% 0.29/0.48  thf(fact_475_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( ord_less_nat @ I @ J )
% 0.29/0.48          & ( ord_less_nat @ K @ L ) )
% 0.29/0.48       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(5)
% 0.29/0.48  thf(fact_476_add__mono__thms__linordered__field_I5_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( ord_less_int @ I @ J )
% 0.29/0.48          & ( ord_less_int @ K @ L ) )
% 0.29/0.48       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(5)
% 0.29/0.48  thf(fact_477_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_real @ K @ L ) )
% 0.29/0.48       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(2)
% 0.29/0.48  thf(fact_478_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_rat @ K @ L ) )
% 0.29/0.48       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(2)
% 0.29/0.48  thf(fact_479_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.48      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_nat @ K @ L ) )
% 0.29/0.48       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(2)
% 0.29/0.48  thf(fact_480_add__mono__thms__linordered__field_I2_J,axiom,
% 0.29/0.48      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.48        ( ( ( I = J )
% 0.29/0.48          & ( ord_less_int @ K @ L ) )
% 0.29/0.48       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(2)
% 0.29/0.48  thf(fact_481_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.48      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.48        ( ( ( ord_less_real @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(1)
% 0.29/0.48  thf(fact_482_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.48      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.48        ( ( ( ord_less_rat @ I @ J )
% 0.29/0.48          & ( K = L ) )
% 0.29/0.48       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.48  
% 0.29/0.48  % add_mono_thms_linordered_field(1)
% 0.29/0.48  thf(fact_483_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.49        ( ( ( ord_less_nat @ I @ J )
% 0.29/0.49          & ( K = L ) )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(1)
% 0.29/0.49  thf(fact_484_add__mono__thms__linordered__field_I1_J,axiom,
% 0.29/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.49        ( ( ( ord_less_int @ I @ J )
% 0.29/0.49          & ( K = L ) )
% 0.29/0.49       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(1)
% 0.29/0.49  thf(fact_485_add__strict__mono,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ( ord_less_real @ C @ D )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_mono
% 0.29/0.49  thf(fact_486_add__strict__mono,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_rat @ C @ D )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_mono
% 0.29/0.49  thf(fact_487_add__strict__mono,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_nat @ C @ D )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_mono
% 0.29/0.49  thf(fact_488_add__strict__mono,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_int @ C @ D )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_mono
% 0.29/0.49  thf(fact_489_add__strict__left__mono,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_left_mono
% 0.29/0.49  thf(fact_490_add__strict__left__mono,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_left_mono
% 0.29/0.49  thf(fact_491_add__strict__left__mono,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_left_mono
% 0.29/0.49  thf(fact_492_add__strict__left__mono,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_left_mono
% 0.29/0.49  thf(fact_493_add__strict__right__mono,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_right_mono
% 0.29/0.49  thf(fact_494_add__strict__right__mono,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_right_mono
% 0.29/0.49  thf(fact_495_add__strict__right__mono,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_right_mono
% 0.29/0.49  thf(fact_496_add__strict__right__mono,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_right_mono
% 0.29/0.49  thf(fact_497_add__less__imp__less__left,axiom,
% 0.29/0.49      ! [C: real,A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.29/0.49       => ( ord_less_real @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_left
% 0.29/0.49  thf(fact_498_add__less__imp__less__left,axiom,
% 0.29/0.49      ! [C: rat,A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.29/0.49       => ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_left
% 0.29/0.49  thf(fact_499_add__less__imp__less__left,axiom,
% 0.29/0.49      ! [C: nat,A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.29/0.49       => ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_left
% 0.29/0.49  thf(fact_500_add__less__imp__less__left,axiom,
% 0.29/0.49      ! [C: int,A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.29/0.49       => ( ord_less_int @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_left
% 0.29/0.49  thf(fact_501_add__less__imp__less__right,axiom,
% 0.29/0.49      ! [A: real,C: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.29/0.49       => ( ord_less_real @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_right
% 0.29/0.49  thf(fact_502_add__less__imp__less__right,axiom,
% 0.29/0.49      ! [A: rat,C: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.49       => ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_right
% 0.29/0.49  thf(fact_503_add__less__imp__less__right,axiom,
% 0.29/0.49      ! [A: nat,C: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.49       => ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_right
% 0.29/0.49  thf(fact_504_add__less__imp__less__right,axiom,
% 0.29/0.49      ! [A: int,C: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.29/0.49       => ( ord_less_int @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_imp_less_right
% 0.29/0.49  thf(fact_505_nat_Odistinct_I1_J,axiom,
% 0.29/0.49      ! [X22: nat] :
% 0.29/0.49        ( zero_zero_nat
% 0.29/0.49       != ( suc @ X22 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nat.distinct(1)
% 0.29/0.49  thf(fact_506_old_Onat_Odistinct_I2_J,axiom,
% 0.29/0.49      ! [Nat2: nat] :
% 0.29/0.49        ( ( suc @ Nat2 )
% 0.29/0.49       != zero_zero_nat ) ).
% 0.29/0.49  
% 0.29/0.49  % old.nat.distinct(2)
% 0.29/0.49  thf(fact_507_old_Onat_Odistinct_I1_J,axiom,
% 0.29/0.49      ! [Nat2: nat] :
% 0.29/0.49        ( zero_zero_nat
% 0.29/0.49       != ( suc @ Nat2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % old.nat.distinct(1)
% 0.29/0.49  thf(fact_508_nat_OdiscI,axiom,
% 0.29/0.49      ! [Nat: nat,X22: nat] :
% 0.29/0.49        ( ( Nat
% 0.29/0.49          = ( suc @ X22 ) )
% 0.29/0.49       => ( Nat != zero_zero_nat ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nat.discI
% 0.29/0.49  thf(fact_509_old_Onat_Oexhaust,axiom,
% 0.29/0.49      ! [Y3: nat] :
% 0.29/0.49        ( ( Y3 != zero_zero_nat )
% 0.29/0.49       => ~ ! [Nat3: nat] :
% 0.29/0.49              ( Y3
% 0.29/0.49             != ( suc @ Nat3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % old.nat.exhaust
% 0.29/0.49  thf(fact_510_nat__induct,axiom,
% 0.29/0.49      ! [P: nat > $o,N2: nat] :
% 0.29/0.49        ( ( P @ zero_zero_nat )
% 0.29/0.49       => ( ! [N3: nat] :
% 0.29/0.49              ( ( P @ N3 )
% 0.29/0.49             => ( P @ ( suc @ N3 ) ) )
% 0.29/0.49         => ( P @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nat_induct
% 0.29/0.49  thf(fact_511_diff__induct,axiom,
% 0.29/0.49      ! [P: nat > nat > $o,M: nat,N2: nat] :
% 0.29/0.49        ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 0.29/0.49       => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( P @ X4 @ Y5 )
% 0.29/0.49               => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 0.29/0.49           => ( P @ M @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % diff_induct
% 0.29/0.49  thf(fact_512_zero__induct,axiom,
% 0.29/0.49      ! [P: nat > $o,K: nat] :
% 0.29/0.49        ( ( P @ K )
% 0.29/0.49       => ( ! [N3: nat] :
% 0.29/0.49              ( ( P @ ( suc @ N3 ) )
% 0.29/0.49             => ( P @ N3 ) )
% 0.29/0.49         => ( P @ zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_induct
% 0.29/0.49  thf(fact_513_Suc__neq__Zero,axiom,
% 0.29/0.49      ! [M: nat] :
% 0.29/0.49        ( ( suc @ M )
% 0.29/0.49       != zero_zero_nat ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_neq_Zero
% 0.29/0.49  thf(fact_514_Zero__neq__Suc,axiom,
% 0.29/0.49      ! [M: nat] :
% 0.29/0.49        ( zero_zero_nat
% 0.29/0.49       != ( suc @ M ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Zero_neq_Suc
% 0.29/0.49  thf(fact_515_Zero__not__Suc,axiom,
% 0.29/0.49      ! [M: nat] :
% 0.29/0.49        ( zero_zero_nat
% 0.29/0.49       != ( suc @ M ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Zero_not_Suc
% 0.29/0.49  thf(fact_516_not0__implies__Suc,axiom,
% 0.29/0.49      ! [N2: nat] :
% 0.29/0.49        ( ( N2 != zero_zero_nat )
% 0.29/0.49       => ? [M3: nat] :
% 0.29/0.49            ( N2
% 0.29/0.49            = ( suc @ M3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not0_implies_Suc
% 0.29/0.49  thf(fact_517_plus__nat_Oadd__0,axiom,
% 0.29/0.49      ! [N2: nat] :
% 0.29/0.49        ( ( plus_plus_nat @ zero_zero_nat @ N2 )
% 0.29/0.49        = N2 ) ).
% 0.29/0.49  
% 0.29/0.49  % plus_nat.add_0
% 0.29/0.49  thf(fact_518_add__eq__self__zero,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ( plus_plus_nat @ M @ N2 )
% 0.29/0.49          = M )
% 0.29/0.49       => ( N2 = zero_zero_nat ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_eq_self_zero
% 0.29/0.49  thf(fact_519_Nat_OlessE,axiom,
% 0.29/0.49      ! [I: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ K )
% 0.29/0.49       => ( ( K
% 0.29/0.49           != ( suc @ I ) )
% 0.29/0.49         => ~ ! [J2: nat] :
% 0.29/0.49                ( ( ord_less_nat @ I @ J2 )
% 0.29/0.49               => ( K
% 0.29/0.49                 != ( suc @ J2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Nat.lessE
% 0.29/0.49  thf(fact_520_Suc__lessD,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( suc @ M ) @ N2 )
% 0.29/0.49       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_lessD
% 0.29/0.49  thf(fact_521_Suc__lessE,axiom,
% 0.29/0.49      ! [I: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 0.29/0.49       => ~ ! [J2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I @ J2 )
% 0.29/0.49             => ( K
% 0.29/0.49               != ( suc @ J2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_lessE
% 0.29/0.49  thf(fact_522_Suc__lessI,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.49       => ( ( ( suc @ M )
% 0.29/0.49           != N2 )
% 0.29/0.49         => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_lessI
% 0.29/0.49  thf(fact_523_less__SucE,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49       => ( ~ ( ord_less_nat @ M @ N2 )
% 0.29/0.49         => ( M = N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_SucE
% 0.29/0.49  thf(fact_524_less__SucI,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.49       => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_SucI
% 0.29/0.49  thf(fact_525_Ex__less__Suc,axiom,
% 0.29/0.49      ! [N2: nat,P: nat > $o] :
% 0.29/0.49        ( ( ? [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 0.29/0.49              & ( P @ I2 ) ) )
% 0.29/0.49        = ( ( P @ N2 )
% 0.29/0.49          | ? [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ N2 )
% 0.29/0.49              & ( P @ I2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Ex_less_Suc
% 0.29/0.49  thf(fact_526_less__Suc__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49        = ( ( ord_less_nat @ M @ N2 )
% 0.29/0.49          | ( M = N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_Suc_eq
% 0.29/0.49  thf(fact_527_not__less__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ~ ( ord_less_nat @ M @ N2 ) )
% 0.29/0.49        = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_eq
% 0.29/0.49  thf(fact_528_All__less__Suc,axiom,
% 0.29/0.49      ! [N2: nat,P: nat > $o] :
% 0.29/0.49        ( ( ! [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 0.29/0.49             => ( P @ I2 ) ) )
% 0.29/0.49        = ( ( P @ N2 )
% 0.29/0.49          & ! [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ N2 )
% 0.29/0.49             => ( P @ I2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % All_less_Suc
% 0.29/0.49  thf(fact_529_Suc__less__eq2,axiom,
% 0.29/0.49      ! [N2: nat,M: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.29/0.49        = ( ? [M4: nat] :
% 0.29/0.49              ( ( M
% 0.29/0.49                = ( suc @ M4 ) )
% 0.29/0.49              & ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_less_eq2
% 0.29/0.49  thf(fact_530_less__antisym,axiom,
% 0.29/0.49      ! [N2: nat,M: nat] :
% 0.29/0.49        ( ~ ( ord_less_nat @ N2 @ M )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.29/0.49         => ( M = N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_antisym
% 0.29/0.49  thf(fact_531_Suc__less__SucD,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.29/0.49       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_less_SucD
% 0.29/0.49  thf(fact_532_less__trans__Suc,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ( ord_less_nat @ J @ K )
% 0.29/0.49         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_trans_Suc
% 0.29/0.49  thf(fact_533_less__Suc__induct,axiom,
% 0.29/0.49      ! [I: nat,J: nat,P: nat > nat > $o] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 0.29/0.49         => ( ! [I3: nat,J2: nat,K2: nat] :
% 0.29/0.49                ( ( ord_less_nat @ I3 @ J2 )
% 0.29/0.49               => ( ( ord_less_nat @ J2 @ K2 )
% 0.29/0.49                 => ( ( P @ I3 @ J2 )
% 0.29/0.49                   => ( ( P @ J2 @ K2 )
% 0.29/0.49                     => ( P @ I3 @ K2 ) ) ) ) )
% 0.29/0.49           => ( P @ I @ J ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_Suc_induct
% 0.29/0.49  thf(fact_534_strict__inc__induct,axiom,
% 0.29/0.49      ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ! [I3: nat] :
% 0.29/0.49              ( ( J
% 0.29/0.49                = ( suc @ I3 ) )
% 0.29/0.49             => ( P @ I3 ) )
% 0.29/0.49         => ( ! [I3: nat] :
% 0.29/0.49                ( ( ord_less_nat @ I3 @ J )
% 0.29/0.49               => ( ( P @ ( suc @ I3 ) )
% 0.29/0.49                 => ( P @ I3 ) ) )
% 0.29/0.49           => ( P @ I ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % strict_inc_induct
% 0.29/0.49  thf(fact_535_not__less__less__Suc__eq,axiom,
% 0.29/0.49      ! [N2: nat,M: nat] :
% 0.29/0.49        ( ~ ( ord_less_nat @ N2 @ M )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.29/0.49          = ( N2 = M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_less_Suc_eq
% 0.29/0.49  thf(fact_536_add__lessD1,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.29/0.49       => ( ord_less_nat @ I @ K ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_lessD1
% 0.29/0.49  thf(fact_537_add__less__mono,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ( ord_less_nat @ K @ L )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_mono
% 0.29/0.49  thf(fact_538_not__add__less1,axiom,
% 0.29/0.49      ! [I: nat,J: nat] :
% 0.29/0.49        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 0.29/0.49  
% 0.29/0.49  % not_add_less1
% 0.29/0.49  thf(fact_539_not__add__less2,axiom,
% 0.29/0.49      ! [J: nat,I: nat] :
% 0.29/0.49        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 0.29/0.49  
% 0.29/0.49  % not_add_less2
% 0.29/0.49  thf(fact_540_add__less__mono1,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_mono1
% 0.29/0.49  thf(fact_541_trans__less__add1,axiom,
% 0.29/0.49      ! [I: nat,J: nat,M: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % trans_less_add1
% 0.29/0.49  thf(fact_542_trans__less__add2,axiom,
% 0.29/0.49      ! [I: nat,J: nat,M: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % trans_less_add2
% 0.29/0.49  thf(fact_543_less__add__eq__less,axiom,
% 0.29/0.49      ! [K: nat,L: nat,M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ K @ L )
% 0.29/0.49       => ( ( ( plus_plus_nat @ M @ L )
% 0.29/0.49            = ( plus_plus_nat @ K @ N2 ) )
% 0.29/0.49         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_add_eq_less
% 0.29/0.49  thf(fact_544_Suc__leD,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.29/0.49       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_leD
% 0.29/0.49  thf(fact_545_le__SucE,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49       => ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49         => ( M
% 0.29/0.49            = ( suc @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_SucE
% 0.29/0.49  thf(fact_546_le__SucI,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49       => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_SucI
% 0.29/0.49  thf(fact_547_Suc__le__D,axiom,
% 0.29/0.49      ! [N2: nat,M5: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
% 0.29/0.49       => ? [M3: nat] :
% 0.29/0.49            ( M5
% 0.29/0.49            = ( suc @ M3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_le_D
% 0.29/0.49  thf(fact_548_le__Suc__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49        = ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49          | ( M
% 0.29/0.49            = ( suc @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_Suc_eq
% 0.29/0.49  thf(fact_549_Suc__n__not__le__n,axiom,
% 0.29/0.49      ! [N2: nat] :
% 0.29/0.49        ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_n_not_le_n
% 0.29/0.49  thf(fact_550_not__less__eq__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
% 0.29/0.49        = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_eq_eq
% 0.29/0.49  thf(fact_551_full__nat__induct,axiom,
% 0.29/0.49      ! [P: nat > $o,N2: nat] :
% 0.29/0.49        ( ! [N3: nat] :
% 0.29/0.49            ( ! [M6: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
% 0.29/0.49               => ( P @ M6 ) )
% 0.29/0.49           => ( P @ N3 ) )
% 0.29/0.49       => ( P @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % full_nat_induct
% 0.29/0.49  thf(fact_552_nat__induct__at__least,axiom,
% 0.29/0.49      ! [M: nat,N2: nat,P: nat > $o] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49       => ( ( P @ M )
% 0.29/0.49         => ( ! [N3: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ M @ N3 )
% 0.29/0.49               => ( ( P @ N3 )
% 0.29/0.49                 => ( P @ ( suc @ N3 ) ) ) )
% 0.29/0.49           => ( P @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nat_induct_at_least
% 0.29/0.49  thf(fact_553_transitive__stepwise__le,axiom,
% 0.29/0.49      ! [M: nat,N2: nat,R: nat > nat > $o] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49       => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat,Z3: nat] :
% 0.29/0.49                ( ( R @ X4 @ Y5 )
% 0.29/0.49               => ( ( R @ Y5 @ Z3 )
% 0.29/0.49                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.49           => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 0.29/0.49             => ( R @ M @ N2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % transitive_stepwise_le
% 0.29/0.49  thf(fact_554_add__leE,axiom,
% 0.29/0.49      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.29/0.49       => ~ ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49           => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_leE
% 0.29/0.49  thf(fact_555_le__add1,axiom,
% 0.29/0.49      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_add1
% 0.29/0.49  thf(fact_556_le__add2,axiom,
% 0.29/0.49      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_add2
% 0.29/0.49  thf(fact_557_add__leD1,axiom,
% 0.29/0.49      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.29/0.49       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_leD1
% 0.29/0.49  thf(fact_558_add__leD2,axiom,
% 0.29/0.49      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 0.29/0.49       => ( ord_less_eq_nat @ K @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_leD2
% 0.29/0.49  thf(fact_559_le__Suc__ex,axiom,
% 0.29/0.49      ! [K: nat,L: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.49       => ? [N3: nat] :
% 0.29/0.49            ( L
% 0.29/0.49            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_Suc_ex
% 0.29/0.49  thf(fact_560_add__le__mono,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_mono
% 0.29/0.49  thf(fact_561_add__le__mono1,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_mono1
% 0.29/0.49  thf(fact_562_trans__le__add1,axiom,
% 0.29/0.49      ! [I: nat,J: nat,M: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % trans_le_add1
% 0.29/0.49  thf(fact_563_trans__le__add2,axiom,
% 0.29/0.49      ! [I: nat,J: nat,M: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % trans_le_add2
% 0.29/0.49  thf(fact_564_nat__le__iff__add,axiom,
% 0.29/0.49      ( ord_less_eq_nat
% 0.29/0.49      = ( ^ [M2: nat,N: nat] :
% 0.29/0.49          ? [K3: nat] :
% 0.29/0.49            ( N
% 0.29/0.49            = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nat_le_iff_add
% 0.29/0.49  thf(fact_565_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.29/0.49        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: product_prod_nat_nat] :
% 0.29/0.49              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.49             => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_566_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_real,B2: set_real] :
% 0.29/0.49        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: real] :
% 0.29/0.49              ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.29/0.49             => ( member_real @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_567_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_set_nat,B2: set_set_nat] :
% 0.29/0.49        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: set_nat] :
% 0.29/0.49              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.29/0.49             => ( member_set_nat @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_568_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 0.29/0.49        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: vEBT_VEBT] :
% 0.29/0.49              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.49             => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_569_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_nat,B2: set_nat] :
% 0.29/0.49        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: nat] :
% 0.29/0.49              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.49             => ( member_nat @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_570_subset__code_I1_J,axiom,
% 0.29/0.49      ! [Xs: list_int,B2: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 0.29/0.49        = ( ! [X: int] :
% 0.29/0.49              ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.49             => ( member_int @ X @ B2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % subset_code(1)
% 0.29/0.49  thf(fact_571_add__decreasing,axiom,
% 0.29/0.49      ! [A: real,C: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.49         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing
% 0.29/0.49  thf(fact_572_add__decreasing,axiom,
% 0.29/0.49      ! [A: rat,C: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_eq_rat @ C @ B )
% 0.29/0.49         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing
% 0.29/0.49  thf(fact_573_add__decreasing,axiom,
% 0.29/0.49      ! [A: nat,C: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing
% 0.29/0.49  thf(fact_574_add__decreasing,axiom,
% 0.29/0.49      ! [A: int,C: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_eq_int @ C @ B )
% 0.29/0.49         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing
% 0.29/0.49  thf(fact_575_add__increasing,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.49         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing
% 0.29/0.49  thf(fact_576_add__increasing,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing
% 0.29/0.49  thf(fact_577_add__increasing,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing
% 0.29/0.49  thf(fact_578_add__increasing,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing
% 0.29/0.49  thf(fact_579_add__decreasing2,axiom,
% 0.29/0.49      ! [C: real,A: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.49         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing2
% 0.29/0.49  thf(fact_580_add__decreasing2,axiom,
% 0.29/0.49      ! [C: rat,A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing2
% 0.29/0.49  thf(fact_581_add__decreasing2,axiom,
% 0.29/0.49      ! [C: nat,A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing2
% 0.29/0.49  thf(fact_582_add__decreasing2,axiom,
% 0.29/0.49      ! [C: int,A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_decreasing2
% 0.29/0.49  thf(fact_583_add__increasing2,axiom,
% 0.29/0.49      ! [C: real,B: real,A: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.49       => ( ( ord_less_eq_real @ B @ A )
% 0.29/0.49         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing2
% 0.29/0.49  thf(fact_584_add__increasing2,axiom,
% 0.29/0.49      ! [C: rat,B: rat,A: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.49         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing2
% 0.29/0.49  thf(fact_585_add__increasing2,axiom,
% 0.29/0.49      ! [C: nat,B: nat,A: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.49         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing2
% 0.29/0.49  thf(fact_586_add__increasing2,axiom,
% 0.29/0.49      ! [C: int,B: int,A: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ A )
% 0.29/0.49         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_increasing2
% 0.29/0.49  thf(fact_587_add__nonneg__nonneg,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.49         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_nonneg
% 0.29/0.49  thf(fact_588_add__nonneg__nonneg,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.49         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_nonneg
% 0.29/0.49  thf(fact_589_add__nonneg__nonneg,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.49         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_nonneg
% 0.29/0.49  thf(fact_590_add__nonneg__nonneg,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.49         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_nonneg
% 0.29/0.49  thf(fact_591_add__nonpos__nonpos,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.49         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_nonpos
% 0.29/0.49  thf(fact_592_add__nonpos__nonpos,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.49         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_nonpos
% 0.29/0.49  thf(fact_593_add__nonpos__nonpos,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.49         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_nonpos
% 0.29/0.49  thf(fact_594_add__nonpos__nonpos,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.49         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_nonpos
% 0.29/0.49  thf(fact_595_add__nonneg__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.49         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_real )
% 0.29/0.49            = ( ( X3 = zero_zero_real )
% 0.29/0.49              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_eq_0_iff
% 0.29/0.49  thf(fact_596_add__nonneg__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.49         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_rat )
% 0.29/0.49            = ( ( X3 = zero_zero_rat )
% 0.29/0.49              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_eq_0_iff
% 0.29/0.49  thf(fact_597_add__nonneg__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.29/0.49         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_nat )
% 0.29/0.49            = ( ( X3 = zero_zero_nat )
% 0.29/0.49              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_eq_0_iff
% 0.29/0.49  thf(fact_598_add__nonneg__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.49         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_int )
% 0.29/0.49            = ( ( X3 = zero_zero_int )
% 0.29/0.49              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_eq_0_iff
% 0.29/0.49  thf(fact_599_add__nonpos__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.29/0.49         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_real )
% 0.29/0.49            = ( ( X3 = zero_zero_real )
% 0.29/0.49              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_eq_0_iff
% 0.29/0.49  thf(fact_600_add__nonpos__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.29/0.49         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_rat )
% 0.29/0.49            = ( ( X3 = zero_zero_rat )
% 0.29/0.49              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_eq_0_iff
% 0.29/0.49  thf(fact_601_add__nonpos__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 0.29/0.49         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_nat )
% 0.29/0.49            = ( ( X3 = zero_zero_nat )
% 0.29/0.49              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_eq_0_iff
% 0.29/0.49  thf(fact_602_add__nonpos__eq__0__iff,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.29/0.49         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 0.29/0.49              = zero_zero_int )
% 0.29/0.49            = ( ( X3 = zero_zero_int )
% 0.29/0.49              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_eq_0_iff
% 0.29/0.49  thf(fact_603_add__less__le__mono,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_le_mono
% 0.29/0.49  thf(fact_604_add__less__le__mono,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_le_mono
% 0.29/0.49  thf(fact_605_add__less__le__mono,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_le_mono
% 0.29/0.49  thf(fact_606_add__less__le__mono,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_less_le_mono
% 0.29/0.49  thf(fact_607_add__le__less__mono,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.49       => ( ( ord_less_real @ C @ D )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_less_mono
% 0.29/0.49  thf(fact_608_add__le__less__mono,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_rat @ C @ D )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_less_mono
% 0.29/0.49  thf(fact_609_add__le__less__mono,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_nat @ C @ D )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_less_mono
% 0.29/0.49  thf(fact_610_add__le__less__mono,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_int @ C @ D )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_le_less_mono
% 0.29/0.49  thf(fact_611_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.49        ( ( ( ord_less_real @ I @ J )
% 0.29/0.49          & ( ord_less_eq_real @ K @ L ) )
% 0.29/0.49       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(3)
% 0.29/0.49  thf(fact_612_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.49        ( ( ( ord_less_rat @ I @ J )
% 0.29/0.49          & ( ord_less_eq_rat @ K @ L ) )
% 0.29/0.49       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(3)
% 0.29/0.49  thf(fact_613_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.49        ( ( ( ord_less_nat @ I @ J )
% 0.29/0.49          & ( ord_less_eq_nat @ K @ L ) )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(3)
% 0.29/0.49  thf(fact_614_add__mono__thms__linordered__field_I3_J,axiom,
% 0.29/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.49        ( ( ( ord_less_int @ I @ J )
% 0.29/0.49          & ( ord_less_eq_int @ K @ L ) )
% 0.29/0.49       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(3)
% 0.29/0.49  thf(fact_615_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.49      ! [I: real,J: real,K: real,L: real] :
% 0.29/0.49        ( ( ( ord_less_eq_real @ I @ J )
% 0.29/0.49          & ( ord_less_real @ K @ L ) )
% 0.29/0.49       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(4)
% 0.29/0.49  thf(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.49      ! [I: rat,J: rat,K: rat,L: rat] :
% 0.29/0.49        ( ( ( ord_less_eq_rat @ I @ J )
% 0.29/0.49          & ( ord_less_rat @ K @ L ) )
% 0.29/0.49       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(4)
% 0.29/0.49  thf(fact_617_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.49      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.49        ( ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49          & ( ord_less_nat @ K @ L ) )
% 0.29/0.49       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(4)
% 0.29/0.49  thf(fact_618_add__mono__thms__linordered__field_I4_J,axiom,
% 0.29/0.49      ! [I: int,J: int,K: int,L: int] :
% 0.29/0.49        ( ( ( ord_less_eq_int @ I @ J )
% 0.29/0.49          & ( ord_less_int @ K @ L ) )
% 0.29/0.49       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_mono_thms_linordered_field(4)
% 0.29/0.49  thf(fact_619_pos__add__strict,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_real @ B @ C )
% 0.29/0.49         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % pos_add_strict
% 0.29/0.49  thf(fact_620_pos__add__strict,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_rat @ B @ C )
% 0.29/0.49         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % pos_add_strict
% 0.29/0.49  thf(fact_621_pos__add__strict,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_nat @ B @ C )
% 0.29/0.49         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % pos_add_strict
% 0.29/0.49  thf(fact_622_pos__add__strict,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_int @ B @ C )
% 0.29/0.49         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % pos_add_strict
% 0.29/0.49  thf(fact_623_canonically__ordered__monoid__add__class_OlessE,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ~ ! [C2: nat] :
% 0.29/0.49              ( ( B
% 0.29/0.49                = ( plus_plus_nat @ A @ C2 ) )
% 0.29/0.49             => ( C2 = zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % canonically_ordered_monoid_add_class.lessE
% 0.29/0.49  thf(fact_624_add__pos__pos,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.49         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_pos
% 0.29/0.49  thf(fact_625_add__pos__pos,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.49         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_pos
% 0.29/0.49  thf(fact_626_add__pos__pos,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.49         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_pos
% 0.29/0.49  thf(fact_627_add__pos__pos,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.49         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_pos
% 0.29/0.49  thf(fact_628_add__neg__neg,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_neg
% 0.29/0.49  thf(fact_629_add__neg__neg,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_neg
% 0.29/0.49  thf(fact_630_add__neg__neg,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_neg
% 0.29/0.49  thf(fact_631_add__neg__neg,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_neg
% 0.29/0.49  thf(fact_632_lift__Suc__mono__less,axiom,
% 0.29/0.49      ! [F: nat > real,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less
% 0.29/0.49  thf(fact_633_lift__Suc__mono__less,axiom,
% 0.29/0.49      ! [F: nat > rat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less
% 0.29/0.49  thf(fact_634_lift__Suc__mono__less,axiom,
% 0.29/0.49      ! [F: nat > num,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less
% 0.29/0.49  thf(fact_635_lift__Suc__mono__less,axiom,
% 0.29/0.49      ! [F: nat > nat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less
% 0.29/0.49  thf(fact_636_lift__Suc__mono__less,axiom,
% 0.29/0.49      ! [F: nat > int,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less
% 0.29/0.49  thf(fact_637_lift__Suc__mono__less__iff,axiom,
% 0.29/0.49      ! [F: nat > real,N2: nat,M: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
% 0.29/0.49          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less_iff
% 0.29/0.49  thf(fact_638_lift__Suc__mono__less__iff,axiom,
% 0.29/0.49      ! [F: nat > rat,N2: nat,M: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
% 0.29/0.49          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less_iff
% 0.29/0.49  thf(fact_639_lift__Suc__mono__less__iff,axiom,
% 0.29/0.49      ! [F: nat > num,N2: nat,M: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
% 0.29/0.49          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less_iff
% 0.29/0.49  thf(fact_640_lift__Suc__mono__less__iff,axiom,
% 0.29/0.49      ! [F: nat > nat,N2: nat,M: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
% 0.29/0.49          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less_iff
% 0.29/0.49  thf(fact_641_lift__Suc__mono__less__iff,axiom,
% 0.29/0.49      ! [F: nat > int,N2: nat,M: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
% 0.29/0.49          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_less_iff
% 0.29/0.49  thf(fact_642_lift__Suc__mono__le,axiom,
% 0.29/0.49      ! [F: nat > set_int,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_le
% 0.29/0.49  thf(fact_643_lift__Suc__mono__le,axiom,
% 0.29/0.49      ! [F: nat > rat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_le
% 0.29/0.49  thf(fact_644_lift__Suc__mono__le,axiom,
% 0.29/0.49      ! [F: nat > num,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_le
% 0.29/0.49  thf(fact_645_lift__Suc__mono__le,axiom,
% 0.29/0.49      ! [F: nat > nat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_le
% 0.29/0.49  thf(fact_646_lift__Suc__mono__le,axiom,
% 0.29/0.49      ! [F: nat > int,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_mono_le
% 0.29/0.49  thf(fact_647_lift__Suc__antimono__le,axiom,
% 0.29/0.49      ! [F: nat > set_int,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_antimono_le
% 0.29/0.49  thf(fact_648_lift__Suc__antimono__le,axiom,
% 0.29/0.49      ! [F: nat > rat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_antimono_le
% 0.29/0.49  thf(fact_649_lift__Suc__antimono__le,axiom,
% 0.29/0.49      ! [F: nat > num,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_antimono_le
% 0.29/0.49  thf(fact_650_lift__Suc__antimono__le,axiom,
% 0.29/0.49      ! [F: nat > nat,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_antimono_le
% 0.29/0.49  thf(fact_651_lift__Suc__antimono__le,axiom,
% 0.29/0.49      ! [F: nat > int,N2: nat,N4: nat] :
% 0.29/0.49        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 0.29/0.49         => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % lift_Suc_antimono_le
% 0.29/0.49  thf(fact_652_less__Suc__eq__0__disj,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49        = ( ( M = zero_zero_nat )
% 0.29/0.49          | ? [J3: nat] :
% 0.29/0.49              ( ( M
% 0.29/0.49                = ( suc @ J3 ) )
% 0.29/0.49              & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_Suc_eq_0_disj
% 0.29/0.49  thf(fact_653_gr0__implies__Suc,axiom,
% 0.29/0.49      ! [N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.49       => ? [M3: nat] :
% 0.29/0.49            ( N2
% 0.29/0.49            = ( suc @ M3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % gr0_implies_Suc
% 0.29/0.49  thf(fact_654_All__less__Suc2,axiom,
% 0.29/0.49      ! [N2: nat,P: nat > $o] :
% 0.29/0.49        ( ( ! [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 0.29/0.49             => ( P @ I2 ) ) )
% 0.29/0.49        = ( ( P @ zero_zero_nat )
% 0.29/0.49          & ! [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ N2 )
% 0.29/0.49             => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % All_less_Suc2
% 0.29/0.49  thf(fact_655_gr0__conv__Suc,axiom,
% 0.29/0.49      ! [N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.49        = ( ? [M2: nat] :
% 0.29/0.49              ( N2
% 0.29/0.49              = ( suc @ M2 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % gr0_conv_Suc
% 0.29/0.49  thf(fact_656_Ex__less__Suc2,axiom,
% 0.29/0.49      ! [N2: nat,P: nat > $o] :
% 0.29/0.49        ( ( ? [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 0.29/0.49              & ( P @ I2 ) ) )
% 0.29/0.49        = ( ( P @ zero_zero_nat )
% 0.29/0.49          | ? [I2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ I2 @ N2 )
% 0.29/0.49              & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Ex_less_Suc2
% 0.29/0.49  thf(fact_657_less__imp__add__positive,axiom,
% 0.29/0.49      ! [I: nat,J: nat] :
% 0.29/0.49        ( ( ord_less_nat @ I @ J )
% 0.29/0.49       => ? [K2: nat] :
% 0.29/0.49            ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 0.29/0.49            & ( ( plus_plus_nat @ I @ K2 )
% 0.29/0.49              = J ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_add_positive
% 0.29/0.49  thf(fact_658_le__imp__less__Suc,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49       => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_imp_less_Suc
% 0.29/0.49  thf(fact_659_less__eq__Suc__le,axiom,
% 0.29/0.49      ( ord_less_nat
% 0.29/0.49      = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_eq_Suc_le
% 0.29/0.49  thf(fact_660_less__Suc__eq__le,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 0.29/0.49        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_Suc_eq_le
% 0.29/0.49  thf(fact_661_le__less__Suc__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.49       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.29/0.49          = ( N2 = M ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_less_Suc_eq
% 0.29/0.49  thf(fact_662_Suc__le__lessD,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.29/0.49       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_le_lessD
% 0.29/0.49  thf(fact_663_inc__induct,axiom,
% 0.29/0.49      ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ( P @ J )
% 0.29/0.49         => ( ! [N3: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ I @ N3 )
% 0.29/0.49               => ( ( ord_less_nat @ N3 @ J )
% 0.29/0.49                 => ( ( P @ ( suc @ N3 ) )
% 0.29/0.49                   => ( P @ N3 ) ) ) )
% 0.29/0.49           => ( P @ I ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % inc_induct
% 0.29/0.49  thf(fact_664_dec__induct,axiom,
% 0.29/0.49      ! [I: nat,J: nat,P: nat > $o] :
% 0.29/0.49        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.49       => ( ( P @ I )
% 0.29/0.49         => ( ! [N3: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ I @ N3 )
% 0.29/0.49               => ( ( ord_less_nat @ N3 @ J )
% 0.29/0.49                 => ( ( P @ N3 )
% 0.29/0.49                   => ( P @ ( suc @ N3 ) ) ) ) )
% 0.29/0.49           => ( P @ J ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dec_induct
% 0.29/0.49  thf(fact_665_Suc__le__eq,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.29/0.49        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_le_eq
% 0.29/0.49  thf(fact_666_Suc__leI,axiom,
% 0.29/0.49      ! [M: nat,N2: nat] :
% 0.29/0.49        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.49       => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Suc_leI
% 0.29/0.49  thf(fact_667_mono__nat__linear__lb,axiom,
% 0.29/0.49      ! [F: nat > nat,M: nat,K: nat] :
% 0.29/0.49        ( ! [M3: nat,N3: nat] :
% 0.29/0.49            ( ( ord_less_nat @ M3 @ N3 )
% 0.29/0.49           => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
% 0.29/0.49       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % mono_nat_linear_lb
% 0.29/0.49  thf(fact_668_add__strict__increasing2,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_real @ B @ C )
% 0.29/0.49         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing2
% 0.29/0.49  thf(fact_669_add__strict__increasing2,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_rat @ B @ C )
% 0.29/0.49         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing2
% 0.29/0.49  thf(fact_670_add__strict__increasing2,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_nat @ B @ C )
% 0.29/0.49         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing2
% 0.29/0.49  thf(fact_671_add__strict__increasing2,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_int @ B @ C )
% 0.29/0.49         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing2
% 0.29/0.49  thf(fact_672_add__strict__increasing,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.49         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing
% 0.29/0.49  thf(fact_673_add__strict__increasing,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing
% 0.29/0.49  thf(fact_674_add__strict__increasing,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing
% 0.29/0.49  thf(fact_675_add__strict__increasing,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_strict_increasing
% 0.29/0.49  thf(fact_676_add__pos__nonneg,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.49         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_nonneg
% 0.29/0.49  thf(fact_677_add__pos__nonneg,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.49         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_nonneg
% 0.29/0.49  thf(fact_678_add__pos__nonneg,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.49         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_nonneg
% 0.29/0.49  thf(fact_679_add__pos__nonneg,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.49         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_pos_nonneg
% 0.29/0.49  thf(fact_680_add__nonpos__neg,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_neg
% 0.29/0.49  thf(fact_681_add__nonpos__neg,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_neg
% 0.29/0.49  thf(fact_682_add__nonpos__neg,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_neg
% 0.29/0.49  thf(fact_683_add__nonpos__neg,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonpos_neg
% 0.29/0.49  thf(fact_684_add__nonneg__pos,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.49       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.49         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_pos
% 0.29/0.49  thf(fact_685_add__nonneg__pos,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.49       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.49         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_pos
% 0.29/0.49  thf(fact_686_add__nonneg__pos,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.49       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.49         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_pos
% 0.29/0.49  thf(fact_687_add__nonneg__pos,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.49       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.49         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_nonneg_pos
% 0.29/0.49  thf(fact_688_add__neg__nonpos,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.49       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.49         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_nonpos
% 0.29/0.49  thf(fact_689_add__neg__nonpos,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.49         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_nonpos
% 0.29/0.49  thf(fact_690_add__neg__nonpos,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.49         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_nonpos
% 0.29/0.49  thf(fact_691_add__neg__nonpos,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.49         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % add_neg_nonpos
% 0.29/0.49  thf(fact_692_ex__least__nat__less,axiom,
% 0.29/0.49      ! [P: nat > $o,N2: nat] :
% 0.29/0.49        ( ( P @ N2 )
% 0.29/0.49       => ( ~ ( P @ zero_zero_nat )
% 0.29/0.49         => ? [K2: nat] :
% 0.29/0.49              ( ( ord_less_nat @ K2 @ N2 )
% 0.29/0.49              & ! [I4: nat] :
% 0.29/0.49                  ( ( ord_less_eq_nat @ I4 @ K2 )
% 0.29/0.49                 => ~ ( P @ I4 ) )
% 0.29/0.49              & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ex_least_nat_less
% 0.29/0.49  thf(fact_693_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,I: nat] :
% 0.29/0.49        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.29/0.49         => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_694_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_real,A2: set_real,X3: real,I: nat] :
% 0.29/0.49        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member_real @ X3 @ A2 )
% 0.29/0.49         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_695_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_set_nat,A2: set_set_nat,X3: set_nat,I: nat] :
% 0.29/0.49        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member_set_nat @ X3 @ A2 )
% 0.29/0.49         => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_696_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
% 0.29/0.49        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member_VEBT_VEBT @ X3 @ A2 )
% 0.29/0.49         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_697_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_nat,A2: set_nat,X3: nat,I: nat] :
% 0.29/0.49        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member_nat @ X3 @ A2 )
% 0.29/0.49         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_698_set__update__subsetI,axiom,
% 0.29/0.49      ! [Xs: list_int,A2: set_int,X3: int,I: nat] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 0.29/0.49       => ( ( member_int @ X3 @ A2 )
% 0.29/0.49         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % set_update_subsetI
% 0.29/0.49  thf(fact_699_nle__le,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.49        = ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.49          & ( B != A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nle_le
% 0.29/0.49  thf(fact_700_nle__le,axiom,
% 0.29/0.49      ! [A: num,B: num] :
% 0.29/0.49        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 0.29/0.49        = ( ( ord_less_eq_num @ B @ A )
% 0.29/0.49          & ( B != A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nle_le
% 0.29/0.49  thf(fact_701_nle__le,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 0.29/0.49        = ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.49          & ( B != A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nle_le
% 0.29/0.49  thf(fact_702_nle__le,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 0.29/0.49        = ( ( ord_less_eq_int @ B @ A )
% 0.29/0.49          & ( B != A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % nle_le
% 0.29/0.49  thf(fact_703_le__cases3,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.49        ( ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49         => ~ ( ord_less_eq_rat @ Y3 @ Z2 ) )
% 0.29/0.49       => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.29/0.49           => ~ ( ord_less_eq_rat @ X3 @ Z2 ) )
% 0.29/0.49         => ( ( ( ord_less_eq_rat @ X3 @ Z2 )
% 0.29/0.49             => ~ ( ord_less_eq_rat @ Z2 @ Y3 ) )
% 0.29/0.49           => ( ( ( ord_less_eq_rat @ Z2 @ Y3 )
% 0.29/0.49               => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
% 0.29/0.49             => ( ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 0.29/0.49                 => ~ ( ord_less_eq_rat @ Z2 @ X3 ) )
% 0.29/0.49               => ~ ( ( ord_less_eq_rat @ Z2 @ X3 )
% 0.29/0.49                   => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_cases3
% 0.29/0.49  thf(fact_704_le__cases3,axiom,
% 0.29/0.49      ! [X3: num,Y3: num,Z2: num] :
% 0.29/0.49        ( ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49         => ~ ( ord_less_eq_num @ Y3 @ Z2 ) )
% 0.29/0.49       => ( ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.29/0.49           => ~ ( ord_less_eq_num @ X3 @ Z2 ) )
% 0.29/0.49         => ( ( ( ord_less_eq_num @ X3 @ Z2 )
% 0.29/0.49             => ~ ( ord_less_eq_num @ Z2 @ Y3 ) )
% 0.29/0.49           => ( ( ( ord_less_eq_num @ Z2 @ Y3 )
% 0.29/0.49               => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
% 0.29/0.49             => ( ( ( ord_less_eq_num @ Y3 @ Z2 )
% 0.29/0.49                 => ~ ( ord_less_eq_num @ Z2 @ X3 ) )
% 0.29/0.49               => ~ ( ( ord_less_eq_num @ Z2 @ X3 )
% 0.29/0.49                   => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_cases3
% 0.29/0.49  thf(fact_705_le__cases3,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.49        ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49         => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
% 0.29/0.49       => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.29/0.49           => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 0.29/0.49         => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 0.29/0.49             => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
% 0.29/0.49           => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
% 0.29/0.49               => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 0.29/0.49             => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 0.29/0.49                 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 0.29/0.49               => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 0.29/0.49                   => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_cases3
% 0.29/0.49  thf(fact_706_le__cases3,axiom,
% 0.29/0.49      ! [X3: int,Y3: int,Z2: int] :
% 0.29/0.49        ( ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49         => ~ ( ord_less_eq_int @ Y3 @ Z2 ) )
% 0.29/0.49       => ( ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.29/0.49           => ~ ( ord_less_eq_int @ X3 @ Z2 ) )
% 0.29/0.49         => ( ( ( ord_less_eq_int @ X3 @ Z2 )
% 0.29/0.49             => ~ ( ord_less_eq_int @ Z2 @ Y3 ) )
% 0.29/0.49           => ( ( ( ord_less_eq_int @ Z2 @ Y3 )
% 0.29/0.49               => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
% 0.29/0.49             => ( ( ( ord_less_eq_int @ Y3 @ Z2 )
% 0.29/0.49                 => ~ ( ord_less_eq_int @ Z2 @ X3 ) )
% 0.29/0.49               => ~ ( ( ord_less_eq_int @ Z2 @ X3 )
% 0.29/0.49                   => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % le_cases3
% 0.29/0.49  thf(fact_707_order__class_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [X: set_int,Y: set_int] :
% 0.29/0.49            ( ( ord_less_eq_set_int @ X @ Y )
% 0.29/0.49            & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_class.order_eq_iff
% 0.29/0.49  thf(fact_708_order__class_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [X: rat,Y: rat] :
% 0.29/0.49            ( ( ord_less_eq_rat @ X @ Y )
% 0.29/0.49            & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_class.order_eq_iff
% 0.29/0.49  thf(fact_709_order__class_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [X: num,Y: num] :
% 0.29/0.49            ( ( ord_less_eq_num @ X @ Y )
% 0.29/0.49            & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_class.order_eq_iff
% 0.29/0.49  thf(fact_710_order__class_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [X: nat,Y: nat] :
% 0.29/0.49            ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.49            & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_class.order_eq_iff
% 0.29/0.49  thf(fact_711_order__class_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [X: int,Y: int] :
% 0.29/0.49            ( ( ord_less_eq_int @ X @ Y )
% 0.29/0.49            & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_class.order_eq_iff
% 0.29/0.49  thf(fact_712_ord__eq__le__trans,axiom,
% 0.29/0.49      ! [A: set_int,B: set_int,C: set_int] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ B @ C )
% 0.29/0.49         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_trans
% 0.29/0.49  thf(fact_713_ord__eq__le__trans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_trans
% 0.29/0.49  thf(fact_714_ord__eq__le__trans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_trans
% 0.29/0.49  thf(fact_715_ord__eq__le__trans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_trans
% 0.29/0.49  thf(fact_716_ord__eq__le__trans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_trans
% 0.29/0.49  thf(fact_717_ord__le__eq__trans,axiom,
% 0.29/0.49      ! [A: set_int,B: set_int,C: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_trans
% 0.29/0.49  thf(fact_718_ord__le__eq__trans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_trans
% 0.29/0.49  thf(fact_719_ord__le__eq__trans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_trans
% 0.29/0.49  thf(fact_720_ord__le__eq__trans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_trans
% 0.29/0.49  thf(fact_721_ord__le__eq__trans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_trans
% 0.29/0.49  thf(fact_722_order__antisym,axiom,
% 0.29/0.49      ! [X3: set_int,Y3: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.29/0.49         => ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym
% 0.29/0.49  thf(fact_723_order__antisym,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.29/0.49         => ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym
% 0.29/0.49  thf(fact_724_order__antisym,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.29/0.49         => ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym
% 0.29/0.49  thf(fact_725_order__antisym,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.29/0.49         => ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym
% 0.29/0.49  thf(fact_726_order__antisym,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.29/0.49         => ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym
% 0.29/0.49  thf(fact_727_order_Otrans,axiom,
% 0.29/0.49      ! [A: set_int,B: set_int,C: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ B @ C )
% 0.29/0.49         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.trans
% 0.29/0.49  thf(fact_728_order_Otrans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.trans
% 0.29/0.49  thf(fact_729_order_Otrans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.trans
% 0.29/0.49  thf(fact_730_order_Otrans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.trans
% 0.29/0.49  thf(fact_731_order_Otrans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.trans
% 0.29/0.49  thf(fact_732_order__trans,axiom,
% 0.29/0.49      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
% 0.29/0.49         => ( ord_less_eq_set_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_trans
% 0.29/0.49  thf(fact_733_order__trans,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 0.29/0.49         => ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_trans
% 0.29/0.49  thf(fact_734_order__trans,axiom,
% 0.29/0.49      ! [X3: num,Y3: num,Z2: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_num @ Y3 @ Z2 )
% 0.29/0.49         => ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_trans
% 0.29/0.49  thf(fact_735_order__trans,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 0.29/0.49         => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_trans
% 0.29/0.49  thf(fact_736_order__trans,axiom,
% 0.29/0.49      ! [X3: int,Y3: int,Z2: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49       => ( ( ord_less_eq_int @ Y3 @ Z2 )
% 0.29/0.49         => ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_trans
% 0.29/0.49  thf(fact_737_linorder__wlog,axiom,
% 0.29/0.49      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.29/0.49        ( ! [A4: rat,B4: rat] :
% 0.29/0.49            ( ( ord_less_eq_rat @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: rat,B4: rat] :
% 0.29/0.49              ( ( P @ B4 @ A4 )
% 0.29/0.49             => ( P @ A4 @ B4 ) )
% 0.29/0.49         => ( P @ A @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_wlog
% 0.29/0.49  thf(fact_738_linorder__wlog,axiom,
% 0.29/0.49      ! [P: num > num > $o,A: num,B: num] :
% 0.29/0.49        ( ! [A4: num,B4: num] :
% 0.29/0.49            ( ( ord_less_eq_num @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: num,B4: num] :
% 0.29/0.49              ( ( P @ B4 @ A4 )
% 0.29/0.49             => ( P @ A4 @ B4 ) )
% 0.29/0.49         => ( P @ A @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_wlog
% 0.29/0.49  thf(fact_739_linorder__wlog,axiom,
% 0.29/0.49      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.49        ( ! [A4: nat,B4: nat] :
% 0.29/0.49            ( ( ord_less_eq_nat @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: nat,B4: nat] :
% 0.29/0.49              ( ( P @ B4 @ A4 )
% 0.29/0.49             => ( P @ A4 @ B4 ) )
% 0.29/0.49         => ( P @ A @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_wlog
% 0.29/0.49  thf(fact_740_linorder__wlog,axiom,
% 0.29/0.49      ! [P: int > int > $o,A: int,B: int] :
% 0.29/0.49        ( ! [A4: int,B4: int] :
% 0.29/0.49            ( ( ord_less_eq_int @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: int,B4: int] :
% 0.29/0.49              ( ( P @ B4 @ A4 )
% 0.29/0.49             => ( P @ A4 @ B4 ) )
% 0.29/0.49         => ( P @ A @ B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_wlog
% 0.29/0.49  thf(fact_741_dual__order_Oeq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: set_int,B3: set_int] :
% 0.29/0.49            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 0.29/0.49            & ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.eq_iff
% 0.29/0.49  thf(fact_742_dual__order_Oeq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: rat,B3: rat] :
% 0.29/0.49            ( ( ord_less_eq_rat @ B3 @ A3 )
% 0.29/0.49            & ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.eq_iff
% 0.29/0.49  thf(fact_743_dual__order_Oeq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: num,B3: num] :
% 0.29/0.49            ( ( ord_less_eq_num @ B3 @ A3 )
% 0.29/0.49            & ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.eq_iff
% 0.29/0.49  thf(fact_744_dual__order_Oeq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.49            ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.29/0.49            & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.eq_iff
% 0.29/0.49  thf(fact_745_dual__order_Oeq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: int,B3: int] :
% 0.29/0.49            ( ( ord_less_eq_int @ B3 @ A3 )
% 0.29/0.49            & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.eq_iff
% 0.29/0.49  thf(fact_746_dual__order_Oantisym,axiom,
% 0.29/0.49      ! [B: set_int,A: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.antisym
% 0.29/0.49  thf(fact_747_dual__order_Oantisym,axiom,
% 0.29/0.49      ! [B: rat,A: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.antisym
% 0.29/0.49  thf(fact_748_dual__order_Oantisym,axiom,
% 0.29/0.49      ! [B: num,A: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.antisym
% 0.29/0.49  thf(fact_749_dual__order_Oantisym,axiom,
% 0.29/0.49      ! [B: nat,A: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.antisym
% 0.29/0.49  thf(fact_750_dual__order_Oantisym,axiom,
% 0.29/0.49      ! [B: int,A: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.antisym
% 0.29/0.49  thf(fact_751_dual__order_Otrans,axiom,
% 0.29/0.49      ! [B: set_int,A: set_int,C: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ C @ B )
% 0.29/0.49         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.trans
% 0.29/0.49  thf(fact_752_dual__order_Otrans,axiom,
% 0.29/0.49      ! [B: rat,A: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_rat @ C @ B )
% 0.29/0.49         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.trans
% 0.29/0.49  thf(fact_753_dual__order_Otrans,axiom,
% 0.29/0.49      ! [B: num,A: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_num @ C @ B )
% 0.29/0.49         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.trans
% 0.29/0.49  thf(fact_754_dual__order_Otrans,axiom,
% 0.29/0.49      ! [B: nat,A: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.49         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.trans
% 0.29/0.49  thf(fact_755_dual__order_Otrans,axiom,
% 0.29/0.49      ! [B: int,A: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.49       => ( ( ord_less_eq_int @ C @ B )
% 0.29/0.49         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.trans
% 0.29/0.49  thf(fact_756_antisym,axiom,
% 0.29/0.49      ! [A: set_int,B: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ B @ A )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym
% 0.29/0.49  thf(fact_757_antisym,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym
% 0.29/0.49  thf(fact_758_antisym,axiom,
% 0.29/0.49      ! [A: num,B: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ A )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym
% 0.29/0.49  thf(fact_759_antisym,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym
% 0.29/0.49  thf(fact_760_antisym,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ A )
% 0.29/0.49         => ( A = B ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym
% 0.29/0.49  thf(fact_761_Orderings_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: set_int,B3: set_int] :
% 0.29/0.49            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 0.29/0.49            & ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Orderings.order_eq_iff
% 0.29/0.49  thf(fact_762_Orderings_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: rat,B3: rat] :
% 0.29/0.49            ( ( ord_less_eq_rat @ A3 @ B3 )
% 0.29/0.49            & ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Orderings.order_eq_iff
% 0.29/0.49  thf(fact_763_Orderings_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: num,B3: num] :
% 0.29/0.49            ( ( ord_less_eq_num @ A3 @ B3 )
% 0.29/0.49            & ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Orderings.order_eq_iff
% 0.29/0.49  thf(fact_764_Orderings_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.49            ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.29/0.49            & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Orderings.order_eq_iff
% 0.29/0.49  thf(fact_765_Orderings_Oorder__eq__iff,axiom,
% 0.29/0.49      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 0.29/0.49      = ( ^ [A3: int,B3: int] :
% 0.29/0.49            ( ( ord_less_eq_int @ A3 @ B3 )
% 0.29/0.49            & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % Orderings.order_eq_iff
% 0.29/0.49  thf(fact_766_order__subst1,axiom,
% 0.29/0.49      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_767_order__subst1,axiom,
% 0.29/0.49      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_768_order__subst1,axiom,
% 0.29/0.49      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_769_order__subst1,axiom,
% 0.29/0.49      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ! [X4: int,Y5: int] :
% 0.29/0.49                ( ( ord_less_eq_int @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_770_order__subst1,axiom,
% 0.29/0.49      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_771_order__subst1,axiom,
% 0.29/0.49      ! [A: num,F: num > num,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_772_order__subst1,axiom,
% 0.29/0.49      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_773_order__subst1,axiom,
% 0.29/0.49      ! [A: num,F: int > num,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.49         => ( ! [X4: int,Y5: int] :
% 0.29/0.49                ( ( ord_less_eq_int @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_774_order__subst1,axiom,
% 0.29/0.49      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_775_order__subst1,axiom,
% 0.29/0.49      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst1
% 0.29/0.49  thf(fact_776_order__subst2,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_777_order__subst2,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_778_order__subst2,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_779_order__subst2,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_780_order__subst2,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_781_order__subst2,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_782_order__subst2,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_783_order__subst2,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_784_order__subst2,axiom,
% 0.29/0.49      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_785_order__subst2,axiom,
% 0.29/0.49      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_subst2
% 0.29/0.49  thf(fact_786_order__eq__refl,axiom,
% 0.29/0.49      ! [X3: set_int,Y3: set_int] :
% 0.29/0.49        ( ( X3 = Y3 )
% 0.29/0.49       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_eq_refl
% 0.29/0.49  thf(fact_787_order__eq__refl,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( X3 = Y3 )
% 0.29/0.49       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_eq_refl
% 0.29/0.49  thf(fact_788_order__eq__refl,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ( X3 = Y3 )
% 0.29/0.49       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_eq_refl
% 0.29/0.49  thf(fact_789_order__eq__refl,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( X3 = Y3 )
% 0.29/0.49       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_eq_refl
% 0.29/0.49  thf(fact_790_order__eq__refl,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( X3 = Y3 )
% 0.29/0.49       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_eq_refl
% 0.29/0.49  thf(fact_791_linorder__linear,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49        | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_linear
% 0.29/0.49  thf(fact_792_linorder__linear,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49        | ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_linear
% 0.29/0.49  thf(fact_793_linorder__linear,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49        | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_linear
% 0.29/0.49  thf(fact_794_linorder__linear,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49        | ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_linear
% 0.29/0.49  thf(fact_795_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_796_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_797_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_798_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_799_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_800_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: num,F: num > num,B: num,C: num] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_801_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_802_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: int,F: num > int,B: num,C: num] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_803_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_804_ord__eq__le__subst,axiom,
% 0.29/0.49      ! [A: num,F: nat > num,B: nat,C: nat] :
% 0.29/0.49        ( ( A
% 0.29/0.49          = ( F @ B ) )
% 0.29/0.49       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_le_subst
% 0.29/0.49  thf(fact_805_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_806_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_807_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_808_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.49                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_809_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_810_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_811_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_812_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: num,B: num,F: num > int,C: int] :
% 0.29/0.49        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: num,Y5: num] :
% 0.29/0.49                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_813_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_814_ord__le__eq__subst,axiom,
% 0.29/0.49      ! [A: nat,B: nat,F: nat > num,C: num] :
% 0.29/0.49        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.49       => ( ( ( F @ B )
% 0.29/0.49            = C )
% 0.29/0.49         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.49                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 0.29/0.49               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.49           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_le_eq_subst
% 0.29/0.49  thf(fact_815_linorder__le__cases,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_le_cases
% 0.29/0.49  thf(fact_816_linorder__le__cases,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ~ ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_le_cases
% 0.29/0.49  thf(fact_817_linorder__le__cases,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_le_cases
% 0.29/0.49  thf(fact_818_linorder__le__cases,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ~ ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_le_cases
% 0.29/0.49  thf(fact_819_order__antisym__conv,axiom,
% 0.29/0.49      ! [Y3: set_int,X3: set_int] :
% 0.29/0.49        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym_conv
% 0.29/0.49  thf(fact_820_order__antisym__conv,axiom,
% 0.29/0.49      ! [Y3: rat,X3: rat] :
% 0.29/0.49        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym_conv
% 0.29/0.49  thf(fact_821_order__antisym__conv,axiom,
% 0.29/0.49      ! [Y3: num,X3: num] :
% 0.29/0.49        ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym_conv
% 0.29/0.49  thf(fact_822_order__antisym__conv,axiom,
% 0.29/0.49      ! [Y3: nat,X3: nat] :
% 0.29/0.49        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym_conv
% 0.29/0.49  thf(fact_823_order__antisym__conv,axiom,
% 0.29/0.49      ! [Y3: int,X3: int] :
% 0.29/0.49        ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.29/0.49       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order_antisym_conv
% 0.29/0.49  thf(fact_824_zero__reorient,axiom,
% 0.29/0.49      ! [X3: complex] :
% 0.29/0.49        ( ( zero_zero_complex = X3 )
% 0.29/0.49        = ( X3 = zero_zero_complex ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_reorient
% 0.29/0.49  thf(fact_825_zero__reorient,axiom,
% 0.29/0.49      ! [X3: real] :
% 0.29/0.49        ( ( zero_zero_real = X3 )
% 0.29/0.49        = ( X3 = zero_zero_real ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_reorient
% 0.29/0.49  thf(fact_826_zero__reorient,axiom,
% 0.29/0.49      ! [X3: rat] :
% 0.29/0.49        ( ( zero_zero_rat = X3 )
% 0.29/0.49        = ( X3 = zero_zero_rat ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_reorient
% 0.29/0.49  thf(fact_827_zero__reorient,axiom,
% 0.29/0.49      ! [X3: nat] :
% 0.29/0.49        ( ( zero_zero_nat = X3 )
% 0.29/0.49        = ( X3 = zero_zero_nat ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_reorient
% 0.29/0.49  thf(fact_828_zero__reorient,axiom,
% 0.29/0.49      ! [X3: int] :
% 0.29/0.49        ( ( zero_zero_int = X3 )
% 0.29/0.49        = ( X3 = zero_zero_int ) ) ).
% 0.29/0.49  
% 0.29/0.49  % zero_reorient
% 0.29/0.49  thf(fact_829_lt__ex,axiom,
% 0.29/0.49      ! [X3: real] :
% 0.29/0.49      ? [Y5: real] : ( ord_less_real @ Y5 @ X3 ) ).
% 0.29/0.49  
% 0.29/0.49  % lt_ex
% 0.29/0.49  thf(fact_830_lt__ex,axiom,
% 0.29/0.49      ! [X3: rat] :
% 0.29/0.49      ? [Y5: rat] : ( ord_less_rat @ Y5 @ X3 ) ).
% 0.29/0.49  
% 0.29/0.49  % lt_ex
% 0.29/0.49  thf(fact_831_lt__ex,axiom,
% 0.29/0.49      ! [X3: int] :
% 0.29/0.49      ? [Y5: int] : ( ord_less_int @ Y5 @ X3 ) ).
% 0.29/0.49  
% 0.29/0.49  % lt_ex
% 0.29/0.49  thf(fact_832_gt__ex,axiom,
% 0.29/0.49      ! [X3: real] :
% 0.29/0.49      ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% 0.29/0.49  
% 0.29/0.49  % gt_ex
% 0.29/0.49  thf(fact_833_gt__ex,axiom,
% 0.29/0.49      ! [X3: rat] :
% 0.29/0.49      ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% 0.29/0.49  
% 0.29/0.49  % gt_ex
% 0.29/0.49  thf(fact_834_gt__ex,axiom,
% 0.29/0.49      ! [X3: nat] :
% 0.29/0.49      ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 0.29/0.49  
% 0.29/0.49  % gt_ex
% 0.29/0.49  thf(fact_835_gt__ex,axiom,
% 0.29/0.49      ! [X3: int] :
% 0.29/0.49      ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 0.29/0.49  
% 0.29/0.49  % gt_ex
% 0.29/0.49  thf(fact_836_dense,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.49       => ? [Z3: real] :
% 0.29/0.49            ( ( ord_less_real @ X3 @ Z3 )
% 0.29/0.49            & ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dense
% 0.29/0.49  thf(fact_837_dense,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.49       => ? [Z3: rat] :
% 0.29/0.49            ( ( ord_less_rat @ X3 @ Z3 )
% 0.29/0.49            & ( ord_less_rat @ Z3 @ Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dense
% 0.29/0.49  thf(fact_838_less__imp__neq,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.49       => ( X3 != Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_neq
% 0.29/0.49  thf(fact_839_less__imp__neq,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.49       => ( X3 != Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_neq
% 0.29/0.49  thf(fact_840_less__imp__neq,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.49       => ( X3 != Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_neq
% 0.29/0.49  thf(fact_841_less__imp__neq,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.49       => ( X3 != Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_neq
% 0.29/0.49  thf(fact_842_less__imp__neq,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.49       => ( X3 != Y3 ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_imp_neq
% 0.29/0.49  thf(fact_843_order_Oasym,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.asym
% 0.29/0.49  thf(fact_844_order_Oasym,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.asym
% 0.29/0.49  thf(fact_845_order_Oasym,axiom,
% 0.29/0.49      ! [A: num,B: num] :
% 0.29/0.49        ( ( ord_less_num @ A @ B )
% 0.29/0.49       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.asym
% 0.29/0.49  thf(fact_846_order_Oasym,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.asym
% 0.29/0.49  thf(fact_847_order_Oasym,axiom,
% 0.29/0.49      ! [A: int,B: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.asym
% 0.29/0.49  thf(fact_848_ord__eq__less__trans,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_real @ B @ C )
% 0.29/0.49         => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_less_trans
% 0.29/0.49  thf(fact_849_ord__eq__less__trans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_rat @ B @ C )
% 0.29/0.49         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_less_trans
% 0.29/0.49  thf(fact_850_ord__eq__less__trans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_num @ B @ C )
% 0.29/0.49         => ( ord_less_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_less_trans
% 0.29/0.49  thf(fact_851_ord__eq__less__trans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_nat @ B @ C )
% 0.29/0.49         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_less_trans
% 0.29/0.49  thf(fact_852_ord__eq__less__trans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( A = B )
% 0.29/0.49       => ( ( ord_less_int @ B @ C )
% 0.29/0.49         => ( ord_less_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_eq_less_trans
% 0.29/0.49  thf(fact_853_ord__less__eq__trans,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_less_eq_trans
% 0.29/0.49  thf(fact_854_ord__less__eq__trans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_less_eq_trans
% 0.29/0.49  thf(fact_855_ord__less__eq__trans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_num @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_less_eq_trans
% 0.29/0.49  thf(fact_856_ord__less__eq__trans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_less_eq_trans
% 0.29/0.49  thf(fact_857_ord__less__eq__trans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ( B = C )
% 0.29/0.49         => ( ord_less_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % ord_less_eq_trans
% 0.29/0.49  thf(fact_858_less__induct,axiom,
% 0.29/0.49      ! [P: nat > $o,A: nat] :
% 0.29/0.49        ( ! [X4: nat] :
% 0.29/0.49            ( ! [Y6: nat] :
% 0.29/0.49                ( ( ord_less_nat @ Y6 @ X4 )
% 0.29/0.49               => ( P @ Y6 ) )
% 0.29/0.49           => ( P @ X4 ) )
% 0.29/0.49       => ( P @ A ) ) ).
% 0.29/0.49  
% 0.29/0.49  % less_induct
% 0.29/0.49  thf(fact_859_antisym__conv3,axiom,
% 0.29/0.49      ! [Y3: real,X3: real] :
% 0.29/0.49        ( ~ ( ord_less_real @ Y3 @ X3 )
% 0.29/0.49       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym_conv3
% 0.29/0.49  thf(fact_860_antisym__conv3,axiom,
% 0.29/0.49      ! [Y3: rat,X3: rat] :
% 0.29/0.49        ( ~ ( ord_less_rat @ Y3 @ X3 )
% 0.29/0.49       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym_conv3
% 0.29/0.49  thf(fact_861_antisym__conv3,axiom,
% 0.29/0.49      ! [Y3: num,X3: num] :
% 0.29/0.49        ( ~ ( ord_less_num @ Y3 @ X3 )
% 0.29/0.49       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym_conv3
% 0.29/0.49  thf(fact_862_antisym__conv3,axiom,
% 0.29/0.49      ! [Y3: nat,X3: nat] :
% 0.29/0.49        ( ~ ( ord_less_nat @ Y3 @ X3 )
% 0.29/0.49       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym_conv3
% 0.29/0.49  thf(fact_863_antisym__conv3,axiom,
% 0.29/0.49      ! [Y3: int,X3: int] :
% 0.29/0.49        ( ~ ( ord_less_int @ Y3 @ X3 )
% 0.29/0.49       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.29/0.49          = ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % antisym_conv3
% 0.29/0.49  thf(fact_864_linorder__cases,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.29/0.49       => ( ( X3 != Y3 )
% 0.29/0.49         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_cases
% 0.29/0.49  thf(fact_865_linorder__cases,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.49       => ( ( X3 != Y3 )
% 0.29/0.49         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_cases
% 0.29/0.49  thf(fact_866_linorder__cases,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.29/0.49       => ( ( X3 != Y3 )
% 0.29/0.49         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_cases
% 0.29/0.49  thf(fact_867_linorder__cases,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.49       => ( ( X3 != Y3 )
% 0.29/0.49         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_cases
% 0.29/0.49  thf(fact_868_linorder__cases,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.29/0.49       => ( ( X3 != Y3 )
% 0.29/0.49         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_cases
% 0.29/0.49  thf(fact_869_dual__order_Oasym,axiom,
% 0.29/0.49      ! [B: real,A: real] :
% 0.29/0.49        ( ( ord_less_real @ B @ A )
% 0.29/0.49       => ~ ( ord_less_real @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.asym
% 0.29/0.49  thf(fact_870_dual__order_Oasym,axiom,
% 0.29/0.49      ! [B: rat,A: rat] :
% 0.29/0.49        ( ( ord_less_rat @ B @ A )
% 0.29/0.49       => ~ ( ord_less_rat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.asym
% 0.29/0.49  thf(fact_871_dual__order_Oasym,axiom,
% 0.29/0.49      ! [B: num,A: num] :
% 0.29/0.49        ( ( ord_less_num @ B @ A )
% 0.29/0.49       => ~ ( ord_less_num @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.asym
% 0.29/0.49  thf(fact_872_dual__order_Oasym,axiom,
% 0.29/0.49      ! [B: nat,A: nat] :
% 0.29/0.49        ( ( ord_less_nat @ B @ A )
% 0.29/0.49       => ~ ( ord_less_nat @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.asym
% 0.29/0.49  thf(fact_873_dual__order_Oasym,axiom,
% 0.29/0.49      ! [B: int,A: int] :
% 0.29/0.49        ( ( ord_less_int @ B @ A )
% 0.29/0.49       => ~ ( ord_less_int @ A @ B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.asym
% 0.29/0.49  thf(fact_874_dual__order_Oirrefl,axiom,
% 0.29/0.49      ! [A: real] :
% 0.29/0.49        ~ ( ord_less_real @ A @ A ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.irrefl
% 0.29/0.49  thf(fact_875_dual__order_Oirrefl,axiom,
% 0.29/0.49      ! [A: rat] :
% 0.29/0.49        ~ ( ord_less_rat @ A @ A ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.irrefl
% 0.29/0.49  thf(fact_876_dual__order_Oirrefl,axiom,
% 0.29/0.49      ! [A: num] :
% 0.29/0.49        ~ ( ord_less_num @ A @ A ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.irrefl
% 0.29/0.49  thf(fact_877_dual__order_Oirrefl,axiom,
% 0.29/0.49      ! [A: nat] :
% 0.29/0.49        ~ ( ord_less_nat @ A @ A ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.irrefl
% 0.29/0.49  thf(fact_878_dual__order_Oirrefl,axiom,
% 0.29/0.49      ! [A: int] :
% 0.29/0.49        ~ ( ord_less_int @ A @ A ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.irrefl
% 0.29/0.49  thf(fact_879_exists__least__iff,axiom,
% 0.29/0.49      ( ( ^ [P2: nat > $o] :
% 0.29/0.49          ? [X6: nat] : ( P2 @ X6 ) )
% 0.29/0.49      = ( ^ [P3: nat > $o] :
% 0.29/0.49          ? [N: nat] :
% 0.29/0.49            ( ( P3 @ N )
% 0.29/0.49            & ! [M2: nat] :
% 0.29/0.49                ( ( ord_less_nat @ M2 @ N )
% 0.29/0.49               => ~ ( P3 @ M2 ) ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % exists_least_iff
% 0.29/0.49  thf(fact_880_linorder__less__wlog,axiom,
% 0.29/0.49      ! [P: real > real > $o,A: real,B: real] :
% 0.29/0.49        ( ! [A4: real,B4: real] :
% 0.29/0.49            ( ( ord_less_real @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: real] : ( P @ A4 @ A4 )
% 0.29/0.49         => ( ! [A4: real,B4: real] :
% 0.29/0.49                ( ( P @ B4 @ A4 )
% 0.29/0.49               => ( P @ A4 @ B4 ) )
% 0.29/0.49           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_less_wlog
% 0.29/0.49  thf(fact_881_linorder__less__wlog,axiom,
% 0.29/0.49      ! [P: rat > rat > $o,A: rat,B: rat] :
% 0.29/0.49        ( ! [A4: rat,B4: rat] :
% 0.29/0.49            ( ( ord_less_rat @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 0.29/0.49         => ( ! [A4: rat,B4: rat] :
% 0.29/0.49                ( ( P @ B4 @ A4 )
% 0.29/0.49               => ( P @ A4 @ B4 ) )
% 0.29/0.49           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_less_wlog
% 0.29/0.49  thf(fact_882_linorder__less__wlog,axiom,
% 0.29/0.49      ! [P: num > num > $o,A: num,B: num] :
% 0.29/0.49        ( ! [A4: num,B4: num] :
% 0.29/0.49            ( ( ord_less_num @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: num] : ( P @ A4 @ A4 )
% 0.29/0.49         => ( ! [A4: num,B4: num] :
% 0.29/0.49                ( ( P @ B4 @ A4 )
% 0.29/0.49               => ( P @ A4 @ B4 ) )
% 0.29/0.49           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_less_wlog
% 0.29/0.49  thf(fact_883_linorder__less__wlog,axiom,
% 0.29/0.49      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.49        ( ! [A4: nat,B4: nat] :
% 0.29/0.49            ( ( ord_less_nat @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 0.29/0.49         => ( ! [A4: nat,B4: nat] :
% 0.29/0.49                ( ( P @ B4 @ A4 )
% 0.29/0.49               => ( P @ A4 @ B4 ) )
% 0.29/0.49           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_less_wlog
% 0.29/0.49  thf(fact_884_linorder__less__wlog,axiom,
% 0.29/0.49      ! [P: int > int > $o,A: int,B: int] :
% 0.29/0.49        ( ! [A4: int,B4: int] :
% 0.29/0.49            ( ( ord_less_int @ A4 @ B4 )
% 0.29/0.49           => ( P @ A4 @ B4 ) )
% 0.29/0.49       => ( ! [A4: int] : ( P @ A4 @ A4 )
% 0.29/0.49         => ( ! [A4: int,B4: int] :
% 0.29/0.49                ( ( P @ B4 @ A4 )
% 0.29/0.49               => ( P @ A4 @ B4 ) )
% 0.29/0.49           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % linorder_less_wlog
% 0.29/0.49  thf(fact_885_order_Ostrict__trans,axiom,
% 0.29/0.49      ! [A: real,B: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( ( ord_less_real @ B @ C )
% 0.29/0.49         => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_trans
% 0.29/0.49  thf(fact_886_order_Ostrict__trans,axiom,
% 0.29/0.49      ! [A: rat,B: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( ( ord_less_rat @ B @ C )
% 0.29/0.49         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_trans
% 0.29/0.49  thf(fact_887_order_Ostrict__trans,axiom,
% 0.29/0.49      ! [A: num,B: num,C: num] :
% 0.29/0.49        ( ( ord_less_num @ A @ B )
% 0.29/0.49       => ( ( ord_less_num @ B @ C )
% 0.29/0.49         => ( ord_less_num @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_trans
% 0.29/0.49  thf(fact_888_order_Ostrict__trans,axiom,
% 0.29/0.49      ! [A: nat,B: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( ( ord_less_nat @ B @ C )
% 0.29/0.49         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_trans
% 0.29/0.49  thf(fact_889_order_Ostrict__trans,axiom,
% 0.29/0.49      ! [A: int,B: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ A @ B )
% 0.29/0.49       => ( ( ord_less_int @ B @ C )
% 0.29/0.49         => ( ord_less_int @ A @ C ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_trans
% 0.29/0.49  thf(fact_890_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49      ! [X3: real,Y3: real] :
% 0.29/0.49        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.29/0.49        = ( ( ord_less_real @ Y3 @ X3 )
% 0.29/0.49          | ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_iff_gr_or_eq
% 0.29/0.49  thf(fact_891_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49      ! [X3: rat,Y3: rat] :
% 0.29/0.49        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.29/0.49        = ( ( ord_less_rat @ Y3 @ X3 )
% 0.29/0.49          | ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_iff_gr_or_eq
% 0.29/0.49  thf(fact_892_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49      ! [X3: num,Y3: num] :
% 0.29/0.49        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.29/0.49        = ( ( ord_less_num @ Y3 @ X3 )
% 0.29/0.49          | ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_iff_gr_or_eq
% 0.29/0.49  thf(fact_893_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49      ! [X3: nat,Y3: nat] :
% 0.29/0.49        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.29/0.49        = ( ( ord_less_nat @ Y3 @ X3 )
% 0.29/0.49          | ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_iff_gr_or_eq
% 0.29/0.49  thf(fact_894_not__less__iff__gr__or__eq,axiom,
% 0.29/0.49      ! [X3: int,Y3: int] :
% 0.29/0.49        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.29/0.49        = ( ( ord_less_int @ Y3 @ X3 )
% 0.29/0.49          | ( X3 = Y3 ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % not_less_iff_gr_or_eq
% 0.29/0.49  thf(fact_895_dual__order_Ostrict__trans,axiom,
% 0.29/0.49      ! [B: real,A: real,C: real] :
% 0.29/0.49        ( ( ord_less_real @ B @ A )
% 0.29/0.49       => ( ( ord_less_real @ C @ B )
% 0.29/0.49         => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.strict_trans
% 0.29/0.49  thf(fact_896_dual__order_Ostrict__trans,axiom,
% 0.29/0.49      ! [B: rat,A: rat,C: rat] :
% 0.29/0.49        ( ( ord_less_rat @ B @ A )
% 0.29/0.49       => ( ( ord_less_rat @ C @ B )
% 0.29/0.49         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.strict_trans
% 0.29/0.49  thf(fact_897_dual__order_Ostrict__trans,axiom,
% 0.29/0.49      ! [B: num,A: num,C: num] :
% 0.29/0.49        ( ( ord_less_num @ B @ A )
% 0.29/0.49       => ( ( ord_less_num @ C @ B )
% 0.29/0.49         => ( ord_less_num @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.strict_trans
% 0.29/0.49  thf(fact_898_dual__order_Ostrict__trans,axiom,
% 0.29/0.49      ! [B: nat,A: nat,C: nat] :
% 0.29/0.49        ( ( ord_less_nat @ B @ A )
% 0.29/0.49       => ( ( ord_less_nat @ C @ B )
% 0.29/0.49         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.strict_trans
% 0.29/0.49  thf(fact_899_dual__order_Ostrict__trans,axiom,
% 0.29/0.49      ! [B: int,A: int,C: int] :
% 0.29/0.49        ( ( ord_less_int @ B @ A )
% 0.29/0.49       => ( ( ord_less_int @ C @ B )
% 0.29/0.49         => ( ord_less_int @ C @ A ) ) ) ).
% 0.29/0.49  
% 0.29/0.49  % dual_order.strict_trans
% 0.29/0.49  thf(fact_900_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49      ! [A: real,B: real] :
% 0.29/0.49        ( ( ord_less_real @ A @ B )
% 0.29/0.49       => ( A != B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_implies_not_eq
% 0.29/0.49  thf(fact_901_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49      ! [A: rat,B: rat] :
% 0.29/0.49        ( ( ord_less_rat @ A @ B )
% 0.29/0.49       => ( A != B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_implies_not_eq
% 0.29/0.49  thf(fact_902_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49      ! [A: num,B: num] :
% 0.29/0.49        ( ( ord_less_num @ A @ B )
% 0.29/0.49       => ( A != B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_implies_not_eq
% 0.29/0.49  thf(fact_903_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.49      ! [A: nat,B: nat] :
% 0.29/0.49        ( ( ord_less_nat @ A @ B )
% 0.29/0.49       => ( A != B ) ) ).
% 0.29/0.49  
% 0.29/0.49  % order.strict_implies_not_eq
% 0.29/0.49  thf(fact_904_order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_not_eq
% 0.29/0.50  thf(fact_905_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [B: real,A: real] :
% 0.29/0.50        ( ( ord_less_real @ B @ A )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_not_eq
% 0.29/0.50  thf(fact_906_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [B: rat,A: rat] :
% 0.29/0.50        ( ( ord_less_rat @ B @ A )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_not_eq
% 0.29/0.50  thf(fact_907_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [B: num,A: num] :
% 0.29/0.50        ( ( ord_less_num @ B @ A )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_not_eq
% 0.29/0.50  thf(fact_908_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [B: nat,A: nat] :
% 0.29/0.50        ( ( ord_less_nat @ B @ A )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_not_eq
% 0.29/0.50  thf(fact_909_dual__order_Ostrict__implies__not__eq,axiom,
% 0.29/0.50      ! [B: int,A: int] :
% 0.29/0.50        ( ( ord_less_int @ B @ A )
% 0.29/0.50       => ( A != B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_not_eq
% 0.29/0.50  thf(fact_910_linorder__neqE,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE
% 0.29/0.50  thf(fact_911_linorder__neqE,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE
% 0.29/0.50  thf(fact_912_linorder__neqE,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE
% 0.29/0.50  thf(fact_913_linorder__neqE,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE
% 0.29/0.50  thf(fact_914_linorder__neqE,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE
% 0.29/0.50  thf(fact_915_order__less__asym,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym
% 0.29/0.50  thf(fact_916_order__less__asym,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym
% 0.29/0.50  thf(fact_917_order__less__asym,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym
% 0.29/0.50  thf(fact_918_order__less__asym,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym
% 0.29/0.50  thf(fact_919_order__less__asym,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym
% 0.29/0.50  thf(fact_920_linorder__neq__iff,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50        = ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50          | ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neq_iff
% 0.29/0.50  thf(fact_921_linorder__neq__iff,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50        = ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50          | ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neq_iff
% 0.29/0.50  thf(fact_922_linorder__neq__iff,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50        = ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50          | ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neq_iff
% 0.29/0.50  thf(fact_923_linorder__neq__iff,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50        = ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50          | ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neq_iff
% 0.29/0.50  thf(fact_924_linorder__neq__iff,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50        = ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50          | ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neq_iff
% 0.29/0.50  thf(fact_925_order__less__asym_H,axiom,
% 0.29/0.50      ! [A: real,B: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ~ ( ord_less_real @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym'
% 0.29/0.50  thf(fact_926_order__less__asym_H,axiom,
% 0.29/0.50      ! [A: rat,B: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ~ ( ord_less_rat @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym'
% 0.29/0.50  thf(fact_927_order__less__asym_H,axiom,
% 0.29/0.50      ! [A: num,B: num] :
% 0.29/0.50        ( ( ord_less_num @ A @ B )
% 0.29/0.50       => ~ ( ord_less_num @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym'
% 0.29/0.50  thf(fact_928_order__less__asym_H,axiom,
% 0.29/0.50      ! [A: nat,B: nat] :
% 0.29/0.50        ( ( ord_less_nat @ A @ B )
% 0.29/0.50       => ~ ( ord_less_nat @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym'
% 0.29/0.50  thf(fact_929_order__less__asym_H,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ~ ( ord_less_int @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_asym'
% 0.29/0.50  thf(fact_930_order__less__trans,axiom,
% 0.29/0.50      ! [X3: real,Y3: real,Z2: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_real @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_trans
% 0.29/0.50  thf(fact_931_order__less__trans,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_rat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_trans
% 0.29/0.50  thf(fact_932_order__less__trans,axiom,
% 0.29/0.50      ! [X3: num,Y3: num,Z2: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_num @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_trans
% 0.29/0.50  thf(fact_933_order__less__trans,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_nat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_trans
% 0.29/0.50  thf(fact_934_order__less__trans,axiom,
% 0.29/0.50      ! [X3: int,Y3: int,Z2: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_int @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_trans
% 0.29/0.50  thf(fact_935_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_936_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_937_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: num,F: real > num,B: real,C: real] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_938_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: nat,F: real > nat,B: real,C: real] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_939_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: int,F: real > int,B: real,C: real] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_940_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_941_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_942_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_943_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_944_ord__eq__less__subst,axiom,
% 0.29/0.50      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.29/0.50        ( ( A
% 0.29/0.50          = ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_eq_less_subst
% 0.29/0.50  thf(fact_945_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_946_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_947_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > num,C: num] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_948_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_949_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > int,C: int] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_950_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_951_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_952_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_953_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_954_ord__less__eq__subst,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ( F @ B )
% 0.29/0.50            = C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % ord_less_eq_subst
% 0.29/0.50  thf(fact_955_order__less__irrefl,axiom,
% 0.29/0.50      ! [X3: real] :
% 0.29/0.50        ~ ( ord_less_real @ X3 @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_irrefl
% 0.29/0.50  thf(fact_956_order__less__irrefl,axiom,
% 0.29/0.50      ! [X3: rat] :
% 0.29/0.50        ~ ( ord_less_rat @ X3 @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_irrefl
% 0.29/0.50  thf(fact_957_order__less__irrefl,axiom,
% 0.29/0.50      ! [X3: num] :
% 0.29/0.50        ~ ( ord_less_num @ X3 @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_irrefl
% 0.29/0.50  thf(fact_958_order__less__irrefl,axiom,
% 0.29/0.50      ! [X3: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ X3 @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_irrefl
% 0.29/0.50  thf(fact_959_order__less__irrefl,axiom,
% 0.29/0.50      ! [X3: int] :
% 0.29/0.50        ~ ( ord_less_int @ X3 @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_irrefl
% 0.29/0.50  thf(fact_960_order__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_961_order__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_962_order__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_963_order__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_nat @ B @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_964_order__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: int > real,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_int @ B @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_965_order__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_966_order__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_967_order__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_968_order__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_nat @ B @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_969_order__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_int @ B @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst1
% 0.29/0.50  thf(fact_970_order__less__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_971_order__less__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_972_order__less__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > num,C: num] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_973_order__less__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_974_order__less__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > int,C: int] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_975_order__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_976_order__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_977_order__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_978_order__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_979_order__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_subst2
% 0.29/0.50  thf(fact_980_order__less__not__sym,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_not_sym
% 0.29/0.50  thf(fact_981_order__less__not__sym,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_not_sym
% 0.29/0.50  thf(fact_982_order__less__not__sym,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_not_sym
% 0.29/0.50  thf(fact_983_order__less__not__sym,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_not_sym
% 0.29/0.50  thf(fact_984_order__less__not__sym,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_not_sym
% 0.29/0.50  thf(fact_985_order__less__imp__triv,axiom,
% 0.29/0.50      ! [X3: real,Y3: real,P: $o] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_real @ Y3 @ X3 )
% 0.29/0.50         => P ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_triv
% 0.29/0.50  thf(fact_986_order__less__imp__triv,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat,P: $o] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_rat @ Y3 @ X3 )
% 0.29/0.50         => P ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_triv
% 0.29/0.50  thf(fact_987_order__less__imp__triv,axiom,
% 0.29/0.50      ! [X3: num,Y3: num,P: $o] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_num @ Y3 @ X3 )
% 0.29/0.50         => P ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_triv
% 0.29/0.50  thf(fact_988_order__less__imp__triv,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat,P: $o] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_nat @ Y3 @ X3 )
% 0.29/0.50         => P ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_triv
% 0.29/0.50  thf(fact_989_order__less__imp__triv,axiom,
% 0.29/0.50      ! [X3: int,Y3: int,P: $o] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_int @ Y3 @ X3 )
% 0.29/0.50         => P ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_triv
% 0.29/0.50  thf(fact_990_linorder__less__linear,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50        | ( X3 = Y3 )
% 0.29/0.50        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_less_linear
% 0.29/0.50  thf(fact_991_linorder__less__linear,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50        | ( X3 = Y3 )
% 0.29/0.50        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_less_linear
% 0.29/0.50  thf(fact_992_linorder__less__linear,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50        | ( X3 = Y3 )
% 0.29/0.50        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_less_linear
% 0.29/0.50  thf(fact_993_linorder__less__linear,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50        | ( X3 = Y3 )
% 0.29/0.50        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_less_linear
% 0.29/0.50  thf(fact_994_linorder__less__linear,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50        | ( X3 = Y3 )
% 0.29/0.50        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_less_linear
% 0.29/0.50  thf(fact_995_order__less__imp__not__eq,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq
% 0.29/0.50  thf(fact_996_order__less__imp__not__eq,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq
% 0.29/0.50  thf(fact_997_order__less__imp__not__eq,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq
% 0.29/0.50  thf(fact_998_order__less__imp__not__eq,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq
% 0.29/0.50  thf(fact_999_order__less__imp__not__eq,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq
% 0.29/0.50  thf(fact_1000_order__less__imp__not__eq2,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( Y3 != X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq2
% 0.29/0.50  thf(fact_1001_order__less__imp__not__eq2,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( Y3 != X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq2
% 0.29/0.50  thf(fact_1002_order__less__imp__not__eq2,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( Y3 != X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq2
% 0.29/0.50  thf(fact_1003_order__less__imp__not__eq2,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( Y3 != X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq2
% 0.29/0.50  thf(fact_1004_order__less__imp__not__eq2,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( Y3 != X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_eq2
% 0.29/0.50  thf(fact_1005_order__less__imp__not__less,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_less
% 0.29/0.50  thf(fact_1006_order__less__imp__not__less,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_less
% 0.29/0.50  thf(fact_1007_order__less__imp__not__less,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_less
% 0.29/0.50  thf(fact_1008_order__less__imp__not__less,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_less
% 0.29/0.50  thf(fact_1009_order__less__imp__not__less,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_not_less
% 0.29/0.50  thf(fact_1010_nat__neq__iff,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( M != N2 )
% 0.29/0.50        = ( ( ord_less_nat @ M @ N2 )
% 0.29/0.50          | ( ord_less_nat @ N2 @ M ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nat_neq_iff
% 0.29/0.50  thf(fact_1011_less__not__refl,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.29/0.50  
% 0.29/0.50  % less_not_refl
% 0.29/0.50  thf(fact_1012_less__not__refl2,axiom,
% 0.29/0.50      ! [N2: nat,M: nat] :
% 0.29/0.50        ( ( ord_less_nat @ N2 @ M )
% 0.29/0.50       => ( M != N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_not_refl2
% 0.29/0.50  thf(fact_1013_less__not__refl3,axiom,
% 0.29/0.50      ! [S: nat,T: nat] :
% 0.29/0.50        ( ( ord_less_nat @ S @ T )
% 0.29/0.50       => ( S != T ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_not_refl3
% 0.29/0.50  thf(fact_1014_less__irrefl__nat,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ N2 @ N2 ) ).
% 0.29/0.50  
% 0.29/0.50  % less_irrefl_nat
% 0.29/0.50  thf(fact_1015_nat__less__induct,axiom,
% 0.29/0.50      ! [P: nat > $o,N2: nat] :
% 0.29/0.50        ( ! [N3: nat] :
% 0.29/0.50            ( ! [M6: nat] :
% 0.29/0.50                ( ( ord_less_nat @ M6 @ N3 )
% 0.29/0.50               => ( P @ M6 ) )
% 0.29/0.50           => ( P @ N3 ) )
% 0.29/0.50       => ( P @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nat_less_induct
% 0.29/0.50  thf(fact_1016_infinite__descent,axiom,
% 0.29/0.50      ! [P: nat > $o,N2: nat] :
% 0.29/0.50        ( ! [N3: nat] :
% 0.29/0.50            ( ~ ( P @ N3 )
% 0.29/0.50           => ? [M6: nat] :
% 0.29/0.50                ( ( ord_less_nat @ M6 @ N3 )
% 0.29/0.50                & ~ ( P @ M6 ) ) )
% 0.29/0.50       => ( P @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % infinite_descent
% 0.29/0.50  thf(fact_1017_linorder__neqE__nat,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( X3 != Y3 )
% 0.29/0.50       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_neqE_nat
% 0.29/0.50  thf(fact_1018_le__refl,axiom,
% 0.29/0.50      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 0.29/0.50  
% 0.29/0.50  % le_refl
% 0.29/0.50  thf(fact_1019_le__trans,axiom,
% 0.29/0.50      ! [I: nat,J: nat,K: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.50       => ( ( ord_less_eq_nat @ J @ K )
% 0.29/0.50         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % le_trans
% 0.29/0.50  thf(fact_1020_eq__imp__le,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( M = N2 )
% 0.29/0.50       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % eq_imp_le
% 0.29/0.50  thf(fact_1021_le__antisym,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.50       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.29/0.50         => ( M = N2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % le_antisym
% 0.29/0.50  thf(fact_1022_nat__le__linear,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.50        | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nat_le_linear
% 0.29/0.50  thf(fact_1023_Nat_Oex__has__greatest__nat,axiom,
% 0.29/0.50      ! [P: nat > $o,K: nat,B: nat] :
% 0.29/0.50        ( ( P @ K )
% 0.29/0.50       => ( ! [Y5: nat] :
% 0.29/0.50              ( ( P @ Y5 )
% 0.29/0.50             => ( ord_less_eq_nat @ Y5 @ B ) )
% 0.29/0.50         => ? [X4: nat] :
% 0.29/0.50              ( ( P @ X4 )
% 0.29/0.50              & ! [Y6: nat] :
% 0.29/0.50                  ( ( P @ Y6 )
% 0.29/0.50                 => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Nat.ex_has_greatest_nat
% 0.29/0.50  thf(fact_1024_size__neq__size__imp__neq,axiom,
% 0.29/0.50      ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 0.29/0.50        ( ( ( size_s6755466524823107622T_VEBT @ X3 )
% 0.29/0.50         != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % size_neq_size_imp_neq
% 0.29/0.50  thf(fact_1025_size__neq__size__imp__neq,axiom,
% 0.29/0.50      ! [X3: list_o,Y3: list_o] :
% 0.29/0.50        ( ( ( size_size_list_o @ X3 )
% 0.29/0.50         != ( size_size_list_o @ Y3 ) )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % size_neq_size_imp_neq
% 0.29/0.50  thf(fact_1026_size__neq__size__imp__neq,axiom,
% 0.29/0.50      ! [X3: list_nat,Y3: list_nat] :
% 0.29/0.50        ( ( ( size_size_list_nat @ X3 )
% 0.29/0.50         != ( size_size_list_nat @ Y3 ) )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % size_neq_size_imp_neq
% 0.29/0.50  thf(fact_1027_size__neq__size__imp__neq,axiom,
% 0.29/0.50      ! [X3: list_int,Y3: list_int] :
% 0.29/0.50        ( ( ( size_size_list_int @ X3 )
% 0.29/0.50         != ( size_size_list_int @ Y3 ) )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % size_neq_size_imp_neq
% 0.29/0.50  thf(fact_1028_size__neq__size__imp__neq,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ( size_size_num @ X3 )
% 0.29/0.50         != ( size_size_num @ Y3 ) )
% 0.29/0.50       => ( X3 != Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % size_neq_size_imp_neq
% 0.29/0.50  thf(fact_1029_zero__le,axiom,
% 0.29/0.50      ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 0.29/0.50  
% 0.29/0.50  % zero_le
% 0.29/0.50  thf(fact_1030_leD,axiom,
% 0.29/0.50      ! [Y3: real,X3: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1031_leD,axiom,
% 0.29/0.50      ! [Y3: set_int,X3: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1032_leD,axiom,
% 0.29/0.50      ! [Y3: rat,X3: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1033_leD,axiom,
% 0.29/0.50      ! [Y3: num,X3: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_num @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1034_leD,axiom,
% 0.29/0.50      ! [Y3: nat,X3: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1035_leD,axiom,
% 0.29/0.50      ! [Y3: int,X3: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ Y3 @ X3 )
% 0.29/0.50       => ~ ( ord_less_int @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leD
% 0.29/0.50  thf(fact_1036_leI,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leI
% 0.29/0.50  thf(fact_1037_leI,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leI
% 0.29/0.50  thf(fact_1038_leI,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leI
% 0.29/0.50  thf(fact_1039_leI,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leI
% 0.29/0.50  thf(fact_1040_leI,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % leI
% 0.29/0.50  thf(fact_1041_nless__le,axiom,
% 0.29/0.50      ! [A: real,B: real] :
% 0.29/0.50        ( ( ~ ( ord_less_real @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_real @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1042_nless__le,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int] :
% 0.29/0.50        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1043_nless__le,axiom,
% 0.29/0.50      ! [A: rat,B: rat] :
% 0.29/0.50        ( ( ~ ( ord_less_rat @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1044_nless__le,axiom,
% 0.29/0.50      ! [A: num,B: num] :
% 0.29/0.50        ( ( ~ ( ord_less_num @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_num @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1045_nless__le,axiom,
% 0.29/0.50      ! [A: nat,B: nat] :
% 0.29/0.50        ( ( ~ ( ord_less_nat @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1046_nless__le,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( ~ ( ord_less_int @ A @ B ) )
% 0.29/0.50        = ( ~ ( ord_less_eq_int @ A @ B )
% 0.29/0.50          | ( A = B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nless_le
% 0.29/0.50  thf(fact_1047_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1048_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int] :
% 0.29/0.50        ( ~ ( ord_less_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1049_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1050_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ~ ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1051_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1052_antisym__conv1,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv1
% 0.29/0.50  thf(fact_1053_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1054_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1055_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1056_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1057_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1058_antisym__conv2,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.29/0.50          = ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % antisym_conv2
% 0.29/0.50  thf(fact_1059_dense__ge,axiom,
% 0.29/0.50      ! [Z2: real,Y3: real] :
% 0.29/0.50        ( ! [X4: real] :
% 0.29/0.50            ( ( ord_less_real @ Z2 @ X4 )
% 0.29/0.50           => ( ord_less_eq_real @ Y3 @ X4 ) )
% 0.29/0.50       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_ge
% 0.29/0.50  thf(fact_1060_dense__ge,axiom,
% 0.29/0.50      ! [Z2: rat,Y3: rat] :
% 0.29/0.50        ( ! [X4: rat] :
% 0.29/0.50            ( ( ord_less_rat @ Z2 @ X4 )
% 0.29/0.50           => ( ord_less_eq_rat @ Y3 @ X4 ) )
% 0.29/0.50       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_ge
% 0.29/0.50  thf(fact_1061_dense__le,axiom,
% 0.29/0.50      ! [Y3: real,Z2: real] :
% 0.29/0.50        ( ! [X4: real] :
% 0.29/0.50            ( ( ord_less_real @ X4 @ Y3 )
% 0.29/0.50           => ( ord_less_eq_real @ X4 @ Z2 ) )
% 0.29/0.50       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_le
% 0.29/0.50  thf(fact_1062_dense__le,axiom,
% 0.29/0.50      ! [Y3: rat,Z2: rat] :
% 0.29/0.50        ( ! [X4: rat] :
% 0.29/0.50            ( ( ord_less_rat @ X4 @ Y3 )
% 0.29/0.50           => ( ord_less_eq_rat @ X4 @ Z2 ) )
% 0.29/0.50       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_le
% 0.29/0.50  thf(fact_1063_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [X: real,Y: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1064_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [X: set_int,Y: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1065_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [X: rat,Y: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1066_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [X: num,Y: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1067_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [X: nat,Y: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1068_less__le__not__le,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [X: int,Y: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ X @ Y )
% 0.29/0.50            & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_le_not_le
% 0.29/0.50  thf(fact_1069_not__le__imp__less,axiom,
% 0.29/0.50      ! [Y3: real,X3: real] :
% 0.29/0.50        ( ~ ( ord_less_eq_real @ Y3 @ X3 )
% 0.29/0.50       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_le_imp_less
% 0.29/0.50  thf(fact_1070_not__le__imp__less,axiom,
% 0.29/0.50      ! [Y3: rat,X3: rat] :
% 0.29/0.50        ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
% 0.29/0.50       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_le_imp_less
% 0.29/0.50  thf(fact_1071_not__le__imp__less,axiom,
% 0.29/0.50      ! [Y3: num,X3: num] :
% 0.29/0.50        ( ~ ( ord_less_eq_num @ Y3 @ X3 )
% 0.29/0.50       => ( ord_less_num @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_le_imp_less
% 0.29/0.50  thf(fact_1072_not__le__imp__less,axiom,
% 0.29/0.50      ! [Y3: nat,X3: nat] :
% 0.29/0.50        ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
% 0.29/0.50       => ( ord_less_nat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_le_imp_less
% 0.29/0.50  thf(fact_1073_not__le__imp__less,axiom,
% 0.29/0.50      ! [Y3: int,X3: int] :
% 0.29/0.50        ( ~ ( ord_less_eq_int @ Y3 @ X3 )
% 0.29/0.50       => ( ord_less_int @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_le_imp_less
% 0.29/0.50  thf(fact_1074_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_real
% 0.29/0.50      = ( ^ [A3: real,B3: real] :
% 0.29/0.50            ( ( ord_less_real @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1075_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_set_int
% 0.29/0.50      = ( ^ [A3: set_int,B3: set_int] :
% 0.29/0.50            ( ( ord_less_set_int @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1076_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_rat
% 0.29/0.50      = ( ^ [A3: rat,B3: rat] :
% 0.29/0.50            ( ( ord_less_rat @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1077_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_num
% 0.29/0.50      = ( ^ [A3: num,B3: num] :
% 0.29/0.50            ( ( ord_less_num @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1078_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_nat
% 0.29/0.50      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.50            ( ( ord_less_nat @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1079_order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_int
% 0.29/0.50      = ( ^ [A3: int,B3: int] :
% 0.29/0.50            ( ( ord_less_int @ A3 @ B3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.order_iff_strict
% 0.29/0.50  thf(fact_1080_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [A3: real,B3: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1081_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [A3: set_int,B3: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1082_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [A3: rat,B3: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1083_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [A3: num,B3: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1084_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1085_order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [A3: int,B3: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ A3 @ B3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_order
% 0.29/0.50  thf(fact_1086_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: real,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1087_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int,C: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_set_int @ B @ C )
% 0.29/0.50         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1088_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: rat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1089_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: num,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_num @ B @ C )
% 0.29/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1090_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: nat,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.50       => ( ( ord_less_nat @ B @ C )
% 0.29/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1091_order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [A: int,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_int @ B @ C )
% 0.29/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans1
% 0.29/0.50  thf(fact_1092_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: real,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ B @ C )
% 0.29/0.50         => ( ord_less_real @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1093_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int,C: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ B @ C )
% 0.29/0.50         => ( ord_less_set_int @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1094_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ord_less_rat @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1095_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: num,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ord_less_num @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1096_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: nat,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_nat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_nat @ B @ C )
% 0.29/0.50         => ( ord_less_nat @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1097_order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [A: int,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_int @ B @ C )
% 0.29/0.50         => ( ord_less_int @ A @ C ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_trans2
% 0.29/0.50  thf(fact_1098_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [A3: real,B3: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1099_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [A3: set_int,B3: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1100_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [A3: rat,B3: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1101_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [A3: num,B3: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1102_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [A3: nat,B3: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1103_order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [A3: int,B3: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ A3 @ B3 )
% 0.29/0.50            & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_iff_not
% 0.29/0.50  thf(fact_1104_dense__ge__bounded,axiom,
% 0.29/0.50      ! [Z2: real,X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ Z2 @ X3 )
% 0.29/0.50       => ( ! [W: real] :
% 0.29/0.50              ( ( ord_less_real @ Z2 @ W )
% 0.29/0.50             => ( ( ord_less_real @ W @ X3 )
% 0.29/0.50               => ( ord_less_eq_real @ Y3 @ W ) ) )
% 0.29/0.50         => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_ge_bounded
% 0.29/0.50  thf(fact_1105_dense__ge__bounded,axiom,
% 0.29/0.50      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ Z2 @ X3 )
% 0.29/0.50       => ( ! [W: rat] :
% 0.29/0.50              ( ( ord_less_rat @ Z2 @ W )
% 0.29/0.50             => ( ( ord_less_rat @ W @ X3 )
% 0.29/0.50               => ( ord_less_eq_rat @ Y3 @ W ) ) )
% 0.29/0.50         => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_ge_bounded
% 0.29/0.50  thf(fact_1106_dense__le__bounded,axiom,
% 0.29/0.50      ! [X3: real,Y3: real,Z2: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ! [W: real] :
% 0.29/0.50              ( ( ord_less_real @ X3 @ W )
% 0.29/0.50             => ( ( ord_less_real @ W @ Y3 )
% 0.29/0.50               => ( ord_less_eq_real @ W @ Z2 ) ) )
% 0.29/0.50         => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_le_bounded
% 0.29/0.50  thf(fact_1107_dense__le__bounded,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ! [W: rat] :
% 0.29/0.50              ( ( ord_less_rat @ X3 @ W )
% 0.29/0.50             => ( ( ord_less_rat @ W @ Y3 )
% 0.29/0.50               => ( ord_less_eq_rat @ W @ Z2 ) ) )
% 0.29/0.50         => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dense_le_bounded
% 0.29/0.50  thf(fact_1108_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_real
% 0.29/0.50      = ( ^ [B3: real,A3: real] :
% 0.29/0.50            ( ( ord_less_real @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1109_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_set_int
% 0.29/0.50      = ( ^ [B3: set_int,A3: set_int] :
% 0.29/0.50            ( ( ord_less_set_int @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1110_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_rat
% 0.29/0.50      = ( ^ [B3: rat,A3: rat] :
% 0.29/0.50            ( ( ord_less_rat @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1111_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_num
% 0.29/0.50      = ( ^ [B3: num,A3: num] :
% 0.29/0.50            ( ( ord_less_num @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1112_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_nat
% 0.29/0.50      = ( ^ [B3: nat,A3: nat] :
% 0.29/0.50            ( ( ord_less_nat @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1113_dual__order_Oorder__iff__strict,axiom,
% 0.29/0.50      ( ord_less_eq_int
% 0.29/0.50      = ( ^ [B3: int,A3: int] :
% 0.29/0.50            ( ( ord_less_int @ B3 @ A3 )
% 0.29/0.50            | ( A3 = B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.order_iff_strict
% 0.29/0.50  thf(fact_1114_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [B3: real,A3: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1115_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [B3: set_int,A3: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1116_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [B3: rat,A3: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1117_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [B3: num,A3: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1118_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [B3: nat,A3: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1119_dual__order_Ostrict__iff__order,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [B3: int,A3: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ B3 @ A3 )
% 0.29/0.50            & ( A3 != B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_order
% 0.29/0.50  thf(fact_1120_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: real,A: real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.50       => ( ( ord_less_real @ C @ B )
% 0.29/0.50         => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1121_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: set_int,A: set_int,C: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ B @ A )
% 0.29/0.50       => ( ( ord_less_set_int @ C @ B )
% 0.29/0.50         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1122_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: rat,A: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.50       => ( ( ord_less_rat @ C @ B )
% 0.29/0.50         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1123_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: num,A: num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ B @ A )
% 0.29/0.50       => ( ( ord_less_num @ C @ B )
% 0.29/0.50         => ( ord_less_num @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1124_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: nat,A: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ B @ A )
% 0.29/0.50       => ( ( ord_less_nat @ C @ B )
% 0.29/0.50         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1125_dual__order_Ostrict__trans1,axiom,
% 0.29/0.50      ! [B: int,A: int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.50       => ( ( ord_less_int @ C @ B )
% 0.29/0.50         => ( ord_less_int @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans1
% 0.29/0.50  thf(fact_1126_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: real,A: real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_real @ C @ B )
% 0.29/0.50         => ( ord_less_real @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1127_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: set_int,A: set_int,C: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ C @ B )
% 0.29/0.50         => ( ord_less_set_int @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1128_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: rat,A: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_rat @ C @ B )
% 0.29/0.50         => ( ord_less_rat @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1129_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: num,A: num,C: num] :
% 0.29/0.50        ( ( ord_less_num @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_num @ C @ B )
% 0.29/0.50         => ( ord_less_num @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1130_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: nat,A: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_nat @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_nat @ C @ B )
% 0.29/0.50         => ( ord_less_nat @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1131_dual__order_Ostrict__trans2,axiom,
% 0.29/0.50      ! [B: int,A: int,C: int] :
% 0.29/0.50        ( ( ord_less_int @ B @ A )
% 0.29/0.50       => ( ( ord_less_eq_int @ C @ B )
% 0.29/0.50         => ( ord_less_int @ C @ A ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_trans2
% 0.29/0.50  thf(fact_1132_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [B3: real,A3: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1133_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [B3: set_int,A3: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1134_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [B3: rat,A3: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1135_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [B3: num,A3: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1136_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [B3: nat,A3: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1137_dual__order_Ostrict__iff__not,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [B3: int,A3: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ B3 @ A3 )
% 0.29/0.50            & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_iff_not
% 0.29/0.50  thf(fact_1138_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: real,B: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1139_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ A @ B )
% 0.29/0.50       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1140_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: rat,B: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1141_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: num,B: num] :
% 0.29/0.50        ( ( ord_less_num @ A @ B )
% 0.29/0.50       => ( ord_less_eq_num @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1142_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: nat,B: nat] :
% 0.29/0.50        ( ( ord_less_nat @ A @ B )
% 0.29/0.50       => ( ord_less_eq_nat @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1143_order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order.strict_implies_order
% 0.29/0.50  thf(fact_1144_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: real,A: real] :
% 0.29/0.50        ( ( ord_less_real @ B @ A )
% 0.29/0.50       => ( ord_less_eq_real @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1145_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: set_int,A: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ B @ A )
% 0.29/0.50       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1146_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: rat,A: rat] :
% 0.29/0.50        ( ( ord_less_rat @ B @ A )
% 0.29/0.50       => ( ord_less_eq_rat @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1147_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: num,A: num] :
% 0.29/0.50        ( ( ord_less_num @ B @ A )
% 0.29/0.50       => ( ord_less_eq_num @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1148_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: nat,A: nat] :
% 0.29/0.50        ( ( ord_less_nat @ B @ A )
% 0.29/0.50       => ( ord_less_eq_nat @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1149_dual__order_Ostrict__implies__order,axiom,
% 0.29/0.50      ! [B: int,A: int] :
% 0.29/0.50        ( ( ord_less_int @ B @ A )
% 0.29/0.50       => ( ord_less_eq_int @ B @ A ) ) ).
% 0.29/0.50  
% 0.29/0.50  % dual_order.strict_implies_order
% 0.29/0.50  thf(fact_1150_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_real
% 0.29/0.50      = ( ^ [X: real,Y: real] :
% 0.29/0.50            ( ( ord_less_real @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1151_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_set_int
% 0.29/0.50      = ( ^ [X: set_int,Y: set_int] :
% 0.29/0.50            ( ( ord_less_set_int @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1152_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_rat
% 0.29/0.50      = ( ^ [X: rat,Y: rat] :
% 0.29/0.50            ( ( ord_less_rat @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1153_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_num
% 0.29/0.50      = ( ^ [X: num,Y: num] :
% 0.29/0.50            ( ( ord_less_num @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1154_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_nat
% 0.29/0.50      = ( ^ [X: nat,Y: nat] :
% 0.29/0.50            ( ( ord_less_nat @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1155_order__le__less,axiom,
% 0.29/0.50      ( ord_less_eq_int
% 0.29/0.50      = ( ^ [X: int,Y: int] :
% 0.29/0.50            ( ( ord_less_int @ X @ Y )
% 0.29/0.50            | ( X = Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less
% 0.29/0.50  thf(fact_1156_order__less__le,axiom,
% 0.29/0.50      ( ord_less_real
% 0.29/0.50      = ( ^ [X: real,Y: real] :
% 0.29/0.50            ( ( ord_less_eq_real @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1157_order__less__le,axiom,
% 0.29/0.50      ( ord_less_set_int
% 0.29/0.50      = ( ^ [X: set_int,Y: set_int] :
% 0.29/0.50            ( ( ord_less_eq_set_int @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1158_order__less__le,axiom,
% 0.29/0.50      ( ord_less_rat
% 0.29/0.50      = ( ^ [X: rat,Y: rat] :
% 0.29/0.50            ( ( ord_less_eq_rat @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1159_order__less__le,axiom,
% 0.29/0.50      ( ord_less_num
% 0.29/0.50      = ( ^ [X: num,Y: num] :
% 0.29/0.50            ( ( ord_less_eq_num @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1160_order__less__le,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [X: nat,Y: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1161_order__less__le,axiom,
% 0.29/0.50      ( ord_less_int
% 0.29/0.50      = ( ^ [X: int,Y: int] :
% 0.29/0.50            ( ( ord_less_eq_int @ X @ Y )
% 0.29/0.50            & ( X != Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le
% 0.29/0.50  thf(fact_1162_linorder__not__le,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_le
% 0.29/0.50  thf(fact_1163_linorder__not__le,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_le
% 0.29/0.50  thf(fact_1164_linorder__not__le,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_le
% 0.29/0.50  thf(fact_1165_linorder__not__le,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_le
% 0.29/0.50  thf(fact_1166_linorder__not__le,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_le
% 0.29/0.50  thf(fact_1167_linorder__not__less,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_less
% 0.29/0.50  thf(fact_1168_linorder__not__less,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_less
% 0.29/0.50  thf(fact_1169_linorder__not__less,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_less
% 0.29/0.50  thf(fact_1170_linorder__not__less,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_less
% 0.29/0.50  thf(fact_1171_linorder__not__less,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 0.29/0.50        = ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_not_less
% 0.29/0.50  thf(fact_1172_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1173_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1174_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1175_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1176_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1177_order__less__imp__le,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_imp_le
% 0.29/0.50  thf(fact_1178_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: real,B: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1179_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1180_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: rat,B: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1181_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: num,B: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_num @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1182_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: nat,B: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1183_order__le__neq__trans,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.50       => ( ( A != B )
% 0.29/0.50         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_neq_trans
% 0.29/0.50  thf(fact_1184_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: real,B: real] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_real @ A @ B )
% 0.29/0.50         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1185_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: set_int,B: set_int] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ A @ B )
% 0.29/0.50         => ( ord_less_set_int @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1186_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: rat,B: rat] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1187_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: num,B: num] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50         => ( ord_less_num @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1188_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: nat,B: nat] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.50         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1189_order__neq__le__trans,axiom,
% 0.29/0.50      ! [A: int,B: int] :
% 0.29/0.50        ( ( A != B )
% 0.29/0.50       => ( ( ord_less_eq_int @ A @ B )
% 0.29/0.50         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_neq_le_trans
% 0.29/0.50  thf(fact_1190_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: real,Y3: real,Z2: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_real @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1191_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_set_int @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1192_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_rat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1193_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: num,Y3: num,Z2: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_num @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1194_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_nat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1195_order__le__less__trans,axiom,
% 0.29/0.50      ! [X3: int,Y3: int,Z2: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_int @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_trans
% 0.29/0.50  thf(fact_1196_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: real,Y3: real,Z2: real] :
% 0.29/0.50        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_real @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1197_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1198_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.29/0.50        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1199_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: num,Y3: num,Z2: num] :
% 0.29/0.50        ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_num @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1200_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1201_order__less__le__trans,axiom,
% 0.29/0.50      ! [X3: int,Y3: int,Z2: int] :
% 0.29/0.50        ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_eq_int @ Y3 @ Z2 )
% 0.29/0.50         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_trans
% 0.29/0.50  thf(fact_1202_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: real > real,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1203_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1204_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1205_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: nat > real,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_nat @ B @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1206_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: real,F: int > real,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_int @ B @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1207_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: real > rat,B: real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_real @ B @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1208_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1209_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1210_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_nat @ B @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1211_order__le__less__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: int > rat,B: int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_int @ B @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst1
% 0.29/0.50  thf(fact_1212_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1213_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1214_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1215_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1216_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1217_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > real,C: real] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1218_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1219_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > num,C: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_num @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1220_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > nat,C: nat] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1221_order__le__less__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > int,C: int] :
% 0.29/0.50        ( ( ord_less_eq_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_int @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_less_subst2
% 0.29/0.50  thf(fact_1222_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: real,F: rat > real,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1223_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1224_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: num,F: rat > num,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1225_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1226_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: int,F: rat > int,B: rat,C: rat] :
% 0.29/0.50        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_rat @ B @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1227_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: real,F: num > real,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_real @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1228_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: rat,F: num > rat,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_rat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1229_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: num,F: num > num,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_num @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1230_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: nat,F: num > nat,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_nat @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1231_order__less__le__subst1,axiom,
% 0.29/0.50      ! [A: int,F: num > int,B: num,C: num] :
% 0.29/0.50        ( ( ord_less_int @ A @ ( F @ B ) )
% 0.29/0.50       => ( ( ord_less_eq_num @ B @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_eq_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst1
% 0.29/0.50  thf(fact_1232_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > real,C: real] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1233_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > real,C: real] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1234_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > real,C: real] :
% 0.29/0.50        ( ( ord_less_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1235_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: nat,B: nat,F: nat > real,C: real] :
% 0.29/0.50        ( ( ord_less_nat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1236_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: int,B: int,F: int > real,C: real] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1237_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: real,B: real,F: real > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_real @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: real,Y5: real] :
% 0.29/0.50                ( ( ord_less_real @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1238_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_rat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: rat,Y5: rat] :
% 0.29/0.50                ( ( ord_less_rat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1239_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: num,B: num,F: num > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_num @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: num,Y5: num] :
% 0.29/0.50                ( ( ord_less_num @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1240_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_nat @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: nat,Y5: nat] :
% 0.29/0.50                ( ( ord_less_nat @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1241_order__less__le__subst2,axiom,
% 0.29/0.50      ! [A: int,B: int,F: int > rat,C: rat] :
% 0.29/0.50        ( ( ord_less_int @ A @ B )
% 0.29/0.50       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 0.29/0.50         => ( ! [X4: int,Y5: int] :
% 0.29/0.50                ( ( ord_less_int @ X4 @ Y5 )
% 0.29/0.50               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 0.29/0.50           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_less_le_subst2
% 0.29/0.50  thf(fact_1242_linorder__le__less__linear,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.50        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_le_less_linear
% 0.29/0.50  thf(fact_1243_linorder__le__less__linear,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.50        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_le_less_linear
% 0.29/0.50  thf(fact_1244_linorder__le__less__linear,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.50        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_le_less_linear
% 0.29/0.50  thf(fact_1245_linorder__le__less__linear,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.50        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_le_less_linear
% 0.29/0.50  thf(fact_1246_linorder__le__less__linear,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.50        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % linorder_le_less_linear
% 0.29/0.50  thf(fact_1247_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1248_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: set_int,Y3: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_set_int @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1249_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1250_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: num,Y3: num] :
% 0.29/0.50        ( ( ord_less_eq_num @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_num @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1251_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: nat,Y3: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_nat @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1252_order__le__imp__less__or__eq,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.29/0.50       => ( ( ord_less_int @ X3 @ Y3 )
% 0.29/0.50          | ( X3 = Y3 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % order_le_imp_less_or_eq
% 0.29/0.50  thf(fact_1253_gr__zeroI,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ( ( N2 != zero_zero_nat )
% 0.29/0.50       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % gr_zeroI
% 0.29/0.50  thf(fact_1254_not__less__zero,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % not_less_zero
% 0.29/0.50  thf(fact_1255_gr__implies__not__zero,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.50       => ( N2 != zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % gr_implies_not_zero
% 0.29/0.50  thf(fact_1256_zero__less__iff__neq__zero,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.50        = ( N2 != zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % zero_less_iff_neq_zero
% 0.29/0.50  thf(fact_1257_bot__nat__0_Oextremum__strict,axiom,
% 0.29/0.50      ! [A: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % bot_nat_0.extremum_strict
% 0.29/0.50  thf(fact_1258_gr0I,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ( ( N2 != zero_zero_nat )
% 0.29/0.50       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % gr0I
% 0.29/0.50  thf(fact_1259_not__gr0,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 0.29/0.50        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_gr0
% 0.29/0.50  thf(fact_1260_not__less0,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % not_less0
% 0.29/0.50  thf(fact_1261_less__zeroE,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % less_zeroE
% 0.29/0.50  thf(fact_1262_gr__implies__not0,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.50       => ( N2 != zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % gr_implies_not0
% 0.29/0.50  thf(fact_1263_infinite__descent0,axiom,
% 0.29/0.50      ! [P: nat > $o,N2: nat] :
% 0.29/0.50        ( ( P @ zero_zero_nat )
% 0.29/0.50       => ( ! [N3: nat] :
% 0.29/0.50              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.29/0.50             => ( ~ ( P @ N3 )
% 0.29/0.50               => ? [M6: nat] :
% 0.29/0.50                    ( ( ord_less_nat @ M6 @ N3 )
% 0.29/0.50                    & ~ ( P @ M6 ) ) ) )
% 0.29/0.50         => ( P @ N2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % infinite_descent0
% 0.29/0.50  thf(fact_1264_less__eq__nat_Osimps_I1_J,axiom,
% 0.29/0.50      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 0.29/0.50  
% 0.29/0.50  % less_eq_nat.simps(1)
% 0.29/0.50  thf(fact_1265_bot__nat__0_Oextremum__unique,axiom,
% 0.29/0.50      ! [A: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.50        = ( A = zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % bot_nat_0.extremum_unique
% 0.29/0.50  thf(fact_1266_bot__nat__0_Oextremum__uniqueI,axiom,
% 0.29/0.50      ! [A: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.50       => ( A = zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % bot_nat_0.extremum_uniqueI
% 0.29/0.50  thf(fact_1267_le__0__eq,axiom,
% 0.29/0.50      ! [N2: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 0.29/0.50        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % le_0_eq
% 0.29/0.50  thf(fact_1268_nat__less__le,axiom,
% 0.29/0.50      ( ord_less_nat
% 0.29/0.50      = ( ^ [M2: nat,N: nat] :
% 0.29/0.50            ( ( ord_less_eq_nat @ M2 @ N )
% 0.29/0.50            & ( M2 != N ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % nat_less_le
% 0.29/0.50  thf(fact_1269_less__imp__le__nat,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.50       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_imp_le_nat
% 0.29/0.50  thf(fact_1270_le__eq__less__or__eq,axiom,
% 0.29/0.50      ( ord_less_eq_nat
% 0.29/0.50      = ( ^ [M2: nat,N: nat] :
% 0.29/0.50            ( ( ord_less_nat @ M2 @ N )
% 0.29/0.50            | ( M2 = N ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % le_eq_less_or_eq
% 0.29/0.50  thf(fact_1271_less__or__eq__imp__le,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ( ord_less_nat @ M @ N2 )
% 0.29/0.50          | ( M = N2 ) )
% 0.29/0.50       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_or_eq_imp_le
% 0.29/0.50  thf(fact_1272_le__neq__implies__less,axiom,
% 0.29/0.50      ! [M: nat,N2: nat] :
% 0.29/0.50        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.50       => ( ( M != N2 )
% 0.29/0.50         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % le_neq_implies_less
% 0.29/0.50  thf(fact_1273_less__mono__imp__le__mono,axiom,
% 0.29/0.50      ! [F: nat > nat,I: nat,J: nat] :
% 0.29/0.50        ( ! [I3: nat,J2: nat] :
% 0.29/0.50            ( ( ord_less_nat @ I3 @ J2 )
% 0.29/0.50           => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 0.29/0.50       => ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.50         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % less_mono_imp_le_mono
% 0.29/0.50  thf(fact_1274_double__eq__0__iff,axiom,
% 0.29/0.50      ! [A: real] :
% 0.29/0.50        ( ( ( plus_plus_real @ A @ A )
% 0.29/0.50          = zero_zero_real )
% 0.29/0.50        = ( A = zero_zero_real ) ) ).
% 0.29/0.50  
% 0.29/0.50  % double_eq_0_iff
% 0.29/0.50  thf(fact_1275_double__eq__0__iff,axiom,
% 0.29/0.50      ! [A: rat] :
% 0.29/0.50        ( ( ( plus_plus_rat @ A @ A )
% 0.29/0.50          = zero_zero_rat )
% 0.29/0.50        = ( A = zero_zero_rat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % double_eq_0_iff
% 0.29/0.50  thf(fact_1276_double__eq__0__iff,axiom,
% 0.29/0.50      ! [A: int] :
% 0.29/0.50        ( ( ( plus_plus_int @ A @ A )
% 0.29/0.50          = zero_zero_int )
% 0.29/0.50        = ( A = zero_zero_int ) ) ).
% 0.29/0.50  
% 0.29/0.50  % double_eq_0_iff
% 0.29/0.50  thf(fact_1277_assms_I1_J,axiom,
% 0.29/0.50      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 0.29/0.50  
% 0.29/0.50  % assms(1)
% 0.29/0.50  thf(fact_1278_mint__corr__help__empty,axiom,
% 0.29/0.50      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.50        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.50       => ( ( ( vEBT_vebt_mint @ T )
% 0.29/0.50            = none_nat )
% 0.29/0.50         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.29/0.50            = bot_bot_set_nat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % mint_corr_help_empty
% 0.29/0.50  thf(fact_1279_maxt__corr__help__empty,axiom,
% 0.29/0.50      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.50        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.50       => ( ( ( vEBT_vebt_maxt @ T )
% 0.29/0.50            = none_nat )
% 0.29/0.50         => ( ( vEBT_VEBT_set_vebt @ T )
% 0.29/0.50            = bot_bot_set_nat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % maxt_corr_help_empty
% 0.29/0.50  thf(fact_1280_field__le__epsilon,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ! [E: real] :
% 0.29/0.50            ( ( ord_less_real @ zero_zero_real @ E )
% 0.29/0.50           => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E ) ) )
% 0.29/0.50       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % field_le_epsilon
% 0.29/0.50  thf(fact_1281_field__le__epsilon,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ! [E: rat] :
% 0.29/0.50            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.29/0.50           => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E ) ) )
% 0.29/0.50       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % field_le_epsilon
% 0.29/0.50  thf(fact_1282_option_Osize__gen_I2_J,axiom,
% 0.29/0.50      ! [X3: nat > nat,X22: nat] :
% 0.29/0.50        ( ( size_option_nat @ X3 @ ( some_nat @ X22 ) )
% 0.29/0.50        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.size_gen(2)
% 0.29/0.50  thf(fact_1283_option_Osize__gen_I2_J,axiom,
% 0.29/0.50      ! [X3: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 0.29/0.50        ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.50        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.size_gen(2)
% 0.29/0.50  thf(fact_1284_option_Osize__gen_I2_J,axiom,
% 0.29/0.50      ! [X3: num > nat,X22: num] :
% 0.29/0.50        ( ( size_option_num @ X3 @ ( some_num @ X22 ) )
% 0.29/0.50        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.size_gen(2)
% 0.29/0.50  thf(fact_1285_add__less__zeroD,axiom,
% 0.29/0.50      ! [X3: real,Y3: real] :
% 0.29/0.50        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 0.29/0.50       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.29/0.50          | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % add_less_zeroD
% 0.29/0.50  thf(fact_1286_add__less__zeroD,axiom,
% 0.29/0.50      ! [X3: rat,Y3: rat] :
% 0.29/0.50        ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
% 0.29/0.50       => ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.29/0.50          | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % add_less_zeroD
% 0.29/0.50  thf(fact_1287_add__less__zeroD,axiom,
% 0.29/0.50      ! [X3: int,Y3: int] :
% 0.29/0.50        ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
% 0.29/0.50       => ( ( ord_less_int @ X3 @ zero_zero_int )
% 0.29/0.50          | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % add_less_zeroD
% 0.29/0.50  thf(fact_1288_empty__subsetI,axiom,
% 0.29/0.50      ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_subsetI
% 0.29/0.50  thf(fact_1289_empty__subsetI,axiom,
% 0.29/0.50      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_subsetI
% 0.29/0.50  thf(fact_1290_empty__subsetI,axiom,
% 0.29/0.50      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_subsetI
% 0.29/0.50  thf(fact_1291_empty__subsetI,axiom,
% 0.29/0.50      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_subsetI
% 0.29/0.50  thf(fact_1292_subset__empty,axiom,
% 0.29/0.50      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.50        ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 0.29/0.50        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subset_empty
% 0.29/0.50  thf(fact_1293_subset__empty,axiom,
% 0.29/0.50      ! [A2: set_real] :
% 0.29/0.50        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 0.29/0.50        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subset_empty
% 0.29/0.50  thf(fact_1294_subset__empty,axiom,
% 0.29/0.50      ! [A2: set_nat] :
% 0.29/0.50        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 0.29/0.50        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subset_empty
% 0.29/0.50  thf(fact_1295_subset__empty,axiom,
% 0.29/0.50      ! [A2: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 0.29/0.50        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subset_empty
% 0.29/0.50  thf(fact_1296_empty__iff,axiom,
% 0.29/0.50      ! [C: set_nat] :
% 0.29/0.50        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_iff
% 0.29/0.50  thf(fact_1297_empty__iff,axiom,
% 0.29/0.50      ! [C: product_prod_nat_nat] :
% 0.29/0.50        ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_iff
% 0.29/0.50  thf(fact_1298_empty__iff,axiom,
% 0.29/0.50      ! [C: real] :
% 0.29/0.50        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_iff
% 0.29/0.50  thf(fact_1299_empty__iff,axiom,
% 0.29/0.50      ! [C: nat] :
% 0.29/0.50        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_iff
% 0.29/0.50  thf(fact_1300_empty__iff,axiom,
% 0.29/0.50      ! [C: int] :
% 0.29/0.50        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_iff
% 0.29/0.50  thf(fact_1301_all__not__in__conv,axiom,
% 0.29/0.50      ! [A2: set_set_nat] :
% 0.29/0.50        ( ( ! [X: set_nat] :
% 0.29/0.50              ~ ( member_set_nat @ X @ A2 ) )
% 0.29/0.50        = ( A2 = bot_bot_set_set_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % all_not_in_conv
% 0.29/0.50  thf(fact_1302_all__not__in__conv,axiom,
% 0.29/0.50      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.50        ( ( ! [X: product_prod_nat_nat] :
% 0.29/0.50              ~ ( member8440522571783428010at_nat @ X @ A2 ) )
% 0.29/0.50        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % all_not_in_conv
% 0.29/0.50  thf(fact_1303_all__not__in__conv,axiom,
% 0.29/0.50      ! [A2: set_real] :
% 0.29/0.50        ( ( ! [X: real] :
% 0.29/0.50              ~ ( member_real @ X @ A2 ) )
% 0.29/0.50        = ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.50  
% 0.29/0.50  % all_not_in_conv
% 0.29/0.50  thf(fact_1304_all__not__in__conv,axiom,
% 0.29/0.50      ! [A2: set_nat] :
% 0.29/0.50        ( ( ! [X: nat] :
% 0.29/0.50              ~ ( member_nat @ X @ A2 ) )
% 0.29/0.50        = ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % all_not_in_conv
% 0.29/0.50  thf(fact_1305_all__not__in__conv,axiom,
% 0.29/0.50      ! [A2: set_int] :
% 0.29/0.50        ( ( ! [X: int] :
% 0.29/0.50              ~ ( member_int @ X @ A2 ) )
% 0.29/0.50        = ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.50  
% 0.29/0.50  % all_not_in_conv
% 0.29/0.50  thf(fact_1306_deg__deg__n,axiom,
% 0.29/0.50      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 0.29/0.50        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.29/0.50       => ( Deg = N2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % deg_deg_n
% 0.29/0.50  thf(fact_1307_deg__SUcn__Node,axiom,
% 0.29/0.50      ! [Tree: vEBT_VEBT,N2: nat] :
% 0.29/0.50        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
% 0.29/0.50       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.29/0.50            ( Tree
% 0.29/0.50            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % deg_SUcn_Node
% 0.29/0.50  thf(fact_1308_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: list_nat > $o] :
% 0.29/0.50        ( ( bot_bot_set_list_nat
% 0.29/0.50          = ( collect_list_nat @ P ) )
% 0.29/0.50        = ( ! [X: list_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1309_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: set_nat > $o] :
% 0.29/0.50        ( ( bot_bot_set_set_nat
% 0.29/0.50          = ( collect_set_nat @ P ) )
% 0.29/0.50        = ( ! [X: set_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1310_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: product_prod_nat_nat > $o] :
% 0.29/0.50        ( ( bot_bo2099793752762293965at_nat
% 0.29/0.50          = ( collec3392354462482085612at_nat @ P ) )
% 0.29/0.50        = ( ! [X: product_prod_nat_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1311_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: real > $o] :
% 0.29/0.50        ( ( bot_bot_set_real
% 0.29/0.50          = ( collect_real @ P ) )
% 0.29/0.50        = ( ! [X: real] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1312_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: nat > $o] :
% 0.29/0.50        ( ( bot_bot_set_nat
% 0.29/0.50          = ( collect_nat @ P ) )
% 0.29/0.50        = ( ! [X: nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1313_empty__Collect__eq,axiom,
% 0.29/0.50      ! [P: int > $o] :
% 0.29/0.50        ( ( bot_bot_set_int
% 0.29/0.50          = ( collect_int @ P ) )
% 0.29/0.50        = ( ! [X: int] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % empty_Collect_eq
% 0.29/0.50  thf(fact_1314_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: list_nat > $o] :
% 0.29/0.50        ( ( ( collect_list_nat @ P )
% 0.29/0.50          = bot_bot_set_list_nat )
% 0.29/0.50        = ( ! [X: list_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1315_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: set_nat > $o] :
% 0.29/0.50        ( ( ( collect_set_nat @ P )
% 0.29/0.50          = bot_bot_set_set_nat )
% 0.29/0.50        = ( ! [X: set_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1316_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: product_prod_nat_nat > $o] :
% 0.29/0.50        ( ( ( collec3392354462482085612at_nat @ P )
% 0.29/0.50          = bot_bo2099793752762293965at_nat )
% 0.29/0.50        = ( ! [X: product_prod_nat_nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1317_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: real > $o] :
% 0.29/0.50        ( ( ( collect_real @ P )
% 0.29/0.50          = bot_bot_set_real )
% 0.29/0.50        = ( ! [X: real] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1318_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: nat > $o] :
% 0.29/0.50        ( ( ( collect_nat @ P )
% 0.29/0.50          = bot_bot_set_nat )
% 0.29/0.50        = ( ! [X: nat] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1319_Collect__empty__eq,axiom,
% 0.29/0.50      ! [P: int > $o] :
% 0.29/0.50        ( ( ( collect_int @ P )
% 0.29/0.50          = bot_bot_set_int )
% 0.29/0.50        = ( ! [X: int] :
% 0.29/0.50              ~ ( P @ X ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % Collect_empty_eq
% 0.29/0.50  thf(fact_1320_psubsetI,axiom,
% 0.29/0.50      ! [A2: set_int,B2: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.50       => ( ( A2 != B2 )
% 0.29/0.50         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % psubsetI
% 0.29/0.50  thf(fact_1321_subset__antisym,axiom,
% 0.29/0.50      ! [A2: set_int,B2: set_int] :
% 0.29/0.50        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.50       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.29/0.50         => ( A2 = B2 ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subset_antisym
% 0.29/0.50  thf(fact_1322_subsetI,axiom,
% 0.29/0.50      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.29/0.50        ( ! [X4: product_prod_nat_nat] :
% 0.29/0.50            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.29/0.50           => ( member8440522571783428010at_nat @ X4 @ B2 ) )
% 0.29/0.50       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subsetI
% 0.29/0.50  thf(fact_1323_subsetI,axiom,
% 0.29/0.50      ! [A2: set_real,B2: set_real] :
% 0.29/0.50        ( ! [X4: real] :
% 0.29/0.50            ( ( member_real @ X4 @ A2 )
% 0.29/0.50           => ( member_real @ X4 @ B2 ) )
% 0.29/0.50       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subsetI
% 0.29/0.50  thf(fact_1324_subsetI,axiom,
% 0.29/0.50      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.29/0.50        ( ! [X4: set_nat] :
% 0.29/0.50            ( ( member_set_nat @ X4 @ A2 )
% 0.29/0.50           => ( member_set_nat @ X4 @ B2 ) )
% 0.29/0.50       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subsetI
% 0.29/0.50  thf(fact_1325_subsetI,axiom,
% 0.29/0.50      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.50        ( ! [X4: nat] :
% 0.29/0.50            ( ( member_nat @ X4 @ A2 )
% 0.29/0.50           => ( member_nat @ X4 @ B2 ) )
% 0.29/0.50       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subsetI
% 0.29/0.50  thf(fact_1326_subsetI,axiom,
% 0.29/0.50      ! [A2: set_int,B2: set_int] :
% 0.29/0.50        ( ! [X4: int] :
% 0.29/0.50            ( ( member_int @ X4 @ A2 )
% 0.29/0.50           => ( member_int @ X4 @ B2 ) )
% 0.29/0.50       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.29/0.50  
% 0.29/0.50  % subsetI
% 0.29/0.50  thf(fact_1327_mi__eq__ma__no__ch,axiom,
% 0.29/0.50      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 0.29/0.50        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 0.29/0.50       => ( ( Mi = Ma )
% 0.29/0.50         => ( ! [X2: vEBT_VEBT] :
% 0.29/0.50                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.50               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
% 0.29/0.50            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % mi_eq_ma_no_ch
% 0.29/0.50  thf(fact_1328_not__Some__eq,axiom,
% 0.29/0.50      ! [X3: option_nat] :
% 0.29/0.50        ( ( ! [Y: nat] :
% 0.29/0.50              ( X3
% 0.29/0.50             != ( some_nat @ Y ) ) )
% 0.29/0.50        = ( X3 = none_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_Some_eq
% 0.29/0.50  thf(fact_1329_not__Some__eq,axiom,
% 0.29/0.50      ! [X3: option4927543243414619207at_nat] :
% 0.29/0.50        ( ( ! [Y: product_prod_nat_nat] :
% 0.29/0.50              ( X3
% 0.29/0.50             != ( some_P7363390416028606310at_nat @ Y ) ) )
% 0.29/0.50        = ( X3 = none_P5556105721700978146at_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_Some_eq
% 0.29/0.50  thf(fact_1330_not__Some__eq,axiom,
% 0.29/0.50      ! [X3: option_num] :
% 0.29/0.50        ( ( ! [Y: num] :
% 0.29/0.50              ( X3
% 0.29/0.50             != ( some_num @ Y ) ) )
% 0.29/0.50        = ( X3 = none_num ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_Some_eq
% 0.29/0.50  thf(fact_1331_not__None__eq,axiom,
% 0.29/0.50      ! [X3: option_nat] :
% 0.29/0.50        ( ( X3 != none_nat )
% 0.29/0.50        = ( ? [Y: nat] :
% 0.29/0.50              ( X3
% 0.29/0.50              = ( some_nat @ Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_None_eq
% 0.29/0.50  thf(fact_1332_not__None__eq,axiom,
% 0.29/0.50      ! [X3: option4927543243414619207at_nat] :
% 0.29/0.50        ( ( X3 != none_P5556105721700978146at_nat )
% 0.29/0.50        = ( ? [Y: product_prod_nat_nat] :
% 0.29/0.50              ( X3
% 0.29/0.50              = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_None_eq
% 0.29/0.50  thf(fact_1333_not__None__eq,axiom,
% 0.29/0.50      ! [X3: option_num] :
% 0.29/0.50        ( ( X3 != none_num )
% 0.29/0.50        = ( ? [Y: num] :
% 0.29/0.50              ( X3
% 0.29/0.50              = ( some_num @ Y ) ) ) ) ).
% 0.29/0.50  
% 0.29/0.50  % not_None_eq
% 0.29/0.50  thf(fact_1334_option_Ocollapse,axiom,
% 0.29/0.50      ! [Option: option_nat] :
% 0.29/0.50        ( ( Option != none_nat )
% 0.29/0.50       => ( ( some_nat @ ( the_nat @ Option ) )
% 0.29/0.50          = Option ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.collapse
% 0.29/0.50  thf(fact_1335_option_Ocollapse,axiom,
% 0.29/0.50      ! [Option: option4927543243414619207at_nat] :
% 0.29/0.50        ( ( Option != none_P5556105721700978146at_nat )
% 0.29/0.50       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 0.29/0.50          = Option ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.collapse
% 0.29/0.50  thf(fact_1336_option_Ocollapse,axiom,
% 0.29/0.50      ! [Option: option_num] :
% 0.29/0.50        ( ( Option != none_num )
% 0.29/0.50       => ( ( some_num @ ( the_num @ Option ) )
% 0.29/0.50          = Option ) ) ).
% 0.29/0.50  
% 0.29/0.50  % option.collapse
% 0.29/0.50  thf(fact_1337_add__def,axiom,
% 0.29/0.50      ( vEBT_VEBT_add
% 0.29/0.50      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 0.29/0.50  
% 0.29/0.50  % add_def
% 0.29/0.50  thf(fact_1338_psubsetE,axiom,
% 0.29/0.50      ! [A2: set_int,B2: set_int] :
% 0.29/0.50        ( ( ord_less_set_int @ A2 @ B2 )
% 0.29/0.51       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetE
% 0.29/0.51  thf(fact_1339_psubset__eq,axiom,
% 0.29/0.51      ( ord_less_set_int
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.29/0.51            & ( A5 != B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubset_eq
% 0.29/0.51  thf(fact_1340_psubset__imp__subset,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int] :
% 0.29/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubset_imp_subset
% 0.29/0.51  thf(fact_1341_psubset__subset__trans,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,C4: set_int] :
% 0.29/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 0.29/0.51         => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubset_subset_trans
% 0.29/0.51  thf(fact_1342_subset__not__subset__eq,axiom,
% 0.29/0.51      ( ord_less_set_int
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.29/0.51            & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_not_subset_eq
% 0.29/0.51  thf(fact_1343_subset__psubset__trans,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,C4: set_int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( ord_less_set_int @ B2 @ C4 )
% 0.29/0.51         => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_psubset_trans
% 0.29/0.51  thf(fact_1344_subset__iff__psubset__eq,axiom,
% 0.29/0.51      ( ord_less_eq_set_int
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51            ( ( ord_less_set_int @ A5 @ B5 )
% 0.29/0.51            | ( A5 = B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff_psubset_eq
% 0.29/0.51  thf(fact_1345_Collect__mono__iff,axiom,
% 0.29/0.51      ! [P: real > $o,Q: real > $o] :
% 0.29/0.51        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 0.29/0.51        = ( ! [X: real] :
% 0.29/0.51              ( ( P @ X )
% 0.29/0.51             => ( Q @ X ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono_iff
% 0.29/0.51  thf(fact_1346_Collect__mono__iff,axiom,
% 0.29/0.51      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.29/0.51        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 0.29/0.51        = ( ! [X: list_nat] :
% 0.29/0.51              ( ( P @ X )
% 0.29/0.51             => ( Q @ X ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono_iff
% 0.29/0.51  thf(fact_1347_Collect__mono__iff,axiom,
% 0.29/0.51      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.29/0.51        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 0.29/0.51        = ( ! [X: set_nat] :
% 0.29/0.51              ( ( P @ X )
% 0.29/0.51             => ( Q @ X ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono_iff
% 0.29/0.51  thf(fact_1348_Collect__mono__iff,axiom,
% 0.29/0.51      ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.51        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 0.29/0.51        = ( ! [X: nat] :
% 0.29/0.51              ( ( P @ X )
% 0.29/0.51             => ( Q @ X ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono_iff
% 0.29/0.51  thf(fact_1349_Collect__mono__iff,axiom,
% 0.29/0.51      ! [P: int > $o,Q: int > $o] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 0.29/0.51        = ( ! [X: int] :
% 0.29/0.51              ( ( P @ X )
% 0.29/0.51             => ( Q @ X ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono_iff
% 0.29/0.51  thf(fact_1350_set__eq__subset,axiom,
% 0.29/0.51      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 0.29/0.51            & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % set_eq_subset
% 0.29/0.51  thf(fact_1351_subset__trans,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,C4: set_int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 0.29/0.51         => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_trans
% 0.29/0.51  thf(fact_1352_Collect__mono,axiom,
% 0.29/0.51      ! [P: real > $o,Q: real > $o] :
% 0.29/0.51        ( ! [X4: real] :
% 0.29/0.51            ( ( P @ X4 )
% 0.29/0.51           => ( Q @ X4 ) )
% 0.29/0.51       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono
% 0.29/0.51  thf(fact_1353_Collect__mono,axiom,
% 0.29/0.51      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.29/0.51        ( ! [X4: list_nat] :
% 0.29/0.51            ( ( P @ X4 )
% 0.29/0.51           => ( Q @ X4 ) )
% 0.29/0.51       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono
% 0.29/0.51  thf(fact_1354_Collect__mono,axiom,
% 0.29/0.51      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.29/0.51        ( ! [X4: set_nat] :
% 0.29/0.51            ( ( P @ X4 )
% 0.29/0.51           => ( Q @ X4 ) )
% 0.29/0.51       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono
% 0.29/0.51  thf(fact_1355_Collect__mono,axiom,
% 0.29/0.51      ! [P: nat > $o,Q: nat > $o] :
% 0.29/0.51        ( ! [X4: nat] :
% 0.29/0.51            ( ( P @ X4 )
% 0.29/0.51           => ( Q @ X4 ) )
% 0.29/0.51       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono
% 0.29/0.51  thf(fact_1356_Collect__mono,axiom,
% 0.29/0.51      ! [P: int > $o,Q: int > $o] :
% 0.29/0.51        ( ! [X4: int] :
% 0.29/0.51            ( ( P @ X4 )
% 0.29/0.51           => ( Q @ X4 ) )
% 0.29/0.51       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_mono
% 0.29/0.51  thf(fact_1357_subset__refl,axiom,
% 0.29/0.51      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_refl
% 0.29/0.51  thf(fact_1358_subset__iff,axiom,
% 0.29/0.51      ( ord_le3146513528884898305at_nat
% 0.29/0.51      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.29/0.51          ! [T2: product_prod_nat_nat] :
% 0.29/0.51            ( ( member8440522571783428010at_nat @ T2 @ A5 )
% 0.29/0.51           => ( member8440522571783428010at_nat @ T2 @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff
% 0.29/0.51  thf(fact_1359_subset__iff,axiom,
% 0.29/0.51      ( ord_less_eq_set_real
% 0.29/0.51      = ( ^ [A5: set_real,B5: set_real] :
% 0.29/0.51          ! [T2: real] :
% 0.29/0.51            ( ( member_real @ T2 @ A5 )
% 0.29/0.51           => ( member_real @ T2 @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff
% 0.29/0.51  thf(fact_1360_subset__iff,axiom,
% 0.29/0.51      ( ord_le6893508408891458716et_nat
% 0.29/0.51      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.29/0.51          ! [T2: set_nat] :
% 0.29/0.51            ( ( member_set_nat @ T2 @ A5 )
% 0.29/0.51           => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff
% 0.29/0.51  thf(fact_1361_subset__iff,axiom,
% 0.29/0.51      ( ord_less_eq_set_nat
% 0.29/0.51      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.29/0.51          ! [T2: nat] :
% 0.29/0.51            ( ( member_nat @ T2 @ A5 )
% 0.29/0.51           => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff
% 0.29/0.51  thf(fact_1362_subset__iff,axiom,
% 0.29/0.51      ( ord_less_eq_set_int
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51          ! [T2: int] :
% 0.29/0.51            ( ( member_int @ T2 @ A5 )
% 0.29/0.51           => ( member_int @ T2 @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_iff
% 0.29/0.51  thf(fact_1363_equalityD2,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int] :
% 0.29/0.51        ( ( A2 = B2 )
% 0.29/0.51       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equalityD2
% 0.29/0.51  thf(fact_1364_equalityD1,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int] :
% 0.29/0.51        ( ( A2 = B2 )
% 0.29/0.51       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equalityD1
% 0.29/0.51  thf(fact_1365_subset__eq,axiom,
% 0.29/0.51      ( ord_le3146513528884898305at_nat
% 0.29/0.51      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.29/0.51          ! [X: product_prod_nat_nat] :
% 0.29/0.51            ( ( member8440522571783428010at_nat @ X @ A5 )
% 0.29/0.51           => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_eq
% 0.29/0.51  thf(fact_1366_subset__eq,axiom,
% 0.29/0.51      ( ord_less_eq_set_real
% 0.29/0.51      = ( ^ [A5: set_real,B5: set_real] :
% 0.29/0.51          ! [X: real] :
% 0.29/0.51            ( ( member_real @ X @ A5 )
% 0.29/0.51           => ( member_real @ X @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_eq
% 0.29/0.51  thf(fact_1367_subset__eq,axiom,
% 0.29/0.51      ( ord_le6893508408891458716et_nat
% 0.29/0.51      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.29/0.51          ! [X: set_nat] :
% 0.29/0.51            ( ( member_set_nat @ X @ A5 )
% 0.29/0.51           => ( member_set_nat @ X @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_eq
% 0.29/0.51  thf(fact_1368_subset__eq,axiom,
% 0.29/0.51      ( ord_less_eq_set_nat
% 0.29/0.51      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.29/0.51          ! [X: nat] :
% 0.29/0.51            ( ( member_nat @ X @ A5 )
% 0.29/0.51           => ( member_nat @ X @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_eq
% 0.29/0.51  thf(fact_1369_subset__eq,axiom,
% 0.29/0.51      ( ord_less_eq_set_int
% 0.29/0.51      = ( ^ [A5: set_int,B5: set_int] :
% 0.29/0.51          ! [X: int] :
% 0.29/0.51            ( ( member_int @ X @ A5 )
% 0.29/0.51           => ( member_int @ X @ B5 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_eq
% 0.29/0.51  thf(fact_1370_equalityE,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int] :
% 0.29/0.51        ( ( A2 = B2 )
% 0.29/0.51       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equalityE
% 0.29/0.51  thf(fact_1371_subsetD,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 0.29/0.51        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.29/0.51         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subsetD
% 0.29/0.51  thf(fact_1372_subsetD,axiom,
% 0.29/0.51      ! [A2: set_real,B2: set_real,C: real] :
% 0.29/0.51        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.29/0.51       => ( ( member_real @ C @ A2 )
% 0.29/0.51         => ( member_real @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subsetD
% 0.29/0.51  thf(fact_1373_subsetD,axiom,
% 0.29/0.51      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.29/0.51        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_set_nat @ C @ A2 )
% 0.29/0.51         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subsetD
% 0.29/0.51  thf(fact_1374_subsetD,axiom,
% 0.29/0.51      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.29/0.51        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_nat @ C @ A2 )
% 0.29/0.51         => ( member_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subsetD
% 0.29/0.51  thf(fact_1375_subsetD,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,C: int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( member_int @ C @ A2 )
% 0.29/0.51         => ( member_int @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subsetD
% 0.29/0.51  thf(fact_1376_in__mono,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
% 0.29/0.51        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.29/0.51         => ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % in_mono
% 0.29/0.51  thf(fact_1377_in__mono,axiom,
% 0.29/0.51      ! [A2: set_real,B2: set_real,X3: real] :
% 0.29/0.51        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.29/0.51       => ( ( member_real @ X3 @ A2 )
% 0.29/0.51         => ( member_real @ X3 @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % in_mono
% 0.29/0.51  thf(fact_1378_in__mono,axiom,
% 0.29/0.51      ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat] :
% 0.29/0.51        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_set_nat @ X3 @ A2 )
% 0.29/0.51         => ( member_set_nat @ X3 @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % in_mono
% 0.29/0.51  thf(fact_1379_in__mono,axiom,
% 0.29/0.51      ! [A2: set_nat,B2: set_nat,X3: nat] :
% 0.29/0.51        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_nat @ X3 @ A2 )
% 0.29/0.51         => ( member_nat @ X3 @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % in_mono
% 0.29/0.51  thf(fact_1380_in__mono,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,X3: int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( member_int @ X3 @ A2 )
% 0.29/0.51         => ( member_int @ X3 @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % in_mono
% 0.29/0.51  thf(fact_1381_bot__set__def,axiom,
% 0.29/0.51      ( bot_bot_set_list_nat
% 0.29/0.51      = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1382_bot__set__def,axiom,
% 0.29/0.51      ( bot_bot_set_set_nat
% 0.29/0.51      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1383_bot__set__def,axiom,
% 0.29/0.51      ( bot_bo2099793752762293965at_nat
% 0.29/0.51      = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1384_bot__set__def,axiom,
% 0.29/0.51      ( bot_bot_set_real
% 0.29/0.51      = ( collect_real @ bot_bot_real_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1385_bot__set__def,axiom,
% 0.29/0.51      ( bot_bot_set_nat
% 0.29/0.51      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1386_bot__set__def,axiom,
% 0.29/0.51      ( bot_bot_set_int
% 0.29/0.51      = ( collect_int @ bot_bot_int_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_set_def
% 0.29/0.51  thf(fact_1387_bot__nat__def,axiom,
% 0.29/0.51      bot_bot_nat = zero_zero_nat ).
% 0.29/0.51  
% 0.29/0.51  % bot_nat_def
% 0.29/0.51  thf(fact_1388_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 0.29/0.51        ( ( ( X3 = none_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: nat,B4: nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1389_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( ( X3 = none_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: nat,B4: product_prod_nat_nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1390_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 0.29/0.51        ( ( ( X3 = none_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_num )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: nat,B4: num] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_num @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1391_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 0.29/0.51        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: product_prod_nat_nat,B4: nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1392_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1393_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 0.29/0.51        ( ( ( X3 = none_P5556105721700978146at_nat )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_num )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: product_prod_nat_nat,B4: num] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_num @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1394_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 0.29/0.51        ( ( ( X3 = none_num )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: num,B4: nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_num @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1395_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( ( X3 = none_num )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: num,B4: product_prod_nat_nat] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_num @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1396_combine__options__cases,axiom,
% 0.29/0.51      ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 0.29/0.51        ( ( ( X3 = none_num )
% 0.29/0.51         => ( P @ X3 @ Y3 ) )
% 0.29/0.51       => ( ( ( Y3 = none_num )
% 0.29/0.51           => ( P @ X3 @ Y3 ) )
% 0.29/0.51         => ( ! [A4: num,B4: num] :
% 0.29/0.51                ( ( X3
% 0.29/0.51                  = ( some_num @ A4 ) )
% 0.29/0.51               => ( ( Y3
% 0.29/0.51                    = ( some_num @ B4 ) )
% 0.29/0.51                 => ( P @ X3 @ Y3 ) ) )
% 0.29/0.51           => ( P @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % combine_options_cases
% 0.29/0.51  thf(fact_1397_split__option__all,axiom,
% 0.29/0.51      ( ( ^ [P2: option_nat > $o] :
% 0.29/0.51          ! [X6: option_nat] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option_nat > $o] :
% 0.29/0.51            ( ( P3 @ none_nat )
% 0.29/0.51            & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_all
% 0.29/0.51  thf(fact_1398_split__option__all,axiom,
% 0.29/0.51      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.29/0.51          ! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.29/0.51            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.29/0.51            & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_all
% 0.29/0.51  thf(fact_1399_split__option__all,axiom,
% 0.29/0.51      ( ( ^ [P2: option_num > $o] :
% 0.29/0.51          ! [X6: option_num] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option_num > $o] :
% 0.29/0.51            ( ( P3 @ none_num )
% 0.29/0.51            & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_all
% 0.29/0.51  thf(fact_1400_split__option__ex,axiom,
% 0.29/0.51      ( ( ^ [P2: option_nat > $o] :
% 0.29/0.51          ? [X6: option_nat] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option_nat > $o] :
% 0.29/0.51            ( ( P3 @ none_nat )
% 0.29/0.51            | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_ex
% 0.29/0.51  thf(fact_1401_split__option__ex,axiom,
% 0.29/0.51      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 0.29/0.51          ? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 0.29/0.51            ( ( P3 @ none_P5556105721700978146at_nat )
% 0.29/0.51            | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_ex
% 0.29/0.51  thf(fact_1402_split__option__ex,axiom,
% 0.29/0.51      ( ( ^ [P2: option_num > $o] :
% 0.29/0.51          ? [X6: option_num] : ( P2 @ X6 ) )
% 0.29/0.51      = ( ^ [P3: option_num > $o] :
% 0.29/0.51            ( ( P3 @ none_num )
% 0.29/0.51            | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % split_option_ex
% 0.29/0.51  thf(fact_1403_option_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: option_nat] :
% 0.29/0.51        ( ( Y3 != none_nat )
% 0.29/0.51       => ~ ! [X23: nat] :
% 0.29/0.51              ( Y3
% 0.29/0.51             != ( some_nat @ X23 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust
% 0.29/0.51  thf(fact_1404_option_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( Y3 != none_P5556105721700978146at_nat )
% 0.29/0.51       => ~ ! [X23: product_prod_nat_nat] :
% 0.29/0.51              ( Y3
% 0.29/0.51             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust
% 0.29/0.51  thf(fact_1405_option_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: option_num] :
% 0.29/0.51        ( ( Y3 != none_num )
% 0.29/0.51       => ~ ! [X23: num] :
% 0.29/0.51              ( Y3
% 0.29/0.51             != ( some_num @ X23 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust
% 0.29/0.51  thf(fact_1406_option_OdiscI,axiom,
% 0.29/0.51      ! [Option: option_nat,X22: nat] :
% 0.29/0.51        ( ( Option
% 0.29/0.51          = ( some_nat @ X22 ) )
% 0.29/0.51       => ( Option != none_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.discI
% 0.29/0.51  thf(fact_1407_option_OdiscI,axiom,
% 0.29/0.51      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 0.29/0.51        ( ( Option
% 0.29/0.51          = ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.51       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.discI
% 0.29/0.51  thf(fact_1408_option_OdiscI,axiom,
% 0.29/0.51      ! [Option: option_num,X22: num] :
% 0.29/0.51        ( ( Option
% 0.29/0.51          = ( some_num @ X22 ) )
% 0.29/0.51       => ( Option != none_num ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.discI
% 0.29/0.51  thf(fact_1409_option_Odistinct_I1_J,axiom,
% 0.29/0.51      ! [X22: nat] :
% 0.29/0.51        ( none_nat
% 0.29/0.51       != ( some_nat @ X22 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.distinct(1)
% 0.29/0.51  thf(fact_1410_option_Odistinct_I1_J,axiom,
% 0.29/0.51      ! [X22: product_prod_nat_nat] :
% 0.29/0.51        ( none_P5556105721700978146at_nat
% 0.29/0.51       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.distinct(1)
% 0.29/0.51  thf(fact_1411_option_Odistinct_I1_J,axiom,
% 0.29/0.51      ! [X22: num] :
% 0.29/0.51        ( none_num
% 0.29/0.51       != ( some_num @ X22 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.distinct(1)
% 0.29/0.51  thf(fact_1412_vebt__maxt_Osimps_I3_J,axiom,
% 0.29/0.51      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.29/0.51        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.29/0.51        = ( some_nat @ Ma ) ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_maxt.simps(3)
% 0.29/0.51  thf(fact_1413_vebt__mint_Osimps_I3_J,axiom,
% 0.29/0.51      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 0.29/0.51        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 0.29/0.51        = ( some_nat @ Mi ) ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_mint.simps(3)
% 0.29/0.51  thf(fact_1414_option_Oexpand,axiom,
% 0.29/0.51      ! [Option: option_nat,Option2: option_nat] :
% 0.29/0.51        ( ( ( Option = none_nat )
% 0.29/0.51          = ( Option2 = none_nat ) )
% 0.29/0.51       => ( ( ( Option != none_nat )
% 0.29/0.51           => ( ( Option2 != none_nat )
% 0.29/0.51             => ( ( the_nat @ Option )
% 0.29/0.51                = ( the_nat @ Option2 ) ) ) )
% 0.29/0.51         => ( Option = Option2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.expand
% 0.29/0.51  thf(fact_1415_option_Oexpand,axiom,
% 0.29/0.51      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( ( Option = none_P5556105721700978146at_nat )
% 0.29/0.51          = ( Option2 = none_P5556105721700978146at_nat ) )
% 0.29/0.51       => ( ( ( Option != none_P5556105721700978146at_nat )
% 0.29/0.51           => ( ( Option2 != none_P5556105721700978146at_nat )
% 0.29/0.51             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 0.29/0.51                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 0.29/0.51         => ( Option = Option2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.expand
% 0.29/0.51  thf(fact_1416_option_Oexpand,axiom,
% 0.29/0.51      ! [Option: option_num,Option2: option_num] :
% 0.29/0.51        ( ( ( Option = none_num )
% 0.29/0.51          = ( Option2 = none_num ) )
% 0.29/0.51       => ( ( ( Option != none_num )
% 0.29/0.51           => ( ( Option2 != none_num )
% 0.29/0.51             => ( ( the_num @ Option )
% 0.29/0.51                = ( the_num @ Option2 ) ) ) )
% 0.29/0.51         => ( Option = Option2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.expand
% 0.29/0.51  thf(fact_1417_option_Osize_I3_J,axiom,
% 0.29/0.51      ( ( size_size_option_nat @ none_nat )
% 0.29/0.51      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(3)
% 0.29/0.51  thf(fact_1418_option_Osize_I3_J,axiom,
% 0.29/0.51      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 0.29/0.51      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(3)
% 0.29/0.51  thf(fact_1419_option_Osize_I3_J,axiom,
% 0.29/0.51      ( ( size_size_option_num @ none_num )
% 0.29/0.51      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(3)
% 0.29/0.51  thf(fact_1420_option_Osize__gen_I1_J,axiom,
% 0.29/0.51      ! [X3: nat > nat] :
% 0.29/0.51        ( ( size_option_nat @ X3 @ none_nat )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size_gen(1)
% 0.29/0.51  thf(fact_1421_option_Osize__gen_I1_J,axiom,
% 0.29/0.51      ! [X3: product_prod_nat_nat > nat] :
% 0.29/0.51        ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size_gen(1)
% 0.29/0.51  thf(fact_1422_option_Osize__gen_I1_J,axiom,
% 0.29/0.51      ! [X3: num > nat] :
% 0.29/0.51        ( ( size_option_num @ X3 @ none_num )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size_gen(1)
% 0.29/0.51  thf(fact_1423_option_Oexhaust__sel,axiom,
% 0.29/0.51      ! [Option: option_nat] :
% 0.29/0.51        ( ( Option != none_nat )
% 0.29/0.51       => ( Option
% 0.29/0.51          = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust_sel
% 0.29/0.51  thf(fact_1424_option_Oexhaust__sel,axiom,
% 0.29/0.51      ! [Option: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( Option != none_P5556105721700978146at_nat )
% 0.29/0.51       => ( Option
% 0.29/0.51          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust_sel
% 0.29/0.51  thf(fact_1425_option_Oexhaust__sel,axiom,
% 0.29/0.51      ! [Option: option_num] :
% 0.29/0.51        ( ( Option != none_num )
% 0.29/0.51       => ( Option
% 0.29/0.51          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.exhaust_sel
% 0.29/0.51  thf(fact_1426_linorder__neqE__linordered__idom,axiom,
% 0.29/0.51      ! [X3: real,Y3: real] :
% 0.29/0.51        ( ( X3 != Y3 )
% 0.29/0.51       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 0.29/0.51         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % linorder_neqE_linordered_idom
% 0.29/0.51  thf(fact_1427_linorder__neqE__linordered__idom,axiom,
% 0.29/0.51      ! [X3: rat,Y3: rat] :
% 0.29/0.51        ( ( X3 != Y3 )
% 0.29/0.51       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.51         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % linorder_neqE_linordered_idom
% 0.29/0.51  thf(fact_1428_linorder__neqE__linordered__idom,axiom,
% 0.29/0.51      ! [X3: int,Y3: int] :
% 0.29/0.51        ( ( X3 != Y3 )
% 0.29/0.51       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 0.29/0.51         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % linorder_neqE_linordered_idom
% 0.29/0.51  thf(fact_1429_linordered__field__no__ub,axiom,
% 0.29/0.51      ! [X2: real] :
% 0.29/0.51      ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 0.29/0.51  
% 0.29/0.51  % linordered_field_no_ub
% 0.29/0.51  thf(fact_1430_linordered__field__no__ub,axiom,
% 0.29/0.51      ! [X2: rat] :
% 0.29/0.51      ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 0.29/0.51  
% 0.29/0.51  % linordered_field_no_ub
% 0.29/0.51  thf(fact_1431_linordered__field__no__lb,axiom,
% 0.29/0.51      ! [X2: real] :
% 0.29/0.51      ? [Y5: real] : ( ord_less_real @ Y5 @ X2 ) ).
% 0.29/0.51  
% 0.29/0.51  % linordered_field_no_lb
% 0.29/0.51  thf(fact_1432_linordered__field__no__lb,axiom,
% 0.29/0.51      ! [X2: rat] :
% 0.29/0.51      ? [Y5: rat] : ( ord_less_rat @ Y5 @ X2 ) ).
% 0.29/0.51  
% 0.29/0.51  % linordered_field_no_lb
% 0.29/0.51  thf(fact_1433_not__psubset__empty,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % not_psubset_empty
% 0.29/0.51  thf(fact_1434_not__psubset__empty,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 0.29/0.51  
% 0.29/0.51  % not_psubset_empty
% 0.29/0.51  thf(fact_1435_not__psubset__empty,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % not_psubset_empty
% 0.29/0.51  thf(fact_1436_not__psubset__empty,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 0.29/0.51  
% 0.29/0.51  % not_psubset_empty
% 0.29/0.51  thf(fact_1437_ex__in__conv,axiom,
% 0.29/0.51      ! [A2: set_set_nat] :
% 0.29/0.51        ( ( ? [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 0.29/0.51        = ( A2 != bot_bot_set_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_in_conv
% 0.29/0.51  thf(fact_1438_ex__in__conv,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
% 0.29/0.51        = ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_in_conv
% 0.29/0.51  thf(fact_1439_ex__in__conv,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ( ( ? [X: real] : ( member_real @ X @ A2 ) )
% 0.29/0.51        = ( A2 != bot_bot_set_real ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_in_conv
% 0.29/0.51  thf(fact_1440_ex__in__conv,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
% 0.29/0.51        = ( A2 != bot_bot_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_in_conv
% 0.29/0.51  thf(fact_1441_ex__in__conv,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ( ? [X: int] : ( member_int @ X @ A2 ) )
% 0.29/0.51        = ( A2 != bot_bot_set_int ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_in_conv
% 0.29/0.51  thf(fact_1442_equals0I,axiom,
% 0.29/0.51      ! [A2: set_set_nat] :
% 0.29/0.51        ( ! [Y5: set_nat] :
% 0.29/0.51            ~ ( member_set_nat @ Y5 @ A2 )
% 0.29/0.51       => ( A2 = bot_bot_set_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0I
% 0.29/0.51  thf(fact_1443_equals0I,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ! [Y5: product_prod_nat_nat] :
% 0.29/0.51            ~ ( member8440522571783428010at_nat @ Y5 @ A2 )
% 0.29/0.51       => ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0I
% 0.29/0.51  thf(fact_1444_equals0I,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ( ! [Y5: real] :
% 0.29/0.51            ~ ( member_real @ Y5 @ A2 )
% 0.29/0.51       => ( A2 = bot_bot_set_real ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0I
% 0.29/0.51  thf(fact_1445_equals0I,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ! [Y5: nat] :
% 0.29/0.51            ~ ( member_nat @ Y5 @ A2 )
% 0.29/0.51       => ( A2 = bot_bot_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0I
% 0.29/0.51  thf(fact_1446_equals0I,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ! [Y5: int] :
% 0.29/0.51            ~ ( member_int @ Y5 @ A2 )
% 0.29/0.51       => ( A2 = bot_bot_set_int ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0I
% 0.29/0.51  thf(fact_1447_equals0D,axiom,
% 0.29/0.51      ! [A2: set_set_nat,A: set_nat] :
% 0.29/0.51        ( ( A2 = bot_bot_set_set_nat )
% 0.29/0.51       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0D
% 0.29/0.51  thf(fact_1448_equals0D,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.29/0.51        ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.29/0.51       => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0D
% 0.29/0.51  thf(fact_1449_equals0D,axiom,
% 0.29/0.51      ! [A2: set_real,A: real] :
% 0.29/0.51        ( ( A2 = bot_bot_set_real )
% 0.29/0.51       => ~ ( member_real @ A @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0D
% 0.29/0.51  thf(fact_1450_equals0D,axiom,
% 0.29/0.51      ! [A2: set_nat,A: nat] :
% 0.29/0.51        ( ( A2 = bot_bot_set_nat )
% 0.29/0.51       => ~ ( member_nat @ A @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0D
% 0.29/0.51  thf(fact_1451_equals0D,axiom,
% 0.29/0.51      ! [A2: set_int,A: int] :
% 0.29/0.51        ( ( A2 = bot_bot_set_int )
% 0.29/0.51       => ~ ( member_int @ A @ A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % equals0D
% 0.29/0.51  thf(fact_1452_emptyE,axiom,
% 0.29/0.51      ! [A: set_nat] :
% 0.29/0.51        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % emptyE
% 0.29/0.51  thf(fact_1453_emptyE,axiom,
% 0.29/0.51      ! [A: product_prod_nat_nat] :
% 0.29/0.51        ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % emptyE
% 0.29/0.51  thf(fact_1454_emptyE,axiom,
% 0.29/0.51      ! [A: real] :
% 0.29/0.51        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 0.29/0.51  
% 0.29/0.51  % emptyE
% 0.29/0.51  thf(fact_1455_emptyE,axiom,
% 0.29/0.51      ! [A: nat] :
% 0.29/0.51        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % emptyE
% 0.29/0.51  thf(fact_1456_emptyE,axiom,
% 0.29/0.51      ! [A: int] :
% 0.29/0.51        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 0.29/0.51  
% 0.29/0.51  % emptyE
% 0.29/0.51  thf(fact_1457_option_Osize_I4_J,axiom,
% 0.29/0.51      ! [X22: nat] :
% 0.29/0.51        ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(4)
% 0.29/0.51  thf(fact_1458_option_Osize_I4_J,axiom,
% 0.29/0.51      ! [X22: product_prod_nat_nat] :
% 0.29/0.51        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(4)
% 0.29/0.51  thf(fact_1459_option_Osize_I4_J,axiom,
% 0.29/0.51      ! [X22: num] :
% 0.29/0.51        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 0.29/0.51        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % option.size(4)
% 0.29/0.51  thf(fact_1460_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 0.29/0.51      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
% 0.29/0.51        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
% 0.29/0.51        = ( ( X3 = Mi )
% 0.29/0.51          | ( X3 = Ma ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.membermima.simps(3)
% 0.29/0.51  thf(fact_1461_vebt__member_Osimps_I4_J,axiom,
% 0.29/0.51      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 0.29/0.51        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_member.simps(4)
% 0.29/0.51  thf(fact_1462_vebt__member_Osimps_I3_J,axiom,
% 0.29/0.51      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 0.29/0.51        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_member.simps(3)
% 0.29/0.51  thf(fact_1463_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 0.29/0.51      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 0.29/0.51        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.minNull.simps(5)
% 0.29/0.51  thf(fact_1464_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 0.29/0.51        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.naive_member.simps(2)
% 0.29/0.51  thf(fact_1465_VEBT_Oinject_I1_J,axiom,
% 0.29/0.51      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 0.29/0.51        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.29/0.51          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 0.29/0.51        = ( ( X11 = Y11 )
% 0.29/0.51          & ( X12 = Y12 )
% 0.29/0.51          & ( X13 = Y13 )
% 0.29/0.51          & ( X14 = Y14 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT.inject(1)
% 0.29/0.51  thf(fact_1466_old_Oprod_Oinject,axiom,
% 0.29/0.51      ! [A: int,B: int,A6: int,B6: int] :
% 0.29/0.51        ( ( ( product_Pair_int_int @ A @ B )
% 0.29/0.51          = ( product_Pair_int_int @ A6 @ B6 ) )
% 0.29/0.51        = ( ( A = A6 )
% 0.29/0.51          & ( B = B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.inject
% 0.29/0.51  thf(fact_1467_old_Oprod_Oinject,axiom,
% 0.29/0.51      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc6137756002093451184nteger @ A @ B )
% 0.29/0.51          = ( produc6137756002093451184nteger @ A6 @ B6 ) )
% 0.29/0.51        = ( ( A = A6 )
% 0.29/0.51          & ( B = B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.inject
% 0.29/0.51  thf(fact_1468_old_Oprod_Oinject,axiom,
% 0.29/0.51      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc8603105652947943368nteger @ A @ B )
% 0.29/0.51          = ( produc8603105652947943368nteger @ A6 @ B6 ) )
% 0.29/0.51        = ( ( A = A6 )
% 0.29/0.51          & ( B = B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.inject
% 0.29/0.51  thf(fact_1469_old_Oprod_Oinject,axiom,
% 0.29/0.51      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 0.29/0.51          = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
% 0.29/0.51        = ( ( A = A6 )
% 0.29/0.51          & ( B = B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.inject
% 0.29/0.51  thf(fact_1470_old_Oprod_Oinject,axiom,
% 0.29/0.51      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 0.29/0.51          = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
% 0.29/0.51        = ( ( A = A6 )
% 0.29/0.51          & ( B = B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.inject
% 0.29/0.51  thf(fact_1471_prod_Oinject,axiom,
% 0.29/0.51      ! [X1: int,X22: int,Y1: int,Y2: int] :
% 0.29/0.51        ( ( ( product_Pair_int_int @ X1 @ X22 )
% 0.29/0.51          = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 0.29/0.51        = ( ( X1 = Y1 )
% 0.29/0.51          & ( X22 = Y2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod.inject
% 0.29/0.51  thf(fact_1472_prod_Oinject,axiom,
% 0.29/0.51      ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
% 0.29/0.51          = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
% 0.29/0.51        = ( ( X1 = Y1 )
% 0.29/0.51          & ( X22 = Y2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod.inject
% 0.29/0.51  thf(fact_1473_prod_Oinject,axiom,
% 0.29/0.51      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
% 0.29/0.51          = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
% 0.29/0.51        = ( ( X1 = Y1 )
% 0.29/0.51          & ( X22 = Y2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod.inject
% 0.29/0.51  thf(fact_1474_prod_Oinject,axiom,
% 0.29/0.51      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
% 0.29/0.51          = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
% 0.29/0.51        = ( ( X1 = Y1 )
% 0.29/0.51          & ( X22 = Y2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod.inject
% 0.29/0.51  thf(fact_1475_prod_Oinject,axiom,
% 0.29/0.51      ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
% 0.29/0.51          = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
% 0.29/0.51        = ( ( X1 = Y1 )
% 0.29/0.51          & ( X22 = Y2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod.inject
% 0.29/0.51  thf(fact_1476_nth__enumerate__eq,axiom,
% 0.29/0.51      ! [M: nat,Xs: list_VEBT_VEBT,N2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.51       => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) @ M )
% 0.29/0.51          = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % nth_enumerate_eq
% 0.29/0.51  thf(fact_1477_nth__enumerate__eq,axiom,
% 0.29/0.51      ! [M: nat,Xs: list_o,N2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 0.29/0.51       => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N2 @ Xs ) @ M )
% 0.29/0.51          = ( product_Pair_nat_o @ ( plus_plus_nat @ N2 @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % nth_enumerate_eq
% 0.29/0.51  thf(fact_1478_nth__enumerate__eq,axiom,
% 0.29/0.51      ! [M: nat,Xs: list_nat,N2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 0.29/0.51       => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M )
% 0.29/0.51          = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % nth_enumerate_eq
% 0.29/0.51  thf(fact_1479_nth__enumerate__eq,axiom,
% 0.29/0.51      ! [M: nat,Xs: list_int,N2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 0.29/0.51       => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M )
% 0.29/0.51          = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % nth_enumerate_eq
% 0.29/0.51  thf(fact_1480_set__vebt__finite,axiom,
% 0.29/0.51      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.51        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.51       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % set_vebt_finite
% 0.29/0.51  thf(fact_1481_Euclid__induct,axiom,
% 0.29/0.51      ! [P: nat > nat > $o,A: nat,B: nat] :
% 0.29/0.51        ( ! [A4: nat,B4: nat] :
% 0.29/0.51            ( ( P @ A4 @ B4 )
% 0.29/0.51            = ( P @ B4 @ A4 ) )
% 0.29/0.51       => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 0.29/0.51         => ( ! [A4: nat,B4: nat] :
% 0.29/0.51                ( ( P @ A4 @ B4 )
% 0.29/0.51               => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 0.29/0.51           => ( P @ A @ B ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Euclid_induct
% 0.29/0.51  thf(fact_1482_List_Ofinite__set,axiom,
% 0.29/0.51      ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % List.finite_set
% 0.29/0.51  thf(fact_1483_List_Ofinite__set,axiom,
% 0.29/0.51      ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % List.finite_set
% 0.29/0.51  thf(fact_1484_List_Ofinite__set,axiom,
% 0.29/0.51      ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % List.finite_set
% 0.29/0.51  thf(fact_1485_List_Ofinite__set,axiom,
% 0.29/0.51      ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % List.finite_set
% 0.29/0.51  thf(fact_1486_List_Ofinite__set,axiom,
% 0.29/0.51      ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % List.finite_set
% 0.29/0.51  thf(fact_1487_length__enumerate,axiom,
% 0.29/0.51      ! [N2: nat,Xs: list_VEBT_VEBT] :
% 0.29/0.51        ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) )
% 0.29/0.51        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % length_enumerate
% 0.29/0.51  thf(fact_1488_length__enumerate,axiom,
% 0.29/0.51      ! [N2: nat,Xs: list_o] :
% 0.29/0.51        ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N2 @ Xs ) )
% 0.29/0.51        = ( size_size_list_o @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % length_enumerate
% 0.29/0.51  thf(fact_1489_length__enumerate,axiom,
% 0.29/0.51      ! [N2: nat,Xs: list_nat] :
% 0.29/0.51        ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
% 0.29/0.51        = ( size_size_list_nat @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % length_enumerate
% 0.29/0.51  thf(fact_1490_length__enumerate,axiom,
% 0.29/0.51      ! [N2: nat,Xs: list_int] :
% 0.29/0.51        ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs ) )
% 0.29/0.51        = ( size_size_list_int @ Xs ) ) ).
% 0.29/0.51  
% 0.29/0.51  % length_enumerate
% 0.29/0.51  thf(fact_1491_psubsetD,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 0.29/0.51        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.29/0.51         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetD
% 0.29/0.51  thf(fact_1492_psubsetD,axiom,
% 0.29/0.51      ! [A2: set_real,B2: set_real,C: real] :
% 0.29/0.51        ( ( ord_less_set_real @ A2 @ B2 )
% 0.29/0.51       => ( ( member_real @ C @ A2 )
% 0.29/0.51         => ( member_real @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetD
% 0.29/0.51  thf(fact_1493_psubsetD,axiom,
% 0.29/0.51      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 0.29/0.51        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_set_nat @ C @ A2 )
% 0.29/0.51         => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetD
% 0.29/0.51  thf(fact_1494_psubsetD,axiom,
% 0.29/0.51      ! [A2: set_nat,B2: set_nat,C: nat] :
% 0.29/0.51        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( member_nat @ C @ A2 )
% 0.29/0.51         => ( member_nat @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetD
% 0.29/0.51  thf(fact_1495_psubsetD,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int,C: int] :
% 0.29/0.51        ( ( ord_less_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( member_int @ C @ A2 )
% 0.29/0.51         => ( member_int @ C @ B2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % psubsetD
% 0.29/0.51  thf(fact_1496_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc8306885398267862888on_nat] :
% 0.29/0.51        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.29/0.51         => ~ ! [F2: nat > nat > nat,A4: nat,B4: nat] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.cases
% 0.29/0.51  thf(fact_1497_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc5542196010084753463at_nat] :
% 0.29/0.51        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.29/0.51         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.cases
% 0.29/0.51  thf(fact_1498_VEBT__internal_Ooption__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc1193250871479095198on_num] :
% 0.29/0.51        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: num > num > num,V2: num] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.29/0.51         => ~ ! [F2: num > num > num,A4: num,B4: num] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.cases
% 0.29/0.51  thf(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc2233624965454879586on_nat] :
% 0.29/0.51        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 0.29/0.51         => ~ ! [F2: nat > nat > $o,X4: nat,Y5: nat] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_comp_shift.cases
% 0.29/0.51  thf(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc5491161045314408544at_nat] :
% 0.29/0.51        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 0.29/0.51         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_comp_shift.cases
% 0.29/0.51  thf(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 0.29/0.51      ! [X3: produc7036089656553540234on_num] :
% 0.29/0.51        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 0.29/0.51       => ( ! [Uw2: num > num > $o,V2: num] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 0.29/0.51         => ~ ! [F2: num > num > $o,X4: num,Y5: num] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y5 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_comp_shift.cases
% 0.29/0.51  thf(fact_1502_finite__list,axiom,
% 0.29/0.51      ! [A2: set_VEBT_VEBT] :
% 0.29/0.51        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.29/0.51       => ? [Xs3: list_VEBT_VEBT] :
% 0.29/0.51            ( ( set_VEBT_VEBT2 @ Xs3 )
% 0.29/0.51            = A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_list
% 0.29/0.51  thf(fact_1503_finite__list,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ? [Xs3: list_nat] :
% 0.29/0.51            ( ( set_nat2 @ Xs3 )
% 0.29/0.51            = A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_list
% 0.29/0.51  thf(fact_1504_finite__list,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ? [Xs3: list_int] :
% 0.29/0.51            ( ( set_int2 @ Xs3 )
% 0.29/0.51            = A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_list
% 0.29/0.51  thf(fact_1505_finite__list,axiom,
% 0.29/0.51      ! [A2: set_complex] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ A2 )
% 0.29/0.51       => ? [Xs3: list_complex] :
% 0.29/0.51            ( ( set_complex2 @ Xs3 )
% 0.29/0.51            = A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_list
% 0.29/0.51  thf(fact_1506_finite__list,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( finite6177210948735845034at_nat @ A2 )
% 0.29/0.51       => ? [Xs3: list_P6011104703257516679at_nat] :
% 0.29/0.51            ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 0.29/0.51            = A2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_list
% 0.29/0.51  thf(fact_1507_vebt__member_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 0.29/0.51        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_member.simps(2)
% 0.29/0.51  thf(fact_1508_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 0.29/0.51      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.minNull.simps(4)
% 0.29/0.51  thf(fact_1509_ex__min__if__finite,axiom,
% 0.29/0.51      ! [S3: set_real] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ? [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ S3 )
% 0.29/0.51              & ~ ? [Xa: real] :
% 0.29/0.51                    ( ( member_real @ Xa @ S3 )
% 0.29/0.51                    & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_min_if_finite
% 0.29/0.51  thf(fact_1510_ex__min__if__finite,axiom,
% 0.29/0.51      ! [S3: set_rat] :
% 0.29/0.51        ( ( finite_finite_rat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_rat )
% 0.29/0.51         => ? [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ S3 )
% 0.29/0.51              & ~ ? [Xa: rat] :
% 0.29/0.51                    ( ( member_rat @ Xa @ S3 )
% 0.29/0.51                    & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_min_if_finite
% 0.29/0.51  thf(fact_1511_ex__min__if__finite,axiom,
% 0.29/0.51      ! [S3: set_num] :
% 0.29/0.51        ( ( finite_finite_num @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_num )
% 0.29/0.51         => ? [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ S3 )
% 0.29/0.51              & ~ ? [Xa: num] :
% 0.29/0.51                    ( ( member_num @ Xa @ S3 )
% 0.29/0.51                    & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_min_if_finite
% 0.29/0.51  thf(fact_1512_ex__min__if__finite,axiom,
% 0.29/0.51      ! [S3: set_nat] :
% 0.29/0.51        ( ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_nat )
% 0.29/0.51         => ? [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ S3 )
% 0.29/0.51              & ~ ? [Xa: nat] :
% 0.29/0.51                    ( ( member_nat @ Xa @ S3 )
% 0.29/0.51                    & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_min_if_finite
% 0.29/0.51  thf(fact_1513_ex__min__if__finite,axiom,
% 0.29/0.51      ! [S3: set_int] :
% 0.29/0.51        ( ( finite_finite_int @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_int )
% 0.29/0.51         => ? [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ S3 )
% 0.29/0.51              & ~ ? [Xa: int] :
% 0.29/0.51                    ( ( member_int @ Xa @ S3 )
% 0.29/0.51                    & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ex_min_if_finite
% 0.29/0.51  thf(fact_1514_infinite__growing,axiom,
% 0.29/0.51      ! [X7: set_real] :
% 0.29/0.51        ( ( X7 != bot_bot_set_real )
% 0.29/0.51       => ( ! [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ X7 )
% 0.29/0.51             => ? [Xa: real] :
% 0.29/0.51                  ( ( member_real @ Xa @ X7 )
% 0.29/0.51                  & ( ord_less_real @ X4 @ Xa ) ) )
% 0.29/0.51         => ~ ( finite_finite_real @ X7 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_growing
% 0.29/0.51  thf(fact_1515_infinite__growing,axiom,
% 0.29/0.51      ! [X7: set_rat] :
% 0.29/0.51        ( ( X7 != bot_bot_set_rat )
% 0.29/0.51       => ( ! [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ X7 )
% 0.29/0.51             => ? [Xa: rat] :
% 0.29/0.51                  ( ( member_rat @ Xa @ X7 )
% 0.29/0.51                  & ( ord_less_rat @ X4 @ Xa ) ) )
% 0.29/0.51         => ~ ( finite_finite_rat @ X7 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_growing
% 0.29/0.51  thf(fact_1516_infinite__growing,axiom,
% 0.29/0.51      ! [X7: set_num] :
% 0.29/0.51        ( ( X7 != bot_bot_set_num )
% 0.29/0.51       => ( ! [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ X7 )
% 0.29/0.51             => ? [Xa: num] :
% 0.29/0.51                  ( ( member_num @ Xa @ X7 )
% 0.29/0.51                  & ( ord_less_num @ X4 @ Xa ) ) )
% 0.29/0.51         => ~ ( finite_finite_num @ X7 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_growing
% 0.29/0.51  thf(fact_1517_infinite__growing,axiom,
% 0.29/0.51      ! [X7: set_nat] :
% 0.29/0.51        ( ( X7 != bot_bot_set_nat )
% 0.29/0.51       => ( ! [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ X7 )
% 0.29/0.51             => ? [Xa: nat] :
% 0.29/0.51                  ( ( member_nat @ Xa @ X7 )
% 0.29/0.51                  & ( ord_less_nat @ X4 @ Xa ) ) )
% 0.29/0.51         => ~ ( finite_finite_nat @ X7 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_growing
% 0.29/0.51  thf(fact_1518_infinite__growing,axiom,
% 0.29/0.51      ! [X7: set_int] :
% 0.29/0.51        ( ( X7 != bot_bot_set_int )
% 0.29/0.51       => ( ! [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ X7 )
% 0.29/0.51             => ? [Xa: int] :
% 0.29/0.51                  ( ( member_int @ Xa @ X7 )
% 0.29/0.51                  & ( ord_less_int @ X4 @ Xa ) ) )
% 0.29/0.51         => ~ ( finite_finite_int @ X7 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_growing
% 0.29/0.51  thf(fact_1519_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 0.29/0.51        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.membermima.simps(2)
% 0.29/0.51  thf(fact_1520_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.29/0.51      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.29/0.51        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 0.29/0.51        = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(3)
% 0.29/0.51  thf(fact_1521_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.29/0.51      ! [F: num > num > num,A: num,B: num] :
% 0.29/0.51        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 0.29/0.51        = ( some_num @ ( F @ A @ B ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(3)
% 0.29/0.51  thf(fact_1522_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 0.29/0.51      ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.29/0.51        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 0.29/0.51        = ( some_nat @ ( F @ A @ B ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(3)
% 0.29/0.51  thf(fact_1523_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.29/0.51      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 0.29/0.51        = none_P5556105721700978146at_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(1)
% 0.29/0.51  thf(fact_1524_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.29/0.51      ! [Uu: num > num > num,Uv: option_num] :
% 0.29/0.51        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 0.29/0.51        = none_num ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(1)
% 0.29/0.51  thf(fact_1525_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 0.29/0.51      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 0.29/0.51        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 0.29/0.51        = none_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(1)
% 0.29/0.51  thf(fact_1526_old_Oprod_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: product_prod_int_int] :
% 0.29/0.51        ~ ! [A4: int,B4: int] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( product_Pair_int_int @ A4 @ B4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.exhaust
% 0.29/0.51  thf(fact_1527_old_Oprod_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: produc8763457246119570046nteger] :
% 0.29/0.51        ~ ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc6137756002093451184nteger @ A4 @ B4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.exhaust
% 0.29/0.51  thf(fact_1528_old_Oprod_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: produc1908205239877642774nteger] :
% 0.29/0.51        ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc8603105652947943368nteger @ A4 @ B4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.exhaust
% 0.29/0.51  thf(fact_1529_old_Oprod_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: produc2285326912895808259nt_int] :
% 0.29/0.51        ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc5700946648718959541nt_int @ A4 @ B4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.exhaust
% 0.29/0.51  thf(fact_1530_old_Oprod_Oexhaust,axiom,
% 0.29/0.51      ! [Y3: produc7773217078559923341nt_int] :
% 0.29/0.51        ~ ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc4305682042979456191nt_int @ A4 @ B4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % old.prod.exhaust
% 0.29/0.51  thf(fact_1531_surj__pair,axiom,
% 0.29/0.51      ! [P4: product_prod_int_int] :
% 0.29/0.51      ? [X4: int,Y5: int] :
% 0.29/0.51        ( P4
% 0.29/0.51        = ( product_Pair_int_int @ X4 @ Y5 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % surj_pair
% 0.29/0.51  thf(fact_1532_surj__pair,axiom,
% 0.29/0.51      ! [P4: produc8763457246119570046nteger] :
% 0.29/0.51      ? [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.29/0.51        ( P4
% 0.29/0.51        = ( produc6137756002093451184nteger @ X4 @ Y5 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % surj_pair
% 0.29/0.51  thf(fact_1533_surj__pair,axiom,
% 0.29/0.51      ! [P4: produc1908205239877642774nteger] :
% 0.29/0.51      ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.29/0.51        ( P4
% 0.29/0.51        = ( produc8603105652947943368nteger @ X4 @ Y5 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % surj_pair
% 0.29/0.51  thf(fact_1534_surj__pair,axiom,
% 0.29/0.51      ! [P4: produc2285326912895808259nt_int] :
% 0.29/0.51      ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.29/0.51        ( P4
% 0.29/0.51        = ( produc5700946648718959541nt_int @ X4 @ Y5 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % surj_pair
% 0.29/0.51  thf(fact_1535_surj__pair,axiom,
% 0.29/0.51      ! [P4: produc7773217078559923341nt_int] :
% 0.29/0.51      ? [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.29/0.51        ( P4
% 0.29/0.51        = ( produc4305682042979456191nt_int @ X4 @ Y5 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % surj_pair
% 0.29/0.51  thf(fact_1536_prod__cases,axiom,
% 0.29/0.51      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 0.29/0.51        ( ! [A4: int,B4: int] : ( P @ ( product_Pair_int_int @ A4 @ B4 ) )
% 0.29/0.51       => ( P @ P4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases
% 0.29/0.51  thf(fact_1537_prod__cases,axiom,
% 0.29/0.51      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 0.29/0.51        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 0.29/0.51       => ( P @ P4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases
% 0.29/0.51  thf(fact_1538_prod__cases,axiom,
% 0.29/0.51      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 0.29/0.51        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A4 @ B4 ) )
% 0.29/0.51       => ( P @ P4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases
% 0.29/0.51  thf(fact_1539_prod__cases,axiom,
% 0.29/0.51      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 0.29/0.51        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ B4 ) )
% 0.29/0.51       => ( P @ P4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases
% 0.29/0.51  thf(fact_1540_prod__cases,axiom,
% 0.29/0.51      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 0.29/0.51        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 0.29/0.51       => ( P @ P4 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases
% 0.29/0.51  thf(fact_1541_Pair__inject,axiom,
% 0.29/0.51      ! [A: int,B: int,A6: int,B6: int] :
% 0.29/0.51        ( ( ( product_Pair_int_int @ A @ B )
% 0.29/0.51          = ( product_Pair_int_int @ A6 @ B6 ) )
% 0.29/0.51       => ~ ( ( A = A6 )
% 0.29/0.51           => ( B != B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Pair_inject
% 0.29/0.51  thf(fact_1542_Pair__inject,axiom,
% 0.29/0.51      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc6137756002093451184nteger @ A @ B )
% 0.29/0.51          = ( produc6137756002093451184nteger @ A6 @ B6 ) )
% 0.29/0.51       => ~ ( ( A = A6 )
% 0.29/0.51           => ( B != B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Pair_inject
% 0.29/0.51  thf(fact_1543_Pair__inject,axiom,
% 0.29/0.51      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( ( produc8603105652947943368nteger @ A @ B )
% 0.29/0.51          = ( produc8603105652947943368nteger @ A6 @ B6 ) )
% 0.29/0.51       => ~ ( ( A = A6 )
% 0.29/0.51           => ( B != B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Pair_inject
% 0.29/0.51  thf(fact_1544_Pair__inject,axiom,
% 0.29/0.51      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 0.29/0.51          = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
% 0.29/0.51       => ~ ( ( A = A6 )
% 0.29/0.51           => ( B != B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Pair_inject
% 0.29/0.51  thf(fact_1545_Pair__inject,axiom,
% 0.29/0.51      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
% 0.29/0.51        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 0.29/0.51          = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
% 0.29/0.51       => ~ ( ( A = A6 )
% 0.29/0.51           => ( B != B6 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Pair_inject
% 0.29/0.51  thf(fact_1546_prod__cases3,axiom,
% 0.29/0.51      ! [Y3: produc8763457246119570046nteger] :
% 0.29/0.51        ~ ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases3
% 0.29/0.51  thf(fact_1547_prod__cases3,axiom,
% 0.29/0.51      ! [Y3: produc1908205239877642774nteger] :
% 0.29/0.51        ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases3
% 0.29/0.51  thf(fact_1548_prod__cases3,axiom,
% 0.29/0.51      ! [Y3: produc2285326912895808259nt_int] :
% 0.29/0.51        ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases3
% 0.29/0.51  thf(fact_1549_prod__cases3,axiom,
% 0.29/0.51      ! [Y3: produc7773217078559923341nt_int] :
% 0.29/0.51        ~ ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] :
% 0.29/0.51            ( Y3
% 0.29/0.51           != ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_cases3
% 0.29/0.51  thf(fact_1550_prod__induct3,axiom,
% 0.29/0.51      ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
% 0.29/0.51        ( ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
% 0.29/0.51       => ( P @ X3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_induct3
% 0.29/0.51  thf(fact_1551_prod__induct3,axiom,
% 0.29/0.51      ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
% 0.29/0.51        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
% 0.29/0.51       => ( P @ X3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_induct3
% 0.29/0.51  thf(fact_1552_prod__induct3,axiom,
% 0.29/0.51      ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
% 0.29/0.51        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
% 0.29/0.51       => ( P @ X3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_induct3
% 0.29/0.51  thf(fact_1553_prod__induct3,axiom,
% 0.29/0.51      ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
% 0.29/0.51        ( ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
% 0.29/0.51       => ( P @ X3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_induct3
% 0.29/0.51  thf(fact_1554_vebt__maxt_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.29/0.51        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.29/0.51        = none_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_maxt.simps(2)
% 0.29/0.51  thf(fact_1555_vebt__mint_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 0.29/0.51        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 0.29/0.51        = none_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_mint.simps(2)
% 0.29/0.51  thf(fact_1556_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 0.29/0.51        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 0.29/0.51        = none_P5556105721700978146at_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(2)
% 0.29/0.51  thf(fact_1557_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uw: num > num > num,V: num] :
% 0.29/0.51        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 0.29/0.51        = none_num ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(2)
% 0.29/0.51  thf(fact_1558_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 0.29/0.51      ! [Uw: nat > nat > nat,V: nat] :
% 0.29/0.51        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 0.29/0.51        = none_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.simps(2)
% 0.29/0.51  thf(fact_1559_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.29/0.51      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 0.29/0.51        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb )
% 0.29/0.51          = Y3 )
% 0.29/0.51       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 0.29/0.51           => ( Y3 != none_P5556105721700978146at_nat ) )
% 0.29/0.51         => ( ( ? [V2: product_prod_nat_nat] :
% 0.29/0.51                  ( Xa2
% 0.29/0.51                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 0.29/0.51             => ( ( Xb = none_P5556105721700978146at_nat )
% 0.29/0.51               => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 0.29/0.51           => ~ ! [A4: product_prod_nat_nat] :
% 0.29/0.51                  ( ( Xa2
% 0.29/0.51                    = ( some_P7363390416028606310at_nat @ A4 ) )
% 0.29/0.51                 => ! [B4: product_prod_nat_nat] :
% 0.29/0.51                      ( ( Xb
% 0.29/0.51                        = ( some_P7363390416028606310at_nat @ B4 ) )
% 0.29/0.51                     => ( Y3
% 0.29/0.51                       != ( some_P7363390416028606310at_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.elims
% 0.29/0.51  thf(fact_1560_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.29/0.51      ! [X3: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 0.29/0.51        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb )
% 0.29/0.51          = Y3 )
% 0.29/0.51       => ( ( ( Xa2 = none_num )
% 0.29/0.51           => ( Y3 != none_num ) )
% 0.29/0.51         => ( ( ? [V2: num] :
% 0.29/0.51                  ( Xa2
% 0.29/0.51                  = ( some_num @ V2 ) )
% 0.29/0.51             => ( ( Xb = none_num )
% 0.29/0.51               => ( Y3 != none_num ) ) )
% 0.29/0.51           => ~ ! [A4: num] :
% 0.29/0.51                  ( ( Xa2
% 0.29/0.51                    = ( some_num @ A4 ) )
% 0.29/0.51                 => ! [B4: num] :
% 0.29/0.51                      ( ( Xb
% 0.29/0.51                        = ( some_num @ B4 ) )
% 0.29/0.51                     => ( Y3
% 0.29/0.51                       != ( some_num @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.elims
% 0.29/0.51  thf(fact_1561_VEBT__internal_Ooption__shift_Oelims,axiom,
% 0.29/0.51      ! [X3: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 0.29/0.51        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb )
% 0.29/0.51          = Y3 )
% 0.29/0.51       => ( ( ( Xa2 = none_nat )
% 0.29/0.51           => ( Y3 != none_nat ) )
% 0.29/0.51         => ( ( ? [V2: nat] :
% 0.29/0.51                  ( Xa2
% 0.29/0.51                  = ( some_nat @ V2 ) )
% 0.29/0.51             => ( ( Xb = none_nat )
% 0.29/0.51               => ( Y3 != none_nat ) ) )
% 0.29/0.51           => ~ ! [A4: nat] :
% 0.29/0.51                  ( ( Xa2
% 0.29/0.51                    = ( some_nat @ A4 ) )
% 0.29/0.51                 => ! [B4: nat] :
% 0.29/0.51                      ( ( Xb
% 0.29/0.51                        = ( some_nat @ B4 ) )
% 0.29/0.51                     => ( Y3
% 0.29/0.51                       != ( some_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % VEBT_internal.option_shift.elims
% 0.29/0.51  thf(fact_1562_vebt__buildup_Ocases,axiom,
% 0.29/0.51      ! [X3: nat] :
% 0.29/0.51        ( ( X3 != zero_zero_nat )
% 0.29/0.51       => ( ( X3
% 0.29/0.51           != ( suc @ zero_zero_nat ) )
% 0.29/0.51         => ~ ! [Va2: nat] :
% 0.29/0.51                ( X3
% 0.29/0.51               != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % vebt_buildup.cases
% 0.29/0.51  thf(fact_1563_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ( ( finite_finite_real @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_real )
% 0.29/0.51         => ? [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: real] :
% 0.29/0.51                  ( ( member_real @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1564_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_set_int] :
% 0.29/0.51        ( ( finite6197958912794628473et_int @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_set_int )
% 0.29/0.51         => ? [X4: set_int] :
% 0.29/0.51              ( ( member_set_int @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: set_int] :
% 0.29/0.51                  ( ( member_set_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1565_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_rat] :
% 0.29/0.51        ( ( finite_finite_rat @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_rat )
% 0.29/0.51         => ? [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: rat] :
% 0.29/0.51                  ( ( member_rat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1566_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_num] :
% 0.29/0.51        ( ( finite_finite_num @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_num )
% 0.29/0.51         => ? [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: num] :
% 0.29/0.51                  ( ( member_num @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1567_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_nat )
% 0.29/0.51         => ? [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: nat] :
% 0.29/0.51                  ( ( member_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1568_finite__has__minimal,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_int )
% 0.29/0.51         => ? [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: int] :
% 0.29/0.51                  ( ( member_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal
% 0.29/0.51  thf(fact_1569_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ( ( finite_finite_real @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_real )
% 0.29/0.51         => ? [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: real] :
% 0.29/0.51                  ( ( member_real @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1570_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_set_int] :
% 0.29/0.51        ( ( finite6197958912794628473et_int @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_set_int )
% 0.29/0.51         => ? [X4: set_int] :
% 0.29/0.51              ( ( member_set_int @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: set_int] :
% 0.29/0.51                  ( ( member_set_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1571_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_rat] :
% 0.29/0.51        ( ( finite_finite_rat @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_rat )
% 0.29/0.51         => ? [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: rat] :
% 0.29/0.51                  ( ( member_rat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1572_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_num] :
% 0.29/0.51        ( ( finite_finite_num @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_num )
% 0.29/0.51         => ? [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: num] :
% 0.29/0.51                  ( ( member_num @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1573_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_nat )
% 0.29/0.51         => ? [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: nat] :
% 0.29/0.51                  ( ( member_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1574_finite__has__maximal,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ( A2 != bot_bot_set_int )
% 0.29/0.51         => ? [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ A2 )
% 0.29/0.51              & ! [Xa: int] :
% 0.29/0.51                  ( ( member_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal
% 0.29/0.51  thf(fact_1575_bot__empty__eq,axiom,
% 0.29/0.51      ( bot_bot_set_nat_o
% 0.29/0.51      = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_empty_eq
% 0.29/0.51  thf(fact_1576_bot__empty__eq,axiom,
% 0.29/0.51      ( bot_bo482883023278783056_nat_o
% 0.29/0.51      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_empty_eq
% 0.29/0.51  thf(fact_1577_bot__empty__eq,axiom,
% 0.29/0.51      ( bot_bot_real_o
% 0.29/0.51      = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_empty_eq
% 0.29/0.51  thf(fact_1578_bot__empty__eq,axiom,
% 0.29/0.51      ( bot_bot_nat_o
% 0.29/0.51      = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_empty_eq
% 0.29/0.51  thf(fact_1579_bot__empty__eq,axiom,
% 0.29/0.51      ( bot_bot_int_o
% 0.29/0.51      = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bot_empty_eq
% 0.29/0.51  thf(fact_1580_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: list_nat > $o] :
% 0.29/0.51        ( ( ( collect_list_nat @ P )
% 0.29/0.51          = bot_bot_set_list_nat )
% 0.29/0.51        = ( P = bot_bot_list_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1581_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: set_nat > $o] :
% 0.29/0.51        ( ( ( collect_set_nat @ P )
% 0.29/0.51          = bot_bot_set_set_nat )
% 0.29/0.51        = ( P = bot_bot_set_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1582_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: product_prod_nat_nat > $o] :
% 0.29/0.51        ( ( ( collec3392354462482085612at_nat @ P )
% 0.29/0.51          = bot_bo2099793752762293965at_nat )
% 0.29/0.51        = ( P = bot_bo482883023278783056_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1583_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: real > $o] :
% 0.29/0.51        ( ( ( collect_real @ P )
% 0.29/0.51          = bot_bot_set_real )
% 0.29/0.51        = ( P = bot_bot_real_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1584_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: nat > $o] :
% 0.29/0.51        ( ( ( collect_nat @ P )
% 0.29/0.51          = bot_bot_set_nat )
% 0.29/0.51        = ( P = bot_bot_nat_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1585_Collect__empty__eq__bot,axiom,
% 0.29/0.51      ! [P: int > $o] :
% 0.29/0.51        ( ( ( collect_int @ P )
% 0.29/0.51          = bot_bot_set_int )
% 0.29/0.51        = ( P = bot_bot_int_o ) ) ).
% 0.29/0.51  
% 0.29/0.51  % Collect_empty_eq_bot
% 0.29/0.51  thf(fact_1586_infinite__nat__iff__unbounded__le,axiom,
% 0.29/0.51      ! [S3: set_nat] :
% 0.29/0.51        ( ( ~ ( finite_finite_nat @ S3 ) )
% 0.29/0.51        = ( ! [M2: nat] :
% 0.29/0.51            ? [N: nat] :
% 0.29/0.51              ( ( ord_less_eq_nat @ M2 @ N )
% 0.29/0.51              & ( member_nat @ N @ S3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_nat_iff_unbounded_le
% 0.29/0.51  thf(fact_1587_finite__nat__set__iff__bounded__le,axiom,
% 0.29/0.51      ( finite_finite_nat
% 0.29/0.51      = ( ^ [N5: set_nat] :
% 0.29/0.51          ? [M2: nat] :
% 0.29/0.51          ! [X: nat] :
% 0.29/0.51            ( ( member_nat @ X @ N5 )
% 0.29/0.51           => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_nat_set_iff_bounded_le
% 0.29/0.51  thf(fact_1588_unbounded__k__infinite,axiom,
% 0.29/0.51      ! [K: nat,S3: set_nat] :
% 0.29/0.51        ( ! [M3: nat] :
% 0.29/0.51            ( ( ord_less_nat @ K @ M3 )
% 0.29/0.51           => ? [N6: nat] :
% 0.29/0.51                ( ( ord_less_nat @ M3 @ N6 )
% 0.29/0.51                & ( member_nat @ N6 @ S3 ) ) )
% 0.29/0.51       => ~ ( finite_finite_nat @ S3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % unbounded_k_infinite
% 0.29/0.51  thf(fact_1589_bounded__nat__set__is__finite,axiom,
% 0.29/0.51      ! [N7: set_nat,N2: nat] :
% 0.29/0.51        ( ! [X4: nat] :
% 0.29/0.51            ( ( member_nat @ X4 @ N7 )
% 0.29/0.51           => ( ord_less_nat @ X4 @ N2 ) )
% 0.29/0.51       => ( finite_finite_nat @ N7 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bounded_nat_set_is_finite
% 0.29/0.51  thf(fact_1590_finite__maxlen,axiom,
% 0.29/0.51      ! [M7: set_list_VEBT_VEBT] :
% 0.29/0.51        ( ( finite3004134309566078307T_VEBT @ M7 )
% 0.29/0.51       => ? [N3: nat] :
% 0.29/0.51          ! [X2: list_VEBT_VEBT] :
% 0.29/0.51            ( ( member2936631157270082147T_VEBT @ X2 @ M7 )
% 0.29/0.51           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_maxlen
% 0.29/0.51  thf(fact_1591_finite__maxlen,axiom,
% 0.29/0.51      ! [M7: set_list_o] :
% 0.29/0.51        ( ( finite_finite_list_o @ M7 )
% 0.29/0.51       => ? [N3: nat] :
% 0.29/0.51          ! [X2: list_o] :
% 0.29/0.51            ( ( member_list_o @ X2 @ M7 )
% 0.29/0.51           => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_maxlen
% 0.29/0.51  thf(fact_1592_finite__maxlen,axiom,
% 0.29/0.51      ! [M7: set_list_nat] :
% 0.29/0.51        ( ( finite8100373058378681591st_nat @ M7 )
% 0.29/0.51       => ? [N3: nat] :
% 0.29/0.51          ! [X2: list_nat] :
% 0.29/0.51            ( ( member_list_nat @ X2 @ M7 )
% 0.29/0.51           => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_maxlen
% 0.29/0.51  thf(fact_1593_finite__maxlen,axiom,
% 0.29/0.51      ! [M7: set_list_int] :
% 0.29/0.51        ( ( finite3922522038869484883st_int @ M7 )
% 0.29/0.51       => ? [N3: nat] :
% 0.29/0.51          ! [X2: list_int] :
% 0.29/0.51            ( ( member_list_int @ X2 @ M7 )
% 0.29/0.51           => ( ord_less_nat @ ( size_size_list_int @ X2 ) @ N3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_maxlen
% 0.29/0.51  thf(fact_1594_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ! [X4: nat,Y5: nat] :
% 0.29/0.51            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1595_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 0.29/0.51        ( ! [X4: int,Y5: int] :
% 0.29/0.51            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1596_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 0.29/0.51        ( ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.29/0.51            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le3216752416896350996nteger @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1597_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 0.29/0.51        ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.29/0.51            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1598_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 0.29/0.51        ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.29/0.51            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1599_subrelI,axiom,
% 0.29/0.51      ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 0.29/0.51        ( ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.29/0.51            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ R2 )
% 0.29/0.51           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ S ) )
% 0.29/0.51       => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subrelI
% 0.29/0.51  thf(fact_1600_bounded__Max__nat,axiom,
% 0.29/0.51      ! [P: nat > $o,X3: nat,M7: nat] :
% 0.29/0.51        ( ( P @ X3 )
% 0.29/0.51       => ( ! [X4: nat] :
% 0.29/0.51              ( ( P @ X4 )
% 0.29/0.51             => ( ord_less_eq_nat @ X4 @ M7 ) )
% 0.29/0.51         => ~ ! [M3: nat] :
% 0.29/0.51                ( ( P @ M3 )
% 0.29/0.51               => ~ ! [X2: nat] :
% 0.29/0.51                      ( ( P @ X2 )
% 0.29/0.51                     => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % bounded_Max_nat
% 0.29/0.51  thf(fact_1601_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_real,A: real] :
% 0.29/0.51        ( ( finite_finite_real @ A2 )
% 0.29/0.51       => ( ( member_real @ A @ A2 )
% 0.29/0.51         => ? [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_real @ X4 @ A )
% 0.29/0.51              & ! [Xa: real] :
% 0.29/0.51                  ( ( member_real @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1602_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_set_nat,A: set_nat] :
% 0.29/0.51        ( ( finite1152437895449049373et_nat @ A2 )
% 0.29/0.51       => ( ( member_set_nat @ A @ A2 )
% 0.29/0.51         => ? [X4: set_nat] :
% 0.29/0.51              ( ( member_set_nat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_set_nat @ X4 @ A )
% 0.29/0.51              & ! [Xa: set_nat] :
% 0.29/0.51                  ( ( member_set_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1603_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_set_int,A: set_int] :
% 0.29/0.51        ( ( finite6197958912794628473et_int @ A2 )
% 0.29/0.51       => ( ( member_set_int @ A @ A2 )
% 0.29/0.51         => ? [X4: set_int] :
% 0.29/0.51              ( ( member_set_int @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_set_int @ X4 @ A )
% 0.29/0.51              & ! [Xa: set_int] :
% 0.29/0.51                  ( ( member_set_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1604_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_rat,A: rat] :
% 0.29/0.51        ( ( finite_finite_rat @ A2 )
% 0.29/0.51       => ( ( member_rat @ A @ A2 )
% 0.29/0.51         => ? [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_rat @ X4 @ A )
% 0.29/0.51              & ! [Xa: rat] :
% 0.29/0.51                  ( ( member_rat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1605_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_num,A: num] :
% 0.29/0.51        ( ( finite_finite_num @ A2 )
% 0.29/0.51       => ( ( member_num @ A @ A2 )
% 0.29/0.51         => ? [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_num @ X4 @ A )
% 0.29/0.51              & ! [Xa: num] :
% 0.29/0.51                  ( ( member_num @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1606_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_nat,A: nat] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ( member_nat @ A @ A2 )
% 0.29/0.51         => ? [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_nat @ X4 @ A )
% 0.29/0.51              & ! [Xa: nat] :
% 0.29/0.51                  ( ( member_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1607_finite__has__minimal2,axiom,
% 0.29/0.51      ! [A2: set_int,A: int] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ( member_int @ A @ A2 )
% 0.29/0.51         => ? [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_int @ X4 @ A )
% 0.29/0.51              & ! [Xa: int] :
% 0.29/0.51                  ( ( member_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_minimal2
% 0.29/0.51  thf(fact_1608_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_real,A: real] :
% 0.29/0.51        ( ( finite_finite_real @ A2 )
% 0.29/0.51       => ( ( member_real @ A @ A2 )
% 0.29/0.51         => ? [X4: real] :
% 0.29/0.51              ( ( member_real @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_real @ A @ X4 )
% 0.29/0.51              & ! [Xa: real] :
% 0.29/0.51                  ( ( member_real @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1609_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_set_nat,A: set_nat] :
% 0.29/0.51        ( ( finite1152437895449049373et_nat @ A2 )
% 0.29/0.51       => ( ( member_set_nat @ A @ A2 )
% 0.29/0.51         => ? [X4: set_nat] :
% 0.29/0.51              ( ( member_set_nat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_set_nat @ A @ X4 )
% 0.29/0.51              & ! [Xa: set_nat] :
% 0.29/0.51                  ( ( member_set_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1610_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_set_int,A: set_int] :
% 0.29/0.51        ( ( finite6197958912794628473et_int @ A2 )
% 0.29/0.51       => ( ( member_set_int @ A @ A2 )
% 0.29/0.51         => ? [X4: set_int] :
% 0.29/0.51              ( ( member_set_int @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_set_int @ A @ X4 )
% 0.29/0.51              & ! [Xa: set_int] :
% 0.29/0.51                  ( ( member_set_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1611_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_rat,A: rat] :
% 0.29/0.51        ( ( finite_finite_rat @ A2 )
% 0.29/0.51       => ( ( member_rat @ A @ A2 )
% 0.29/0.51         => ? [X4: rat] :
% 0.29/0.51              ( ( member_rat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_rat @ A @ X4 )
% 0.29/0.51              & ! [Xa: rat] :
% 0.29/0.51                  ( ( member_rat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1612_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_num,A: num] :
% 0.29/0.51        ( ( finite_finite_num @ A2 )
% 0.29/0.51       => ( ( member_num @ A @ A2 )
% 0.29/0.51         => ? [X4: num] :
% 0.29/0.51              ( ( member_num @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_num @ A @ X4 )
% 0.29/0.51              & ! [Xa: num] :
% 0.29/0.51                  ( ( member_num @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1613_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_nat,A: nat] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ( member_nat @ A @ A2 )
% 0.29/0.51         => ? [X4: nat] :
% 0.29/0.51              ( ( member_nat @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_nat @ A @ X4 )
% 0.29/0.51              & ! [Xa: nat] :
% 0.29/0.51                  ( ( member_nat @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1614_finite__has__maximal2,axiom,
% 0.29/0.51      ! [A2: set_int,A: int] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ( member_int @ A @ A2 )
% 0.29/0.51         => ? [X4: int] :
% 0.29/0.51              ( ( member_int @ X4 @ A2 )
% 0.29/0.51              & ( ord_less_eq_int @ A @ X4 )
% 0.29/0.51              & ! [Xa: int] :
% 0.29/0.51                  ( ( member_int @ Xa @ A2 )
% 0.29/0.51                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 0.29/0.51                   => ( X4 = Xa ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_has_maximal2
% 0.29/0.51  thf(fact_1615_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
% 0.29/0.51        ( ( finite1152437895449049373et_nat @ A2 )
% 0.29/0.51       => ( ! [X4: set_nat] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: set_nat] :
% 0.29/0.51                  ( ( member_set_nat @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: set_nat] :
% 0.29/0.51                      ( ( member_set_nat @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1616_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_complex,R: complex > complex > $o] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ A2 )
% 0.29/0.51       => ( ! [X4: complex] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: complex,Y5: complex,Z3: complex] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: complex] :
% 0.29/0.51                  ( ( member_complex @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: complex] :
% 0.29/0.51                      ( ( member_complex @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bot_set_complex ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1617_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.29/0.51        ( ( finite6177210948735845034at_nat @ A2 )
% 0.29/0.51       => ( ! [X4: product_prod_nat_nat] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat,Z3: product_prod_nat_nat] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: product_prod_nat_nat] :
% 0.29/0.51                  ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: product_prod_nat_nat] :
% 0.29/0.51                      ( ( member8440522571783428010at_nat @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1618_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_real,R: real > real > $o] :
% 0.29/0.51        ( ( finite_finite_real @ A2 )
% 0.29/0.51       => ( ! [X4: real] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: real,Y5: real,Z3: real] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: real] :
% 0.29/0.51                  ( ( member_real @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: real] :
% 0.29/0.51                      ( ( member_real @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bot_set_real ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1619_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_nat,R: nat > nat > $o] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ! [X4: nat] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: nat,Y5: nat,Z3: nat] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: nat] :
% 0.29/0.51                  ( ( member_nat @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: nat] :
% 0.29/0.51                      ( ( member_nat @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1620_finite__transitivity__chain,axiom,
% 0.29/0.51      ! [A2: set_int,R: int > int > $o] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ! [X4: int] :
% 0.29/0.51              ~ ( R @ X4 @ X4 )
% 0.29/0.51         => ( ! [X4: int,Y5: int,Z3: int] :
% 0.29/0.51                ( ( R @ X4 @ Y5 )
% 0.29/0.51               => ( ( R @ Y5 @ Z3 )
% 0.29/0.51                 => ( R @ X4 @ Z3 ) ) )
% 0.29/0.51           => ( ! [X4: int] :
% 0.29/0.51                  ( ( member_int @ X4 @ A2 )
% 0.29/0.51                 => ? [Y6: int] :
% 0.29/0.51                      ( ( member_int @ Y6 @ A2 )
% 0.29/0.51                      & ( R @ X4 @ Y6 ) ) )
% 0.29/0.51             => ( A2 = bot_bot_set_int ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_transitivity_chain
% 0.29/0.51  thf(fact_1621_infinite__imp__nonempty,axiom,
% 0.29/0.51      ! [S3: set_complex] :
% 0.29/0.51        ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( S3 != bot_bot_set_complex ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_imp_nonempty
% 0.29/0.51  thf(fact_1622_infinite__imp__nonempty,axiom,
% 0.29/0.51      ! [S3: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ~ ( finite6177210948735845034at_nat @ S3 )
% 0.29/0.51       => ( S3 != bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_imp_nonempty
% 0.29/0.51  thf(fact_1623_infinite__imp__nonempty,axiom,
% 0.29/0.51      ! [S3: set_real] :
% 0.29/0.51        ( ~ ( finite_finite_real @ S3 )
% 0.29/0.51       => ( S3 != bot_bot_set_real ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_imp_nonempty
% 0.29/0.51  thf(fact_1624_infinite__imp__nonempty,axiom,
% 0.29/0.51      ! [S3: set_nat] :
% 0.29/0.51        ( ~ ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( S3 != bot_bot_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_imp_nonempty
% 0.29/0.51  thf(fact_1625_infinite__imp__nonempty,axiom,
% 0.29/0.51      ! [S3: set_int] :
% 0.29/0.51        ( ~ ( finite_finite_int @ S3 )
% 0.29/0.51       => ( S3 != bot_bot_set_int ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_imp_nonempty
% 0.29/0.51  thf(fact_1626_finite_OemptyI,axiom,
% 0.29/0.51      finite3207457112153483333omplex @ bot_bot_set_complex ).
% 0.29/0.51  
% 0.29/0.51  % finite.emptyI
% 0.29/0.51  thf(fact_1627_finite_OemptyI,axiom,
% 0.29/0.51      finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 0.29/0.51  
% 0.29/0.51  % finite.emptyI
% 0.29/0.51  thf(fact_1628_finite_OemptyI,axiom,
% 0.29/0.51      finite_finite_real @ bot_bot_set_real ).
% 0.29/0.51  
% 0.29/0.51  % finite.emptyI
% 0.29/0.51  thf(fact_1629_finite_OemptyI,axiom,
% 0.29/0.51      finite_finite_nat @ bot_bot_set_nat ).
% 0.29/0.51  
% 0.29/0.51  % finite.emptyI
% 0.29/0.51  thf(fact_1630_finite_OemptyI,axiom,
% 0.29/0.51      finite_finite_int @ bot_bot_set_int ).
% 0.29/0.51  
% 0.29/0.51  % finite.emptyI
% 0.29/0.51  thf(fact_1631_rev__finite__subset,axiom,
% 0.29/0.51      ! [B2: set_nat,A2: set_nat] :
% 0.29/0.51        ( ( finite_finite_nat @ B2 )
% 0.29/0.51       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.51         => ( finite_finite_nat @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % rev_finite_subset
% 0.29/0.51  thf(fact_1632_rev__finite__subset,axiom,
% 0.29/0.51      ! [B2: set_complex,A2: set_complex] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ B2 )
% 0.29/0.51       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.29/0.51         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % rev_finite_subset
% 0.29/0.51  thf(fact_1633_rev__finite__subset,axiom,
% 0.29/0.51      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( finite6177210948735845034at_nat @ B2 )
% 0.29/0.51       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.29/0.51         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % rev_finite_subset
% 0.29/0.51  thf(fact_1634_rev__finite__subset,axiom,
% 0.29/0.51      ! [B2: set_int,A2: set_int] :
% 0.29/0.51        ( ( finite_finite_int @ B2 )
% 0.29/0.51       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51         => ( finite_finite_int @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % rev_finite_subset
% 0.29/0.51  thf(fact_1635_infinite__super,axiom,
% 0.29/0.51      ! [S3: set_nat,T3: set_nat] :
% 0.29/0.51        ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.29/0.51       => ( ~ ( finite_finite_nat @ S3 )
% 0.29/0.51         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_super
% 0.29/0.51  thf(fact_1636_infinite__super,axiom,
% 0.29/0.51      ! [S3: set_complex,T3: set_complex] :
% 0.29/0.51        ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.29/0.51       => ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_super
% 0.29/0.51  thf(fact_1637_infinite__super,axiom,
% 0.29/0.51      ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( ord_le3146513528884898305at_nat @ S3 @ T3 )
% 0.29/0.51       => ( ~ ( finite6177210948735845034at_nat @ S3 )
% 0.29/0.51         => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_super
% 0.29/0.51  thf(fact_1638_infinite__super,axiom,
% 0.29/0.51      ! [S3: set_int,T3: set_int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.29/0.51       => ( ~ ( finite_finite_int @ S3 )
% 0.29/0.51         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_super
% 0.29/0.51  thf(fact_1639_finite__subset,axiom,
% 0.29/0.51      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.51        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( finite_finite_nat @ B2 )
% 0.29/0.51         => ( finite_finite_nat @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_subset
% 0.29/0.51  thf(fact_1640_finite__subset,axiom,
% 0.29/0.51      ! [A2: set_complex,B2: set_complex] :
% 0.29/0.51        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.29/0.51       => ( ( finite3207457112153483333omplex @ B2 )
% 0.29/0.51         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_subset
% 0.29/0.51  thf(fact_1641_finite__subset,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.29/0.51       => ( ( finite6177210948735845034at_nat @ B2 )
% 0.29/0.51         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_subset
% 0.29/0.51  thf(fact_1642_finite__subset,axiom,
% 0.29/0.51      ! [A2: set_int,B2: set_int] :
% 0.29/0.51        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.29/0.51       => ( ( finite_finite_int @ B2 )
% 0.29/0.51         => ( finite_finite_int @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_subset
% 0.29/0.51  thf(fact_1643_finite__psubset__induct,axiom,
% 0.29/0.51      ! [A2: set_nat,P: set_nat > $o] :
% 0.29/0.51        ( ( finite_finite_nat @ A2 )
% 0.29/0.51       => ( ! [A7: set_nat] :
% 0.29/0.51              ( ( finite_finite_nat @ A7 )
% 0.29/0.51             => ( ! [B7: set_nat] :
% 0.29/0.51                    ( ( ord_less_set_nat @ B7 @ A7 )
% 0.29/0.51                   => ( P @ B7 ) )
% 0.29/0.51               => ( P @ A7 ) ) )
% 0.29/0.51         => ( P @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_psubset_induct
% 0.29/0.51  thf(fact_1644_finite__psubset__induct,axiom,
% 0.29/0.51      ! [A2: set_int,P: set_int > $o] :
% 0.29/0.51        ( ( finite_finite_int @ A2 )
% 0.29/0.51       => ( ! [A7: set_int] :
% 0.29/0.51              ( ( finite_finite_int @ A7 )
% 0.29/0.51             => ( ! [B7: set_int] :
% 0.29/0.51                    ( ( ord_less_set_int @ B7 @ A7 )
% 0.29/0.51                   => ( P @ B7 ) )
% 0.29/0.51               => ( P @ A7 ) ) )
% 0.29/0.51         => ( P @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_psubset_induct
% 0.29/0.51  thf(fact_1645_finite__psubset__induct,axiom,
% 0.29/0.51      ! [A2: set_complex,P: set_complex > $o] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ A2 )
% 0.29/0.51       => ( ! [A7: set_complex] :
% 0.29/0.51              ( ( finite3207457112153483333omplex @ A7 )
% 0.29/0.51             => ( ! [B7: set_complex] :
% 0.29/0.51                    ( ( ord_less_set_complex @ B7 @ A7 )
% 0.29/0.51                   => ( P @ B7 ) )
% 0.29/0.51               => ( P @ A7 ) ) )
% 0.29/0.51         => ( P @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_psubset_induct
% 0.29/0.51  thf(fact_1646_finite__psubset__induct,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.29/0.51        ( ( finite6177210948735845034at_nat @ A2 )
% 0.29/0.51       => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 0.29/0.51              ( ( finite6177210948735845034at_nat @ A7 )
% 0.29/0.51             => ( ! [B7: set_Pr1261947904930325089at_nat] :
% 0.29/0.51                    ( ( ord_le7866589430770878221at_nat @ B7 @ A7 )
% 0.29/0.51                   => ( P @ B7 ) )
% 0.29/0.51               => ( P @ A7 ) ) )
% 0.29/0.51         => ( P @ A2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_psubset_induct
% 0.29/0.51  thf(fact_1647_finite__nat__set__iff__bounded,axiom,
% 0.29/0.51      ( finite_finite_nat
% 0.29/0.51      = ( ^ [N5: set_nat] :
% 0.29/0.51          ? [M2: nat] :
% 0.29/0.51          ! [X: nat] :
% 0.29/0.51            ( ( member_nat @ X @ N5 )
% 0.29/0.51           => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % finite_nat_set_iff_bounded
% 0.29/0.51  thf(fact_1648_infinite__nat__iff__unbounded,axiom,
% 0.29/0.51      ! [S3: set_nat] :
% 0.29/0.51        ( ( ~ ( finite_finite_nat @ S3 ) )
% 0.29/0.51        = ( ! [M2: nat] :
% 0.29/0.51            ? [N: nat] :
% 0.29/0.51              ( ( ord_less_nat @ M2 @ N )
% 0.29/0.51              & ( member_nat @ N @ S3 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % infinite_nat_iff_unbounded
% 0.29/0.51  thf(fact_1649_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_complex,F: complex > real] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ~ ? [X2: complex] :
% 0.29/0.51                ( ( member_complex @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1650_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_real,F: real > real] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ~ ? [X2: real] :
% 0.29/0.51                ( ( member_real @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1651_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_nat,F: nat > real] :
% 0.29/0.51        ( ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_nat )
% 0.29/0.51         => ~ ? [X2: nat] :
% 0.29/0.51                ( ( member_nat @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1652_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_int,F: int > real] :
% 0.29/0.51        ( ( finite_finite_int @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_int )
% 0.29/0.51         => ~ ? [X2: int] :
% 0.29/0.51                ( ( member_int @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1653_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_complex,F: complex > rat] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ~ ? [X2: complex] :
% 0.29/0.51                ( ( member_complex @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1654_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_real,F: real > rat] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ~ ? [X2: real] :
% 0.29/0.51                ( ( member_real @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1655_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_nat,F: nat > rat] :
% 0.29/0.51        ( ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_nat )
% 0.29/0.51         => ~ ? [X2: nat] :
% 0.29/0.51                ( ( member_nat @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1656_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_int,F: int > rat] :
% 0.29/0.51        ( ( finite_finite_int @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_int )
% 0.29/0.51         => ~ ? [X2: int] :
% 0.29/0.51                ( ( member_int @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1657_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_complex,F: complex > num] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ~ ? [X2: complex] :
% 0.29/0.51                ( ( member_complex @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1658_arg__min__if__finite_I2_J,axiom,
% 0.29/0.51      ! [S3: set_real,F: real > num] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ~ ? [X2: real] :
% 0.29/0.51                ( ( member_real @ X2 @ S3 )
% 0.29/0.51                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_if_finite(2)
% 0.29/0.51  thf(fact_1659_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_complex,Y3: complex,F: complex > rat] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ( ( member_complex @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1660_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_real,Y3: real,F: real > rat] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ( ( member_real @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1661_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_nat,Y3: nat,F: nat > rat] :
% 0.29/0.51        ( ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_nat )
% 0.29/0.51         => ( ( member_nat @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1662_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_int,Y3: int,F: int > rat] :
% 0.29/0.51        ( ( finite_finite_int @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_int )
% 0.29/0.51         => ( ( member_int @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1663_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_complex,Y3: complex,F: complex > num] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ( ( member_complex @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1664_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_real,Y3: real,F: real > num] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ( ( member_real @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1665_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_nat,Y3: nat,F: nat > num] :
% 0.29/0.51        ( ( finite_finite_nat @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_nat )
% 0.29/0.51         => ( ( member_nat @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1666_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_int,Y3: int,F: int > num] :
% 0.29/0.51        ( ( finite_finite_int @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_int )
% 0.29/0.51         => ( ( member_int @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1667_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_complex,Y3: complex,F: complex > nat] :
% 0.29/0.51        ( ( finite3207457112153483333omplex @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_complex )
% 0.29/0.51         => ( ( member_complex @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1668_arg__min__least,axiom,
% 0.29/0.51      ! [S3: set_real,Y3: real,F: real > nat] :
% 0.29/0.51        ( ( finite_finite_real @ S3 )
% 0.29/0.51       => ( ( S3 != bot_bot_set_real )
% 0.29/0.51         => ( ( member_real @ Y3 @ S3 )
% 0.29/0.51           => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % arg_min_least
% 0.29/0.51  thf(fact_1669_nat__descend__induct,axiom,
% 0.29/0.51      ! [N2: nat,P: nat > $o,M: nat] :
% 0.29/0.51        ( ! [K2: nat] :
% 0.29/0.51            ( ( ord_less_nat @ N2 @ K2 )
% 0.29/0.51           => ( P @ K2 ) )
% 0.29/0.51       => ( ! [K2: nat] :
% 0.29/0.51              ( ( ord_less_eq_nat @ K2 @ N2 )
% 0.29/0.51             => ( ! [I4: nat] :
% 0.29/0.51                    ( ( ord_less_nat @ K2 @ I4 )
% 0.29/0.51                   => ( P @ I4 ) )
% 0.29/0.51               => ( P @ K2 ) ) )
% 0.29/0.51         => ( P @ M ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % nat_descend_induct
% 0.29/0.51  thf(fact_1670_subset__emptyI,axiom,
% 0.29/0.51      ! [A2: set_set_nat] :
% 0.29/0.51        ( ! [X4: set_nat] :
% 0.29/0.51            ~ ( member_set_nat @ X4 @ A2 )
% 0.29/0.51       => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_emptyI
% 0.29/0.51  thf(fact_1671_subset__emptyI,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ! [X4: product_prod_nat_nat] :
% 0.29/0.51            ~ ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.29/0.51       => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_emptyI
% 0.29/0.51  thf(fact_1672_subset__emptyI,axiom,
% 0.29/0.51      ! [A2: set_real] :
% 0.29/0.51        ( ! [X4: real] :
% 0.29/0.51            ~ ( member_real @ X4 @ A2 )
% 0.29/0.51       => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_emptyI
% 0.29/0.51  thf(fact_1673_subset__emptyI,axiom,
% 0.29/0.51      ! [A2: set_nat] :
% 0.29/0.51        ( ! [X4: nat] :
% 0.29/0.51            ~ ( member_nat @ X4 @ A2 )
% 0.29/0.51       => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_emptyI
% 0.29/0.51  thf(fact_1674_subset__emptyI,axiom,
% 0.29/0.51      ! [A2: set_int] :
% 0.29/0.51        ( ! [X4: int] :
% 0.29/0.51            ~ ( member_int @ X4 @ A2 )
% 0.29/0.51       => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% 0.29/0.51  
% 0.29/0.51  % subset_emptyI
% 0.29/0.51  thf(fact_1675_exists__least__lemma,axiom,
% 0.29/0.51      ! [P: nat > $o] :
% 0.29/0.51        ( ~ ( P @ zero_zero_nat )
% 0.29/0.51       => ( ? [X_12: nat] : ( P @ X_12 )
% 0.29/0.51         => ? [N3: nat] :
% 0.29/0.51              ( ~ ( P @ N3 )
% 0.29/0.51              & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % exists_least_lemma
% 0.29/0.51  thf(fact_1676_list__decode_Ocases,axiom,
% 0.29/0.51      ! [X3: nat] :
% 0.29/0.51        ( ( X3 != zero_zero_nat )
% 0.29/0.51       => ~ ! [N3: nat] :
% 0.29/0.51              ( X3
% 0.29/0.51             != ( suc @ N3 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % list_decode.cases
% 0.29/0.51  thf(fact_1677_verit__sum__simplify,axiom,
% 0.29/0.51      ! [A: complex] :
% 0.29/0.51        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 0.29/0.51        = A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_sum_simplify
% 0.29/0.51  thf(fact_1678_verit__sum__simplify,axiom,
% 0.29/0.51      ! [A: real] :
% 0.29/0.51        ( ( plus_plus_real @ A @ zero_zero_real )
% 0.29/0.51        = A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_sum_simplify
% 0.29/0.51  thf(fact_1679_verit__sum__simplify,axiom,
% 0.29/0.51      ! [A: rat] :
% 0.29/0.51        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 0.29/0.51        = A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_sum_simplify
% 0.29/0.51  thf(fact_1680_verit__sum__simplify,axiom,
% 0.29/0.51      ! [A: nat] :
% 0.29/0.51        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 0.29/0.51        = A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_sum_simplify
% 0.29/0.51  thf(fact_1681_verit__sum__simplify,axiom,
% 0.29/0.51      ! [A: int] :
% 0.29/0.51        ( ( plus_plus_int @ A @ zero_zero_int )
% 0.29/0.51        = A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_sum_simplify
% 0.29/0.51  thf(fact_1682_add__0__iff,axiom,
% 0.29/0.51      ! [B: complex,A: complex] :
% 0.29/0.51        ( ( B
% 0.29/0.51          = ( plus_plus_complex @ B @ A ) )
% 0.29/0.51        = ( A = zero_zero_complex ) ) ).
% 0.29/0.51  
% 0.29/0.51  % add_0_iff
% 0.29/0.51  thf(fact_1683_add__0__iff,axiom,
% 0.29/0.51      ! [B: real,A: real] :
% 0.29/0.51        ( ( B
% 0.29/0.51          = ( plus_plus_real @ B @ A ) )
% 0.29/0.51        = ( A = zero_zero_real ) ) ).
% 0.29/0.51  
% 0.29/0.51  % add_0_iff
% 0.29/0.51  thf(fact_1684_add__0__iff,axiom,
% 0.29/0.51      ! [B: rat,A: rat] :
% 0.29/0.51        ( ( B
% 0.29/0.51          = ( plus_plus_rat @ B @ A ) )
% 0.29/0.51        = ( A = zero_zero_rat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % add_0_iff
% 0.29/0.51  thf(fact_1685_add__0__iff,axiom,
% 0.29/0.51      ! [B: nat,A: nat] :
% 0.29/0.51        ( ( B
% 0.29/0.51          = ( plus_plus_nat @ B @ A ) )
% 0.29/0.51        = ( A = zero_zero_nat ) ) ).
% 0.29/0.51  
% 0.29/0.51  % add_0_iff
% 0.29/0.51  thf(fact_1686_add__0__iff,axiom,
% 0.29/0.51      ! [B: int,A: int] :
% 0.29/0.51        ( ( B
% 0.29/0.51          = ( plus_plus_int @ B @ A ) )
% 0.29/0.51        = ( A = zero_zero_int ) ) ).
% 0.29/0.51  
% 0.29/0.51  % add_0_iff
% 0.29/0.51  thf(fact_1687_field__lbound__gt__zero,axiom,
% 0.29/0.51      ! [D1: real,D2: real] :
% 0.29/0.51        ( ( ord_less_real @ zero_zero_real @ D1 )
% 0.29/0.51       => ( ( ord_less_real @ zero_zero_real @ D2 )
% 0.29/0.51         => ? [E: real] :
% 0.29/0.51              ( ( ord_less_real @ zero_zero_real @ E )
% 0.29/0.51              & ( ord_less_real @ E @ D1 )
% 0.29/0.51              & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % field_lbound_gt_zero
% 0.29/0.51  thf(fact_1688_field__lbound__gt__zero,axiom,
% 0.29/0.51      ! [D1: rat,D2: rat] :
% 0.29/0.51        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 0.29/0.51       => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 0.29/0.51         => ? [E: rat] :
% 0.29/0.51              ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.29/0.51              & ( ord_less_rat @ E @ D1 )
% 0.29/0.51              & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % field_lbound_gt_zero
% 0.29/0.51  thf(fact_1689_less__by__empty,axiom,
% 0.29/0.51      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.29/0.51        ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.29/0.51       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % less_by_empty
% 0.29/0.51  thf(fact_1690_verit__comp__simplify1_I2_J,axiom,
% 0.29/0.51      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(2)
% 0.29/0.51  thf(fact_1691_verit__comp__simplify1_I2_J,axiom,
% 0.29/0.51      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(2)
% 0.29/0.51  thf(fact_1692_verit__comp__simplify1_I2_J,axiom,
% 0.29/0.51      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(2)
% 0.29/0.51  thf(fact_1693_verit__comp__simplify1_I2_J,axiom,
% 0.29/0.51      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(2)
% 0.29/0.51  thf(fact_1694_verit__comp__simplify1_I2_J,axiom,
% 0.29/0.51      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(2)
% 0.29/0.51  thf(fact_1695_verit__la__disequality,axiom,
% 0.29/0.51      ! [A: rat,B: rat] :
% 0.29/0.51        ( ( A = B )
% 0.29/0.51        | ~ ( ord_less_eq_rat @ A @ B )
% 0.29/0.51        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_la_disequality
% 0.29/0.51  thf(fact_1696_verit__la__disequality,axiom,
% 0.29/0.51      ! [A: num,B: num] :
% 0.29/0.51        ( ( A = B )
% 0.29/0.51        | ~ ( ord_less_eq_num @ A @ B )
% 0.29/0.51        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_la_disequality
% 0.29/0.51  thf(fact_1697_verit__la__disequality,axiom,
% 0.29/0.51      ! [A: nat,B: nat] :
% 0.29/0.51        ( ( A = B )
% 0.29/0.51        | ~ ( ord_less_eq_nat @ A @ B )
% 0.29/0.51        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_la_disequality
% 0.29/0.51  thf(fact_1698_verit__la__disequality,axiom,
% 0.29/0.51      ! [A: int,B: int] :
% 0.29/0.51        ( ( A = B )
% 0.29/0.51        | ~ ( ord_less_eq_int @ A @ B )
% 0.29/0.51        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_la_disequality
% 0.29/0.51  thf(fact_1699_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.51      ! [A: real] :
% 0.29/0.51        ~ ( ord_less_real @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(1)
% 0.29/0.51  thf(fact_1700_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.51      ! [A: rat] :
% 0.29/0.51        ~ ( ord_less_rat @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(1)
% 0.29/0.51  thf(fact_1701_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.51      ! [A: num] :
% 0.29/0.51        ~ ( ord_less_num @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(1)
% 0.29/0.51  thf(fact_1702_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.51      ! [A: nat] :
% 0.29/0.51        ~ ( ord_less_nat @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(1)
% 0.29/0.51  thf(fact_1703_verit__comp__simplify1_I1_J,axiom,
% 0.29/0.51      ! [A: int] :
% 0.29/0.51        ~ ( ord_less_int @ A @ A ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(1)
% 0.29/0.51  thf(fact_1704_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
% 0.29/0.51        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1705_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
% 0.29/0.51        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1706_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S4: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1707_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S4: produc8923325533196201883nteger] :
% 0.29/0.51        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1708_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S4: product_prod_int_int] :
% 0.29/0.51        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1709_ssubst__Pair__rhs,axiom,
% 0.29/0.51      ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S4: product_prod_int_int] :
% 0.29/0.51        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R )
% 0.29/0.51       => ( ( S4 = S )
% 0.29/0.51         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S4 ) @ R ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % ssubst_Pair_rhs
% 0.29/0.51  thf(fact_1710_prod__decode__aux_Ocases,axiom,
% 0.29/0.51      ! [X3: product_prod_nat_nat] :
% 0.29/0.51        ~ ! [K2: nat,M3: nat] :
% 0.29/0.51            ( X3
% 0.29/0.51           != ( product_Pair_nat_nat @ K2 @ M3 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % prod_decode_aux.cases
% 0.29/0.51  thf(fact_1711_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.51      ! [B6: real,A6: real] :
% 0.29/0.51        ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 0.29/0.51        = ( ord_less_real @ A6 @ B6 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(3)
% 0.29/0.51  thf(fact_1712_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.51      ! [B6: rat,A6: rat] :
% 0.29/0.51        ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 0.29/0.51        = ( ord_less_rat @ A6 @ B6 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(3)
% 0.29/0.51  thf(fact_1713_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.51      ! [B6: num,A6: num] :
% 0.29/0.51        ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 0.29/0.51        = ( ord_less_num @ A6 @ B6 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(3)
% 0.29/0.51  thf(fact_1714_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.51      ! [B6: nat,A6: nat] :
% 0.29/0.51        ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 0.29/0.51        = ( ord_less_nat @ A6 @ B6 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(3)
% 0.29/0.51  thf(fact_1715_verit__comp__simplify1_I3_J,axiom,
% 0.29/0.51      ! [B6: int,A6: int] :
% 0.29/0.51        ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 0.29/0.51        = ( ord_less_int @ A6 @ B6 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % verit_comp_simplify1(3)
% 0.29/0.51  thf(fact_1716_triangle__Suc,axiom,
% 0.29/0.51      ! [N2: nat] :
% 0.29/0.51        ( ( nat_triangle @ ( suc @ N2 ) )
% 0.29/0.51        = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% 0.29/0.51  
% 0.29/0.51  % triangle_Suc
% 0.29/0.51  thf(fact_1717_less__numeral__extra_I3_J,axiom,
% 0.29/0.51      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 0.29/0.51  
% 0.29/0.51  % less_numeral_extra(3)
% 0.29/0.51  thf(fact_1718_less__numeral__extra_I3_J,axiom,
% 0.29/0.51      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 0.29/0.51  
% 0.29/0.51  % less_numeral_extra(3)
% 0.29/0.51  thf(fact_1719_less__numeral__extra_I3_J,axiom,
% 0.29/0.51      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 0.29/0.51  
% 0.29/0.51  % less_numeral_extra(3)
% 0.29/0.51  thf(fact_1720_less__numeral__extra_I3_J,axiom,
% 0.29/0.51      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.29/0.51  
% 0.29/0.51  % less_numeral_extra(3)
% 0.29/0.51  thf(fact_1721_minf_I8_J,axiom,
% 0.29/0.51      ! [T: real] :
% 0.29/0.51      ? [Z3: real] :
% 0.29/0.51      ! [X2: real] :
% 0.29/0.51        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.51       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(8)
% 0.29/0.51  thf(fact_1722_minf_I8_J,axiom,
% 0.29/0.51      ! [T: rat] :
% 0.29/0.51      ? [Z3: rat] :
% 0.29/0.51      ! [X2: rat] :
% 0.29/0.51        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.51       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(8)
% 0.29/0.51  thf(fact_1723_minf_I8_J,axiom,
% 0.29/0.51      ! [T: num] :
% 0.29/0.51      ? [Z3: num] :
% 0.29/0.51      ! [X2: num] :
% 0.29/0.51        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.51       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(8)
% 0.29/0.51  thf(fact_1724_minf_I8_J,axiom,
% 0.29/0.51      ! [T: nat] :
% 0.29/0.51      ? [Z3: nat] :
% 0.29/0.51      ! [X2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.51       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(8)
% 0.29/0.51  thf(fact_1725_minf_I8_J,axiom,
% 0.29/0.51      ! [T: int] :
% 0.29/0.51      ? [Z3: int] :
% 0.29/0.51      ! [X2: int] :
% 0.29/0.51        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.51       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(8)
% 0.29/0.51  thf(fact_1726_minf_I6_J,axiom,
% 0.29/0.51      ! [T: real] :
% 0.29/0.51      ? [Z3: real] :
% 0.29/0.51      ! [X2: real] :
% 0.29/0.51        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.51       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(6)
% 0.29/0.51  thf(fact_1727_minf_I6_J,axiom,
% 0.29/0.51      ! [T: rat] :
% 0.29/0.51      ? [Z3: rat] :
% 0.29/0.51      ! [X2: rat] :
% 0.29/0.51        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.51       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(6)
% 0.29/0.51  thf(fact_1728_minf_I6_J,axiom,
% 0.29/0.51      ! [T: num] :
% 0.29/0.51      ? [Z3: num] :
% 0.29/0.51      ! [X2: num] :
% 0.29/0.51        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.51       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(6)
% 0.29/0.51  thf(fact_1729_minf_I6_J,axiom,
% 0.29/0.51      ! [T: nat] :
% 0.29/0.51      ? [Z3: nat] :
% 0.29/0.51      ! [X2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.51       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(6)
% 0.29/0.51  thf(fact_1730_minf_I6_J,axiom,
% 0.29/0.51      ! [T: int] :
% 0.29/0.51      ? [Z3: int] :
% 0.29/0.51      ! [X2: int] :
% 0.29/0.51        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.51       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.29/0.51  
% 0.29/0.51  % minf(6)
% 0.29/0.51  thf(fact_1731_pinf_I8_J,axiom,
% 0.29/0.51      ! [T: real] :
% 0.29/0.51      ? [Z3: real] :
% 0.29/0.51      ! [X2: real] :
% 0.29/0.51        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.51       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % pinf(8)
% 0.29/0.51  thf(fact_1732_pinf_I8_J,axiom,
% 0.29/0.51      ! [T: rat] :
% 0.29/0.51      ? [Z3: rat] :
% 0.29/0.51      ! [X2: rat] :
% 0.29/0.51        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.51       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % pinf(8)
% 0.29/0.51  thf(fact_1733_pinf_I8_J,axiom,
% 0.29/0.51      ! [T: num] :
% 0.29/0.51      ? [Z3: num] :
% 0.29/0.51      ! [X2: num] :
% 0.29/0.51        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.51       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % pinf(8)
% 0.29/0.51  thf(fact_1734_pinf_I8_J,axiom,
% 0.29/0.51      ! [T: nat] :
% 0.29/0.51      ? [Z3: nat] :
% 0.29/0.51      ! [X2: nat] :
% 0.29/0.51        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.51       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 0.29/0.51  
% 0.29/0.51  % pinf(8)
% 0.29/0.51  thf(fact_1735_pinf_I8_J,axiom,
% 0.29/0.51      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(8)
% 0.29/0.52  thf(fact_1736_pinf_I6_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(6)
% 0.29/0.52  thf(fact_1737_pinf_I6_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(6)
% 0.29/0.52  thf(fact_1738_pinf_I6_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(6)
% 0.29/0.52  thf(fact_1739_pinf_I6_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(6)
% 0.29/0.52  thf(fact_1740_pinf_I6_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(6)
% 0.29/0.52  thf(fact_1741_complete__interval,axiom,
% 0.29/0.52      ! [A: real,B: real,P: real > $o] :
% 0.29/0.52        ( ( ord_less_real @ A @ B )
% 0.29/0.52       => ( ( P @ A )
% 0.29/0.52         => ( ~ ( P @ B )
% 0.29/0.52           => ? [C2: real] :
% 0.29/0.52                ( ( ord_less_eq_real @ A @ C2 )
% 0.29/0.52                & ( ord_less_eq_real @ C2 @ B )
% 0.29/0.52                & ! [X2: real] :
% 0.29/0.52                    ( ( ( ord_less_eq_real @ A @ X2 )
% 0.29/0.52                      & ( ord_less_real @ X2 @ C2 ) )
% 0.29/0.52                   => ( P @ X2 ) )
% 0.29/0.52                & ! [D3: real] :
% 0.29/0.52                    ( ! [X4: real] :
% 0.29/0.52                        ( ( ( ord_less_eq_real @ A @ X4 )
% 0.29/0.52                          & ( ord_less_real @ X4 @ D3 ) )
% 0.29/0.52                       => ( P @ X4 ) )
% 0.29/0.52                   => ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % complete_interval
% 0.29/0.52  thf(fact_1742_complete__interval,axiom,
% 0.29/0.52      ! [A: nat,B: nat,P: nat > $o] :
% 0.29/0.52        ( ( ord_less_nat @ A @ B )
% 0.29/0.52       => ( ( P @ A )
% 0.29/0.52         => ( ~ ( P @ B )
% 0.29/0.52           => ? [C2: nat] :
% 0.29/0.52                ( ( ord_less_eq_nat @ A @ C2 )
% 0.29/0.52                & ( ord_less_eq_nat @ C2 @ B )
% 0.29/0.52                & ! [X2: nat] :
% 0.29/0.52                    ( ( ( ord_less_eq_nat @ A @ X2 )
% 0.29/0.52                      & ( ord_less_nat @ X2 @ C2 ) )
% 0.29/0.52                   => ( P @ X2 ) )
% 0.29/0.52                & ! [D3: nat] :
% 0.29/0.52                    ( ! [X4: nat] :
% 0.29/0.52                        ( ( ( ord_less_eq_nat @ A @ X4 )
% 0.29/0.52                          & ( ord_less_nat @ X4 @ D3 ) )
% 0.29/0.52                       => ( P @ X4 ) )
% 0.29/0.52                   => ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % complete_interval
% 0.29/0.52  thf(fact_1743_complete__interval,axiom,
% 0.29/0.52      ! [A: int,B: int,P: int > $o] :
% 0.29/0.52        ( ( ord_less_int @ A @ B )
% 0.29/0.52       => ( ( P @ A )
% 0.29/0.52         => ( ~ ( P @ B )
% 0.29/0.52           => ? [C2: int] :
% 0.29/0.52                ( ( ord_less_eq_int @ A @ C2 )
% 0.29/0.52                & ( ord_less_eq_int @ C2 @ B )
% 0.29/0.52                & ! [X2: int] :
% 0.29/0.52                    ( ( ( ord_less_eq_int @ A @ X2 )
% 0.29/0.52                      & ( ord_less_int @ X2 @ C2 ) )
% 0.29/0.52                   => ( P @ X2 ) )
% 0.29/0.52                & ! [D3: int] :
% 0.29/0.52                    ( ! [X4: int] :
% 0.29/0.52                        ( ( ( ord_less_eq_int @ A @ X4 )
% 0.29/0.52                          & ( ord_less_int @ X4 @ D3 ) )
% 0.29/0.52                       => ( P @ X4 ) )
% 0.29/0.52                   => ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % complete_interval
% 0.29/0.52  thf(fact_1744_le__numeral__extra_I3_J,axiom,
% 0.29/0.52      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 0.29/0.52  
% 0.29/0.52  % le_numeral_extra(3)
% 0.29/0.52  thf(fact_1745_le__numeral__extra_I3_J,axiom,
% 0.29/0.52      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 0.29/0.52  
% 0.29/0.52  % le_numeral_extra(3)
% 0.29/0.52  thf(fact_1746_le__numeral__extra_I3_J,axiom,
% 0.29/0.52      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 0.29/0.52  
% 0.29/0.52  % le_numeral_extra(3)
% 0.29/0.52  thf(fact_1747_le__numeral__extra_I3_J,axiom,
% 0.29/0.52      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 0.29/0.52  
% 0.29/0.52  % le_numeral_extra(3)
% 0.29/0.52  thf(fact_1748_find__Some__iff,axiom,
% 0.29/0.52      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
% 0.29/0.52        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.29/0.52          = ( some_P7363390416028606310at_nat @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1749_find__Some__iff,axiom,
% 0.29/0.52      ! [P: num > $o,Xs: list_num,X3: num] :
% 0.29/0.52        ( ( ( find_num @ P @ Xs )
% 0.29/0.52          = ( some_num @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_num @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_num @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1750_find__Some__iff,axiom,
% 0.29/0.52      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.29/0.52          = ( some_VEBT_VEBT @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1751_find__Some__iff,axiom,
% 0.29/0.52      ! [P: $o > $o,Xs: list_o,X3: $o] :
% 0.29/0.52        ( ( ( find_o @ P @ Xs )
% 0.29/0.52          = ( some_o @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_o @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_o @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1752_find__Some__iff,axiom,
% 0.29/0.52      ! [P: nat > $o,Xs: list_nat,X3: nat] :
% 0.29/0.52        ( ( ( find_nat @ P @ Xs )
% 0.29/0.52          = ( some_nat @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_nat @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_nat @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1753_find__Some__iff,axiom,
% 0.29/0.52      ! [P: int > $o,Xs: list_int,X3: int] :
% 0.29/0.52        ( ( ( find_int @ P @ Xs )
% 0.29/0.52          = ( some_int @ X3 ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_int @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_int @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff
% 0.29/0.52  thf(fact_1754_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.52        ( ( ( some_P7363390416028606310at_nat @ X3 )
% 0.29/0.52          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1755_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: num,P: num > $o,Xs: list_num] :
% 0.29/0.52        ( ( ( some_num @ X3 )
% 0.29/0.52          = ( find_num @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_num @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_num @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1756_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.29/0.52        ( ( ( some_VEBT_VEBT @ X3 )
% 0.29/0.52          = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1757_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: $o,P: $o > $o,Xs: list_o] :
% 0.29/0.52        ( ( ( some_o @ X3 )
% 0.29/0.52          = ( find_o @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_o @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_o @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1758_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: nat,P: nat > $o,Xs: list_nat] :
% 0.29/0.52        ( ( ( some_nat @ X3 )
% 0.29/0.52          = ( find_nat @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_nat @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_nat @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1759_find__Some__iff2,axiom,
% 0.29/0.52      ! [X3: int,P: int > $o,Xs: list_int] :
% 0.29/0.52        ( ( ( some_int @ X3 )
% 0.29/0.52          = ( find_int @ P @ Xs ) )
% 0.29/0.52        = ( ? [I2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 0.29/0.52              & ( P @ ( nth_int @ Xs @ I2 ) )
% 0.29/0.52              & ( X3
% 0.29/0.52                = ( nth_int @ Xs @ I2 ) )
% 0.29/0.52              & ! [J3: nat] :
% 0.29/0.52                  ( ( ord_less_nat @ J3 @ I2 )
% 0.29/0.52                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_Some_iff2
% 0.29/0.52  thf(fact_1760_triangle__0,axiom,
% 0.29/0.52      ( ( nat_triangle @ zero_zero_nat )
% 0.29/0.52      = zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % triangle_0
% 0.29/0.52  thf(fact_1761_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: product_prod_nat_nat] :
% 0.29/0.52              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.29/0.52            = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1762_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: real] :
% 0.29/0.52              ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_real @ P @ Xs )
% 0.29/0.52            = ( find_real @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1763_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: set_nat] :
% 0.29/0.52              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_set_nat @ P @ Xs )
% 0.29/0.52            = ( find_set_nat @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1764_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: vEBT_VEBT] :
% 0.29/0.52              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_VEBT_VEBT @ P @ Xs )
% 0.29/0.52            = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1765_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: int] :
% 0.29/0.52              ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_int @ P @ Xs )
% 0.29/0.52            = ( find_int @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1766_find__cong,axiom,
% 0.29/0.52      ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
% 0.29/0.52        ( ( Xs = Ys )
% 0.29/0.52       => ( ! [X4: nat] :
% 0.29/0.52              ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
% 0.29/0.52             => ( ( P @ X4 )
% 0.29/0.52                = ( Q @ X4 ) ) )
% 0.29/0.52         => ( ( find_nat @ P @ Xs )
% 0.29/0.52            = ( find_nat @ Q @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_cong
% 0.29/0.52  thf(fact_1767_ex__gt__or__lt,axiom,
% 0.29/0.52      ! [A: real] :
% 0.29/0.52      ? [B4: real] :
% 0.29/0.52        ( ( ord_less_real @ A @ B4 )
% 0.29/0.52        | ( ord_less_real @ B4 @ A ) ) ).
% 0.29/0.52  
% 0.29/0.52  % ex_gt_or_lt
% 0.29/0.52  thf(fact_1768_pinf_I1_J,axiom,
% 0.29/0.52      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 0.29/0.52        ( ? [Z4: real] :
% 0.29/0.52          ! [X4: real] :
% 0.29/0.52            ( ( ord_less_real @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: real] :
% 0.29/0.52            ! [X4: real] :
% 0.29/0.52              ( ( ord_less_real @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: real] :
% 0.29/0.52            ! [X2: real] :
% 0.29/0.52              ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(1)
% 0.29/0.52  thf(fact_1769_pinf_I1_J,axiom,
% 0.29/0.52      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.29/0.52        ( ? [Z4: rat] :
% 0.29/0.52          ! [X4: rat] :
% 0.29/0.52            ( ( ord_less_rat @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: rat] :
% 0.29/0.52            ! [X4: rat] :
% 0.29/0.52              ( ( ord_less_rat @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: rat] :
% 0.29/0.52            ! [X2: rat] :
% 0.29/0.52              ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(1)
% 0.29/0.52  thf(fact_1770_pinf_I1_J,axiom,
% 0.29/0.52      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 0.29/0.52        ( ? [Z4: num] :
% 0.29/0.52          ! [X4: num] :
% 0.29/0.52            ( ( ord_less_num @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: num] :
% 0.29/0.52            ! [X4: num] :
% 0.29/0.52              ( ( ord_less_num @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: num] :
% 0.29/0.52            ! [X2: num] :
% 0.29/0.52              ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(1)
% 0.29/0.52  thf(fact_1771_pinf_I1_J,axiom,
% 0.29/0.52      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.29/0.52        ( ? [Z4: nat] :
% 0.29/0.52          ! [X4: nat] :
% 0.29/0.52            ( ( ord_less_nat @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: nat] :
% 0.29/0.52            ! [X4: nat] :
% 0.29/0.52              ( ( ord_less_nat @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: nat] :
% 0.29/0.52            ! [X2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(1)
% 0.29/0.52  thf(fact_1772_pinf_I1_J,axiom,
% 0.29/0.52      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 0.29/0.52        ( ? [Z4: int] :
% 0.29/0.52          ! [X4: int] :
% 0.29/0.52            ( ( ord_less_int @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: int] :
% 0.29/0.52            ! [X4: int] :
% 0.29/0.52              ( ( ord_less_int @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: int] :
% 0.29/0.52            ! [X2: int] :
% 0.29/0.52              ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(1)
% 0.29/0.52  thf(fact_1773_pinf_I2_J,axiom,
% 0.29/0.52      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 0.29/0.52        ( ? [Z4: real] :
% 0.29/0.52          ! [X4: real] :
% 0.29/0.52            ( ( ord_less_real @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: real] :
% 0.29/0.52            ! [X4: real] :
% 0.29/0.52              ( ( ord_less_real @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: real] :
% 0.29/0.52            ! [X2: real] :
% 0.29/0.52              ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(2)
% 0.29/0.52  thf(fact_1774_pinf_I2_J,axiom,
% 0.29/0.52      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.29/0.52        ( ? [Z4: rat] :
% 0.29/0.52          ! [X4: rat] :
% 0.29/0.52            ( ( ord_less_rat @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: rat] :
% 0.29/0.52            ! [X4: rat] :
% 0.29/0.52              ( ( ord_less_rat @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: rat] :
% 0.29/0.52            ! [X2: rat] :
% 0.29/0.52              ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(2)
% 0.29/0.52  thf(fact_1775_pinf_I2_J,axiom,
% 0.29/0.52      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 0.29/0.52        ( ? [Z4: num] :
% 0.29/0.52          ! [X4: num] :
% 0.29/0.52            ( ( ord_less_num @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: num] :
% 0.29/0.52            ! [X4: num] :
% 0.29/0.52              ( ( ord_less_num @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: num] :
% 0.29/0.52            ! [X2: num] :
% 0.29/0.52              ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(2)
% 0.29/0.52  thf(fact_1776_pinf_I2_J,axiom,
% 0.29/0.52      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.29/0.52        ( ? [Z4: nat] :
% 0.29/0.52          ! [X4: nat] :
% 0.29/0.52            ( ( ord_less_nat @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: nat] :
% 0.29/0.52            ! [X4: nat] :
% 0.29/0.52              ( ( ord_less_nat @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: nat] :
% 0.29/0.52            ! [X2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(2)
% 0.29/0.52  thf(fact_1777_pinf_I2_J,axiom,
% 0.29/0.52      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 0.29/0.52        ( ? [Z4: int] :
% 0.29/0.52          ! [X4: int] :
% 0.29/0.52            ( ( ord_less_int @ Z4 @ X4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: int] :
% 0.29/0.52            ! [X4: int] :
% 0.29/0.52              ( ( ord_less_int @ Z4 @ X4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: int] :
% 0.29/0.52            ! [X2: int] :
% 0.29/0.52              ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(2)
% 0.29/0.52  thf(fact_1778_pinf_I3_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(3)
% 0.29/0.52  thf(fact_1779_pinf_I3_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(3)
% 0.29/0.52  thf(fact_1780_pinf_I3_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(3)
% 0.29/0.52  thf(fact_1781_pinf_I3_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(3)
% 0.29/0.52  thf(fact_1782_pinf_I3_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(3)
% 0.29/0.52  thf(fact_1783_pinf_I4_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(4)
% 0.29/0.52  thf(fact_1784_pinf_I4_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(4)
% 0.29/0.52  thf(fact_1785_pinf_I4_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(4)
% 0.29/0.52  thf(fact_1786_pinf_I4_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(4)
% 0.29/0.52  thf(fact_1787_pinf_I4_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(4)
% 0.29/0.52  thf(fact_1788_pinf_I5_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(5)
% 0.29/0.52  thf(fact_1789_pinf_I5_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(5)
% 0.29/0.52  thf(fact_1790_pinf_I5_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(5)
% 0.29/0.52  thf(fact_1791_pinf_I5_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(5)
% 0.29/0.52  thf(fact_1792_pinf_I5_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(5)
% 0.29/0.52  thf(fact_1793_pinf_I7_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_real @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(7)
% 0.29/0.52  thf(fact_1794_pinf_I7_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_rat @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(7)
% 0.29/0.52  thf(fact_1795_pinf_I7_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_num @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(7)
% 0.29/0.52  thf(fact_1796_pinf_I7_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_nat @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(7)
% 0.29/0.52  thf(fact_1797_pinf_I7_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ Z3 @ X2 )
% 0.29/0.52       => ( ord_less_int @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pinf(7)
% 0.29/0.52  thf(fact_1798_minf_I1_J,axiom,
% 0.29/0.52      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 0.29/0.52        ( ? [Z4: real] :
% 0.29/0.52          ! [X4: real] :
% 0.29/0.52            ( ( ord_less_real @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: real] :
% 0.29/0.52            ! [X4: real] :
% 0.29/0.52              ( ( ord_less_real @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: real] :
% 0.29/0.52            ! [X2: real] :
% 0.29/0.52              ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(1)
% 0.29/0.52  thf(fact_1799_minf_I1_J,axiom,
% 0.29/0.52      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.29/0.52        ( ? [Z4: rat] :
% 0.29/0.52          ! [X4: rat] :
% 0.29/0.52            ( ( ord_less_rat @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: rat] :
% 0.29/0.52            ! [X4: rat] :
% 0.29/0.52              ( ( ord_less_rat @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: rat] :
% 0.29/0.52            ! [X2: rat] :
% 0.29/0.52              ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(1)
% 0.29/0.52  thf(fact_1800_minf_I1_J,axiom,
% 0.29/0.52      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 0.29/0.52        ( ? [Z4: num] :
% 0.29/0.52          ! [X4: num] :
% 0.29/0.52            ( ( ord_less_num @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: num] :
% 0.29/0.52            ! [X4: num] :
% 0.29/0.52              ( ( ord_less_num @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: num] :
% 0.29/0.52            ! [X2: num] :
% 0.29/0.52              ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(1)
% 0.29/0.52  thf(fact_1801_minf_I1_J,axiom,
% 0.29/0.52      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.29/0.52        ( ? [Z4: nat] :
% 0.29/0.52          ! [X4: nat] :
% 0.29/0.52            ( ( ord_less_nat @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: nat] :
% 0.29/0.52            ! [X4: nat] :
% 0.29/0.52              ( ( ord_less_nat @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: nat] :
% 0.29/0.52            ! [X2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(1)
% 0.29/0.52  thf(fact_1802_minf_I1_J,axiom,
% 0.29/0.52      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 0.29/0.52        ( ? [Z4: int] :
% 0.29/0.52          ! [X4: int] :
% 0.29/0.52            ( ( ord_less_int @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: int] :
% 0.29/0.52            ! [X4: int] :
% 0.29/0.52              ( ( ord_less_int @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: int] :
% 0.29/0.52            ! [X2: int] :
% 0.29/0.52              ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  & ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(1)
% 0.29/0.52  thf(fact_1803_minf_I2_J,axiom,
% 0.29/0.52      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 0.29/0.52        ( ? [Z4: real] :
% 0.29/0.52          ! [X4: real] :
% 0.29/0.52            ( ( ord_less_real @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: real] :
% 0.29/0.52            ! [X4: real] :
% 0.29/0.52              ( ( ord_less_real @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: real] :
% 0.29/0.52            ! [X2: real] :
% 0.29/0.52              ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(2)
% 0.29/0.52  thf(fact_1804_minf_I2_J,axiom,
% 0.29/0.52      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 0.29/0.52        ( ? [Z4: rat] :
% 0.29/0.52          ! [X4: rat] :
% 0.29/0.52            ( ( ord_less_rat @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: rat] :
% 0.29/0.52            ! [X4: rat] :
% 0.29/0.52              ( ( ord_less_rat @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: rat] :
% 0.29/0.52            ! [X2: rat] :
% 0.29/0.52              ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(2)
% 0.29/0.52  thf(fact_1805_minf_I2_J,axiom,
% 0.29/0.52      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 0.29/0.52        ( ? [Z4: num] :
% 0.29/0.52          ! [X4: num] :
% 0.29/0.52            ( ( ord_less_num @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: num] :
% 0.29/0.52            ! [X4: num] :
% 0.29/0.52              ( ( ord_less_num @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: num] :
% 0.29/0.52            ! [X2: num] :
% 0.29/0.52              ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(2)
% 0.29/0.52  thf(fact_1806_minf_I2_J,axiom,
% 0.29/0.52      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 0.29/0.52        ( ? [Z4: nat] :
% 0.29/0.52          ! [X4: nat] :
% 0.29/0.52            ( ( ord_less_nat @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: nat] :
% 0.29/0.52            ! [X4: nat] :
% 0.29/0.52              ( ( ord_less_nat @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: nat] :
% 0.29/0.52            ! [X2: nat] :
% 0.29/0.52              ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(2)
% 0.29/0.52  thf(fact_1807_minf_I2_J,axiom,
% 0.29/0.52      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 0.29/0.52        ( ? [Z4: int] :
% 0.29/0.52          ! [X4: int] :
% 0.29/0.52            ( ( ord_less_int @ X4 @ Z4 )
% 0.29/0.52           => ( ( P @ X4 )
% 0.29/0.52              = ( P5 @ X4 ) ) )
% 0.29/0.52       => ( ? [Z4: int] :
% 0.29/0.52            ! [X4: int] :
% 0.29/0.52              ( ( ord_less_int @ X4 @ Z4 )
% 0.29/0.52             => ( ( Q @ X4 )
% 0.29/0.52                = ( Q3 @ X4 ) ) )
% 0.29/0.52         => ? [Z3: int] :
% 0.29/0.52            ! [X2: int] :
% 0.29/0.52              ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52             => ( ( ( P @ X2 )
% 0.29/0.52                  | ( Q @ X2 ) )
% 0.29/0.52                = ( ( P5 @ X2 )
% 0.29/0.52                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(2)
% 0.29/0.52  thf(fact_1808_minf_I3_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(3)
% 0.29/0.52  thf(fact_1809_minf_I3_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(3)
% 0.29/0.52  thf(fact_1810_minf_I3_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(3)
% 0.29/0.52  thf(fact_1811_minf_I3_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(3)
% 0.29/0.52  thf(fact_1812_minf_I3_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(3)
% 0.29/0.52  thf(fact_1813_minf_I4_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(4)
% 0.29/0.52  thf(fact_1814_minf_I4_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(4)
% 0.29/0.52  thf(fact_1815_minf_I4_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(4)
% 0.29/0.52  thf(fact_1816_minf_I4_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(4)
% 0.29/0.52  thf(fact_1817_minf_I4_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52       => ( X2 != T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(4)
% 0.29/0.52  thf(fact_1818_minf_I5_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52       => ( ord_less_real @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(5)
% 0.29/0.52  thf(fact_1819_minf_I5_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52       => ( ord_less_rat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(5)
% 0.29/0.52  thf(fact_1820_minf_I5_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52       => ( ord_less_num @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(5)
% 0.29/0.52  thf(fact_1821_minf_I5_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52       => ( ord_less_nat @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(5)
% 0.29/0.52  thf(fact_1822_minf_I5_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52       => ( ord_less_int @ X2 @ T ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(5)
% 0.29/0.52  thf(fact_1823_minf_I7_J,axiom,
% 0.29/0.52      ! [T: real] :
% 0.29/0.52      ? [Z3: real] :
% 0.29/0.52      ! [X2: real] :
% 0.29/0.52        ( ( ord_less_real @ X2 @ Z3 )
% 0.29/0.52       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(7)
% 0.29/0.52  thf(fact_1824_minf_I7_J,axiom,
% 0.29/0.52      ! [T: rat] :
% 0.29/0.52      ? [Z3: rat] :
% 0.29/0.52      ! [X2: rat] :
% 0.29/0.52        ( ( ord_less_rat @ X2 @ Z3 )
% 0.29/0.52       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(7)
% 0.29/0.52  thf(fact_1825_minf_I7_J,axiom,
% 0.29/0.52      ! [T: num] :
% 0.29/0.52      ? [Z3: num] :
% 0.29/0.52      ! [X2: num] :
% 0.29/0.52        ( ( ord_less_num @ X2 @ Z3 )
% 0.29/0.52       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(7)
% 0.29/0.52  thf(fact_1826_minf_I7_J,axiom,
% 0.29/0.52      ! [T: nat] :
% 0.29/0.52      ? [Z3: nat] :
% 0.29/0.52      ! [X2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X2 @ Z3 )
% 0.29/0.52       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(7)
% 0.29/0.52  thf(fact_1827_minf_I7_J,axiom,
% 0.29/0.52      ! [T: int] :
% 0.29/0.52      ? [Z3: int] :
% 0.29/0.52      ! [X2: int] :
% 0.29/0.52        ( ( ord_less_int @ X2 @ Z3 )
% 0.29/0.52       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % minf(7)
% 0.29/0.52  thf(fact_1828_is__num__normalize_I1_J,axiom,
% 0.29/0.52      ! [A: real,B: real,C: real] :
% 0.29/0.52        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.52        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % is_num_normalize(1)
% 0.29/0.52  thf(fact_1829_is__num__normalize_I1_J,axiom,
% 0.29/0.52      ! [A: rat,B: rat,C: rat] :
% 0.29/0.52        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.52        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % is_num_normalize(1)
% 0.29/0.52  thf(fact_1830_is__num__normalize_I1_J,axiom,
% 0.29/0.52      ! [A: int,B: int,C: int] :
% 0.29/0.52        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.52        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % is_num_normalize(1)
% 0.29/0.52  thf(fact_1831_find__None__iff2,axiom,
% 0.29/0.52      ! [P: real > $o,Xs: list_real] :
% 0.29/0.52        ( ( none_real
% 0.29/0.52          = ( find_real @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: real] :
% 0.29/0.52                ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1832_find__None__iff2,axiom,
% 0.29/0.52      ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.29/0.52        ( ( none_set_nat
% 0.29/0.52          = ( find_set_nat @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: set_nat] :
% 0.29/0.52                ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1833_find__None__iff2,axiom,
% 0.29/0.52      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.29/0.52        ( ( none_VEBT_VEBT
% 0.29/0.52          = ( find_VEBT_VEBT @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: vEBT_VEBT] :
% 0.29/0.52                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1834_find__None__iff2,axiom,
% 0.29/0.52      ! [P: int > $o,Xs: list_int] :
% 0.29/0.52        ( ( none_int
% 0.29/0.52          = ( find_int @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: int] :
% 0.29/0.52                ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1835_find__None__iff2,axiom,
% 0.29/0.52      ! [P: nat > $o,Xs: list_nat] :
% 0.29/0.52        ( ( none_nat
% 0.29/0.52          = ( find_nat @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: nat] :
% 0.29/0.52                ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1836_find__None__iff2,axiom,
% 0.29/0.52      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.52        ( ( none_P5556105721700978146at_nat
% 0.29/0.52          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: product_prod_nat_nat] :
% 0.29/0.52                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1837_find__None__iff2,axiom,
% 0.29/0.52      ! [P: num > $o,Xs: list_num] :
% 0.29/0.52        ( ( none_num
% 0.29/0.52          = ( find_num @ P @ Xs ) )
% 0.29/0.52        = ( ~ ? [X: num] :
% 0.29/0.52                ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff2
% 0.29/0.52  thf(fact_1838_find__None__iff,axiom,
% 0.29/0.52      ! [P: real > $o,Xs: list_real] :
% 0.29/0.52        ( ( ( find_real @ P @ Xs )
% 0.29/0.52          = none_real )
% 0.29/0.52        = ( ~ ? [X: real] :
% 0.29/0.52                ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1839_find__None__iff,axiom,
% 0.29/0.52      ! [P: set_nat > $o,Xs: list_set_nat] :
% 0.29/0.52        ( ( ( find_set_nat @ P @ Xs )
% 0.29/0.52          = none_set_nat )
% 0.29/0.52        = ( ~ ? [X: set_nat] :
% 0.29/0.52                ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1840_find__None__iff,axiom,
% 0.29/0.52      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 0.29/0.52        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 0.29/0.52          = none_VEBT_VEBT )
% 0.29/0.52        = ( ~ ? [X: vEBT_VEBT] :
% 0.29/0.52                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1841_find__None__iff,axiom,
% 0.29/0.52      ! [P: int > $o,Xs: list_int] :
% 0.29/0.52        ( ( ( find_int @ P @ Xs )
% 0.29/0.52          = none_int )
% 0.29/0.52        = ( ~ ? [X: int] :
% 0.29/0.52                ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1842_find__None__iff,axiom,
% 0.29/0.52      ! [P: nat > $o,Xs: list_nat] :
% 0.29/0.52        ( ( ( find_nat @ P @ Xs )
% 0.29/0.52          = none_nat )
% 0.29/0.52        = ( ~ ? [X: nat] :
% 0.29/0.52                ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1843_find__None__iff,axiom,
% 0.29/0.52      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.52        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 0.29/0.52          = none_P5556105721700978146at_nat )
% 0.29/0.52        = ( ~ ? [X: product_prod_nat_nat] :
% 0.29/0.52                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1844_find__None__iff,axiom,
% 0.29/0.52      ! [P: num > $o,Xs: list_num] :
% 0.29/0.52        ( ( ( find_num @ P @ Xs )
% 0.29/0.52          = none_num )
% 0.29/0.52        = ( ~ ? [X: num] :
% 0.29/0.52                ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 0.29/0.52                & ( P @ X ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % find_None_iff
% 0.29/0.52  thf(fact_1845_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1846_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1847_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1848_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1849_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1850_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_o,Ys: list_o] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.29/0.52         => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( product_Pair_o_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1851_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_o,Ys: list_nat] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( product_Pair_o_nat @ ( nth_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1852_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_o,Ys: list_int] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1853_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1854_nth__zip,axiom,
% 0.29/0.52      ! [I: nat,Xs: list_nat,Ys: list_o] :
% 0.29/0.52        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 0.29/0.52       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 0.29/0.52         => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys ) @ I )
% 0.29/0.52            = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % nth_zip
% 0.29/0.52  thf(fact_1855__C5_C_I10_J,axiom,
% 0.29/0.52      ( ( mi != ma )
% 0.29/0.52     => ! [I4: nat] :
% 0.29/0.52          ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.29/0.52         => ( ( ( ( vEBT_VEBT_high @ ma @ n )
% 0.29/0.52                = I4 )
% 0.29/0.52             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ n ) ) )
% 0.29/0.52            & ! [X2: nat] :
% 0.29/0.52                ( ( ( ( vEBT_VEBT_high @ X2 @ n )
% 0.29/0.52                    = I4 )
% 0.29/0.52                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) )
% 0.29/0.52               => ( ( ord_less_nat @ mi @ X2 )
% 0.29/0.52                  & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % "5"(10)
% 0.29/0.52  thf(fact_1856_vebt__member_Ocases,axiom,
% 0.29/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.29/0.52        ( ! [A4: $o,B4: $o,X4: nat] :
% 0.29/0.52            ( X3
% 0.29/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 0.29/0.52       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 0.29/0.52         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 0.29/0.52           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                  ( X3
% 0.29/0.52                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 0.29/0.52             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                    ( X3
% 0.29/0.52                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % vebt_member.cases
% 0.29/0.52  thf(fact_1857_VEBT__internal_Omembermima_Ocases,axiom,
% 0.29/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.29/0.52        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 0.29/0.52            ( X3
% 0.29/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 0.29/0.52       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 0.29/0.52         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 0.29/0.52           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                  ( X3
% 0.29/0.52                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 0.29/0.52             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 0.29/0.52                    ( X3
% 0.29/0.52                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X4 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.membermima.cases
% 0.29/0.52  thf(fact_1858_count__notin,axiom,
% 0.29/0.52      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.52        ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.52       => ( ( count_4203492906077236349at_nat @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1859_count__notin,axiom,
% 0.29/0.52      ! [X3: real,Xs: list_real] :
% 0.29/0.52        ( ~ ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52       => ( ( count_list_real @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1860_count__notin,axiom,
% 0.29/0.52      ! [X3: set_nat,Xs: list_set_nat] :
% 0.29/0.52        ( ~ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 0.29/0.52       => ( ( count_list_set_nat @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1861_count__notin,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.29/0.52        ( ~ ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52       => ( ( count_list_VEBT_VEBT @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1862_count__notin,axiom,
% 0.29/0.52      ! [X3: int,Xs: list_int] :
% 0.29/0.52        ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.52       => ( ( count_list_int @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1863_count__notin,axiom,
% 0.29/0.52      ! [X3: nat,Xs: list_nat] :
% 0.29/0.52        ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 0.29/0.52       => ( ( count_list_nat @ Xs @ X3 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_notin
% 0.29/0.52  thf(fact_1864_listrel1__iff__update,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 0.29/0.52        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 0.29/0.52        = ( ? [Y: vEBT_VEBT,N: nat] :
% 0.29/0.52              ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ Y ) @ R2 )
% 0.29/0.52              & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 0.29/0.52              & ( Ys
% 0.29/0.52                = ( list_u1324408373059187874T_VEBT @ Xs @ N @ Y ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_iff_update
% 0.29/0.52  thf(fact_1865_listrel1__iff__update,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 0.29/0.52        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 0.29/0.52        = ( ? [Y: $o,N: nat] :
% 0.29/0.52              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N ) @ Y ) @ R2 )
% 0.29/0.52              & ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 0.29/0.52              & ( Ys
% 0.29/0.52                = ( list_update_o @ Xs @ N @ Y ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_iff_update
% 0.29/0.52  thf(fact_1866_listrel1__iff__update,axiom,
% 0.29/0.52      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.29/0.52        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 0.29/0.52        = ( ? [Y: nat,N: nat] :
% 0.29/0.52              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ Y ) @ R2 )
% 0.29/0.52              & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 0.29/0.52              & ( Ys
% 0.29/0.52                = ( list_update_nat @ Xs @ N @ Y ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_iff_update
% 0.29/0.52  thf(fact_1867_listrel1__iff__update,axiom,
% 0.29/0.52      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 0.29/0.52        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 0.29/0.52        = ( ? [Y: int,N: nat] :
% 0.29/0.52              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ Y ) @ R2 )
% 0.29/0.52              & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 0.29/0.52              & ( Ys
% 0.29/0.52                = ( list_update_int @ Xs @ N @ Y ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_iff_update
% 0.29/0.52  thf(fact_1868_pair__lessI2,axiom,
% 0.29/0.52      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.29/0.52        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.52       => ( ( ord_less_nat @ S @ T )
% 0.29/0.52         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pair_lessI2
% 0.29/0.52  thf(fact_1869_pair__less__iff1,axiom,
% 0.29/0.52      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.52        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ X3 @ Z2 ) ) @ fun_pair_less )
% 0.29/0.52        = ( ord_less_nat @ Y3 @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pair_less_iff1
% 0.29/0.52  thf(fact_1870_gen__length__def,axiom,
% 0.29/0.52      ( gen_length_VEBT_VEBT
% 0.29/0.52      = ( ^ [N: nat,Xs2: list_VEBT_VEBT] : ( plus_plus_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % gen_length_def
% 0.29/0.52  thf(fact_1871_gen__length__def,axiom,
% 0.29/0.52      ( gen_length_o
% 0.29/0.52      = ( ^ [N: nat,Xs2: list_o] : ( plus_plus_nat @ N @ ( size_size_list_o @ Xs2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % gen_length_def
% 0.29/0.52  thf(fact_1872_gen__length__def,axiom,
% 0.29/0.52      ( gen_length_nat
% 0.29/0.52      = ( ^ [N: nat,Xs2: list_nat] : ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % gen_length_def
% 0.29/0.52  thf(fact_1873_gen__length__def,axiom,
% 0.29/0.52      ( gen_length_int
% 0.29/0.52      = ( ^ [N: nat,Xs2: list_int] : ( plus_plus_nat @ N @ ( size_size_list_int @ Xs2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % gen_length_def
% 0.29/0.52  thf(fact_1874_set__encode__empty,axiom,
% 0.29/0.52      ( ( nat_set_encode @ bot_bot_set_nat )
% 0.29/0.52      = zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % set_encode_empty
% 0.29/0.52  thf(fact_1875_Leaf__0__not,axiom,
% 0.29/0.52      ! [A: $o,B: $o] :
% 0.29/0.52        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % Leaf_0_not
% 0.29/0.52  thf(fact_1876_power__shift,axiom,
% 0.29/0.52      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.52        ( ( ( power_power_nat @ X3 @ Y3 )
% 0.29/0.52          = Z2 )
% 0.29/0.52        = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.29/0.52          = ( some_nat @ Z2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_shift
% 0.29/0.52  thf(fact_1877__C5_C_I3_J,axiom,
% 0.29/0.52      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 0.29/0.52      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 0.29/0.52  
% 0.29/0.52  % "5"(3)
% 0.29/0.52  thf(fact_1878__C5_C_I9_J,axiom,
% 0.29/0.52      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 0.29/0.52  
% 0.29/0.52  % "5"(9)
% 0.29/0.52  thf(fact_1879_VEBT_Oinject_I2_J,axiom,
% 0.29/0.52      ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 0.29/0.52        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 0.29/0.52          = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 0.29/0.52        = ( ( X21 = Y21 )
% 0.29/0.52          & ( X222 = Y22 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT.inject(2)
% 0.29/0.52  thf(fact_1880_local_Opower__def,axiom,
% 0.29/0.52      ( vEBT_VEBT_power
% 0.29/0.52      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % local.power_def
% 0.29/0.52  thf(fact_1881_high__bound__aux,axiom,
% 0.29/0.52      ! [Ma: nat,N2: nat,M: nat] :
% 0.29/0.52        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.29/0.52       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % high_bound_aux
% 0.29/0.52  thf(fact_1882_member__bound,axiom,
% 0.29/0.52      ! [Tree: vEBT_VEBT,X3: nat,N2: nat] :
% 0.29/0.52        ( ( vEBT_vebt_member @ Tree @ X3 )
% 0.29/0.52       => ( ( vEBT_invar_vebt @ Tree @ N2 )
% 0.29/0.52         => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % member_bound
% 0.29/0.52  thf(fact_1883__C5_C_I6_J,axiom,
% 0.29/0.52      ! [I4: nat] :
% 0.29/0.52        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 0.29/0.52       => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X5 ) )
% 0.29/0.52          = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % "5"(6)
% 0.29/0.52  thf(fact_1884_numeral__le__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.52        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_le_iff
% 0.29/0.52  thf(fact_1885_numeral__le__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.29/0.52        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_le_iff
% 0.29/0.52  thf(fact_1886_numeral__le__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.29/0.52        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_le_iff
% 0.29/0.52  thf(fact_1887_numeral__le__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.52        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_le_iff
% 0.29/0.52  thf(fact_1888_numeral__le__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.29/0.52        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_le_iff
% 0.29/0.52  thf(fact_1889_numeral__less__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.52        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_less_iff
% 0.29/0.52  thf(fact_1890_numeral__less__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.29/0.52        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_less_iff
% 0.29/0.52  thf(fact_1891_numeral__less__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.29/0.52        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_less_iff
% 0.29/0.52  thf(fact_1892_numeral__less__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.52        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_less_iff
% 0.29/0.52  thf(fact_1893_numeral__less__iff,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.29/0.52        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_less_iff
% 0.29/0.52  thf(fact_1894_numeral__plus__numeral,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.52        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_plus_numeral
% 0.29/0.52  thf(fact_1895_numeral__plus__numeral,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 0.29/0.52        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_plus_numeral
% 0.29/0.52  thf(fact_1896_numeral__plus__numeral,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 0.29/0.52        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_plus_numeral
% 0.29/0.52  thf(fact_1897_numeral__plus__numeral,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.52        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_plus_numeral
% 0.29/0.52  thf(fact_1898_numeral__plus__numeral,axiom,
% 0.29/0.52      ! [M: num,N2: num] :
% 0.29/0.52        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.29/0.52        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_plus_numeral
% 0.29/0.52  thf(fact_1899_add__numeral__left,axiom,
% 0.29/0.52      ! [V: num,W2: num,Z2: extended_enat] :
% 0.29/0.52        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
% 0.29/0.52        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_numeral_left
% 0.29/0.52  thf(fact_1900_add__numeral__left,axiom,
% 0.29/0.52      ! [V: num,W2: num,Z2: real] :
% 0.29/0.52        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
% 0.29/0.52        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_numeral_left
% 0.29/0.52  thf(fact_1901_add__numeral__left,axiom,
% 0.29/0.52      ! [V: num,W2: num,Z2: rat] :
% 0.29/0.52        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
% 0.29/0.52        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_numeral_left
% 0.29/0.52  thf(fact_1902_add__numeral__left,axiom,
% 0.29/0.52      ! [V: num,W2: num,Z2: nat] :
% 0.29/0.52        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
% 0.29/0.52        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_numeral_left
% 0.29/0.52  thf(fact_1903_add__numeral__left,axiom,
% 0.29/0.52      ! [V: num,W2: num,Z2: int] :
% 0.29/0.52        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
% 0.29/0.52        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_numeral_left
% 0.29/0.52  thf(fact_1904_mi__ma__2__deg,axiom,
% 0.29/0.52      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 0.29/0.52        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.29/0.52       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.29/0.52          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % mi_ma_2_deg
% 0.29/0.52  thf(fact_1905_Suc__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( suc @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.52        = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % Suc_numeral
% 0.29/0.52  thf(fact_1906_add__2__eq__Suc_H,axiom,
% 0.29/0.52      ! [N2: nat] :
% 0.29/0.52        ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52        = ( suc @ ( suc @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_2_eq_Suc'
% 0.29/0.52  thf(fact_1907_add__2__eq__Suc,axiom,
% 0.29/0.52      ! [N2: nat] :
% 0.29/0.52        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.29/0.52        = ( suc @ ( suc @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % add_2_eq_Suc
% 0.29/0.52  thf(fact_1908_numeral__Bit0,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
% 0.29/0.52        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_Bit0
% 0.29/0.52  thf(fact_1909_numeral__Bit0,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 0.29/0.52        = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_Bit0
% 0.29/0.52  thf(fact_1910_numeral__Bit0,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 0.29/0.52        = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_Bit0
% 0.29/0.52  thf(fact_1911_numeral__Bit0,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.29/0.52        = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_Bit0
% 0.29/0.52  thf(fact_1912_numeral__Bit0,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 0.29/0.52        = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_Bit0
% 0.29/0.52  thf(fact_1913_numeral__2__eq__2,axiom,
% 0.29/0.52      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.29/0.52      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_2_eq_2
% 0.29/0.52  thf(fact_1914_less__2__cases,axiom,
% 0.29/0.52      ! [N2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52       => ( ( N2 = zero_zero_nat )
% 0.29/0.52          | ( N2
% 0.29/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % less_2_cases
% 0.29/0.52  thf(fact_1915_less__2__cases__iff,axiom,
% 0.29/0.52      ! [N2: nat] :
% 0.29/0.52        ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52        = ( ( N2 = zero_zero_nat )
% 0.29/0.52          | ( N2
% 0.29/0.52            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % less_2_cases_iff
% 0.29/0.52  thf(fact_1916_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 0.29/0.52      ! [X3: nat,N2: nat,M: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.29/0.52       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.52           => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.exp_split_high_low(1)
% 0.29/0.52  thf(fact_1917_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 0.29/0.52      ! [X3: nat,N2: nat,M: nat] :
% 0.29/0.52        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 0.29/0.52       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.52           => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.exp_split_high_low(2)
% 0.29/0.52  thf(fact_1918_numeral__1__eq__Suc__0,axiom,
% 0.29/0.52      ( ( numeral_numeral_nat @ one )
% 0.29/0.52      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % numeral_1_eq_Suc_0
% 0.29/0.52  thf(fact_1919_Suc__nat__number__of__add,axiom,
% 0.29/0.52      ! [V: num,N2: nat] :
% 0.29/0.52        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
% 0.29/0.52        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % Suc_nat_number_of_add
% 0.29/0.52  thf(fact_1920_VEBT_Osize_I4_J,axiom,
% 0.29/0.52      ! [X21: $o,X222: $o] :
% 0.29/0.52        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.29/0.52        = zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT.size(4)
% 0.29/0.52  thf(fact_1921_VEBT__internal_Ovalid_H_Ocases,axiom,
% 0.29/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.29/0.52        ( ! [Uu2: $o,Uv2: $o,D4: nat] :
% 0.29/0.52            ( X3
% 0.29/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
% 0.29/0.52       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.valid'.cases
% 0.29/0.52  thf(fact_1922_VEBT_Odistinct_I1_J,axiom,
% 0.29/0.52      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 0.29/0.52        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 0.29/0.52       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT.distinct(1)
% 0.29/0.52  thf(fact_1923_VEBT_Oexhaust,axiom,
% 0.29/0.52      ! [Y3: vEBT_VEBT] :
% 0.29/0.52        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 0.29/0.52            ( Y3
% 0.29/0.52           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 0.29/0.52       => ~ ! [X212: $o,X223: $o] :
% 0.29/0.52              ( Y3
% 0.29/0.52             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT.exhaust
% 0.29/0.52  thf(fact_1924_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_zero_complex
% 0.29/0.52       != ( numera6690914467698888265omplex @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1925_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_z5237406670263579293d_enat
% 0.29/0.52       != ( numera1916890842035813515d_enat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1926_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_zero_real
% 0.29/0.52       != ( numeral_numeral_real @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1927_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_zero_rat
% 0.29/0.52       != ( numeral_numeral_rat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1928_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_zero_nat
% 0.29/0.52       != ( numeral_numeral_nat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1929_zero__neq__numeral,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ( zero_zero_int
% 0.29/0.52       != ( numeral_numeral_int @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_neq_numeral
% 0.29/0.52  thf(fact_1930_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 0.29/0.52      ! [Uu: $o,Uv: $o,Uw: nat] :
% 0.29/0.52        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.membermima.simps(1)
% 0.29/0.52  thf(fact_1931_num_Osize_I4_J,axiom,
% 0.29/0.52      ( ( size_size_num @ one )
% 0.29/0.52      = zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % num.size(4)
% 0.29/0.52  thf(fact_1932_listrel1__eq__len,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 0.29/0.52        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 0.29/0.52       => ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_eq_len
% 0.29/0.52  thf(fact_1933_listrel1__eq__len,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 0.29/0.52        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 0.29/0.52       => ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_eq_len
% 0.29/0.52  thf(fact_1934_listrel1__eq__len,axiom,
% 0.29/0.52      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 0.29/0.52        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 0.29/0.52       => ( ( size_size_list_nat @ Xs )
% 0.29/0.52          = ( size_size_list_nat @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_eq_len
% 0.29/0.52  thf(fact_1935_listrel1__eq__len,axiom,
% 0.29/0.52      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 0.29/0.52        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 0.29/0.52       => ( ( size_size_list_int @ Xs )
% 0.29/0.52          = ( size_size_list_int @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % listrel1_eq_len
% 0.29/0.52  thf(fact_1936_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 0.29/0.52      ! [Uu: $o] :
% 0.29/0.52        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.simps(3)
% 0.29/0.52  thf(fact_1937_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 0.29/0.52      ! [Uv: $o] :
% 0.29/0.52        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.simps(2)
% 0.29/0.52  thf(fact_1938_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 0.29/0.52      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.simps(1)
% 0.29/0.52  thf(fact_1939_invar__vebt_Ointros_I2_J,axiom,
% 0.29/0.52      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.29/0.52        ( ! [X4: vEBT_VEBT] :
% 0.29/0.52            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.29/0.52       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.52         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.52              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52           => ( ( M = N2 )
% 0.29/0.52             => ( ( Deg
% 0.29/0.52                  = ( plus_plus_nat @ N2 @ M ) )
% 0.29/0.52               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.29/0.52                 => ( ! [X4: vEBT_VEBT] :
% 0.29/0.52                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.intros(2)
% 0.29/0.52  thf(fact_1940_invar__vebt_Ointros_I3_J,axiom,
% 0.29/0.52      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 0.29/0.52        ( ! [X4: vEBT_VEBT] :
% 0.29/0.52            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.29/0.52       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.52         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.52              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52           => ( ( M
% 0.29/0.52                = ( suc @ N2 ) )
% 0.29/0.52             => ( ( Deg
% 0.29/0.52                  = ( plus_plus_nat @ N2 @ M ) )
% 0.29/0.52               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 0.29/0.52                 => ( ! [X4: vEBT_VEBT] :
% 0.29/0.52                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.intros(3)
% 0.29/0.52  thf(fact_1941_zip__same,axiom,
% 0.29/0.52      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 0.29/0.52        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1942_zip__same,axiom,
% 0.29/0.52      ! [A: real,B: real,Xs: list_real] :
% 0.29/0.52        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member_real @ A @ ( set_real2 @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1943_zip__same,axiom,
% 0.29/0.52      ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
% 0.29/0.52        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1944_zip__same,axiom,
% 0.29/0.52      ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 0.29/0.52        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1945_zip__same,axiom,
% 0.29/0.52      ! [A: nat,B: nat,Xs: list_nat] :
% 0.29/0.52        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1946_zip__same,axiom,
% 0.29/0.52      ! [A: int,B: int,Xs: list_int] :
% 0.29/0.52        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
% 0.29/0.52        = ( ( member_int @ A @ ( set_int2 @ Xs ) )
% 0.29/0.52          & ( A = B ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_same
% 0.29/0.52  thf(fact_1947_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: real,Y3: real,Xs: list_real,Ys: list_real] :
% 0.29/0.52        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y3 ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1948_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: real,Y3: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1949_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: real,Y3: int,Xs: list_real,Ys: list_int] :
% 0.29/0.52        ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52           => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1950_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: real,Y3: nat,Xs: list_real,Ys: list_nat] :
% 0.29/0.52        ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52           => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1951_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Y3: real,Xs: list_VEBT_VEBT,Ys: list_real] :
% 0.29/0.52        ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X3 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1952_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1953_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Y3: int,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.29/0.52        ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52           => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1954_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Y3: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.29/0.52        ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52           => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1955_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: int,Y3: real,Xs: list_int,Ys: list_real] :
% 0.29/0.52        ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X3 @ Y3 ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.52           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1956_in__set__zipE,axiom,
% 0.29/0.52      ! [X3: int,Y3: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
% 0.29/0.52        ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X3 @ Y3 ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
% 0.29/0.52       => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 0.29/0.52           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_zipE
% 0.29/0.52  thf(fact_1957_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
% 0.29/0.52        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1958_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
% 0.29/0.52        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member_int @ X3 @ ( set_int2 @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1959_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 0.29/0.52        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member1535805642427569193e_term @ X3 @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1960_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 0.29/0.52        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member4242434998011752849e_term @ X3 @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1961_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 0.29/0.52        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member3222579708246209666e_term @ X3 @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1962_set__zip__leftD,axiom,
% 0.29/0.52      ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 0.29/0.52        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member8845023287901829240e_term @ X3 @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_leftD
% 0.29/0.52  thf(fact_1963_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
% 0.29/0.52        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1964_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
% 0.29/0.52        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1965_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 0.29/0.52        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1966_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 0.29/0.52        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1967_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 0.29/0.52        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1968_set__zip__rightD,axiom,
% 0.29/0.52      ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 0.29/0.52        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 0.29/0.52       => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_zip_rightD
% 0.29/0.52  thf(fact_1969_zip__update,axiom,
% 0.29/0.52      ! [Xs: list_int,I: nat,X3: int,Ys: list_int,Y3: int] :
% 0.29/0.52        ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X3 ) @ ( list_update_int @ Ys @ I @ Y3 ) )
% 0.29/0.52        = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X3 @ Y3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_update
% 0.29/0.52  thf(fact_1970_zip__update,axiom,
% 0.29/0.52      ! [Xs: list_C878401137130745250e_term,I: nat,X3: code_integer > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
% 0.29/0.52        ( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
% 0.29/0.52        = ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) @ I @ ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_update
% 0.29/0.52  thf(fact_1971_zip__update,axiom,
% 0.29/0.52      ! [Xs: list_P1316552470764441098e_term,I: nat,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
% 0.29/0.52        ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
% 0.29/0.52        = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_update
% 0.29/0.52  thf(fact_1972_zip__update,axiom,
% 0.29/0.52      ! [Xs: list_P1743416141875011707e_term,I: nat,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
% 0.29/0.52        ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
% 0.29/0.52        = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_update
% 0.29/0.52  thf(fact_1973_zip__update,axiom,
% 0.29/0.52      ! [Xs: list_i8448526496819171953e_term,I: nat,X3: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
% 0.29/0.52        ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
% 0.29/0.52        = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zip_update
% 0.29/0.52  thf(fact_1974_zero__le__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_le_numeral
% 0.29/0.52  thf(fact_1975_zero__le__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_le_numeral
% 0.29/0.52  thf(fact_1976_zero__le__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_le_numeral
% 0.29/0.52  thf(fact_1977_zero__le__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_le_numeral
% 0.29/0.52  thf(fact_1978_zero__le__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_le_numeral
% 0.29/0.52  thf(fact_1979_not__numeral__le__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_le_zero
% 0.29/0.52  thf(fact_1980_not__numeral__le__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_le_zero
% 0.29/0.52  thf(fact_1981_not__numeral__le__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_le_zero
% 0.29/0.52  thf(fact_1982_not__numeral__le__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_le_zero
% 0.29/0.52  thf(fact_1983_not__numeral__le__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_le_zero
% 0.29/0.52  thf(fact_1984_not__numeral__less__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_less_zero
% 0.29/0.52  thf(fact_1985_not__numeral__less__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_less_zero
% 0.29/0.52  thf(fact_1986_not__numeral__less__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_less_zero
% 0.29/0.52  thf(fact_1987_not__numeral__less__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_less_zero
% 0.29/0.52  thf(fact_1988_not__numeral__less__zero,axiom,
% 0.29/0.52      ! [N2: num] :
% 0.29/0.52        ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 0.29/0.52  
% 0.29/0.52  % not_numeral_less_zero
% 0.29/0.52  thf(fact_1989_zero__less__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_numeral
% 0.29/0.52  thf(fact_1990_zero__less__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_numeral
% 0.29/0.52  thf(fact_1991_zero__less__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_numeral
% 0.29/0.52  thf(fact_1992_zero__less__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_numeral
% 0.29/0.52  thf(fact_1993_zero__less__numeral,axiom,
% 0.29/0.52      ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_numeral
% 0.29/0.52  thf(fact_1994_set__encode__eq,axiom,
% 0.29/0.52      ! [A2: set_nat,B2: set_nat] :
% 0.29/0.52        ( ( finite_finite_nat @ A2 )
% 0.29/0.52       => ( ( finite_finite_nat @ B2 )
% 0.29/0.52         => ( ( ( nat_set_encode @ A2 )
% 0.29/0.52              = ( nat_set_encode @ B2 ) )
% 0.29/0.52            = ( A2 = B2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_encode_eq
% 0.29/0.52  thf(fact_1995_vebt__buildup_Osimps_I1_J,axiom,
% 0.29/0.52      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 0.29/0.52      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.29/0.52  
% 0.29/0.52  % vebt_buildup.simps(1)
% 0.29/0.52  thf(fact_1996_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_real,Y3: real] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_real @ Ys ) )
% 0.29/0.52       => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: vEBT_VEBT] :
% 0.29/0.52                ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_1997_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: vEBT_VEBT] :
% 0.29/0.52                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_1998_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_o,Y3: $o] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) )
% 0.29/0.52       => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: vEBT_VEBT] :
% 0.29/0.52                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_1999_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y3: nat] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_nat @ Ys ) )
% 0.29/0.52       => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: vEBT_VEBT] :
% 0.29/0.52                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2000_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_int,Y3: int] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_int @ Ys ) )
% 0.29/0.52       => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: vEBT_VEBT] :
% 0.29/0.52                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2001_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_real,Y3: real] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_real @ Ys ) )
% 0.29/0.52       => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: $o] :
% 0.29/0.52                ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y3 ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2002_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: $o] :
% 0.29/0.52                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X4 @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2003_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_o,Y3: $o] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) )
% 0.29/0.52       => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: $o] :
% 0.29/0.52                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2004_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_nat,Y3: nat] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_nat @ Ys ) )
% 0.29/0.52       => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: $o] :
% 0.29/0.52                ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X4 @ Y3 ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2005_in__set__impl__in__set__zip2,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_int,Y3: int] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_int @ Ys ) )
% 0.29/0.52       => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
% 0.29/0.52         => ~ ! [X4: $o] :
% 0.29/0.52                ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X4 @ Y3 ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip2
% 0.29/0.52  thf(fact_2006_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_real,Ys: list_VEBT_VEBT,X3: real] :
% 0.29/0.52        ( ( ( size_size_list_real @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: vEBT_VEBT] :
% 0.29/0.52                ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y5 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2007_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_real,Ys: list_o,X3: real] :
% 0.29/0.52        ( ( ( size_size_list_real @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) )
% 0.29/0.52       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: $o] :
% 0.29/0.52                ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X3 @ Y5 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2008_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_real,Ys: list_nat,X3: real] :
% 0.29/0.52        ( ( ( size_size_list_real @ Xs )
% 0.29/0.52          = ( size_size_list_nat @ Ys ) )
% 0.29/0.52       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: nat] :
% 0.29/0.52                ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y5 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2009_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_real,Ys: list_int,X3: real] :
% 0.29/0.52        ( ( ( size_size_list_real @ Xs )
% 0.29/0.52          = ( size_size_list_int @ Ys ) )
% 0.29/0.52       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: int] :
% 0.29/0.52                ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y5 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2010_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: vEBT_VEBT] :
% 0.29/0.52                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y5 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2011_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_o,X3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: $o] :
% 0.29/0.52                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X3 @ Y5 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2012_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_nat,X3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_nat @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: nat] :
% 0.29/0.52                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y5 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2013_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,Ys: list_int,X3: vEBT_VEBT] :
% 0.29/0.52        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 0.29/0.52          = ( size_size_list_int @ Ys ) )
% 0.29/0.52       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: int] :
% 0.29/0.52                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y5 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2014_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_VEBT_VEBT,X3: $o] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 0.29/0.52       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: vEBT_VEBT] :
% 0.29/0.52                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X3 @ Y5 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2015_in__set__impl__in__set__zip1,axiom,
% 0.29/0.52      ! [Xs: list_o,Ys: list_o,X3: $o] :
% 0.29/0.52        ( ( ( size_size_list_o @ Xs )
% 0.29/0.52          = ( size_size_list_o @ Ys ) )
% 0.29/0.52       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 0.29/0.52         => ~ ! [Y5: $o] :
% 0.29/0.52                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y5 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % in_set_impl_in_set_zip1
% 0.29/0.52  thf(fact_2016_VEBT__internal_Onaive__member_Ocases,axiom,
% 0.29/0.52      ! [X3: produc9072475918466114483BT_nat] :
% 0.29/0.52        ( ! [A4: $o,B4: $o,X4: nat] :
% 0.29/0.52            ( X3
% 0.29/0.52           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 0.29/0.52       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 0.29/0.52         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X4 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.naive_member.cases
% 0.29/0.52  thf(fact_2017_invar__vebt_Ointros_I1_J,axiom,
% 0.29/0.52      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.intros(1)
% 0.29/0.52  thf(fact_2018_vebt__buildup_Osimps_I2_J,axiom,
% 0.29/0.52      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 0.29/0.52      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 0.29/0.52  
% 0.29/0.52  % vebt_buildup.simps(2)
% 0.29/0.52  thf(fact_2019_invar__vebt_Osimps,axiom,
% 0.29/0.52      ( vEBT_invar_vebt
% 0.29/0.52      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 0.29/0.52            ( ( ? [A3: $o,B3: $o] :
% 0.29/0.52                  ( A1
% 0.29/0.52                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 0.29/0.52              & ( A22
% 0.29/0.52                = ( suc @ zero_zero_nat ) ) )
% 0.29/0.52            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 0.29/0.52                ( ( A1
% 0.29/0.52                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ X @ N ) )
% 0.29/0.52                & ( vEBT_invar_vebt @ Summary3 @ N )
% 0.29/0.52                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.52                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.52                & ( A22
% 0.29/0.52                  = ( plus_plus_nat @ N @ N ) )
% 0.29/0.52                & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.29/0.52            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 0.29/0.52                ( ( A1
% 0.29/0.52                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ X @ N ) )
% 0.29/0.52                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 0.29/0.52                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.52                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.29/0.52                & ( A22
% 0.29/0.52                  = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 0.29/0.52                & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.29/0.52            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.29/0.52                ( ( A1
% 0.29/0.52                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ X @ N ) )
% 0.29/0.52                & ( vEBT_invar_vebt @ Summary3 @ N )
% 0.29/0.52                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.52                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.52                & ( A22
% 0.29/0.52                  = ( plus_plus_nat @ N @ N ) )
% 0.29/0.52                & ! [I2: nat] :
% 0.29/0.52                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.52                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
% 0.29/0.52                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.29/0.52                & ( ( Mi3 = Ma3 )
% 0.29/0.52                 => ! [X: vEBT_VEBT] :
% 0.29/0.52                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.29/0.52                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.52                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.29/0.52                & ( ( Mi3 != Ma3 )
% 0.29/0.52                 => ! [I2: nat] :
% 0.29/0.52                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 0.29/0.52                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 0.29/0.52                            = I2 )
% 0.29/0.52                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 0.29/0.52                        & ! [X: nat] :
% 0.29/0.52                            ( ( ( ( vEBT_VEBT_high @ X @ N )
% 0.29/0.52                                = I2 )
% 0.29/0.52                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
% 0.29/0.52                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.29/0.52                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 0.29/0.52            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 0.29/0.52                ( ( A1
% 0.29/0.52                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 0.29/0.52                & ! [X: vEBT_VEBT] :
% 0.29/0.52                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                   => ( vEBT_invar_vebt @ X @ N ) )
% 0.29/0.52                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 0.29/0.52                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 0.29/0.52                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.29/0.52                & ( A22
% 0.29/0.52                  = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 0.29/0.52                & ! [I2: nat] :
% 0.29/0.52                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.29/0.52                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
% 0.29/0.52                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 0.29/0.52                & ( ( Mi3 = Ma3 )
% 0.29/0.52                 => ! [X: vEBT_VEBT] :
% 0.29/0.52                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 0.29/0.52                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.29/0.52                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.29/0.52                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 0.29/0.52                & ( ( Mi3 != Ma3 )
% 0.29/0.52                 => ! [I2: nat] :
% 0.29/0.52                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 0.29/0.52                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 0.29/0.52                            = I2 )
% 0.29/0.52                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 0.29/0.52                        & ! [X: nat] :
% 0.29/0.52                            ( ( ( ( vEBT_VEBT_high @ X @ N )
% 0.29/0.52                                = I2 )
% 0.29/0.52                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
% 0.29/0.52                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.29/0.52                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.simps
% 0.29/0.52  thf(fact_2020_invar__vebt_Ocases,axiom,
% 0.29/0.52      ! [A12: vEBT_VEBT,A23: nat] :
% 0.29/0.52        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 0.29/0.52       => ( ( ? [A4: $o,B4: $o] :
% 0.29/0.52                ( A12
% 0.29/0.52                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.29/0.52           => ( A23
% 0.29/0.52             != ( suc @ zero_zero_nat ) ) )
% 0.29/0.52         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
% 0.29/0.52                ( ( A12
% 0.29/0.52                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.52               => ( ( A23 = Deg2 )
% 0.29/0.52                 => ( ! [X2: vEBT_VEBT] :
% 0.29/0.52                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.29/0.52                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 0.29/0.52                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.52                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                       => ( ( M3 = N3 )
% 0.29/0.52                         => ( ( Deg2
% 0.29/0.52                              = ( plus_plus_nat @ N3 @ M3 ) )
% 0.29/0.52                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.29/0.52                             => ~ ! [X2: vEBT_VEBT] :
% 0.29/0.52                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.29/0.52           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
% 0.29/0.52                  ( ( A12
% 0.29/0.52                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.52                 => ( ( A23 = Deg2 )
% 0.29/0.52                   => ( ! [X2: vEBT_VEBT] :
% 0.29/0.52                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.29/0.52                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 0.29/0.52                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.52                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                         => ( ( M3
% 0.29/0.52                              = ( suc @ N3 ) )
% 0.29/0.52                           => ( ( Deg2
% 0.29/0.52                                = ( plus_plus_nat @ N3 @ M3 ) )
% 0.29/0.52                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 0.29/0.52                               => ~ ! [X2: vEBT_VEBT] :
% 0.29/0.52                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 0.29/0.52             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.29/0.52                    ( ( A12
% 0.29/0.52                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.52                   => ( ( A23 = Deg2 )
% 0.29/0.52                     => ( ! [X2: vEBT_VEBT] :
% 0.29/0.52                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.29/0.52                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 0.29/0.52                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.52                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                           => ( ( M3 = N3 )
% 0.29/0.52                             => ( ( Deg2
% 0.29/0.52                                  = ( plus_plus_nat @ N3 @ M3 ) )
% 0.29/0.52                               => ( ! [I4: nat] :
% 0.29/0.52                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
% 0.29/0.52                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.29/0.52                                 => ( ( ( Mi2 = Ma2 )
% 0.29/0.52                                     => ! [X2: vEBT_VEBT] :
% 0.29/0.52                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 0.29/0.52                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.29/0.52                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.52                                       => ~ ( ( Mi2 != Ma2 )
% 0.29/0.52                                           => ! [I4: nat] :
% 0.29/0.52                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.29/0.52                                                      = I4 )
% 0.29/0.52                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.29/0.52                                                  & ! [X2: nat] :
% 0.29/0.52                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 0.29/0.52                                                          = I4 )
% 0.29/0.52                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 0.29/0.52                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
% 0.29/0.52                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 0.29/0.52               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 0.29/0.52                      ( ( A12
% 0.29/0.52                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.29/0.52                     => ( ( A23 = Deg2 )
% 0.29/0.52                       => ( ! [X2: vEBT_VEBT] :
% 0.29/0.52                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                             => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 0.29/0.52                         => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 0.29/0.52                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.29/0.52                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                             => ( ( M3
% 0.29/0.52                                  = ( suc @ N3 ) )
% 0.29/0.52                               => ( ( Deg2
% 0.29/0.52                                    = ( plus_plus_nat @ N3 @ M3 ) )
% 0.29/0.52                                 => ( ! [I4: nat] :
% 0.29/0.52                                        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                                       => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
% 0.29/0.52                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 0.29/0.52                                   => ( ( ( Mi2 = Ma2 )
% 0.29/0.52                                       => ! [X2: vEBT_VEBT] :
% 0.29/0.52                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.29/0.52                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 0.29/0.52                                     => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 0.29/0.52                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.29/0.52                                         => ~ ( ( Mi2 != Ma2 )
% 0.29/0.52                                             => ! [I4: nat] :
% 0.29/0.52                                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 0.29/0.52                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 0.29/0.52                                                        = I4 )
% 0.29/0.52                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 0.29/0.52                                                    & ! [X2: nat] :
% 0.29/0.52                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 0.29/0.52                                                            = I4 )
% 0.29/0.52                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 0.29/0.52                                                       => ( ( ord_less_nat @ Mi2 @ X2 )
% 0.29/0.52                                                          & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.cases
% 0.29/0.52  thf(fact_2021_VEBT__internal_OminNull_Ocases,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT] :
% 0.29/0.52        ( ( X3
% 0.29/0.52         != ( vEBT_Leaf @ $false @ $false ) )
% 0.29/0.52       => ( ! [Uv2: $o] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.29/0.52         => ( ! [Uu2: $o] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.29/0.52           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.52                  ( X3
% 0.29/0.52                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.29/0.52             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.29/0.52                    ( X3
% 0.29/0.52                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.cases
% 0.29/0.52  thf(fact_2022_num_Osize_I5_J,axiom,
% 0.29/0.52      ! [X22: num] :
% 0.29/0.52        ( ( size_size_num @ ( bit0 @ X22 ) )
% 0.29/0.52        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % num.size(5)
% 0.29/0.52  thf(fact_2023_invar__vebt_Ointros_I4_J,axiom,
% 0.29/0.52      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.29/0.52        ( ! [X4: vEBT_VEBT] :
% 0.29/0.52            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.29/0.52       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.52         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.52              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52           => ( ( M = N2 )
% 0.29/0.52             => ( ( Deg
% 0.29/0.52                  = ( plus_plus_nat @ N2 @ M ) )
% 0.29/0.52               => ( ! [I3: nat] :
% 0.29/0.52                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
% 0.29/0.52                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.29/0.52                 => ( ( ( Mi = Ma )
% 0.29/0.52                     => ! [X4: vEBT_VEBT] :
% 0.29/0.52                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.29/0.52                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.29/0.52                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.52                       => ( ( ( Mi != Ma )
% 0.29/0.52                           => ! [I3: nat] :
% 0.29/0.52                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 0.29/0.52                                      = I3 )
% 0.29/0.52                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 0.29/0.52                                  & ! [X4: nat] :
% 0.29/0.52                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 0.29/0.52                                          = I3 )
% 0.29/0.52                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 0.29/0.52                                     => ( ( ord_less_nat @ Mi @ X4 )
% 0.29/0.52                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.29/0.52                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.intros(4)
% 0.29/0.52  thf(fact_2024_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT] :
% 0.29/0.52        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 0.29/0.52       => ( ! [Uv2: $o] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.29/0.52         => ( ! [Uu2: $o] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.29/0.52           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.29/0.52                  ( X3
% 0.29/0.52                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.elims(3)
% 0.29/0.52  thf(fact_2025_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT] :
% 0.29/0.52        ( ( vEBT_VEBT_minNull @ X3 )
% 0.29/0.52       => ( ( X3
% 0.29/0.52           != ( vEBT_Leaf @ $false @ $false ) )
% 0.29/0.52         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.elims(2)
% 0.29/0.52  thf(fact_2026_invar__vebt_Ointros_I5_J,axiom,
% 0.29/0.52      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 0.29/0.52        ( ! [X4: vEBT_VEBT] :
% 0.29/0.52            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.29/0.52       => ( ( vEBT_invar_vebt @ Summary @ M )
% 0.29/0.52         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.52              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52           => ( ( M
% 0.29/0.52                = ( suc @ N2 ) )
% 0.29/0.52             => ( ( Deg
% 0.29/0.52                  = ( plus_plus_nat @ N2 @ M ) )
% 0.29/0.52               => ( ! [I3: nat] :
% 0.29/0.52                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
% 0.29/0.52                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 0.29/0.52                 => ( ( ( Mi = Ma )
% 0.29/0.52                     => ! [X4: vEBT_VEBT] :
% 0.29/0.52                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.52                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 0.29/0.52                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 0.29/0.52                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.52                       => ( ( ( Mi != Ma )
% 0.29/0.52                           => ! [I3: nat] :
% 0.29/0.52                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.52                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 0.29/0.52                                      = I3 )
% 0.29/0.52                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 0.29/0.52                                  & ! [X4: nat] :
% 0.29/0.52                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 0.29/0.52                                          = I3 )
% 0.29/0.52                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 0.29/0.52                                     => ( ( ord_less_nat @ Mi @ X4 )
% 0.29/0.52                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 0.29/0.52                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % invar_vebt.intros(5)
% 0.29/0.52  thf(fact_2027_set__encode__inf,axiom,
% 0.29/0.52      ! [A2: set_nat] :
% 0.29/0.52        ( ~ ( finite_finite_nat @ A2 )
% 0.29/0.52       => ( ( nat_set_encode @ A2 )
% 0.29/0.52          = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % set_encode_inf
% 0.29/0.52  thf(fact_2028_pair__lessI1,axiom,
% 0.29/0.52      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.29/0.52        ( ( ord_less_nat @ A @ B )
% 0.29/0.52       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 0.29/0.52  
% 0.29/0.52  % pair_lessI1
% 0.29/0.52  thf(fact_2029_count__le__length,axiom,
% 0.29/0.52      ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X3 ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_le_length
% 0.29/0.52  thf(fact_2030_count__le__length,axiom,
% 0.29/0.52      ! [Xs: list_o,X3: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X3 ) @ ( size_size_list_o @ Xs ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_le_length
% 0.29/0.52  thf(fact_2031_count__le__length,axiom,
% 0.29/0.52      ! [Xs: list_nat,X3: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X3 ) @ ( size_size_list_nat @ Xs ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_le_length
% 0.29/0.52  thf(fact_2032_count__le__length,axiom,
% 0.29/0.52      ! [Xs: list_int,X3: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X3 ) @ ( size_size_list_int @ Xs ) ) ).
% 0.29/0.52  
% 0.29/0.52  % count_le_length
% 0.29/0.52  thf(fact_2033_vebt__mint_Ocases,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT] :
% 0.29/0.52        ( ! [A4: $o,B4: $o] :
% 0.29/0.52            ( X3
% 0.29/0.52           != ( vEBT_Leaf @ A4 @ B4 ) )
% 0.29/0.52       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.52              ( X3
% 0.29/0.52             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.29/0.52         => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.29/0.52                ( X3
% 0.29/0.52               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % vebt_mint.cases
% 0.29/0.52  thf(fact_2034_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 0.29/0.52      ! [X3: vEBT_VEBT,Y3: $o] :
% 0.29/0.52        ( ( ( vEBT_VEBT_minNull @ X3 )
% 0.29/0.52          = Y3 )
% 0.29/0.52       => ( ( ( X3
% 0.29/0.52              = ( vEBT_Leaf @ $false @ $false ) )
% 0.29/0.52           => ~ Y3 )
% 0.29/0.52         => ( ( ? [Uv2: $o] :
% 0.29/0.52                  ( X3
% 0.29/0.52                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.29/0.52             => Y3 )
% 0.29/0.52           => ( ( ? [Uu2: $o] :
% 0.29/0.52                    ( X3
% 0.29/0.52                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.29/0.52               => Y3 )
% 0.29/0.52             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.29/0.52                      ( X3
% 0.29/0.52                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.29/0.52                 => ~ Y3 )
% 0.29/0.52               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.29/0.52                        ( X3
% 0.29/0.52                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.29/0.52                   => Y3 ) ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % VEBT_internal.minNull.elims(1)
% 0.29/0.52  thf(fact_2035_length__code,axiom,
% 0.29/0.52      ( size_s6755466524823107622T_VEBT
% 0.29/0.52      = ( gen_length_VEBT_VEBT @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % length_code
% 0.29/0.52  thf(fact_2036_length__code,axiom,
% 0.29/0.52      ( size_size_list_o
% 0.29/0.52      = ( gen_length_o @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % length_code
% 0.29/0.52  thf(fact_2037_length__code,axiom,
% 0.29/0.52      ( size_size_list_nat
% 0.29/0.52      = ( gen_length_nat @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % length_code
% 0.29/0.52  thf(fact_2038_length__code,axiom,
% 0.29/0.52      ( size_size_list_int
% 0.29/0.52      = ( gen_length_int @ zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % length_code
% 0.29/0.52  thf(fact_2039_sum__power2__eq__zero__iff,axiom,
% 0.29/0.52      ! [X3: rat,Y3: rat] :
% 0.29/0.52        ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52          = zero_zero_rat )
% 0.29/0.52        = ( ( X3 = zero_zero_rat )
% 0.29/0.52          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % sum_power2_eq_zero_iff
% 0.29/0.52  thf(fact_2040_sum__power2__eq__zero__iff,axiom,
% 0.29/0.52      ! [X3: real,Y3: real] :
% 0.29/0.52        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52          = zero_zero_real )
% 0.29/0.52        = ( ( X3 = zero_zero_real )
% 0.29/0.52          & ( Y3 = zero_zero_real ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % sum_power2_eq_zero_iff
% 0.29/0.52  thf(fact_2041_sum__power2__eq__zero__iff,axiom,
% 0.29/0.52      ! [X3: int,Y3: int] :
% 0.29/0.52        ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52          = zero_zero_int )
% 0.29/0.52        = ( ( X3 = zero_zero_int )
% 0.29/0.52          & ( Y3 = zero_zero_int ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % sum_power2_eq_zero_iff
% 0.29/0.52  thf(fact_2042_zero__less__power2,axiom,
% 0.29/0.52      ! [A: real] :
% 0.29/0.52        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52        = ( A != zero_zero_real ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_power2
% 0.29/0.52  thf(fact_2043_zero__less__power2,axiom,
% 0.29/0.52      ! [A: rat] :
% 0.29/0.52        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52        = ( A != zero_zero_rat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_power2
% 0.29/0.52  thf(fact_2044_zero__less__power2,axiom,
% 0.29/0.52      ! [A: int] :
% 0.29/0.52        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52        = ( A != zero_zero_int ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_less_power2
% 0.29/0.52  thf(fact_2045_power2__less__eq__zero__iff,axiom,
% 0.29/0.52      ! [A: real] :
% 0.29/0.52        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 0.29/0.52        = ( A = zero_zero_real ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_less_eq_zero_iff
% 0.29/0.52  thf(fact_2046_power2__less__eq__zero__iff,axiom,
% 0.29/0.52      ! [A: rat] :
% 0.29/0.52        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 0.29/0.52        = ( A = zero_zero_rat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_less_eq_zero_iff
% 0.29/0.52  thf(fact_2047_power2__less__eq__zero__iff,axiom,
% 0.29/0.52      ! [A: int] :
% 0.29/0.52        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.29/0.52        = ( A = zero_zero_int ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_less_eq_zero_iff
% 0.29/0.52  thf(fact_2048_power2__eq__iff__nonneg,axiom,
% 0.29/0.52      ! [X3: real,Y3: real] :
% 0.29/0.52        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.52         => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52              = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_eq_iff_nonneg
% 0.29/0.52  thf(fact_2049_power2__eq__iff__nonneg,axiom,
% 0.29/0.52      ! [X3: rat,Y3: rat] :
% 0.29/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.52         => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52              = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_eq_iff_nonneg
% 0.29/0.52  thf(fact_2050_power2__eq__iff__nonneg,axiom,
% 0.29/0.52      ! [X3: nat,Y3: nat] :
% 0.29/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.29/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.29/0.52         => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52              = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_eq_iff_nonneg
% 0.29/0.52  thf(fact_2051_power2__eq__iff__nonneg,axiom,
% 0.29/0.52      ! [X3: int,Y3: int] :
% 0.29/0.52        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.52         => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52              = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.52            = ( X3 = Y3 ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power2_eq_iff_nonneg
% 0.29/0.52  thf(fact_2052_zero__eq__power2,axiom,
% 0.29/0.52      ! [A: rat] :
% 0.29/0.52        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52          = zero_zero_rat )
% 0.29/0.52        = ( A = zero_zero_rat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_eq_power2
% 0.29/0.52  thf(fact_2053_zero__eq__power2,axiom,
% 0.29/0.52      ! [A: nat] :
% 0.29/0.52        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52          = zero_zero_nat )
% 0.29/0.52        = ( A = zero_zero_nat ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_eq_power2
% 0.29/0.52  thf(fact_2054_zero__eq__power2,axiom,
% 0.29/0.52      ! [A: real] :
% 0.29/0.52        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52          = zero_zero_real )
% 0.29/0.52        = ( A = zero_zero_real ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_eq_power2
% 0.29/0.52  thf(fact_2055_zero__eq__power2,axiom,
% 0.29/0.52      ! [A: int] :
% 0.29/0.52        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52          = zero_zero_int )
% 0.29/0.52        = ( A = zero_zero_int ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_eq_power2
% 0.29/0.52  thf(fact_2056_zero__eq__power2,axiom,
% 0.29/0.52      ! [A: complex] :
% 0.29/0.52        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.52          = zero_zero_complex )
% 0.29/0.52        = ( A = zero_zero_complex ) ) ).
% 0.29/0.52  
% 0.29/0.52  % zero_eq_power2
% 0.29/0.52  thf(fact_2057_power__mono__iff,axiom,
% 0.29/0.52      ! [A: real,B: real,N2: nat] :
% 0.29/0.52        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.52       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 0.29/0.52              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_mono_iff
% 0.29/0.52  thf(fact_2058_power__mono__iff,axiom,
% 0.29/0.52      ! [A: rat,B: rat,N2: nat] :
% 0.29/0.52        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.52       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 0.29/0.52              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_mono_iff
% 0.29/0.52  thf(fact_2059_power__mono__iff,axiom,
% 0.29/0.52      ! [A: nat,B: nat,N2: nat] :
% 0.29/0.52        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.52       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.29/0.52              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_mono_iff
% 0.29/0.52  thf(fact_2060_power__mono__iff,axiom,
% 0.29/0.52      ! [A: int,B: int,N2: nat] :
% 0.29/0.52        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.52       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.52         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.52           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.29/0.52              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_mono_iff
% 0.29/0.52  thf(fact_2061_power__eq__0__iff,axiom,
% 0.29/0.52      ! [A: rat,N2: nat] :
% 0.29/0.52        ( ( ( power_power_rat @ A @ N2 )
% 0.29/0.52          = zero_zero_rat )
% 0.29/0.52        = ( ( A = zero_zero_rat )
% 0.29/0.52          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_eq_0_iff
% 0.29/0.52  thf(fact_2062_power__eq__0__iff,axiom,
% 0.29/0.52      ! [A: nat,N2: nat] :
% 0.29/0.52        ( ( ( power_power_nat @ A @ N2 )
% 0.29/0.52          = zero_zero_nat )
% 0.29/0.52        = ( ( A = zero_zero_nat )
% 0.29/0.52          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_eq_0_iff
% 0.29/0.52  thf(fact_2063_power__eq__0__iff,axiom,
% 0.29/0.52      ! [A: real,N2: nat] :
% 0.29/0.52        ( ( ( power_power_real @ A @ N2 )
% 0.29/0.52          = zero_zero_real )
% 0.29/0.52        = ( ( A = zero_zero_real )
% 0.29/0.52          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_eq_0_iff
% 0.29/0.52  thf(fact_2064_power__eq__0__iff,axiom,
% 0.29/0.52      ! [A: int,N2: nat] :
% 0.29/0.52        ( ( ( power_power_int @ A @ N2 )
% 0.29/0.52          = zero_zero_int )
% 0.29/0.52        = ( ( A = zero_zero_int )
% 0.29/0.52          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.52  
% 0.29/0.52  % power_eq_0_iff
% 0.29/0.52  thf(fact_2065_power__eq__0__iff,axiom,
% 0.29/0.52      ! [A: complex,N2: nat] :
% 0.29/0.52        ( ( ( power_power_complex @ A @ N2 )
% 0.29/0.52          = zero_zero_complex )
% 0.29/0.52        = ( ( A = zero_zero_complex )
% 0.29/0.52          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_0_iff
% 0.29/0.53  thf(fact_2066_member__inv,axiom,
% 0.29/0.53      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.29/0.53        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 0.29/0.53          & ( ( X3 = Mi )
% 0.29/0.53            | ( X3 = Ma )
% 0.29/0.53            | ( ( ord_less_nat @ X3 @ Ma )
% 0.29/0.53              & ( ord_less_nat @ Mi @ X3 )
% 0.29/0.53              & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.29/0.53              & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % member_inv
% 0.29/0.53  thf(fact_2067_both__member__options__ding,axiom,
% 0.29/0.53      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] :
% 0.29/0.53        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 0.29/0.53       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.29/0.53         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % both_member_options_ding
% 0.29/0.53  thf(fact_2068_nat__zero__less__power__iff,axiom,
% 0.29/0.53      ! [X3: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N2 ) )
% 0.29/0.53        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.29/0.53          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % nat_zero_less_power_iff
% 0.29/0.53  thf(fact_2069_pow__sum,axiom,
% 0.29/0.53      ! [A: nat,B: nat] :
% 0.29/0.53        ( ( 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 ) )
% 0.29/0.53        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 0.29/0.53  
% 0.29/0.53  % pow_sum
% 0.29/0.53  thf(fact_2070_high__def,axiom,
% 0.29/0.53      ( vEBT_VEBT_high
% 0.29/0.53      = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % high_def
% 0.29/0.53  thf(fact_2071_division__ring__divide__zero,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.29/0.53        = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % division_ring_divide_zero
% 0.29/0.53  thf(fact_2072_division__ring__divide__zero,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.29/0.53        = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % division_ring_divide_zero
% 0.29/0.53  thf(fact_2073_division__ring__divide__zero,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.29/0.53        = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % division_ring_divide_zero
% 0.29/0.53  thf(fact_2074_divide__cancel__right,axiom,
% 0.29/0.53      ! [A: complex,C: complex,B: complex] :
% 0.29/0.53        ( ( ( divide1717551699836669952omplex @ A @ C )
% 0.29/0.53          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.53        = ( ( C = zero_zero_complex )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_right
% 0.29/0.53  thf(fact_2075_divide__cancel__right,axiom,
% 0.29/0.53      ! [A: real,C: real,B: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ A @ C )
% 0.29/0.53          = ( divide_divide_real @ B @ C ) )
% 0.29/0.53        = ( ( C = zero_zero_real )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_right
% 0.29/0.53  thf(fact_2076_divide__cancel__right,axiom,
% 0.29/0.53      ! [A: rat,C: rat,B: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ A @ C )
% 0.29/0.53          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.53        = ( ( C = zero_zero_rat )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_right
% 0.29/0.53  thf(fact_2077_divide__cancel__left,axiom,
% 0.29/0.53      ! [C: complex,A: complex,B: complex] :
% 0.29/0.53        ( ( ( divide1717551699836669952omplex @ C @ A )
% 0.29/0.53          = ( divide1717551699836669952omplex @ C @ B ) )
% 0.29/0.53        = ( ( C = zero_zero_complex )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_left
% 0.29/0.53  thf(fact_2078_divide__cancel__left,axiom,
% 0.29/0.53      ! [C: real,A: real,B: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ C @ A )
% 0.29/0.53          = ( divide_divide_real @ C @ B ) )
% 0.29/0.53        = ( ( C = zero_zero_real )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_left
% 0.29/0.53  thf(fact_2079_divide__cancel__left,axiom,
% 0.29/0.53      ! [C: rat,A: rat,B: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ C @ A )
% 0.29/0.53          = ( divide_divide_rat @ C @ B ) )
% 0.29/0.53        = ( ( C = zero_zero_rat )
% 0.29/0.53          | ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_cancel_left
% 0.29/0.53  thf(fact_2080_div__by__0,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 0.29/0.53        = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2081_div__by__0,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.29/0.53        = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2082_div__by__0,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.29/0.53        = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2083_div__by__0,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( divide_divide_real @ A @ zero_zero_real )
% 0.29/0.53        = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2084_div__by__0,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 0.29/0.53        = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2085_div__by__0,axiom,
% 0.29/0.53      ! [A: code_integer] :
% 0.29/0.53        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.53        = zero_z3403309356797280102nteger ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_0
% 0.29/0.53  thf(fact_2086_divide__eq__0__iff,axiom,
% 0.29/0.53      ! [A: complex,B: complex] :
% 0.29/0.53        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.29/0.53          = zero_zero_complex )
% 0.29/0.53        = ( ( A = zero_zero_complex )
% 0.29/0.53          | ( B = zero_zero_complex ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_0_iff
% 0.29/0.53  thf(fact_2087_divide__eq__0__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ A @ B )
% 0.29/0.53          = zero_zero_real )
% 0.29/0.53        = ( ( A = zero_zero_real )
% 0.29/0.53          | ( B = zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_0_iff
% 0.29/0.53  thf(fact_2088_divide__eq__0__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ A @ B )
% 0.29/0.53          = zero_zero_rat )
% 0.29/0.53        = ( ( A = zero_zero_rat )
% 0.29/0.53          | ( B = zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_0_iff
% 0.29/0.53  thf(fact_2089_div__0,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 0.29/0.53        = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2090_div__0,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.29/0.53        = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2091_div__0,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.29/0.53        = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2092_div__0,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( divide_divide_real @ zero_zero_real @ A )
% 0.29/0.53        = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2093_div__0,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 0.29/0.53        = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2094_div__0,axiom,
% 0.29/0.53      ! [A: code_integer] :
% 0.29/0.53        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.53        = zero_z3403309356797280102nteger ) ).
% 0.29/0.53  
% 0.29/0.53  % div_0
% 0.29/0.53  thf(fact_2095_power__0__Suc,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
% 0.29/0.53        = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_Suc
% 0.29/0.53  thf(fact_2096_power__0__Suc,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.29/0.53        = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_Suc
% 0.29/0.53  thf(fact_2097_power__0__Suc,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
% 0.29/0.53        = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_Suc
% 0.29/0.53  thf(fact_2098_power__0__Suc,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
% 0.29/0.53        = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_Suc
% 0.29/0.53  thf(fact_2099_power__0__Suc,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
% 0.29/0.53        = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_Suc
% 0.29/0.53  thf(fact_2100_power__zero__numeral,axiom,
% 0.29/0.53      ! [K: num] :
% 0.29/0.53        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.53        = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_zero_numeral
% 0.29/0.53  thf(fact_2101_power__zero__numeral,axiom,
% 0.29/0.53      ! [K: num] :
% 0.29/0.53        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 0.29/0.53        = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_zero_numeral
% 0.29/0.53  thf(fact_2102_power__zero__numeral,axiom,
% 0.29/0.53      ! [K: num] :
% 0.29/0.53        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 0.29/0.53        = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % power_zero_numeral
% 0.29/0.53  thf(fact_2103_power__zero__numeral,axiom,
% 0.29/0.53      ! [K: num] :
% 0.29/0.53        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 0.29/0.53        = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % power_zero_numeral
% 0.29/0.53  thf(fact_2104_power__zero__numeral,axiom,
% 0.29/0.53      ! [K: num] :
% 0.29/0.53        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 0.29/0.53        = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % power_zero_numeral
% 0.29/0.53  thf(fact_2105_power__Suc0__right,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % power_Suc0_right
% 0.29/0.53  thf(fact_2106_power__Suc0__right,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % power_Suc0_right
% 0.29/0.53  thf(fact_2107_power__Suc0__right,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % power_Suc0_right
% 0.29/0.53  thf(fact_2108_power__Suc0__right,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % power_Suc0_right
% 0.29/0.53  thf(fact_2109_power__Suc__0,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.29/0.53        = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_Suc_0
% 0.29/0.53  thf(fact_2110_nat__power__eq__Suc__0__iff,axiom,
% 0.29/0.53      ! [X3: nat,M: nat] :
% 0.29/0.53        ( ( ( power_power_nat @ X3 @ M )
% 0.29/0.53          = ( suc @ zero_zero_nat ) )
% 0.29/0.53        = ( ( M = zero_zero_nat )
% 0.29/0.53          | ( X3
% 0.29/0.53            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % nat_power_eq_Suc_0_iff
% 0.29/0.53  thf(fact_2111_add__divide__distrib,axiom,
% 0.29/0.53      ! [A: real,B: real,C: real] :
% 0.29/0.53        ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.53        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_divide_distrib
% 0.29/0.53  thf(fact_2112_add__divide__distrib,axiom,
% 0.29/0.53      ! [A: rat,B: rat,C: rat] :
% 0.29/0.53        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.53        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_divide_distrib
% 0.29/0.53  thf(fact_2113_divide__right__mono__neg,axiom,
% 0.29/0.53      ! [A: real,B: real,C: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_right_mono_neg
% 0.29/0.53  thf(fact_2114_divide__right__mono__neg,axiom,
% 0.29/0.53      ! [A: rat,B: rat,C: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_right_mono_neg
% 0.29/0.53  thf(fact_2115_divide__nonpos__nonpos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_nonpos
% 0.29/0.53  thf(fact_2116_divide__nonpos__nonpos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_nonpos
% 0.29/0.53  thf(fact_2117_divide__nonpos__nonneg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_nonneg
% 0.29/0.53  thf(fact_2118_divide__nonpos__nonneg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_nonneg
% 0.29/0.53  thf(fact_2119_divide__nonneg__nonpos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_nonpos
% 0.29/0.53  thf(fact_2120_divide__nonneg__nonpos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_nonpos
% 0.29/0.53  thf(fact_2121_divide__nonneg__nonneg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_nonneg
% 0.29/0.53  thf(fact_2122_divide__nonneg__nonneg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_nonneg
% 0.29/0.53  thf(fact_2123_zero__le__divide__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.29/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_divide_iff
% 0.29/0.53  thf(fact_2124_zero__le__divide__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.29/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_divide_iff
% 0.29/0.53  thf(fact_2125_divide__right__mono,axiom,
% 0.29/0.53      ! [A: real,B: real,C: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_right_mono
% 0.29/0.53  thf(fact_2126_divide__right__mono,axiom,
% 0.29/0.53      ! [A: rat,B: rat,C: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_right_mono
% 0.29/0.53  thf(fact_2127_divide__le__0__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.29/0.53        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.29/0.53          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_0_iff
% 0.29/0.53  thf(fact_2128_divide__le__0__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.53        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.29/0.53          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_0_iff
% 0.29/0.53  thf(fact_2129_divide__neg__neg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_neg_neg
% 0.29/0.53  thf(fact_2130_divide__neg__neg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_neg_neg
% 0.29/0.53  thf(fact_2131_divide__neg__pos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_neg_pos
% 0.29/0.53  thf(fact_2132_divide__neg__pos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_neg_pos
% 0.29/0.53  thf(fact_2133_divide__pos__neg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_pos_neg
% 0.29/0.53  thf(fact_2134_divide__pos__neg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_pos_neg
% 0.29/0.53  thf(fact_2135_divide__pos__pos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_pos_pos
% 0.29/0.53  thf(fact_2136_divide__pos__pos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_pos_pos
% 0.29/0.53  thf(fact_2137_divide__less__0__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_0_iff
% 0.29/0.53  thf(fact_2138_divide__less__0__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_0_iff
% 0.29/0.53  thf(fact_2139_divide__less__cancel,axiom,
% 0.29/0.53      ! [A: real,C: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.53           => ( ord_less_real @ A @ B ) )
% 0.29/0.53          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.53           => ( ord_less_real @ B @ A ) )
% 0.29/0.53          & ( C != zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_cancel
% 0.29/0.53  thf(fact_2140_divide__less__cancel,axiom,
% 0.29/0.53      ! [A: rat,C: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.53           => ( ord_less_rat @ A @ B ) )
% 0.29/0.53          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.53           => ( ord_less_rat @ B @ A ) )
% 0.29/0.53          & ( C != zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_cancel
% 0.29/0.53  thf(fact_2141_zero__less__divide__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_divide_iff
% 0.29/0.53  thf(fact_2142_zero__less__divide__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_divide_iff
% 0.29/0.53  thf(fact_2143_divide__strict__right__mono,axiom,
% 0.29/0.53      ! [A: real,B: real,C: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ B )
% 0.29/0.53       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.53         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_strict_right_mono
% 0.29/0.53  thf(fact_2144_divide__strict__right__mono,axiom,
% 0.29/0.53      ! [A: rat,B: rat,C: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ B )
% 0.29/0.53       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.53         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_strict_right_mono
% 0.29/0.53  thf(fact_2145_divide__strict__right__mono__neg,axiom,
% 0.29/0.53      ! [B: real,A: real,C: real] :
% 0.29/0.53        ( ( ord_less_real @ B @ A )
% 0.29/0.53       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.53         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_strict_right_mono_neg
% 0.29/0.53  thf(fact_2146_divide__strict__right__mono__neg,axiom,
% 0.29/0.53      ! [B: rat,A: rat,C: rat] :
% 0.29/0.53        ( ( ord_less_rat @ B @ A )
% 0.29/0.53       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_strict_right_mono_neg
% 0.29/0.53  thf(fact_2147_frac__le,axiom,
% 0.29/0.53      ! [Y3: real,X3: real,W2: real,Z2: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.29/0.53           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_le
% 0.29/0.53  thf(fact_2148_frac__le,axiom,
% 0.29/0.53      ! [Y3: rat,X3: rat,W2: rat,Z2: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.29/0.53           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_le
% 0.29/0.53  thf(fact_2149_frac__less,axiom,
% 0.29/0.53      ! [X3: real,Y3: real,W2: real,Z2: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.29/0.53           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_less
% 0.29/0.53  thf(fact_2150_frac__less,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.29/0.53           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_less
% 0.29/0.53  thf(fact_2151_frac__less2,axiom,
% 0.29/0.53      ! [X3: real,Y3: real,W2: real,Z2: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 0.29/0.53           => ( ( ord_less_real @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_less2
% 0.29/0.53  thf(fact_2152_frac__less2,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.29/0.53         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 0.29/0.53           => ( ( ord_less_rat @ W2 @ Z2 )
% 0.29/0.53             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % frac_less2
% 0.29/0.53  thf(fact_2153_divide__le__cancel,axiom,
% 0.29/0.53      ! [A: real,C: real,B: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.53           => ( ord_less_eq_real @ A @ B ) )
% 0.29/0.53          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.53           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_cancel
% 0.29/0.53  thf(fact_2154_divide__le__cancel,axiom,
% 0.29/0.53      ! [A: rat,C: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.53           => ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.53          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.53           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_cancel
% 0.29/0.53  thf(fact_2155_divide__nonneg__neg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_neg
% 0.29/0.53  thf(fact_2156_divide__nonneg__neg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_neg
% 0.29/0.53  thf(fact_2157_divide__nonneg__pos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_pos
% 0.29/0.53  thf(fact_2158_divide__nonneg__pos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonneg_pos
% 0.29/0.53  thf(fact_2159_divide__nonpos__neg,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 0.29/0.53         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_neg
% 0.29/0.53  thf(fact_2160_divide__nonpos__neg,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_neg
% 0.29/0.53  thf(fact_2161_divide__nonpos__pos,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_pos
% 0.29/0.53  thf(fact_2162_divide__nonpos__pos,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_nonpos_pos
% 0.29/0.53  thf(fact_2163_field__sum__of__halves,axiom,
% 0.29/0.53      ! [X3: real] :
% 0.29/0.53        ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.53        = X3 ) ).
% 0.29/0.53  
% 0.29/0.53  % field_sum_of_halves
% 0.29/0.53  thf(fact_2164_field__sum__of__halves,axiom,
% 0.29/0.53      ! [X3: rat] :
% 0.29/0.53        ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.29/0.53        = X3 ) ).
% 0.29/0.53  
% 0.29/0.53  % field_sum_of_halves
% 0.29/0.53  thf(fact_2165_half__gt__zero,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % half_gt_zero
% 0.29/0.53  thf(fact_2166_half__gt__zero,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % half_gt_zero
% 0.29/0.53  thf(fact_2167_half__gt__zero__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.29/0.53        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % half_gt_zero_iff
% 0.29/0.53  thf(fact_2168_half__gt__zero__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.29/0.53        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % half_gt_zero_iff
% 0.29/0.53  thf(fact_2169_field__less__half__sum,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ X3 @ Y3 )
% 0.29/0.53       => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % field_less_half_sum
% 0.29/0.53  thf(fact_2170_field__less__half__sum,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ X3 @ Y3 )
% 0.29/0.53       => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % field_less_half_sum
% 0.29/0.53  thf(fact_2171_power__not__zero,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( A != zero_zero_rat )
% 0.29/0.53       => ( ( power_power_rat @ A @ N2 )
% 0.29/0.53         != zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_not_zero
% 0.29/0.53  thf(fact_2172_power__not__zero,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( A != zero_zero_nat )
% 0.29/0.53       => ( ( power_power_nat @ A @ N2 )
% 0.29/0.53         != zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_not_zero
% 0.29/0.53  thf(fact_2173_power__not__zero,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( A != zero_zero_real )
% 0.29/0.53       => ( ( power_power_real @ A @ N2 )
% 0.29/0.53         != zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_not_zero
% 0.29/0.53  thf(fact_2174_power__not__zero,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( A != zero_zero_int )
% 0.29/0.53       => ( ( power_power_int @ A @ N2 )
% 0.29/0.53         != zero_zero_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_not_zero
% 0.29/0.53  thf(fact_2175_power__not__zero,axiom,
% 0.29/0.53      ! [A: complex,N2: nat] :
% 0.29/0.53        ( ( A != zero_zero_complex )
% 0.29/0.53       => ( ( power_power_complex @ A @ N2 )
% 0.29/0.53         != zero_zero_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_not_zero
% 0.29/0.53  thf(fact_2176_zero__le__power,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power
% 0.29/0.53  thf(fact_2177_zero__le__power,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power
% 0.29/0.53  thf(fact_2178_zero__le__power,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power
% 0.29/0.53  thf(fact_2179_zero__le__power,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power
% 0.29/0.53  thf(fact_2180_power__mono,axiom,
% 0.29/0.53      ! [A: real,B: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_mono
% 0.29/0.53  thf(fact_2181_power__mono,axiom,
% 0.29/0.53      ! [A: rat,B: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_mono
% 0.29/0.53  thf(fact_2182_power__mono,axiom,
% 0.29/0.53      ! [A: nat,B: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_mono
% 0.29/0.53  thf(fact_2183_power__mono,axiom,
% 0.29/0.53      ! [A: int,B: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_mono
% 0.29/0.53  thf(fact_2184_zero__less__power,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_power
% 0.29/0.53  thf(fact_2185_zero__less__power,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_power
% 0.29/0.53  thf(fact_2186_zero__less__power,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.53       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_power
% 0.29/0.53  thf(fact_2187_zero__less__power,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.53       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_power
% 0.29/0.53  thf(fact_2188_nat__power__less__imp__less,axiom,
% 0.29/0.53      ! [I: nat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ I )
% 0.29/0.53       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
% 0.29/0.53         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % nat_power_less_imp_less
% 0.29/0.53  thf(fact_2189_power__less__imp__less__base,axiom,
% 0.29/0.53      ! [A: real,N2: nat,B: real] :
% 0.29/0.53        ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.53         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_base
% 0.29/0.53  thf(fact_2190_power__less__imp__less__base,axiom,
% 0.29/0.53      ! [A: rat,N2: nat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.53         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_base
% 0.29/0.53  thf(fact_2191_power__less__imp__less__base,axiom,
% 0.29/0.53      ! [A: nat,N2: nat,B: nat] :
% 0.29/0.53        ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.53         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_base
% 0.29/0.53  thf(fact_2192_power__less__imp__less__base,axiom,
% 0.29/0.53      ! [A: int,N2: nat,B: int] :
% 0.29/0.53        ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.53         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_base
% 0.29/0.53  thf(fact_2193_power__le__imp__le__base,axiom,
% 0.29/0.53      ! [A: real,N2: nat,B: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.53         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_imp_le_base
% 0.29/0.53  thf(fact_2194_power__le__imp__le__base,axiom,
% 0.29/0.53      ! [A: rat,N2: nat,B: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.53         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_imp_le_base
% 0.29/0.53  thf(fact_2195_power__le__imp__le__base,axiom,
% 0.29/0.53      ! [A: nat,N2: nat,B: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.53         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_imp_le_base
% 0.29/0.53  thf(fact_2196_power__le__imp__le__base,axiom,
% 0.29/0.53      ! [A: int,N2: nat,B: int] :
% 0.29/0.53        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.53         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_imp_le_base
% 0.29/0.53  thf(fact_2197_power__inject__base,axiom,
% 0.29/0.53      ! [A: real,N2: nat,B: real] :
% 0.29/0.53        ( ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.29/0.53          = ( power_power_real @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.53           => ( A = B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_base
% 0.29/0.53  thf(fact_2198_power__inject__base,axiom,
% 0.29/0.53      ! [A: rat,N2: nat,B: rat] :
% 0.29/0.53        ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.29/0.53          = ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.53           => ( A = B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_base
% 0.29/0.53  thf(fact_2199_power__inject__base,axiom,
% 0.29/0.53      ! [A: nat,N2: nat,B: nat] :
% 0.29/0.53        ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.29/0.53          = ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.53           => ( A = B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_base
% 0.29/0.53  thf(fact_2200_power__inject__base,axiom,
% 0.29/0.53      ! [A: int,N2: nat,B: int] :
% 0.29/0.53        ( ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.29/0.53          = ( power_power_int @ B @ ( suc @ N2 ) ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.53           => ( A = B ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_base
% 0.29/0.53  thf(fact_2201_zero__power,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.29/0.53          = zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power
% 0.29/0.53  thf(fact_2202_zero__power,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.29/0.53          = zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power
% 0.29/0.53  thf(fact_2203_zero__power,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.29/0.53          = zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power
% 0.29/0.53  thf(fact_2204_zero__power,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.29/0.53          = zero_zero_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power
% 0.29/0.53  thf(fact_2205_zero__power,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.29/0.53          = zero_zero_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power
% 0.29/0.53  thf(fact_2206_power__gt__expt,axiom,
% 0.29/0.53      ! [N2: nat,K: nat] :
% 0.29/0.53        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.29/0.53       => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_gt_expt
% 0.29/0.53  thf(fact_2207_nat__one__le__power,axiom,
% 0.29/0.53      ! [I: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 0.29/0.53       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % nat_one_le_power
% 0.29/0.53  thf(fact_2208_power__eq__iff__eq__base,axiom,
% 0.29/0.53      ! [N2: nat,A: real,B: real] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.53           => ( ( ( power_power_real @ A @ N2 )
% 0.29/0.53                = ( power_power_real @ B @ N2 ) )
% 0.29/0.53              = ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_iff_eq_base
% 0.29/0.53  thf(fact_2209_power__eq__iff__eq__base,axiom,
% 0.29/0.53      ! [N2: nat,A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.53           => ( ( ( power_power_rat @ A @ N2 )
% 0.29/0.53                = ( power_power_rat @ B @ N2 ) )
% 0.29/0.53              = ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_iff_eq_base
% 0.29/0.53  thf(fact_2210_power__eq__iff__eq__base,axiom,
% 0.29/0.53      ! [N2: nat,A: nat,B: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.53           => ( ( ( power_power_nat @ A @ N2 )
% 0.29/0.53                = ( power_power_nat @ B @ N2 ) )
% 0.29/0.53              = ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_iff_eq_base
% 0.29/0.53  thf(fact_2211_power__eq__iff__eq__base,axiom,
% 0.29/0.53      ! [N2: nat,A: int,B: int] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.53           => ( ( ( power_power_int @ A @ N2 )
% 0.29/0.53                = ( power_power_int @ B @ N2 ) )
% 0.29/0.53              = ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_iff_eq_base
% 0.29/0.53  thf(fact_2212_power__eq__imp__eq__base,axiom,
% 0.29/0.53      ! [A: real,N2: nat,B: real] :
% 0.29/0.53        ( ( ( power_power_real @ A @ N2 )
% 0.29/0.53          = ( power_power_real @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53             => ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_imp_eq_base
% 0.29/0.53  thf(fact_2213_power__eq__imp__eq__base,axiom,
% 0.29/0.53      ! [A: rat,N2: nat,B: rat] :
% 0.29/0.53        ( ( ( power_power_rat @ A @ N2 )
% 0.29/0.53          = ( power_power_rat @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53             => ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_imp_eq_base
% 0.29/0.53  thf(fact_2214_power__eq__imp__eq__base,axiom,
% 0.29/0.53      ! [A: nat,N2: nat,B: nat] :
% 0.29/0.53        ( ( ( power_power_nat @ A @ N2 )
% 0.29/0.53          = ( power_power_nat @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53             => ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_imp_eq_base
% 0.29/0.53  thf(fact_2215_power__eq__imp__eq__base,axiom,
% 0.29/0.53      ! [A: int,N2: nat,B: int] :
% 0.29/0.53        ( ( ( power_power_int @ A @ N2 )
% 0.29/0.53          = ( power_power_int @ B @ N2 ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.53           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53             => ( A = B ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_eq_imp_eq_base
% 0.29/0.53  thf(fact_2216_zero__power2,axiom,
% 0.29/0.53      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power2
% 0.29/0.53  thf(fact_2217_zero__power2,axiom,
% 0.29/0.53      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power2
% 0.29/0.53  thf(fact_2218_zero__power2,axiom,
% 0.29/0.53      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power2
% 0.29/0.53  thf(fact_2219_zero__power2,axiom,
% 0.29/0.53      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power2
% 0.29/0.53  thf(fact_2220_zero__power2,axiom,
% 0.29/0.53      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_power2
% 0.29/0.53  thf(fact_2221_less__exp,axiom,
% 0.29/0.53      ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_exp
% 0.29/0.53  thf(fact_2222_power2__nat__le__imp__le,axiom,
% 0.29/0.53      ! [M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 0.29/0.53       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_nat_le_imp_le
% 0.29/0.53  thf(fact_2223_power2__nat__le__eq__le,axiom,
% 0.29/0.53      ! [M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_nat_le_eq_le
% 0.29/0.53  thf(fact_2224_self__le__ge2__pow,axiom,
% 0.29/0.53      ! [K: nat,M: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.29/0.53       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % self_le_ge2_pow
% 0.29/0.53  thf(fact_2225_power2__le__imp__le,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_le_imp_le
% 0.29/0.53  thf(fact_2226_power2__le__imp__le,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_le_imp_le
% 0.29/0.53  thf(fact_2227_power2__le__imp__le,axiom,
% 0.29/0.53      ! [X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.29/0.53         => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_le_imp_le
% 0.29/0.53  thf(fact_2228_power2__le__imp__le,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.53         => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_le_imp_le
% 0.29/0.53  thf(fact_2229_power2__eq__imp__eq,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.53         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_eq_imp_eq
% 0.29/0.53  thf(fact_2230_power2__eq__imp__eq,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.53         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_eq_imp_eq
% 0.29/0.53  thf(fact_2231_power2__eq__imp__eq,axiom,
% 0.29/0.53      ! [X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53          = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.29/0.53         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.29/0.53           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_eq_imp_eq
% 0.29/0.53  thf(fact_2232_power2__eq__imp__eq,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.53         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.53           => ( X3 = Y3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_eq_imp_eq
% 0.29/0.53  thf(fact_2233_zero__le__power2,axiom,
% 0.29/0.53      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power2
% 0.29/0.53  thf(fact_2234_zero__le__power2,axiom,
% 0.29/0.53      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power2
% 0.29/0.53  thf(fact_2235_zero__le__power2,axiom,
% 0.29/0.53      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_power2
% 0.29/0.53  thf(fact_2236_power__strict__mono,axiom,
% 0.29/0.53      ! [A: real,B: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53           => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_mono
% 0.29/0.53  thf(fact_2237_power__strict__mono,axiom,
% 0.29/0.53      ! [A: rat,B: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53           => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_mono
% 0.29/0.53  thf(fact_2238_power__strict__mono,axiom,
% 0.29/0.53      ! [A: nat,B: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53           => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_mono
% 0.29/0.53  thf(fact_2239_power__strict__mono,axiom,
% 0.29/0.53      ! [A: int,B: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ A @ B )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53           => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_mono
% 0.29/0.53  thf(fact_2240_power2__less__0,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_0
% 0.29/0.53  thf(fact_2241_power2__less__0,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_0
% 0.29/0.53  thf(fact_2242_power2__less__0,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_0
% 0.29/0.53  thf(fact_2243_power2__less__imp__less,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.53         => ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_imp_less
% 0.29/0.53  thf(fact_2244_power2__less__imp__less,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.53         => ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_imp_less
% 0.29/0.53  thf(fact_2245_power2__less__imp__less,axiom,
% 0.29/0.53      ! [X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 0.29/0.53         => ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_imp_less
% 0.29/0.53  thf(fact_2246_power2__less__imp__less,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.29/0.53       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.53         => ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power2_less_imp_less
% 0.29/0.53  thf(fact_2247_sum__power2__le__zero__iff,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 0.29/0.53        = ( ( X3 = zero_zero_real )
% 0.29/0.53          & ( Y3 = zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_le_zero_iff
% 0.29/0.53  thf(fact_2248_sum__power2__le__zero__iff,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 0.29/0.53        = ( ( X3 = zero_zero_rat )
% 0.29/0.53          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_le_zero_iff
% 0.29/0.53  thf(fact_2249_sum__power2__le__zero__iff,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 0.29/0.53        = ( ( X3 = zero_zero_int )
% 0.29/0.53          & ( Y3 = zero_zero_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_le_zero_iff
% 0.29/0.53  thf(fact_2250_sum__power2__ge__zero,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_ge_zero
% 0.29/0.53  thf(fact_2251_sum__power2__ge__zero,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_ge_zero
% 0.29/0.53  thf(fact_2252_sum__power2__ge__zero,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_ge_zero
% 0.29/0.53  thf(fact_2253_not__sum__power2__lt__zero,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % not_sum_power2_lt_zero
% 0.29/0.53  thf(fact_2254_not__sum__power2__lt__zero,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_sum_power2_lt_zero
% 0.29/0.53  thf(fact_2255_not__sum__power2__lt__zero,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % not_sum_power2_lt_zero
% 0.29/0.53  thf(fact_2256_sum__power2__gt__zero__iff,axiom,
% 0.29/0.53      ! [X3: real,Y3: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53        = ( ( X3 != zero_zero_real )
% 0.29/0.53          | ( Y3 != zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_gt_zero_iff
% 0.29/0.53  thf(fact_2257_sum__power2__gt__zero__iff,axiom,
% 0.29/0.53      ! [X3: rat,Y3: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53        = ( ( X3 != zero_zero_rat )
% 0.29/0.53          | ( Y3 != zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_gt_zero_iff
% 0.29/0.53  thf(fact_2258_sum__power2__gt__zero__iff,axiom,
% 0.29/0.53      ! [X3: int,Y3: int] :
% 0.29/0.53        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53        = ( ( X3 != zero_zero_int )
% 0.29/0.53          | ( Y3 != zero_zero_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % sum_power2_gt_zero_iff
% 0.29/0.53  thf(fact_2259_both__member__options__from__chilf__to__complete__tree,axiom,
% 0.29/0.53      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 0.29/0.53        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.29/0.53       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.29/0.53         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % both_member_options_from_chilf_to_complete_tree
% 0.29/0.53  thf(fact_2260_add__self__div__2,axiom,
% 0.29/0.53      ! [M: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53        = M ) ).
% 0.29/0.53  
% 0.29/0.53  % add_self_div_2
% 0.29/0.53  thf(fact_2261_div2__Suc__Suc,axiom,
% 0.29/0.53      ! [M: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div2_Suc_Suc
% 0.29/0.53  thf(fact_2262_both__member__options__from__complete__tree__to__child,axiom,
% 0.29/0.53      ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 0.29/0.53       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 0.29/0.53         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.53            | ( X3 = Mi )
% 0.29/0.53            | ( X3 = Ma ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % both_member_options_from_complete_tree_to_child
% 0.29/0.53  thf(fact_2263_semiring__norm_I76_J,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % semiring_norm(76)
% 0.29/0.53  thf(fact_2264_div__less,axiom,
% 0.29/0.53      ! [M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ M @ N2 )
% 0.29/0.53       => ( ( divide_divide_nat @ M @ N2 )
% 0.29/0.53          = zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_less
% 0.29/0.53  thf(fact_2265_div__by__Suc__0,axiom,
% 0.29/0.53      ! [M: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.29/0.53        = M ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_Suc_0
% 0.29/0.53  thf(fact_2266_set__n__deg__not__0,axiom,
% 0.29/0.53      ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
% 0.29/0.53        ( ! [X4: vEBT_VEBT] :
% 0.29/0.53            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.29/0.53           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 0.29/0.53       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.29/0.53            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.53         => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % set_n_deg_not_0
% 0.29/0.53  thf(fact_2267_Suc__n__div__2__gt__zero,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % Suc_n_div_2_gt_zero
% 0.29/0.53  thf(fact_2268_div__2__gt__zero,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.29/0.53       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_2_gt_zero
% 0.29/0.53  thf(fact_2269_deg1Leaf,axiom,
% 0.29/0.53      ! [T: vEBT_VEBT] :
% 0.29/0.53        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.29/0.53        = ( ? [A3: $o,B3: $o] :
% 0.29/0.53              ( T
% 0.29/0.53              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % deg1Leaf
% 0.29/0.53  thf(fact_2270_deg__1__Leaf,axiom,
% 0.29/0.53      ! [T: vEBT_VEBT] :
% 0.29/0.53        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 0.29/0.53       => ? [A4: $o,B4: $o] :
% 0.29/0.53            ( T
% 0.29/0.53            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % deg_1_Leaf
% 0.29/0.53  thf(fact_2271_deg__1__Leafy,axiom,
% 0.29/0.53      ! [T: vEBT_VEBT,N2: nat] :
% 0.29/0.53        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.29/0.53       => ( ( N2 = one_one_nat )
% 0.29/0.53         => ? [A4: $o,B4: $o] :
% 0.29/0.53              ( T
% 0.29/0.53              = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % deg_1_Leafy
% 0.29/0.53  thf(fact_2272_div__by__1,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2273_div__by__1,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( divide_divide_nat @ A @ one_one_nat )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2274_div__by__1,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( divide_divide_int @ A @ one_one_int )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2275_div__by__1,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( divide_divide_real @ A @ one_one_real )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2276_div__by__1,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( divide_divide_rat @ A @ one_one_rat )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2277_div__by__1,axiom,
% 0.29/0.53      ! [A: code_integer] :
% 0.29/0.53        ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
% 0.29/0.53        = A ) ).
% 0.29/0.53  
% 0.29/0.53  % div_by_1
% 0.29/0.53  thf(fact_2278_semiring__norm_I78_J,axiom,
% 0.29/0.53      ! [M: num,N2: num] :
% 0.29/0.53        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ M @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % semiring_norm(78)
% 0.29/0.53  thf(fact_2279_semiring__norm_I75_J,axiom,
% 0.29/0.53      ! [M: num] :
% 0.29/0.53        ~ ( ord_less_num @ M @ one ) ).
% 0.29/0.53  
% 0.29/0.53  % semiring_norm(75)
% 0.29/0.53  thf(fact_2280_divide__eq__1__iff,axiom,
% 0.29/0.53      ! [A: complex,B: complex] :
% 0.29/0.53        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.29/0.53          = one_one_complex )
% 0.29/0.53        = ( ( B != zero_zero_complex )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_1_iff
% 0.29/0.53  thf(fact_2281_divide__eq__1__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ A @ B )
% 0.29/0.53          = one_one_real )
% 0.29/0.53        = ( ( B != zero_zero_real )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_1_iff
% 0.29/0.53  thf(fact_2282_divide__eq__1__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ A @ B )
% 0.29/0.53          = one_one_rat )
% 0.29/0.53        = ( ( B != zero_zero_rat )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_1_iff
% 0.29/0.53  thf(fact_2283_div__self,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( A != zero_zero_complex )
% 0.29/0.53       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.29/0.53          = one_one_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2284_div__self,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( A != zero_zero_nat )
% 0.29/0.53       => ( ( divide_divide_nat @ A @ A )
% 0.29/0.53          = one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2285_div__self,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( A != zero_zero_int )
% 0.29/0.53       => ( ( divide_divide_int @ A @ A )
% 0.29/0.53          = one_one_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2286_div__self,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( A != zero_zero_real )
% 0.29/0.53       => ( ( divide_divide_real @ A @ A )
% 0.29/0.53          = one_one_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2287_div__self,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( A != zero_zero_rat )
% 0.29/0.53       => ( ( divide_divide_rat @ A @ A )
% 0.29/0.53          = one_one_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2288_div__self,axiom,
% 0.29/0.53      ! [A: code_integer] :
% 0.29/0.53        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.53       => ( ( divide6298287555418463151nteger @ A @ A )
% 0.29/0.53          = one_one_Code_integer ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_self
% 0.29/0.53  thf(fact_2289_one__eq__divide__iff,axiom,
% 0.29/0.53      ! [A: complex,B: complex] :
% 0.29/0.53        ( ( one_one_complex
% 0.29/0.53          = ( divide1717551699836669952omplex @ A @ B ) )
% 0.29/0.53        = ( ( B != zero_zero_complex )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_eq_divide_iff
% 0.29/0.53  thf(fact_2290_one__eq__divide__iff,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( one_one_real
% 0.29/0.53          = ( divide_divide_real @ A @ B ) )
% 0.29/0.53        = ( ( B != zero_zero_real )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_eq_divide_iff
% 0.29/0.53  thf(fact_2291_one__eq__divide__iff,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( one_one_rat
% 0.29/0.53          = ( divide_divide_rat @ A @ B ) )
% 0.29/0.53        = ( ( B != zero_zero_rat )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_eq_divide_iff
% 0.29/0.53  thf(fact_2292_divide__self,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( A != zero_zero_complex )
% 0.29/0.53       => ( ( divide1717551699836669952omplex @ A @ A )
% 0.29/0.53          = one_one_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self
% 0.29/0.53  thf(fact_2293_divide__self,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( A != zero_zero_real )
% 0.29/0.53       => ( ( divide_divide_real @ A @ A )
% 0.29/0.53          = one_one_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self
% 0.29/0.53  thf(fact_2294_divide__self,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( A != zero_zero_rat )
% 0.29/0.53       => ( ( divide_divide_rat @ A @ A )
% 0.29/0.53          = one_one_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self
% 0.29/0.53  thf(fact_2295_divide__self__if,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( ( A = zero_zero_complex )
% 0.29/0.53         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.29/0.53            = zero_zero_complex ) )
% 0.29/0.53        & ( ( A != zero_zero_complex )
% 0.29/0.53         => ( ( divide1717551699836669952omplex @ A @ A )
% 0.29/0.53            = one_one_complex ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self_if
% 0.29/0.53  thf(fact_2296_divide__self__if,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ( A = zero_zero_real )
% 0.29/0.53         => ( ( divide_divide_real @ A @ A )
% 0.29/0.53            = zero_zero_real ) )
% 0.29/0.53        & ( ( A != zero_zero_real )
% 0.29/0.53         => ( ( divide_divide_real @ A @ A )
% 0.29/0.53            = one_one_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self_if
% 0.29/0.53  thf(fact_2297_divide__self__if,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ( A = zero_zero_rat )
% 0.29/0.53         => ( ( divide_divide_rat @ A @ A )
% 0.29/0.53            = zero_zero_rat ) )
% 0.29/0.53        & ( ( A != zero_zero_rat )
% 0.29/0.53         => ( ( divide_divide_rat @ A @ A )
% 0.29/0.53            = one_one_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_self_if
% 0.29/0.53  thf(fact_2298_divide__eq__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ B @ A )
% 0.29/0.53          = one_one_real )
% 0.29/0.53        = ( ( A != zero_zero_real )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_eq_1
% 0.29/0.53  thf(fact_2299_divide__eq__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ B @ A )
% 0.29/0.53          = one_one_rat )
% 0.29/0.53        = ( ( A != zero_zero_rat )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_eq_eq_1
% 0.29/0.53  thf(fact_2300_eq__divide__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( one_one_real
% 0.29/0.53          = ( divide_divide_real @ B @ A ) )
% 0.29/0.53        = ( ( A != zero_zero_real )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % eq_divide_eq_1
% 0.29/0.53  thf(fact_2301_eq__divide__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( one_one_rat
% 0.29/0.53          = ( divide_divide_rat @ B @ A ) )
% 0.29/0.53        = ( ( A != zero_zero_rat )
% 0.29/0.53          & ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % eq_divide_eq_1
% 0.29/0.53  thf(fact_2302_one__divide__eq__0__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ( divide_divide_real @ one_one_real @ A )
% 0.29/0.53          = zero_zero_real )
% 0.29/0.53        = ( A = zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_divide_eq_0_iff
% 0.29/0.53  thf(fact_2303_one__divide__eq__0__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 0.29/0.53          = zero_zero_rat )
% 0.29/0.53        = ( A = zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_divide_eq_0_iff
% 0.29/0.53  thf(fact_2304_zero__eq__1__divide__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( zero_zero_real
% 0.29/0.53          = ( divide_divide_real @ one_one_real @ A ) )
% 0.29/0.53        = ( A = zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_eq_1_divide_iff
% 0.29/0.53  thf(fact_2305_zero__eq__1__divide__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( zero_zero_rat
% 0.29/0.53          = ( divide_divide_rat @ one_one_rat @ A ) )
% 0.29/0.53        = ( A = zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_eq_1_divide_iff
% 0.29/0.53  thf(fact_2306_power__inject__exp,axiom,
% 0.29/0.53      ! [A: real,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.53       => ( ( ( power_power_real @ A @ M )
% 0.29/0.53            = ( power_power_real @ A @ N2 ) )
% 0.29/0.53          = ( M = N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_exp
% 0.29/0.53  thf(fact_2307_power__inject__exp,axiom,
% 0.29/0.53      ! [A: rat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.53       => ( ( ( power_power_rat @ A @ M )
% 0.29/0.53            = ( power_power_rat @ A @ N2 ) )
% 0.29/0.53          = ( M = N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_exp
% 0.29/0.53  thf(fact_2308_power__inject__exp,axiom,
% 0.29/0.53      ! [A: nat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.53       => ( ( ( power_power_nat @ A @ M )
% 0.29/0.53            = ( power_power_nat @ A @ N2 ) )
% 0.29/0.53          = ( M = N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_exp
% 0.29/0.53  thf(fact_2309_power__inject__exp,axiom,
% 0.29/0.53      ! [A: int,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.53       => ( ( ( power_power_int @ A @ M )
% 0.29/0.53            = ( power_power_int @ A @ N2 ) )
% 0.29/0.53          = ( M = N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_inject_exp
% 0.29/0.53  thf(fact_2310_less__one,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ N2 @ one_one_nat )
% 0.29/0.53        = ( N2 = zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_one
% 0.29/0.53  thf(fact_2311_divide__le__0__1__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.29/0.53        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_0_1_iff
% 0.29/0.53  thf(fact_2312_divide__le__0__1__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.29/0.53        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_0_1_iff
% 0.29/0.53  thf(fact_2313_zero__le__divide__1__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.29/0.53        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_divide_1_iff
% 0.29/0.53  thf(fact_2314_zero__le__divide__1__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.29/0.53        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_le_divide_1_iff
% 0.29/0.53  thf(fact_2315_zero__less__divide__1__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 0.29/0.53        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_divide_1_iff
% 0.29/0.53  thf(fact_2316_zero__less__divide__1__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 0.29/0.53        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_divide_1_iff
% 0.29/0.53  thf(fact_2317_less__divide__eq__1__pos,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53          = ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1_pos
% 0.29/0.53  thf(fact_2318_less__divide__eq__1__pos,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1_pos
% 0.29/0.53  thf(fact_2319_less__divide__eq__1__neg,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53          = ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1_neg
% 0.29/0.53  thf(fact_2320_less__divide__eq__1__neg,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1_neg
% 0.29/0.53  thf(fact_2321_divide__less__eq__1__pos,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53          = ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1_pos
% 0.29/0.53  thf(fact_2322_divide__less__eq__1__pos,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1_pos
% 0.29/0.53  thf(fact_2323_divide__less__eq__1__neg,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53          = ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1_neg
% 0.29/0.53  thf(fact_2324_divide__less__eq__1__neg,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1_neg
% 0.29/0.53  thf(fact_2325_divide__less__0__1__iff,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 0.29/0.53        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_0_1_iff
% 0.29/0.53  thf(fact_2326_divide__less__0__1__iff,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 0.29/0.53        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_0_1_iff
% 0.29/0.53  thf(fact_2327_power__strict__increasing__iff,axiom,
% 0.29/0.53      ! [B: real,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.53       => ( ( ord_less_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing_iff
% 0.29/0.53  thf(fact_2328_power__strict__increasing__iff,axiom,
% 0.29/0.53      ! [B: rat,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ B )
% 0.29/0.53       => ( ( ord_less_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing_iff
% 0.29/0.53  thf(fact_2329_power__strict__increasing__iff,axiom,
% 0.29/0.53      ! [B: nat,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ B )
% 0.29/0.53       => ( ( ord_less_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing_iff
% 0.29/0.53  thf(fact_2330_power__strict__increasing__iff,axiom,
% 0.29/0.53      ! [B: int,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ B )
% 0.29/0.53       => ( ( ord_less_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing_iff
% 0.29/0.53  thf(fact_2331_le__divide__eq__1__pos,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1_pos
% 0.29/0.53  thf(fact_2332_le__divide__eq__1__pos,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1_pos
% 0.29/0.53  thf(fact_2333_le__divide__eq__1__neg,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1_neg
% 0.29/0.53  thf(fact_2334_le__divide__eq__1__neg,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1_neg
% 0.29/0.53  thf(fact_2335_divide__le__eq__1__pos,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_eq_1_pos
% 0.29/0.53  thf(fact_2336_divide__le__eq__1__pos,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_eq_1_pos
% 0.29/0.53  thf(fact_2337_divide__le__eq__1__neg,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_eq_1_neg
% 0.29/0.53  thf(fact_2338_divide__le__eq__1__neg,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_eq_1_neg
% 0.29/0.53  thf(fact_2339_one__add__one,axiom,
% 0.29/0.53      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 0.29/0.53      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2340_one__add__one,axiom,
% 0.29/0.53      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
% 0.29/0.53      = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2341_one__add__one,axiom,
% 0.29/0.53      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 0.29/0.53      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2342_one__add__one,axiom,
% 0.29/0.53      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 0.29/0.53      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2343_one__add__one,axiom,
% 0.29/0.53      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.29/0.53      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2344_one__add__one,axiom,
% 0.29/0.53      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 0.29/0.53      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_add_one
% 0.29/0.53  thf(fact_2345_power__strict__decreasing__iff,axiom,
% 0.29/0.53      ! [B: real,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.53       => ( ( ord_less_real @ B @ one_one_real )
% 0.29/0.53         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_decreasing_iff
% 0.29/0.53  thf(fact_2346_power__strict__decreasing__iff,axiom,
% 0.29/0.53      ! [B: rat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.53       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.29/0.53         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_decreasing_iff
% 0.29/0.53  thf(fact_2347_power__strict__decreasing__iff,axiom,
% 0.29/0.53      ! [B: nat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.53       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.29/0.53         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_decreasing_iff
% 0.29/0.53  thf(fact_2348_power__strict__decreasing__iff,axiom,
% 0.29/0.53      ! [B: int,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.53       => ( ( ord_less_int @ B @ one_one_int )
% 0.29/0.53         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_decreasing_iff
% 0.29/0.53  thf(fact_2349_power__increasing__iff,axiom,
% 0.29/0.53      ! [B: real,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ B )
% 0.29/0.53       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing_iff
% 0.29/0.53  thf(fact_2350_power__increasing__iff,axiom,
% 0.29/0.53      ! [B: rat,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ B )
% 0.29/0.53       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing_iff
% 0.29/0.53  thf(fact_2351_power__increasing__iff,axiom,
% 0.29/0.53      ! [B: nat,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ B )
% 0.29/0.53       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing_iff
% 0.29/0.53  thf(fact_2352_power__increasing__iff,axiom,
% 0.29/0.53      ! [B: int,X3: nat,Y3: nat] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ B )
% 0.29/0.53       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
% 0.29/0.53          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing_iff
% 0.29/0.53  thf(fact_2353_Suc__1,axiom,
% 0.29/0.53      ( ( suc @ one_one_nat )
% 0.29/0.53      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % Suc_1
% 0.29/0.53  thf(fact_2354_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
% 0.29/0.53        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2355_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.53        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2356_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 0.29/0.53        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2357_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.29/0.53        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2358_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.53        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2359_one__plus__numeral,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 0.29/0.53        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral
% 0.29/0.53  thf(fact_2360_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
% 0.29/0.53        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2361_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
% 0.29/0.53        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2362_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 0.29/0.53        = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2363_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 0.29/0.53        = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2364_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 0.29/0.53        = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2365_numeral__plus__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 0.29/0.53        = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_plus_one
% 0.29/0.53  thf(fact_2366_numeral__le__one__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
% 0.29/0.53        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_le_one_iff
% 0.29/0.53  thf(fact_2367_numeral__le__one__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 0.29/0.53        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_le_one_iff
% 0.29/0.53  thf(fact_2368_numeral__le__one__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 0.29/0.53        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_le_one_iff
% 0.29/0.53  thf(fact_2369_numeral__le__one__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 0.29/0.53        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_le_one_iff
% 0.29/0.53  thf(fact_2370_numeral__le__one__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 0.29/0.53        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 0.29/0.53  
% 0.29/0.53  % numeral_le_one_iff
% 0.29/0.53  thf(fact_2371_one__less__numeral__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ one @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_less_numeral_iff
% 0.29/0.53  thf(fact_2372_one__less__numeral__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ one @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_less_numeral_iff
% 0.29/0.53  thf(fact_2373_one__less__numeral__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ one @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_less_numeral_iff
% 0.29/0.53  thf(fact_2374_one__less__numeral__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ one @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_less_numeral_iff
% 0.29/0.53  thf(fact_2375_one__less__numeral__iff,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 0.29/0.53        = ( ord_less_num @ one @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_less_numeral_iff
% 0.29/0.53  thf(fact_2376_one__div__two__eq__zero,axiom,
% 0.29/0.53      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % one_div_two_eq_zero
% 0.29/0.53  thf(fact_2377_one__div__two__eq__zero,axiom,
% 0.29/0.53      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % one_div_two_eq_zero
% 0.29/0.53  thf(fact_2378_one__div__two__eq__zero,axiom,
% 0.29/0.53      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.53      = zero_z3403309356797280102nteger ) ).
% 0.29/0.53  
% 0.29/0.53  % one_div_two_eq_zero
% 0.29/0.53  thf(fact_2379_power__decreasing__iff,axiom,
% 0.29/0.53      ! [B: real,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.53       => ( ( ord_less_real @ B @ one_one_real )
% 0.29/0.53         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_decreasing_iff
% 0.29/0.53  thf(fact_2380_power__decreasing__iff,axiom,
% 0.29/0.53      ! [B: rat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.53       => ( ( ord_less_rat @ B @ one_one_rat )
% 0.29/0.53         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_decreasing_iff
% 0.29/0.53  thf(fact_2381_power__decreasing__iff,axiom,
% 0.29/0.53      ! [B: nat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.53       => ( ( ord_less_nat @ B @ one_one_nat )
% 0.29/0.53         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_decreasing_iff
% 0.29/0.53  thf(fact_2382_power__decreasing__iff,axiom,
% 0.29/0.53      ! [B: int,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.53       => ( ( ord_less_int @ B @ one_one_int )
% 0.29/0.53         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 0.29/0.53            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_decreasing_iff
% 0.29/0.53  thf(fact_2383_one__reorient,axiom,
% 0.29/0.53      ! [X3: complex] :
% 0.29/0.53        ( ( one_one_complex = X3 )
% 0.29/0.53        = ( X3 = one_one_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_reorient
% 0.29/0.53  thf(fact_2384_one__reorient,axiom,
% 0.29/0.53      ! [X3: real] :
% 0.29/0.53        ( ( one_one_real = X3 )
% 0.29/0.53        = ( X3 = one_one_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_reorient
% 0.29/0.53  thf(fact_2385_one__reorient,axiom,
% 0.29/0.53      ! [X3: rat] :
% 0.29/0.53        ( ( one_one_rat = X3 )
% 0.29/0.53        = ( X3 = one_one_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_reorient
% 0.29/0.53  thf(fact_2386_one__reorient,axiom,
% 0.29/0.53      ! [X3: nat] :
% 0.29/0.53        ( ( one_one_nat = X3 )
% 0.29/0.53        = ( X3 = one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_reorient
% 0.29/0.53  thf(fact_2387_one__reorient,axiom,
% 0.29/0.53      ! [X3: int] :
% 0.29/0.53        ( ( one_one_int = X3 )
% 0.29/0.53        = ( X3 = one_one_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_reorient
% 0.29/0.53  thf(fact_2388_le__numeral__extra_I4_J,axiom,
% 0.29/0.53      ord_less_eq_real @ one_one_real @ one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % le_numeral_extra(4)
% 0.29/0.53  thf(fact_2389_le__numeral__extra_I4_J,axiom,
% 0.29/0.53      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % le_numeral_extra(4)
% 0.29/0.53  thf(fact_2390_le__numeral__extra_I4_J,axiom,
% 0.29/0.53      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % le_numeral_extra(4)
% 0.29/0.53  thf(fact_2391_le__numeral__extra_I4_J,axiom,
% 0.29/0.53      ord_less_eq_int @ one_one_int @ one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % le_numeral_extra(4)
% 0.29/0.53  thf(fact_2392_zero__neq__one,axiom,
% 0.29/0.53      zero_zero_complex != one_one_complex ).
% 0.29/0.53  
% 0.29/0.53  % zero_neq_one
% 0.29/0.53  thf(fact_2393_zero__neq__one,axiom,
% 0.29/0.53      zero_zero_real != one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % zero_neq_one
% 0.29/0.53  thf(fact_2394_zero__neq__one,axiom,
% 0.29/0.53      zero_zero_rat != one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % zero_neq_one
% 0.29/0.53  thf(fact_2395_zero__neq__one,axiom,
% 0.29/0.53      zero_zero_nat != one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % zero_neq_one
% 0.29/0.53  thf(fact_2396_zero__neq__one,axiom,
% 0.29/0.53      zero_zero_int != one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % zero_neq_one
% 0.29/0.53  thf(fact_2397_less__numeral__extra_I4_J,axiom,
% 0.29/0.53      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(4)
% 0.29/0.53  thf(fact_2398_less__numeral__extra_I4_J,axiom,
% 0.29/0.53      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(4)
% 0.29/0.53  thf(fact_2399_less__numeral__extra_I4_J,axiom,
% 0.29/0.53      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(4)
% 0.29/0.53  thf(fact_2400_less__numeral__extra_I4_J,axiom,
% 0.29/0.53      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(4)
% 0.29/0.53  thf(fact_2401_div__add__self2,axiom,
% 0.29/0.53      ! [B: nat,A: nat] :
% 0.29/0.53        ( ( B != zero_zero_nat )
% 0.29/0.53       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.29/0.53          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self2
% 0.29/0.53  thf(fact_2402_div__add__self2,axiom,
% 0.29/0.53      ! [B: int,A: int] :
% 0.29/0.53        ( ( B != zero_zero_int )
% 0.29/0.53       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.29/0.53          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self2
% 0.29/0.53  thf(fact_2403_div__add__self2,axiom,
% 0.29/0.53      ! [B: code_integer,A: code_integer] :
% 0.29/0.53        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.53       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.29/0.53          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self2
% 0.29/0.53  thf(fact_2404_div__add__self1,axiom,
% 0.29/0.53      ! [B: nat,A: nat] :
% 0.29/0.53        ( ( B != zero_zero_nat )
% 0.29/0.53       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.29/0.53          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self1
% 0.29/0.53  thf(fact_2405_div__add__self1,axiom,
% 0.29/0.53      ! [B: int,A: int] :
% 0.29/0.53        ( ( B != zero_zero_int )
% 0.29/0.53       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.29/0.53          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self1
% 0.29/0.53  thf(fact_2406_div__add__self1,axiom,
% 0.29/0.53      ! [B: code_integer,A: code_integer] :
% 0.29/0.53        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.53       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.29/0.53          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_add_self1
% 0.29/0.53  thf(fact_2407_zero__less__one__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one_class.zero_le_one
% 0.29/0.53  thf(fact_2408_zero__less__one__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one_class.zero_le_one
% 0.29/0.53  thf(fact_2409_zero__less__one__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one_class.zero_le_one
% 0.29/0.53  thf(fact_2410_zero__less__one__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one_class.zero_le_one
% 0.29/0.53  thf(fact_2411_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.29/0.53  thf(fact_2412_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.29/0.53  thf(fact_2413_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.29/0.53  thf(fact_2414_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 0.29/0.53      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % linordered_nonzero_semiring_class.zero_le_one
% 0.29/0.53  thf(fact_2415_not__one__le__zero,axiom,
% 0.29/0.53      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_le_zero
% 0.29/0.53  thf(fact_2416_not__one__le__zero,axiom,
% 0.29/0.53      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_le_zero
% 0.29/0.53  thf(fact_2417_not__one__le__zero,axiom,
% 0.29/0.53      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_le_zero
% 0.29/0.53  thf(fact_2418_not__one__le__zero,axiom,
% 0.29/0.53      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_le_zero
% 0.29/0.53  thf(fact_2419_not__one__less__zero,axiom,
% 0.29/0.53      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_less_zero
% 0.29/0.53  thf(fact_2420_not__one__less__zero,axiom,
% 0.29/0.53      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_less_zero
% 0.29/0.53  thf(fact_2421_not__one__less__zero,axiom,
% 0.29/0.53      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_less_zero
% 0.29/0.53  thf(fact_2422_not__one__less__zero,axiom,
% 0.29/0.53      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 0.29/0.53  
% 0.29/0.53  % not_one_less_zero
% 0.29/0.53  thf(fact_2423_zero__less__one,axiom,
% 0.29/0.53      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one
% 0.29/0.53  thf(fact_2424_zero__less__one,axiom,
% 0.29/0.53      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one
% 0.29/0.53  thf(fact_2425_zero__less__one,axiom,
% 0.29/0.53      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one
% 0.29/0.53  thf(fact_2426_zero__less__one,axiom,
% 0.29/0.53      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_one
% 0.29/0.53  thf(fact_2427_less__numeral__extra_I1_J,axiom,
% 0.29/0.53      ord_less_real @ zero_zero_real @ one_one_real ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(1)
% 0.29/0.53  thf(fact_2428_less__numeral__extra_I1_J,axiom,
% 0.29/0.53      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(1)
% 0.29/0.53  thf(fact_2429_less__numeral__extra_I1_J,axiom,
% 0.29/0.53      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(1)
% 0.29/0.53  thf(fact_2430_less__numeral__extra_I1_J,axiom,
% 0.29/0.53      ord_less_int @ zero_zero_int @ one_one_int ).
% 0.29/0.53  
% 0.29/0.53  % less_numeral_extra(1)
% 0.29/0.53  thf(fact_2431_one__le__numeral,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_numeral
% 0.29/0.53  thf(fact_2432_one__le__numeral,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_numeral
% 0.29/0.53  thf(fact_2433_one__le__numeral,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_numeral
% 0.29/0.53  thf(fact_2434_one__le__numeral,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_numeral
% 0.29/0.53  thf(fact_2435_one__le__numeral,axiom,
% 0.29/0.53      ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_numeral
% 0.29/0.53  thf(fact_2436_not__numeral__less__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_numeral_less_one
% 0.29/0.53  thf(fact_2437_not__numeral__less__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% 0.29/0.53  
% 0.29/0.53  % not_numeral_less_one
% 0.29/0.53  thf(fact_2438_not__numeral__less__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_numeral_less_one
% 0.29/0.53  thf(fact_2439_not__numeral__less__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % not_numeral_less_one
% 0.29/0.53  thf(fact_2440_not__numeral__less__one,axiom,
% 0.29/0.53      ! [N2: num] :
% 0.29/0.53        ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% 0.29/0.53  
% 0.29/0.53  % not_numeral_less_one
% 0.29/0.53  thf(fact_2441_add__mono1,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ B )
% 0.29/0.53       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_mono1
% 0.29/0.53  thf(fact_2442_add__mono1,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ B )
% 0.29/0.53       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_mono1
% 0.29/0.53  thf(fact_2443_add__mono1,axiom,
% 0.29/0.53      ! [A: nat,B: nat] :
% 0.29/0.53        ( ( ord_less_nat @ A @ B )
% 0.29/0.53       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_mono1
% 0.29/0.53  thf(fact_2444_add__mono1,axiom,
% 0.29/0.53      ! [A: int,B: int] :
% 0.29/0.53        ( ( ord_less_int @ A @ B )
% 0.29/0.53       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % add_mono1
% 0.29/0.53  thf(fact_2445_less__add__one,axiom,
% 0.29/0.53      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_add_one
% 0.29/0.53  thf(fact_2446_less__add__one,axiom,
% 0.29/0.53      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_add_one
% 0.29/0.53  thf(fact_2447_less__add__one,axiom,
% 0.29/0.53      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_add_one
% 0.29/0.53  thf(fact_2448_less__add__one,axiom,
% 0.29/0.53      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_add_one
% 0.29/0.53  thf(fact_2449_right__inverse__eq,axiom,
% 0.29/0.53      ! [B: complex,A: complex] :
% 0.29/0.53        ( ( B != zero_zero_complex )
% 0.29/0.53       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.29/0.53            = one_one_complex )
% 0.29/0.53          = ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % right_inverse_eq
% 0.29/0.53  thf(fact_2450_right__inverse__eq,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( B != zero_zero_real )
% 0.29/0.53       => ( ( ( divide_divide_real @ A @ B )
% 0.29/0.53            = one_one_real )
% 0.29/0.53          = ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % right_inverse_eq
% 0.29/0.53  thf(fact_2451_right__inverse__eq,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( B != zero_zero_rat )
% 0.29/0.53       => ( ( ( divide_divide_rat @ A @ B )
% 0.29/0.53            = one_one_rat )
% 0.29/0.53          = ( A = B ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % right_inverse_eq
% 0.29/0.53  thf(fact_2452_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
% 0.29/0.53        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2453_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 0.29/0.53        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2454_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 0.29/0.53        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2455_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 0.29/0.53        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2456_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 0.29/0.53        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2457_one__plus__numeral__commute,axiom,
% 0.29/0.53      ! [X3: num] :
% 0.29/0.53        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 0.29/0.53        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_plus_numeral_commute
% 0.29/0.53  thf(fact_2458_one__le__power,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.29/0.53       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_power
% 0.29/0.53  thf(fact_2459_one__le__power,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.29/0.53       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_power
% 0.29/0.53  thf(fact_2460_one__le__power,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.29/0.53       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_power
% 0.29/0.53  thf(fact_2461_one__le__power,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.29/0.53       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % one_le_power
% 0.29/0.53  thf(fact_2462_power__0,axiom,
% 0.29/0.53      ! [A: rat] :
% 0.29/0.53        ( ( power_power_rat @ A @ zero_zero_nat )
% 0.29/0.53        = one_one_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0
% 0.29/0.53  thf(fact_2463_power__0,axiom,
% 0.29/0.53      ! [A: nat] :
% 0.29/0.53        ( ( power_power_nat @ A @ zero_zero_nat )
% 0.29/0.53        = one_one_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0
% 0.29/0.53  thf(fact_2464_power__0,axiom,
% 0.29/0.53      ! [A: real] :
% 0.29/0.53        ( ( power_power_real @ A @ zero_zero_nat )
% 0.29/0.53        = one_one_real ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0
% 0.29/0.53  thf(fact_2465_power__0,axiom,
% 0.29/0.53      ! [A: int] :
% 0.29/0.53        ( ( power_power_int @ A @ zero_zero_nat )
% 0.29/0.53        = one_one_int ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0
% 0.29/0.53  thf(fact_2466_power__0,axiom,
% 0.29/0.53      ! [A: complex] :
% 0.29/0.53        ( ( power_power_complex @ A @ zero_zero_nat )
% 0.29/0.53        = one_one_complex ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0
% 0.29/0.53  thf(fact_2467_div__eq__dividend__iff,axiom,
% 0.29/0.53      ! [M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.53       => ( ( ( divide_divide_nat @ M @ N2 )
% 0.29/0.53            = M )
% 0.29/0.53          = ( N2 = one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_eq_dividend_iff
% 0.29/0.53  thf(fact_2468_div__less__dividend,axiom,
% 0.29/0.53      ! [N2: nat,M: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.29/0.53       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.53         => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % div_less_dividend
% 0.29/0.53  thf(fact_2469_One__nat__def,axiom,
% 0.29/0.53      ( one_one_nat
% 0.29/0.53      = ( suc @ zero_zero_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % One_nat_def
% 0.29/0.53  thf(fact_2470_Suc__eq__plus1,axiom,
% 0.29/0.53      ( suc
% 0.29/0.53      = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % Suc_eq_plus1
% 0.29/0.53  thf(fact_2471_plus__1__eq__Suc,axiom,
% 0.29/0.53      ( ( plus_plus_nat @ one_one_nat )
% 0.29/0.53      = suc ) ).
% 0.29/0.53  
% 0.29/0.53  % plus_1_eq_Suc
% 0.29/0.53  thf(fact_2472_Suc__eq__plus1__left,axiom,
% 0.29/0.53      ( suc
% 0.29/0.53      = ( plus_plus_nat @ one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % Suc_eq_plus1_left
% 0.29/0.53  thf(fact_2473_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 0.29/0.53      ! [Uu: $o,Uv: $o,D: nat] :
% 0.29/0.53        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 0.29/0.53        = ( D = one_one_nat ) ) ).
% 0.29/0.53  
% 0.29/0.53  % VEBT_internal.valid'.simps(1)
% 0.29/0.53  thf(fact_2474_zero__less__two,axiom,
% 0.29/0.53      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_two
% 0.29/0.53  thf(fact_2475_zero__less__two,axiom,
% 0.29/0.53      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_two
% 0.29/0.53  thf(fact_2476_zero__less__two,axiom,
% 0.29/0.53      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_two
% 0.29/0.53  thf(fact_2477_zero__less__two,axiom,
% 0.29/0.53      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 0.29/0.53  
% 0.29/0.53  % zero_less_two
% 0.29/0.53  thf(fact_2478_power__le__one,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.53       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.53         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_one
% 0.29/0.53  thf(fact_2479_power__le__one,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.53       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.53         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_one
% 0.29/0.53  thf(fact_2480_power__le__one,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.53       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.29/0.53         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_one
% 0.29/0.53  thf(fact_2481_power__le__one,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.53       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.29/0.53         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_le_one
% 0.29/0.53  thf(fact_2482_less__divide__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_real @ A @ B ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1
% 0.29/0.53  thf(fact_2483_less__divide__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_rat @ A @ B ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_divide_eq_1
% 0.29/0.53  thf(fact_2484_divide__less__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_real @ B @ A ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_real @ A @ B ) )
% 0.29/0.53          | ( A = zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1
% 0.29/0.53  thf(fact_2485_divide__less__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_rat @ B @ A ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_rat @ A @ B ) )
% 0.29/0.53          | ( A = zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_less_eq_1
% 0.29/0.53  thf(fact_2486_less__half__sum,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ B )
% 0.29/0.53       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_half_sum
% 0.29/0.53  thf(fact_2487_less__half__sum,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ B )
% 0.29/0.53       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % less_half_sum
% 0.29/0.53  thf(fact_2488_gt__half__sum,axiom,
% 0.29/0.53      ! [A: real,B: real] :
% 0.29/0.53        ( ( ord_less_real @ A @ B )
% 0.29/0.53       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 0.29/0.53  
% 0.29/0.53  % gt_half_sum
% 0.29/0.53  thf(fact_2489_gt__half__sum,axiom,
% 0.29/0.53      ! [A: rat,B: rat] :
% 0.29/0.53        ( ( ord_less_rat @ A @ B )
% 0.29/0.53       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 0.29/0.53  
% 0.29/0.53  % gt_half_sum
% 0.29/0.53  thf(fact_2490_power__0__left,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ( N2 = zero_zero_nat )
% 0.29/0.53         => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.29/0.53            = one_one_rat ) )
% 0.29/0.53        & ( ( N2 != zero_zero_nat )
% 0.29/0.53         => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 0.29/0.53            = zero_zero_rat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_left
% 0.29/0.53  thf(fact_2491_power__0__left,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ( N2 = zero_zero_nat )
% 0.29/0.53         => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.29/0.53            = one_one_nat ) )
% 0.29/0.53        & ( ( N2 != zero_zero_nat )
% 0.29/0.53         => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 0.29/0.53            = zero_zero_nat ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_left
% 0.29/0.53  thf(fact_2492_power__0__left,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ( N2 = zero_zero_nat )
% 0.29/0.53         => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.29/0.53            = one_one_real ) )
% 0.29/0.53        & ( ( N2 != zero_zero_nat )
% 0.29/0.53         => ( ( power_power_real @ zero_zero_real @ N2 )
% 0.29/0.53            = zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_left
% 0.29/0.53  thf(fact_2493_power__0__left,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ( N2 = zero_zero_nat )
% 0.29/0.53         => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.29/0.53            = one_one_int ) )
% 0.29/0.53        & ( ( N2 != zero_zero_nat )
% 0.29/0.53         => ( ( power_power_int @ zero_zero_int @ N2 )
% 0.29/0.53            = zero_zero_int ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_left
% 0.29/0.53  thf(fact_2494_power__0__left,axiom,
% 0.29/0.53      ! [N2: nat] :
% 0.29/0.53        ( ( ( N2 = zero_zero_nat )
% 0.29/0.53         => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.29/0.53            = one_one_complex ) )
% 0.29/0.53        & ( ( N2 != zero_zero_nat )
% 0.29/0.53         => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 0.29/0.53            = zero_zero_complex ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_0_left
% 0.29/0.53  thf(fact_2495_power__gt1,axiom,
% 0.29/0.53      ! [A: real,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.53       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_gt1
% 0.29/0.53  thf(fact_2496_power__gt1,axiom,
% 0.29/0.53      ! [A: rat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.53       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_gt1
% 0.29/0.53  thf(fact_2497_power__gt1,axiom,
% 0.29/0.53      ! [A: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.53       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_gt1
% 0.29/0.53  thf(fact_2498_power__gt1,axiom,
% 0.29/0.53      ! [A: int,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.53       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_gt1
% 0.29/0.53  thf(fact_2499_power__less__imp__less__exp,axiom,
% 0.29/0.53      ! [A: real,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.53       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 0.29/0.53         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_exp
% 0.29/0.53  thf(fact_2500_power__less__imp__less__exp,axiom,
% 0.29/0.53      ! [A: rat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.53       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 0.29/0.53         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_exp
% 0.29/0.53  thf(fact_2501_power__less__imp__less__exp,axiom,
% 0.29/0.53      ! [A: nat,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.53       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.29/0.53         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_exp
% 0.29/0.53  thf(fact_2502_power__less__imp__less__exp,axiom,
% 0.29/0.53      ! [A: int,M: nat,N2: nat] :
% 0.29/0.53        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.53       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.29/0.53         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_less_imp_less_exp
% 0.29/0.53  thf(fact_2503_power__strict__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: real] :
% 0.29/0.53        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.53         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing
% 0.29/0.53  thf(fact_2504_power__strict__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: rat] :
% 0.29/0.53        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.53         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing
% 0.29/0.53  thf(fact_2505_power__strict__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: nat] :
% 0.29/0.53        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.53         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing
% 0.29/0.53  thf(fact_2506_power__strict__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: int] :
% 0.29/0.53        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.53         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_strict_increasing
% 0.29/0.53  thf(fact_2507_power__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: real] :
% 0.29/0.53        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_eq_real @ one_one_real @ A )
% 0.29/0.53         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing
% 0.29/0.53  thf(fact_2508_power__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: rat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.29/0.53         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing
% 0.29/0.53  thf(fact_2509_power__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: nat] :
% 0.29/0.53        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.29/0.53         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing
% 0.29/0.53  thf(fact_2510_power__increasing,axiom,
% 0.29/0.53      ! [N2: nat,N7: nat,A: int] :
% 0.29/0.53        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.53       => ( ( ord_less_eq_int @ one_one_int @ A )
% 0.29/0.53         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % power_increasing
% 0.29/0.53  thf(fact_2511_nat__induct__non__zero,axiom,
% 0.29/0.53      ! [N2: nat,P: nat > $o] :
% 0.29/0.53        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.53       => ( ( P @ one_one_nat )
% 0.29/0.53         => ( ! [N3: nat] :
% 0.29/0.53                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.29/0.53               => ( ( P @ N3 )
% 0.29/0.53                 => ( P @ ( suc @ N3 ) ) ) )
% 0.29/0.53           => ( P @ N2 ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % nat_induct_non_zero
% 0.29/0.53  thf(fact_2512_vebt__member_Osimps_I1_J,axiom,
% 0.29/0.53      ! [A: $o,B: $o,X3: nat] :
% 0.29/0.53        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X3 )
% 0.29/0.53        = ( ( ( X3 = zero_zero_nat )
% 0.29/0.53           => A )
% 0.29/0.53          & ( ( X3 != zero_zero_nat )
% 0.29/0.53           => ( ( ( X3 = one_one_nat )
% 0.29/0.53               => B )
% 0.29/0.53              & ( X3 = one_one_nat ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % vebt_member.simps(1)
% 0.29/0.53  thf(fact_2513_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 0.29/0.53      ! [A: $o,B: $o,X3: nat] :
% 0.29/0.53        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X3 )
% 0.29/0.53        = ( ( ( X3 = zero_zero_nat )
% 0.29/0.53           => A )
% 0.29/0.53          & ( ( X3 != zero_zero_nat )
% 0.29/0.53           => ( ( ( X3 = one_one_nat )
% 0.29/0.53               => B )
% 0.29/0.53              & ( X3 = one_one_nat ) ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % VEBT_internal.naive_member.simps(1)
% 0.29/0.53  thf(fact_2514_le__divide__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_eq_real @ A @ B ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1
% 0.29/0.53  thf(fact_2515_le__divide__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % le_divide_eq_1
% 0.29/0.53  thf(fact_2516_divide__le__eq__1,axiom,
% 0.29/0.53      ! [B: real,A: real] :
% 0.29/0.53        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 0.29/0.53        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.53            & ( ord_less_eq_real @ B @ A ) )
% 0.29/0.53          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.53            & ( ord_less_eq_real @ A @ B ) )
% 0.29/0.53          | ( A = zero_zero_real ) ) ) ).
% 0.29/0.53  
% 0.29/0.53  % divide_le_eq_1
% 0.29/0.53  thf(fact_2517_divide__le__eq__1,axiom,
% 0.29/0.53      ! [B: rat,A: rat] :
% 0.29/0.53        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 0.29/0.53        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.53            & ( ord_less_eq_rat @ B @ A ) )
% 0.29/0.53          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.53            & ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.53          | ( A = zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_le_eq_1
% 0.29/0.54  thf(fact_2518_power__Suc__le__self,axiom,
% 0.29/0.54      ! [A: real,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_le_self
% 0.29/0.54  thf(fact_2519_power__Suc__le__self,axiom,
% 0.29/0.54      ! [A: rat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_le_self
% 0.29/0.54  thf(fact_2520_power__Suc__le__self,axiom,
% 0.29/0.54      ! [A: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.29/0.54         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_le_self
% 0.29/0.54  thf(fact_2521_power__Suc__le__self,axiom,
% 0.29/0.54      ! [A: int,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_le_self
% 0.29/0.54  thf(fact_2522_power__Suc__less__one,axiom,
% 0.29/0.54      ! [A: real,N2: nat] :
% 0.29/0.54        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_real @ A @ one_one_real )
% 0.29/0.54         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_less_one
% 0.29/0.54  thf(fact_2523_power__Suc__less__one,axiom,
% 0.29/0.54      ! [A: rat,N2: nat] :
% 0.29/0.54        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.29/0.54         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_less_one
% 0.29/0.54  thf(fact_2524_power__Suc__less__one,axiom,
% 0.29/0.54      ! [A: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.29/0.54         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_less_one
% 0.29/0.54  thf(fact_2525_power__Suc__less__one,axiom,
% 0.29/0.54      ! [A: int,N2: nat] :
% 0.29/0.54        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_int @ A @ one_one_int )
% 0.29/0.54         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_Suc_less_one
% 0.29/0.54  thf(fact_2526_power__strict__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: real] :
% 0.29/0.54        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ( ord_less_real @ A @ one_one_real )
% 0.29/0.54           => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_strict_decreasing
% 0.29/0.54  thf(fact_2527_power__strict__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: rat] :
% 0.29/0.54        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54         => ( ( ord_less_rat @ A @ one_one_rat )
% 0.29/0.54           => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_strict_decreasing
% 0.29/0.54  thf(fact_2528_power__strict__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: nat] :
% 0.29/0.54        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.54         => ( ( ord_less_nat @ A @ one_one_nat )
% 0.29/0.54           => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_strict_decreasing
% 0.29/0.54  thf(fact_2529_power__strict__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: int] :
% 0.29/0.54        ( ( ord_less_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.54         => ( ( ord_less_int @ A @ one_one_int )
% 0.29/0.54           => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_strict_decreasing
% 0.29/0.54  thf(fact_2530_power__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: real] :
% 0.29/0.54        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54         => ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.54           => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_decreasing
% 0.29/0.54  thf(fact_2531_power__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: rat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.54           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_decreasing
% 0.29/0.54  thf(fact_2532_power__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.29/0.54           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_decreasing
% 0.29/0.54  thf(fact_2533_power__decreasing,axiom,
% 0.29/0.54      ! [N2: nat,N7: nat,A: int] :
% 0.29/0.54        ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54         => ( ( ord_less_eq_int @ A @ one_one_int )
% 0.29/0.54           => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_decreasing
% 0.29/0.54  thf(fact_2534_power__le__imp__le__exp,axiom,
% 0.29/0.54      ! [A: real,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_le_imp_le_exp
% 0.29/0.54  thf(fact_2535_power__le__imp__le__exp,axiom,
% 0.29/0.54      ! [A: rat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_le_imp_le_exp
% 0.29/0.54  thf(fact_2536_power__le__imp__le__exp,axiom,
% 0.29/0.54      ! [A: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.54       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_le_imp_le_exp
% 0.29/0.54  thf(fact_2537_power__le__imp__le__exp,axiom,
% 0.29/0.54      ! [A: int,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % power_le_imp_le_exp
% 0.29/0.54  thf(fact_2538_self__le__power,axiom,
% 0.29/0.54      ! [A: real,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_real @ one_one_real @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % self_le_power
% 0.29/0.54  thf(fact_2539_self__le__power,axiom,
% 0.29/0.54      ! [A: rat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % self_le_power
% 0.29/0.54  thf(fact_2540_self__le__power,axiom,
% 0.29/0.54      ! [A: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % self_le_power
% 0.29/0.54  thf(fact_2541_self__le__power,axiom,
% 0.29/0.54      ! [A: int,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_int @ one_one_int @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % self_le_power
% 0.29/0.54  thf(fact_2542_one__less__power,axiom,
% 0.29/0.54      ! [A: real,N2: nat] :
% 0.29/0.54        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_less_power
% 0.29/0.54  thf(fact_2543_one__less__power,axiom,
% 0.29/0.54      ! [A: rat,N2: nat] :
% 0.29/0.54        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_less_power
% 0.29/0.54  thf(fact_2544_one__less__power,axiom,
% 0.29/0.54      ! [A: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_less_power
% 0.29/0.54  thf(fact_2545_one__less__power,axiom,
% 0.29/0.54      ! [A: int,N2: nat] :
% 0.29/0.54        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.54       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_less_power
% 0.29/0.54  thf(fact_2546_nat__1__add__1,axiom,
% 0.29/0.54      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 0.29/0.54      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_1_add_1
% 0.29/0.54  thf(fact_2547_vebt__maxt_Osimps_I1_J,axiom,
% 0.29/0.54      ! [B: $o,A: $o] :
% 0.29/0.54        ( ( B
% 0.29/0.54         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54            = ( some_nat @ one_one_nat ) ) )
% 0.29/0.54        & ( ~ B
% 0.29/0.54         => ( ( A
% 0.29/0.54             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54                = ( some_nat @ zero_zero_nat ) ) )
% 0.29/0.54            & ( ~ A
% 0.29/0.54             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54                = none_nat ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % vebt_maxt.simps(1)
% 0.29/0.54  thf(fact_2548_vebt__mint_Osimps_I1_J,axiom,
% 0.29/0.54      ! [A: $o,B: $o] :
% 0.29/0.54        ( ( A
% 0.29/0.54         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54            = ( some_nat @ zero_zero_nat ) ) )
% 0.29/0.54        & ( ~ A
% 0.29/0.54         => ( ( B
% 0.29/0.54             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54                = ( some_nat @ one_one_nat ) ) )
% 0.29/0.54            & ( ~ B
% 0.29/0.54             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 0.29/0.54                = none_nat ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % vebt_mint.simps(1)
% 0.29/0.54  thf(fact_2549_div__le__mono,axiom,
% 0.29/0.54      ! [M: nat,N2: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.54       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_le_mono
% 0.29/0.54  thf(fact_2550_div__le__dividend,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% 0.29/0.54  
% 0.29/0.54  % div_le_dividend
% 0.29/0.54  thf(fact_2551_ex__power__ivl2,axiom,
% 0.29/0.54      ! [B: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.29/0.54         => ? [N3: nat] :
% 0.29/0.54              ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 0.29/0.54              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ex_power_ivl2
% 0.29/0.54  thf(fact_2552_ex__power__ivl1,axiom,
% 0.29/0.54      ! [B: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 0.29/0.54         => ? [N3: nat] :
% 0.29/0.54              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 0.29/0.54              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ex_power_ivl1
% 0.29/0.54  thf(fact_2553_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( divide_divide_nat @ M @ N2 )
% 0.29/0.54          = zero_zero_nat )
% 0.29/0.54        = ( ( ord_less_nat @ M @ N2 )
% 0.29/0.54          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Euclidean_Division.div_eq_0_iff
% 0.29/0.54  thf(fact_2554_Suc__div__le__mono,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_div_le_mono
% 0.29/0.54  thf(fact_2555_vebt__maxt_Oelims,axiom,
% 0.29/0.54      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.29/0.54        ( ( ( vEBT_vebt_maxt @ X3 )
% 0.29/0.54          = Y3 )
% 0.29/0.54       => ( ! [A4: $o,B4: $o] :
% 0.29/0.54              ( ( X3
% 0.29/0.54                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.29/0.54             => ~ ( ( B4
% 0.29/0.54                   => ( Y3
% 0.29/0.54                      = ( some_nat @ one_one_nat ) ) )
% 0.29/0.54                  & ( ~ B4
% 0.29/0.54                   => ( ( A4
% 0.29/0.54                       => ( Y3
% 0.29/0.54                          = ( some_nat @ zero_zero_nat ) ) )
% 0.29/0.54                      & ( ~ A4
% 0.29/0.54                       => ( Y3 = none_nat ) ) ) ) ) )
% 0.29/0.54         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.54                  ( X3
% 0.29/0.54                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.29/0.54             => ( Y3 != none_nat ) )
% 0.29/0.54           => ~ ! [Mi2: nat,Ma2: nat] :
% 0.29/0.54                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.29/0.54                      ( X3
% 0.29/0.54                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.29/0.54                 => ( Y3
% 0.29/0.54                   != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % vebt_maxt.elims
% 0.29/0.54  thf(fact_2556_vebt__mint_Oelims,axiom,
% 0.29/0.54      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.29/0.54        ( ( ( vEBT_vebt_mint @ X3 )
% 0.29/0.54          = Y3 )
% 0.29/0.54       => ( ! [A4: $o,B4: $o] :
% 0.29/0.54              ( ( X3
% 0.29/0.54                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.29/0.54             => ~ ( ( A4
% 0.29/0.54                   => ( Y3
% 0.29/0.54                      = ( some_nat @ zero_zero_nat ) ) )
% 0.29/0.54                  & ( ~ A4
% 0.29/0.54                   => ( ( B4
% 0.29/0.54                       => ( Y3
% 0.29/0.54                          = ( some_nat @ one_one_nat ) ) )
% 0.29/0.54                      & ( ~ B4
% 0.29/0.54                       => ( Y3 = none_nat ) ) ) ) ) )
% 0.29/0.54         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.29/0.54                  ( X3
% 0.29/0.54                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.29/0.54             => ( Y3 != none_nat ) )
% 0.29/0.54           => ~ ! [Mi2: nat] :
% 0.29/0.54                  ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.29/0.54                      ( X3
% 0.29/0.54                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.29/0.54                 => ( Y3
% 0.29/0.54                   != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % vebt_mint.elims
% 0.29/0.54  thf(fact_2557_div__greater__zero__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
% 0.29/0.54        = ( ( ord_less_eq_nat @ N2 @ M )
% 0.29/0.54          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_greater_zero_iff
% 0.29/0.54  thf(fact_2558_div__le__mono2,axiom,
% 0.29/0.54      ! [M: nat,N2: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.54       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.29/0.54         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_le_mono2
% 0.29/0.54  thf(fact_2559_bits__1__div__2,axiom,
% 0.29/0.54      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.54      = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_1_div_2
% 0.29/0.54  thf(fact_2560_bits__1__div__2,axiom,
% 0.29/0.54      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.54      = zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_1_div_2
% 0.29/0.54  thf(fact_2561_bits__1__div__2,axiom,
% 0.29/0.54      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.29/0.54      = zero_z3403309356797280102nteger ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_1_div_2
% 0.29/0.54  thf(fact_2562_nat__induct2,axiom,
% 0.29/0.54      ! [P: nat > $o,N2: nat] :
% 0.29/0.54        ( ( P @ zero_zero_nat )
% 0.29/0.54       => ( ( P @ one_one_nat )
% 0.29/0.54         => ( ! [N3: nat] :
% 0.29/0.54                ( ( P @ N3 )
% 0.29/0.54               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.29/0.54           => ( P @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_induct2
% 0.29/0.54  thf(fact_2563_div__exp__eq,axiom,
% 0.29/0.54      ! [A: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.29/0.54        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_exp_eq
% 0.29/0.54  thf(fact_2564_div__exp__eq,axiom,
% 0.29/0.54      ! [A: int,M: nat,N2: nat] :
% 0.29/0.54        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.29/0.54        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_exp_eq
% 0.29/0.54  thf(fact_2565_div__exp__eq,axiom,
% 0.29/0.54      ! [A: code_integer,M: nat,N2: nat] :
% 0.29/0.54        ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.29/0.54        = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_exp_eq
% 0.29/0.54  thf(fact_2566_exp__add__not__zero__imp__right,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.54         != zero_zero_nat )
% 0.29/0.54       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.29/0.54         != zero_zero_nat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % exp_add_not_zero_imp_right
% 0.29/0.54  thf(fact_2567_exp__add__not__zero__imp__right,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.54         != zero_zero_int )
% 0.29/0.54       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.29/0.54         != zero_zero_int ) ) ).
% 0.29/0.54  
% 0.29/0.54  % exp_add_not_zero_imp_right
% 0.29/0.54  thf(fact_2568_exp__add__not__zero__imp__left,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.54         != zero_zero_nat )
% 0.29/0.54       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.29/0.54         != zero_zero_nat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % exp_add_not_zero_imp_left
% 0.29/0.54  thf(fact_2569_exp__add__not__zero__imp__left,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.54         != zero_zero_int )
% 0.29/0.54       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.29/0.54         != zero_zero_int ) ) ).
% 0.29/0.54  
% 0.29/0.54  % exp_add_not_zero_imp_left
% 0.29/0.54  thf(fact_2570_bit__concat__def,axiom,
% 0.29/0.54      ( vEBT_VEBT_bit_concat
% 0.29/0.54      = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % bit_concat_def
% 0.29/0.54  thf(fact_2571_low__inv,axiom,
% 0.29/0.54      ! [X3: nat,N2: nat,Y3: nat] :
% 0.29/0.54        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.29/0.54       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
% 0.29/0.54          = X3 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % low_inv
% 0.29/0.54  thf(fact_2572_high__inv,axiom,
% 0.29/0.54      ! [X3: nat,N2: nat,Y3: nat] :
% 0.29/0.54        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.29/0.54       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
% 0.29/0.54          = Y3 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % high_inv
% 0.29/0.54  thf(fact_2573_enat__ord__number_I1_J,axiom,
% 0.29/0.54      ! [M: num,N2: num] :
% 0.29/0.54        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.54        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % enat_ord_number(1)
% 0.29/0.54  thf(fact_2574_enat__ord__number_I2_J,axiom,
% 0.29/0.54      ! [M: num,N2: num] :
% 0.29/0.54        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 0.29/0.54        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % enat_ord_number(2)
% 0.29/0.54  thf(fact_2575_mul__shift,axiom,
% 0.29/0.54      ! [X3: nat,Y3: nat,Z2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ X3 @ Y3 )
% 0.29/0.54          = Z2 )
% 0.29/0.54        = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 0.29/0.54          = ( some_nat @ Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mul_shift
% 0.29/0.54  thf(fact_2576_mul__def,axiom,
% 0.29/0.54      ( vEBT_VEBT_mul
% 0.29/0.54      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mul_def
% 0.29/0.54  thf(fact_2577_i0__less,axiom,
% 0.29/0.54      ! [N2: extended_enat] :
% 0.29/0.54        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 0.29/0.54        = ( N2 != zero_z5237406670263579293d_enat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % i0_less
% 0.29/0.54  thf(fact_2578_mult__cancel__right,axiom,
% 0.29/0.54      ! [A: complex,C: complex,B: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ A @ C )
% 0.29/0.54          = ( times_times_complex @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right
% 0.29/0.54  thf(fact_2579_mult__cancel__right,axiom,
% 0.29/0.54      ! [A: real,C: real,B: real] :
% 0.29/0.54        ( ( ( times_times_real @ A @ C )
% 0.29/0.54          = ( times_times_real @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right
% 0.29/0.54  thf(fact_2580_mult__cancel__right,axiom,
% 0.29/0.54      ! [A: rat,C: rat,B: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ A @ C )
% 0.29/0.54          = ( times_times_rat @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right
% 0.29/0.54  thf(fact_2581_mult__cancel__right,axiom,
% 0.29/0.54      ! [A: nat,C: nat,B: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ A @ C )
% 0.29/0.54          = ( times_times_nat @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_nat )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right
% 0.29/0.54  thf(fact_2582_mult__cancel__right,axiom,
% 0.29/0.54      ! [A: int,C: int,B: int] :
% 0.29/0.54        ( ( ( times_times_int @ A @ C )
% 0.29/0.54          = ( times_times_int @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right
% 0.29/0.54  thf(fact_2583_mult__cancel__left,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ C @ A )
% 0.29/0.54          = ( times_times_complex @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left
% 0.29/0.54  thf(fact_2584_mult__cancel__left,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ( times_times_real @ C @ A )
% 0.29/0.54          = ( times_times_real @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left
% 0.29/0.54  thf(fact_2585_mult__cancel__left,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ C @ A )
% 0.29/0.54          = ( times_times_rat @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left
% 0.29/0.54  thf(fact_2586_mult__cancel__left,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ C @ A )
% 0.29/0.54          = ( times_times_nat @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_nat )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left
% 0.29/0.54  thf(fact_2587_mult__cancel__left,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ( times_times_int @ C @ A )
% 0.29/0.54          = ( times_times_int @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left
% 0.29/0.54  thf(fact_2588_mult__eq__0__iff,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ A @ B )
% 0.29/0.54          = zero_zero_complex )
% 0.29/0.54        = ( ( A = zero_zero_complex )
% 0.29/0.54          | ( B = zero_zero_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_0_iff
% 0.29/0.54  thf(fact_2589_mult__eq__0__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( times_times_real @ A @ B )
% 0.29/0.54          = zero_zero_real )
% 0.29/0.54        = ( ( A = zero_zero_real )
% 0.29/0.54          | ( B = zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_0_iff
% 0.29/0.54  thf(fact_2590_mult__eq__0__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ A @ B )
% 0.29/0.54          = zero_zero_rat )
% 0.29/0.54        = ( ( A = zero_zero_rat )
% 0.29/0.54          | ( B = zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_0_iff
% 0.29/0.54  thf(fact_2591_mult__eq__0__iff,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ A @ B )
% 0.29/0.54          = zero_zero_nat )
% 0.29/0.54        = ( ( A = zero_zero_nat )
% 0.29/0.54          | ( B = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_0_iff
% 0.29/0.54  thf(fact_2592_mult__eq__0__iff,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( times_times_int @ A @ B )
% 0.29/0.54          = zero_zero_int )
% 0.29/0.54        = ( ( A = zero_zero_int )
% 0.29/0.54          | ( B = zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_0_iff
% 0.29/0.54  thf(fact_2593_mult__zero__right,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ zero_zero_complex )
% 0.29/0.54        = zero_zero_complex ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_right
% 0.29/0.54  thf(fact_2594_mult__zero__right,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ A @ zero_zero_real )
% 0.29/0.54        = zero_zero_real ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_right
% 0.29/0.54  thf(fact_2595_mult__zero__right,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ zero_zero_rat )
% 0.29/0.54        = zero_zero_rat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_right
% 0.29/0.54  thf(fact_2596_mult__zero__right,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ A @ zero_zero_nat )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_right
% 0.29/0.54  thf(fact_2597_mult__zero__right,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ A @ zero_zero_int )
% 0.29/0.54        = zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_right
% 0.29/0.54  thf(fact_2598_mult__zero__left,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ zero_zero_complex @ A )
% 0.29/0.54        = zero_zero_complex ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_left
% 0.29/0.54  thf(fact_2599_mult__zero__left,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ zero_zero_real @ A )
% 0.29/0.54        = zero_zero_real ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_left
% 0.29/0.54  thf(fact_2600_mult__zero__left,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ zero_zero_rat @ A )
% 0.29/0.54        = zero_zero_rat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_left
% 0.29/0.54  thf(fact_2601_mult__zero__left,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ zero_zero_nat @ A )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_left
% 0.29/0.54  thf(fact_2602_mult__zero__left,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ zero_zero_int @ A )
% 0.29/0.54        = zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_zero_left
% 0.29/0.54  thf(fact_2603_bits__div__by__0,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_by_0
% 0.29/0.54  thf(fact_2604_bits__div__by__0,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( divide_divide_int @ A @ zero_zero_int )
% 0.29/0.54        = zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_by_0
% 0.29/0.54  thf(fact_2605_bits__div__by__0,axiom,
% 0.29/0.54      ! [A: code_integer] :
% 0.29/0.54        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 0.29/0.54        = zero_z3403309356797280102nteger ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_by_0
% 0.29/0.54  thf(fact_2606_bits__div__0,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_0
% 0.29/0.54  thf(fact_2607_bits__div__0,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( divide_divide_int @ zero_zero_int @ A )
% 0.29/0.54        = zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_0
% 0.29/0.54  thf(fact_2608_bits__div__0,axiom,
% 0.29/0.54      ! [A: code_integer] :
% 0.29/0.54        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 0.29/0.54        = zero_z3403309356797280102nteger ) ).
% 0.29/0.54  
% 0.29/0.54  % bits_div_0
% 0.29/0.54  thf(fact_2609_mult_Oright__neutral,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ one_one_complex )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.right_neutral
% 0.29/0.54  thf(fact_2610_mult_Oright__neutral,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ A @ one_one_real )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.right_neutral
% 0.29/0.54  thf(fact_2611_mult_Oright__neutral,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ one_one_rat )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.right_neutral
% 0.29/0.54  thf(fact_2612_mult_Oright__neutral,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ A @ one_one_nat )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.right_neutral
% 0.29/0.54  thf(fact_2613_mult_Oright__neutral,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ A @ one_one_int )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.right_neutral
% 0.29/0.54  thf(fact_2614_mult__1,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ one_one_complex @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_1
% 0.29/0.54  thf(fact_2615_mult__1,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ one_one_real @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_1
% 0.29/0.54  thf(fact_2616_mult__1,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ one_one_rat @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_1
% 0.29/0.54  thf(fact_2617_mult__1,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ one_one_nat @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_1
% 0.29/0.54  thf(fact_2618_mult__1,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ one_one_int @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_1
% 0.29/0.54  thf(fact_2619_half__negative__int__iff,axiom,
% 0.29/0.54      ! [K: int] :
% 0.29/0.54        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 0.29/0.54        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.29/0.54  
% 0.29/0.54  % half_negative_int_iff
% 0.29/0.54  thf(fact_2620_times__divide__eq__left,axiom,
% 0.29/0.54      ! [B: complex,C: complex,A: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 0.29/0.54        = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_left
% 0.29/0.54  thf(fact_2621_times__divide__eq__left,axiom,
% 0.29/0.54      ! [B: real,C: real,A: real] :
% 0.29/0.54        ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.54        = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_left
% 0.29/0.54  thf(fact_2622_times__divide__eq__left,axiom,
% 0.29/0.54      ! [B: rat,C: rat,A: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.54        = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_left
% 0.29/0.54  thf(fact_2623_divide__divide__eq__left,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.29/0.54        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left
% 0.29/0.54  thf(fact_2624_divide__divide__eq__left,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.29/0.54        = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left
% 0.29/0.54  thf(fact_2625_divide__divide__eq__left,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.29/0.54        = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left
% 0.29/0.54  thf(fact_2626_divide__divide__eq__right,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.54        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_right
% 0.29/0.54  thf(fact_2627_divide__divide__eq__right,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.54        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_right
% 0.29/0.54  thf(fact_2628_divide__divide__eq__right,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.54        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_right
% 0.29/0.54  thf(fact_2629_times__divide__eq__right,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.54        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_right
% 0.29/0.54  thf(fact_2630_times__divide__eq__right,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.54        = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_right
% 0.29/0.54  thf(fact_2631_times__divide__eq__right,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.54        = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_eq_right
% 0.29/0.54  thf(fact_2632_mult__cancel2,axiom,
% 0.29/0.54      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ M @ K )
% 0.29/0.54          = ( times_times_nat @ N2 @ K ) )
% 0.29/0.54        = ( ( M = N2 )
% 0.29/0.54          | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel2
% 0.29/0.54  thf(fact_2633_mult__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ K @ M )
% 0.29/0.54          = ( times_times_nat @ K @ N2 ) )
% 0.29/0.54        = ( ( M = N2 )
% 0.29/0.54          | ( K = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel1
% 0.29/0.54  thf(fact_2634_mult__0__right,axiom,
% 0.29/0.54      ! [M: nat] :
% 0.29/0.54        ( ( times_times_nat @ M @ zero_zero_nat )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_0_right
% 0.29/0.54  thf(fact_2635_mult__is__0,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ M @ N2 )
% 0.29/0.54          = zero_zero_nat )
% 0.29/0.54        = ( ( M = zero_zero_nat )
% 0.29/0.54          | ( N2 = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_is_0
% 0.29/0.54  thf(fact_2636_nat__1__eq__mult__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( one_one_nat
% 0.29/0.54          = ( times_times_nat @ M @ N2 ) )
% 0.29/0.54        = ( ( M = one_one_nat )
% 0.29/0.54          & ( N2 = one_one_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_1_eq_mult_iff
% 0.29/0.54  thf(fact_2637_nat__mult__eq__1__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ M @ N2 )
% 0.29/0.54          = one_one_nat )
% 0.29/0.54        = ( ( M = one_one_nat )
% 0.29/0.54          & ( N2 = one_one_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_eq_1_iff
% 0.29/0.54  thf(fact_2638_mult__cancel__right2,axiom,
% 0.29/0.54      ! [A: complex,C: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ A @ C )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( A = one_one_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right2
% 0.29/0.54  thf(fact_2639_mult__cancel__right2,axiom,
% 0.29/0.54      ! [A: real,C: real] :
% 0.29/0.54        ( ( ( times_times_real @ A @ C )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( A = one_one_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right2
% 0.29/0.54  thf(fact_2640_mult__cancel__right2,axiom,
% 0.29/0.54      ! [A: rat,C: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ A @ C )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( A = one_one_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right2
% 0.29/0.54  thf(fact_2641_mult__cancel__right2,axiom,
% 0.29/0.54      ! [A: int,C: int] :
% 0.29/0.54        ( ( ( times_times_int @ A @ C )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( A = one_one_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right2
% 0.29/0.54  thf(fact_2642_mult__cancel__right1,axiom,
% 0.29/0.54      ! [C: complex,B: complex] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_complex @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( B = one_one_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right1
% 0.29/0.54  thf(fact_2643_mult__cancel__right1,axiom,
% 0.29/0.54      ! [C: real,B: real] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_real @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( B = one_one_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right1
% 0.29/0.54  thf(fact_2644_mult__cancel__right1,axiom,
% 0.29/0.54      ! [C: rat,B: rat] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_rat @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( B = one_one_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right1
% 0.29/0.54  thf(fact_2645_mult__cancel__right1,axiom,
% 0.29/0.54      ! [C: int,B: int] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_int @ B @ C ) )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( B = one_one_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_right1
% 0.29/0.54  thf(fact_2646_mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: complex,A: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ C @ A )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( A = one_one_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left2
% 0.29/0.54  thf(fact_2647_mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: real,A: real] :
% 0.29/0.54        ( ( ( times_times_real @ C @ A )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( A = one_one_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left2
% 0.29/0.54  thf(fact_2648_mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: rat,A: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ C @ A )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( A = one_one_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left2
% 0.29/0.54  thf(fact_2649_mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: int,A: int] :
% 0.29/0.54        ( ( ( times_times_int @ C @ A )
% 0.29/0.54          = C )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( A = one_one_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left2
% 0.29/0.54  thf(fact_2650_mult__cancel__left1,axiom,
% 0.29/0.54      ! [C: complex,B: complex] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_complex @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_complex )
% 0.29/0.54          | ( B = one_one_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left1
% 0.29/0.54  thf(fact_2651_mult__cancel__left1,axiom,
% 0.29/0.54      ! [C: real,B: real] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_real @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_real )
% 0.29/0.54          | ( B = one_one_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left1
% 0.29/0.54  thf(fact_2652_mult__cancel__left1,axiom,
% 0.29/0.54      ! [C: rat,B: rat] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_rat @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_rat )
% 0.29/0.54          | ( B = one_one_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left1
% 0.29/0.54  thf(fact_2653_mult__cancel__left1,axiom,
% 0.29/0.54      ! [C: int,B: int] :
% 0.29/0.54        ( ( C
% 0.29/0.54          = ( times_times_int @ C @ B ) )
% 0.29/0.54        = ( ( C = zero_zero_int )
% 0.29/0.54          | ( B = one_one_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_cancel_left1
% 0.29/0.54  thf(fact_2654_sum__squares__eq__zero__iff,axiom,
% 0.29/0.54      ! [X3: real,Y3: real] :
% 0.29/0.54        ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.29/0.54          = zero_zero_real )
% 0.29/0.54        = ( ( X3 = zero_zero_real )
% 0.29/0.54          & ( Y3 = zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_eq_zero_iff
% 0.29/0.54  thf(fact_2655_sum__squares__eq__zero__iff,axiom,
% 0.29/0.54      ! [X3: rat,Y3: rat] :
% 0.29/0.54        ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.29/0.54          = zero_zero_rat )
% 0.29/0.54        = ( ( X3 = zero_zero_rat )
% 0.29/0.54          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_eq_zero_iff
% 0.29/0.54  thf(fact_2656_sum__squares__eq__zero__iff,axiom,
% 0.29/0.54      ! [X3: int,Y3: int] :
% 0.29/0.54        ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.29/0.54          = zero_zero_int )
% 0.29/0.54        = ( ( X3 = zero_zero_int )
% 0.29/0.54          & ( Y3 = zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % sum_squares_eq_zero_iff
% 0.29/0.54  thf(fact_2657_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right2
% 0.29/0.54  thf(fact_2658_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right2
% 0.29/0.54  thf(fact_2659_nonzero__mult__divide__mult__cancel__right2,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right2
% 0.29/0.54  thf(fact_2660_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: complex,A: complex] :
% 0.29/0.54        ( ( B != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2661_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: nat,A: nat] :
% 0.29/0.54        ( ( B != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2662_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: int,A: int] :
% 0.29/0.54        ( ( B != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2663_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: real,A: real] :
% 0.29/0.54        ( ( B != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2664_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: rat,A: rat] :
% 0.29/0.54        ( ( B != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2665_nonzero__mult__div__cancel__right,axiom,
% 0.29/0.54      ! [B: code_integer,A: code_integer] :
% 0.29/0.54        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.29/0.54          = A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_right
% 0.29/0.54  thf(fact_2666_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2667_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2668_nonzero__mult__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2669_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left2
% 0.29/0.54  thf(fact_2670_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 0.29/0.54          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left2
% 0.29/0.54  thf(fact_2671_nonzero__mult__divide__mult__cancel__left2,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.54          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left2
% 0.29/0.54  thf(fact_2672_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( A != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2673_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( A != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2674_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( A != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2675_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( A != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2676_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( A != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2677_nonzero__mult__div__cancel__left,axiom,
% 0.29/0.54      ! [A: code_integer,B: code_integer] :
% 0.29/0.54        ( ( A != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
% 0.29/0.54          = B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_div_cancel_left
% 0.29/0.54  thf(fact_2678_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2679_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2680_nonzero__mult__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_mult_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2681_mult__divide__mult__cancel__left__if,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( ( C = zero_zero_complex )
% 0.29/0.54         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.29/0.54            = zero_zero_complex ) )
% 0.29/0.54        & ( ( C != zero_zero_complex )
% 0.29/0.54         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.29/0.54            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_divide_mult_cancel_left_if
% 0.29/0.54  thf(fact_2682_mult__divide__mult__cancel__left__if,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( ( C = zero_zero_real )
% 0.29/0.54         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54            = zero_zero_real ) )
% 0.29/0.54        & ( ( C != zero_zero_real )
% 0.29/0.54         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.54            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_divide_mult_cancel_left_if
% 0.29/0.54  thf(fact_2683_mult__divide__mult__cancel__left__if,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( ( C = zero_zero_rat )
% 0.29/0.54         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54            = zero_zero_rat ) )
% 0.29/0.54        & ( ( C != zero_zero_rat )
% 0.29/0.54         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.54            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_divide_mult_cancel_left_if
% 0.29/0.54  thf(fact_2684_div__mult__mult1__if,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( ( C = zero_zero_nat )
% 0.29/0.54         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.54            = zero_zero_nat ) )
% 0.29/0.54        & ( ( C != zero_zero_nat )
% 0.29/0.54         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.54            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1_if
% 0.29/0.54  thf(fact_2685_div__mult__mult1__if,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( ( C = zero_zero_int )
% 0.29/0.54         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54            = zero_zero_int ) )
% 0.29/0.54        & ( ( C != zero_zero_int )
% 0.29/0.54         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1_if
% 0.29/0.54  thf(fact_2686_div__mult__mult1__if,axiom,
% 0.29/0.54      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.54        ( ( ( C = zero_z3403309356797280102nteger )
% 0.29/0.54         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.29/0.54            = zero_z3403309356797280102nteger ) )
% 0.29/0.54        & ( ( C != zero_z3403309356797280102nteger )
% 0.29/0.54         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.29/0.54            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1_if
% 0.29/0.54  thf(fact_2687_div__mult__mult2,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( C != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.54          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult2
% 0.29/0.54  thf(fact_2688_div__mult__mult2,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( C != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.54          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult2
% 0.29/0.54  thf(fact_2689_div__mult__mult2,axiom,
% 0.29/0.54      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.54        ( ( C != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.29/0.54          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult2
% 0.29/0.54  thf(fact_2690_div__mult__mult1,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( C != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.54          = ( divide_divide_nat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1
% 0.29/0.54  thf(fact_2691_div__mult__mult1,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( C != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( divide_divide_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1
% 0.29/0.54  thf(fact_2692_div__mult__mult1,axiom,
% 0.29/0.54      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.29/0.54        ( ( C != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.29/0.54          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_mult1
% 0.29/0.54  thf(fact_2693_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: complex,B: complex,V: num] :
% 0.29/0.54        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2694_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: extended_enat,B: extended_enat,V: num] :
% 0.29/0.54        ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
% 0.29/0.54        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2695_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: real,B: real,V: num] :
% 0.29/0.54        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2696_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: rat,B: rat,V: num] :
% 0.29/0.54        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2697_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: nat,B: nat,V: num] :
% 0.29/0.54        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2698_distrib__right__numeral,axiom,
% 0.29/0.54      ! [A: int,B: int,V: num] :
% 0.29/0.54        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right_numeral
% 0.29/0.54  thf(fact_2699_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2700_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: extended_enat,C: extended_enat] :
% 0.29/0.54        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
% 0.29/0.54        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2701_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2702_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2703_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2704_distrib__left__numeral,axiom,
% 0.29/0.54      ! [V: num,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left_numeral
% 0.29/0.54  thf(fact_2705_mult__eq__1__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ M @ N2 )
% 0.29/0.54          = ( suc @ zero_zero_nat ) )
% 0.29/0.54        = ( ( M
% 0.29/0.54            = ( suc @ zero_zero_nat ) )
% 0.29/0.54          & ( N2
% 0.29/0.54            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_eq_1_iff
% 0.29/0.54  thf(fact_2706_one__eq__mult__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ( suc @ zero_zero_nat )
% 0.29/0.54          = ( times_times_nat @ M @ N2 ) )
% 0.29/0.54        = ( ( M
% 0.29/0.54            = ( suc @ zero_zero_nat ) )
% 0.29/0.54          & ( N2
% 0.29/0.54            = ( suc @ zero_zero_nat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_eq_mult_iff
% 0.29/0.54  thf(fact_2707_mult__less__cancel2,axiom,
% 0.29/0.54      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.29/0.54        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54          & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_cancel2
% 0.29/0.54  thf(fact_2708_nat__0__less__mult__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 0.29/0.54        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.29/0.54          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_0_less_mult_iff
% 0.29/0.54  thf(fact_2709_nat__mult__less__cancel__disj,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.54        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54          & ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_less_cancel_disj
% 0.29/0.54  thf(fact_2710_mult__Suc__right,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( times_times_nat @ M @ ( suc @ N2 ) )
% 0.29/0.54        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_Suc_right
% 0.29/0.54  thf(fact_2711_nat__mult__div__cancel__disj,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ( K = zero_zero_nat )
% 0.29/0.54         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.54            = zero_zero_nat ) )
% 0.29/0.54        & ( ( K != zero_zero_nat )
% 0.29/0.54         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.54            = ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_div_cancel_disj
% 0.29/0.54  thf(fact_2712_divide__le__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: real,W2: num,A: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 0.29/0.54        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_le_eq_numeral1(1)
% 0.29/0.54  thf(fact_2713_divide__le__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: rat,W2: num,A: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 0.29/0.54        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_le_eq_numeral1(1)
% 0.29/0.54  thf(fact_2714_le__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: real,B: real,W2: num] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.29/0.54        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2715_le__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,W2: num] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.29/0.54        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2716_divide__eq__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: complex,W2: num,A: complex] :
% 0.29/0.54        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.54             != zero_zero_complex )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.29/0.54          & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.54              = zero_zero_complex )
% 0.29/0.54           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral1(1)
% 0.29/0.54  thf(fact_2717_divide__eq__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: real,W2: num,A: real] :
% 0.29/0.54        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( ( numeral_numeral_real @ W2 )
% 0.29/0.54             != zero_zero_real )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.29/0.54          & ( ( ( numeral_numeral_real @ W2 )
% 0.29/0.54              = zero_zero_real )
% 0.29/0.54           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral1(1)
% 0.29/0.54  thf(fact_2718_divide__eq__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: rat,W2: num,A: rat] :
% 0.29/0.54        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
% 0.29/0.54          = A )
% 0.29/0.54        = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.29/0.54             != zero_zero_rat )
% 0.29/0.54           => ( B
% 0.29/0.54              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.29/0.54          & ( ( ( numeral_numeral_rat @ W2 )
% 0.29/0.54              = zero_zero_rat )
% 0.29/0.54           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_eq_eq_numeral1(1)
% 0.29/0.54  thf(fact_2719_eq__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: complex,B: complex,W2: num] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.29/0.54        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.54             != zero_zero_complex )
% 0.29/0.54           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.54              = zero_zero_complex )
% 0.29/0.54           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2720_eq__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: real,B: real,W2: num] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.29/0.54        = ( ( ( ( numeral_numeral_real @ W2 )
% 0.29/0.54             != zero_zero_real )
% 0.29/0.54           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( ( numeral_numeral_real @ W2 )
% 0.29/0.54              = zero_zero_real )
% 0.29/0.54           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2721_eq__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,W2: num] :
% 0.29/0.54        ( ( A
% 0.29/0.54          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.29/0.54        = ( ( ( ( numeral_numeral_rat @ W2 )
% 0.29/0.54             != zero_zero_rat )
% 0.29/0.54           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
% 0.29/0.54              = B ) )
% 0.29/0.54          & ( ( ( numeral_numeral_rat @ W2 )
% 0.29/0.54              = zero_zero_rat )
% 0.29/0.54           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % eq_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2722_less__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: real,B: real,W2: num] :
% 0.29/0.54        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 0.29/0.54        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2723_less__divide__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,W2: num] :
% 0.29/0.54        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 0.29/0.54        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 0.29/0.54  
% 0.29/0.54  % less_divide_eq_numeral1(1)
% 0.29/0.54  thf(fact_2724_divide__less__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: real,W2: num,A: real] :
% 0.29/0.54        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 0.29/0.54        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_less_eq_numeral1(1)
% 0.29/0.54  thf(fact_2725_divide__less__eq__numeral1_I1_J,axiom,
% 0.29/0.54      ! [B: rat,W2: num,A: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 0.29/0.54        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_less_eq_numeral1(1)
% 0.29/0.54  thf(fact_2726_nonzero__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( A != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2727_nonzero__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( A != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 0.29/0.54          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2728_nonzero__divide__mult__cancel__left,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( A != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 0.29/0.54          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_left
% 0.29/0.54  thf(fact_2729_nonzero__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [B: complex,A: complex] :
% 0.29/0.54        ( ( B != zero_zero_complex )
% 0.29/0.54       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 0.29/0.54          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2730_nonzero__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [B: real,A: real] :
% 0.29/0.54        ( ( B != zero_zero_real )
% 0.29/0.54       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 0.29/0.54          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2731_nonzero__divide__mult__cancel__right,axiom,
% 0.29/0.54      ! [B: rat,A: rat] :
% 0.29/0.54        ( ( B != zero_zero_rat )
% 0.29/0.54       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 0.29/0.54          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nonzero_divide_mult_cancel_right
% 0.29/0.54  thf(fact_2732_div__mult__self1,axiom,
% 0.29/0.54      ! [B: nat,A: nat,C: nat] :
% 0.29/0.54        ( ( B != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.29/0.54          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self1
% 0.29/0.54  thf(fact_2733_div__mult__self1,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( B != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.29/0.54          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self1
% 0.29/0.54  thf(fact_2734_div__mult__self1,axiom,
% 0.29/0.54      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.54        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.29/0.54          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self1
% 0.29/0.54  thf(fact_2735_div__mult__self2,axiom,
% 0.29/0.54      ! [B: nat,A: nat,C: nat] :
% 0.29/0.54        ( ( B != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.29/0.54          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self2
% 0.29/0.54  thf(fact_2736_div__mult__self2,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( B != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.29/0.54          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self2
% 0.29/0.54  thf(fact_2737_div__mult__self2,axiom,
% 0.29/0.54      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.29/0.54        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.29/0.54          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self2
% 0.29/0.54  thf(fact_2738_div__mult__self3,axiom,
% 0.29/0.54      ! [B: nat,C: nat,A: nat] :
% 0.29/0.54        ( ( B != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.29/0.54          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self3
% 0.29/0.54  thf(fact_2739_div__mult__self3,axiom,
% 0.29/0.54      ! [B: int,C: int,A: int] :
% 0.29/0.54        ( ( B != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.29/0.54          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self3
% 0.29/0.54  thf(fact_2740_div__mult__self3,axiom,
% 0.29/0.54      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.29/0.54        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.29/0.54          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self3
% 0.29/0.54  thf(fact_2741_div__mult__self4,axiom,
% 0.29/0.54      ! [B: nat,C: nat,A: nat] :
% 0.29/0.54        ( ( B != zero_zero_nat )
% 0.29/0.54       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.29/0.54          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self4
% 0.29/0.54  thf(fact_2742_div__mult__self4,axiom,
% 0.29/0.54      ! [B: int,C: int,A: int] :
% 0.29/0.54        ( ( B != zero_zero_int )
% 0.29/0.54       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.29/0.54          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self4
% 0.29/0.54  thf(fact_2743_div__mult__self4,axiom,
% 0.29/0.54      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.29/0.54        ( ( B != zero_z3403309356797280102nteger )
% 0.29/0.54       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.29/0.54          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self4
% 0.29/0.54  thf(fact_2744_one__le__mult__iff,axiom,
% 0.29/0.54      ! [M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
% 0.29/0.54        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.54          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % one_le_mult_iff
% 0.29/0.54  thf(fact_2745_mult__le__cancel2,axiom,
% 0.29/0.54      ! [M: nat,K: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 0.29/0.54        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_cancel2
% 0.29/0.54  thf(fact_2746_nat__mult__le__cancel__disj,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.54        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.54         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_le_cancel_disj
% 0.29/0.54  thf(fact_2747_div__mult__self__is__m,axiom,
% 0.29/0.54      ! [N2: nat,M: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
% 0.29/0.54          = M ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self_is_m
% 0.29/0.54  thf(fact_2748_div__mult__self1__is__m,axiom,
% 0.29/0.54      ! [N2: nat,M: nat] :
% 0.29/0.54        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.54       => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
% 0.29/0.54          = M ) ) ).
% 0.29/0.54  
% 0.29/0.54  % div_mult_self1_is_m
% 0.29/0.54  thf(fact_2749_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.54  thf(fact_2750_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.54  thf(fact_2751_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.54  thf(fact_2752_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.54  thf(fact_2753_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ab_semigroup_mult_class.mult_ac(1)
% 0.29/0.54  thf(fact_2754_mult_Oassoc,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.assoc
% 0.29/0.54  thf(fact_2755_mult_Oassoc,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.assoc
% 0.29/0.54  thf(fact_2756_mult_Oassoc,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.assoc
% 0.29/0.54  thf(fact_2757_mult_Oassoc,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.assoc
% 0.29/0.54  thf(fact_2758_mult_Oassoc,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 0.29/0.54        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.assoc
% 0.29/0.54  thf(fact_2759_mult_Ocommute,axiom,
% 0.29/0.54      ( times_times_complex
% 0.29/0.54      = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.commute
% 0.29/0.54  thf(fact_2760_mult_Ocommute,axiom,
% 0.29/0.54      ( times_times_real
% 0.29/0.54      = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.commute
% 0.29/0.54  thf(fact_2761_mult_Ocommute,axiom,
% 0.29/0.54      ( times_times_rat
% 0.29/0.54      = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.commute
% 0.29/0.54  thf(fact_2762_mult_Ocommute,axiom,
% 0.29/0.54      ( times_times_nat
% 0.29/0.54      = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.commute
% 0.29/0.54  thf(fact_2763_mult_Ocommute,axiom,
% 0.29/0.54      ( times_times_int
% 0.29/0.54      = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.commute
% 0.29/0.54  thf(fact_2764_mult_Oleft__commute,axiom,
% 0.29/0.54      ! [B: complex,A: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 0.29/0.54        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.left_commute
% 0.29/0.54  thf(fact_2765_mult_Oleft__commute,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 0.29/0.54        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.left_commute
% 0.29/0.54  thf(fact_2766_mult_Oleft__commute,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 0.29/0.54        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.left_commute
% 0.29/0.54  thf(fact_2767_mult_Oleft__commute,axiom,
% 0.29/0.54      ! [B: nat,A: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 0.29/0.54        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.left_commute
% 0.29/0.54  thf(fact_2768_mult_Oleft__commute,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 0.29/0.54        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.left_commute
% 0.29/0.54  thf(fact_2769_not__iless0,axiom,
% 0.29/0.54      ! [N2: extended_enat] :
% 0.29/0.54        ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% 0.29/0.54  
% 0.29/0.54  % not_iless0
% 0.29/0.54  thf(fact_2770_enat__less__induct,axiom,
% 0.29/0.54      ! [P: extended_enat > $o,N2: extended_enat] :
% 0.29/0.54        ( ! [N3: extended_enat] :
% 0.29/0.54            ( ! [M6: extended_enat] :
% 0.29/0.54                ( ( ord_le72135733267957522d_enat @ M6 @ N3 )
% 0.29/0.54               => ( P @ M6 ) )
% 0.29/0.54           => ( P @ N3 ) )
% 0.29/0.54       => ( P @ N2 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % enat_less_induct
% 0.29/0.54  thf(fact_2771_mult__right__cancel,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( ( times_times_complex @ A @ C )
% 0.29/0.54            = ( times_times_complex @ B @ C ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_cancel
% 0.29/0.54  thf(fact_2772_mult__right__cancel,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( ( times_times_real @ A @ C )
% 0.29/0.54            = ( times_times_real @ B @ C ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_cancel
% 0.29/0.54  thf(fact_2773_mult__right__cancel,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( ( times_times_rat @ A @ C )
% 0.29/0.54            = ( times_times_rat @ B @ C ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_cancel
% 0.29/0.54  thf(fact_2774_mult__right__cancel,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( C != zero_zero_nat )
% 0.29/0.54       => ( ( ( times_times_nat @ A @ C )
% 0.29/0.54            = ( times_times_nat @ B @ C ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_cancel
% 0.29/0.54  thf(fact_2775_mult__right__cancel,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( C != zero_zero_int )
% 0.29/0.54       => ( ( ( times_times_int @ A @ C )
% 0.29/0.54            = ( times_times_int @ B @ C ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_cancel
% 0.29/0.54  thf(fact_2776_mult__left__cancel,axiom,
% 0.29/0.54      ! [C: complex,A: complex,B: complex] :
% 0.29/0.54        ( ( C != zero_zero_complex )
% 0.29/0.54       => ( ( ( times_times_complex @ C @ A )
% 0.29/0.54            = ( times_times_complex @ C @ B ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_cancel
% 0.29/0.54  thf(fact_2777_mult__left__cancel,axiom,
% 0.29/0.54      ! [C: real,A: real,B: real] :
% 0.29/0.54        ( ( C != zero_zero_real )
% 0.29/0.54       => ( ( ( times_times_real @ C @ A )
% 0.29/0.54            = ( times_times_real @ C @ B ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_cancel
% 0.29/0.54  thf(fact_2778_mult__left__cancel,axiom,
% 0.29/0.54      ! [C: rat,A: rat,B: rat] :
% 0.29/0.54        ( ( C != zero_zero_rat )
% 0.29/0.54       => ( ( ( times_times_rat @ C @ A )
% 0.29/0.54            = ( times_times_rat @ C @ B ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_cancel
% 0.29/0.54  thf(fact_2779_mult__left__cancel,axiom,
% 0.29/0.54      ! [C: nat,A: nat,B: nat] :
% 0.29/0.54        ( ( C != zero_zero_nat )
% 0.29/0.54       => ( ( ( times_times_nat @ C @ A )
% 0.29/0.54            = ( times_times_nat @ C @ B ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_cancel
% 0.29/0.54  thf(fact_2780_mult__left__cancel,axiom,
% 0.29/0.54      ! [C: int,A: int,B: int] :
% 0.29/0.54        ( ( C != zero_zero_int )
% 0.29/0.54       => ( ( ( times_times_int @ C @ A )
% 0.29/0.54            = ( times_times_int @ C @ B ) )
% 0.29/0.54          = ( A = B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_cancel
% 0.29/0.54  thf(fact_2781_no__zero__divisors,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( A != zero_zero_complex )
% 0.29/0.54       => ( ( B != zero_zero_complex )
% 0.29/0.54         => ( ( times_times_complex @ A @ B )
% 0.29/0.54           != zero_zero_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % no_zero_divisors
% 0.29/0.54  thf(fact_2782_no__zero__divisors,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( A != zero_zero_real )
% 0.29/0.54       => ( ( B != zero_zero_real )
% 0.29/0.54         => ( ( times_times_real @ A @ B )
% 0.29/0.54           != zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % no_zero_divisors
% 0.29/0.54  thf(fact_2783_no__zero__divisors,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( A != zero_zero_rat )
% 0.29/0.54       => ( ( B != zero_zero_rat )
% 0.29/0.54         => ( ( times_times_rat @ A @ B )
% 0.29/0.54           != zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % no_zero_divisors
% 0.29/0.54  thf(fact_2784_no__zero__divisors,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( A != zero_zero_nat )
% 0.29/0.54       => ( ( B != zero_zero_nat )
% 0.29/0.54         => ( ( times_times_nat @ A @ B )
% 0.29/0.54           != zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % no_zero_divisors
% 0.29/0.54  thf(fact_2785_no__zero__divisors,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( A != zero_zero_int )
% 0.29/0.54       => ( ( B != zero_zero_int )
% 0.29/0.54         => ( ( times_times_int @ A @ B )
% 0.29/0.54           != zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % no_zero_divisors
% 0.29/0.54  thf(fact_2786_divisors__zero,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ A @ B )
% 0.29/0.54          = zero_zero_complex )
% 0.29/0.54       => ( ( A = zero_zero_complex )
% 0.29/0.54          | ( B = zero_zero_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divisors_zero
% 0.29/0.54  thf(fact_2787_divisors__zero,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( times_times_real @ A @ B )
% 0.29/0.54          = zero_zero_real )
% 0.29/0.54       => ( ( A = zero_zero_real )
% 0.29/0.54          | ( B = zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divisors_zero
% 0.29/0.54  thf(fact_2788_divisors__zero,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ A @ B )
% 0.29/0.54          = zero_zero_rat )
% 0.29/0.54       => ( ( A = zero_zero_rat )
% 0.29/0.54          | ( B = zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divisors_zero
% 0.29/0.54  thf(fact_2789_divisors__zero,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ A @ B )
% 0.29/0.54          = zero_zero_nat )
% 0.29/0.54       => ( ( A = zero_zero_nat )
% 0.29/0.54          | ( B = zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divisors_zero
% 0.29/0.54  thf(fact_2790_divisors__zero,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( times_times_int @ A @ B )
% 0.29/0.54          = zero_zero_int )
% 0.29/0.54       => ( ( A = zero_zero_int )
% 0.29/0.54          | ( B = zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divisors_zero
% 0.29/0.54  thf(fact_2791_mult__not__zero,axiom,
% 0.29/0.54      ! [A: complex,B: complex] :
% 0.29/0.54        ( ( ( times_times_complex @ A @ B )
% 0.29/0.54         != zero_zero_complex )
% 0.29/0.54       => ( ( A != zero_zero_complex )
% 0.29/0.54          & ( B != zero_zero_complex ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_not_zero
% 0.29/0.54  thf(fact_2792_mult__not__zero,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( times_times_real @ A @ B )
% 0.29/0.54         != zero_zero_real )
% 0.29/0.54       => ( ( A != zero_zero_real )
% 0.29/0.54          & ( B != zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_not_zero
% 0.29/0.54  thf(fact_2793_mult__not__zero,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( times_times_rat @ A @ B )
% 0.29/0.54         != zero_zero_rat )
% 0.29/0.54       => ( ( A != zero_zero_rat )
% 0.29/0.54          & ( B != zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_not_zero
% 0.29/0.54  thf(fact_2794_mult__not__zero,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ A @ B )
% 0.29/0.54         != zero_zero_nat )
% 0.29/0.54       => ( ( A != zero_zero_nat )
% 0.29/0.54          & ( B != zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_not_zero
% 0.29/0.54  thf(fact_2795_mult__not__zero,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( times_times_int @ A @ B )
% 0.29/0.54         != zero_zero_int )
% 0.29/0.54       => ( ( A != zero_zero_int )
% 0.29/0.54          & ( B != zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_not_zero
% 0.29/0.54  thf(fact_2796_comm__monoid__mult__class_Omult__1,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ one_one_complex @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_monoid_mult_class.mult_1
% 0.29/0.54  thf(fact_2797_comm__monoid__mult__class_Omult__1,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ one_one_real @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_monoid_mult_class.mult_1
% 0.29/0.54  thf(fact_2798_comm__monoid__mult__class_Omult__1,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ one_one_rat @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_monoid_mult_class.mult_1
% 0.29/0.54  thf(fact_2799_comm__monoid__mult__class_Omult__1,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ one_one_nat @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_monoid_mult_class.mult_1
% 0.29/0.54  thf(fact_2800_comm__monoid__mult__class_Omult__1,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ one_one_int @ A )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_monoid_mult_class.mult_1
% 0.29/0.54  thf(fact_2801_mult_Ocomm__neutral,axiom,
% 0.29/0.54      ! [A: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ one_one_complex )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.comm_neutral
% 0.29/0.54  thf(fact_2802_mult_Ocomm__neutral,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ( ( times_times_real @ A @ one_one_real )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.comm_neutral
% 0.29/0.54  thf(fact_2803_mult_Ocomm__neutral,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ one_one_rat )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.comm_neutral
% 0.29/0.54  thf(fact_2804_mult_Ocomm__neutral,axiom,
% 0.29/0.54      ! [A: nat] :
% 0.29/0.54        ( ( times_times_nat @ A @ one_one_nat )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.comm_neutral
% 0.29/0.54  thf(fact_2805_mult_Ocomm__neutral,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ( ( times_times_int @ A @ one_one_int )
% 0.29/0.54        = A ) ).
% 0.29/0.54  
% 0.29/0.54  % mult.comm_neutral
% 0.29/0.54  thf(fact_2806_combine__common__factor,axiom,
% 0.29/0.54      ! [A: complex,E2: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % combine_common_factor
% 0.29/0.54  thf(fact_2807_combine__common__factor,axiom,
% 0.29/0.54      ! [A: real,E2: real,B: real,C: real] :
% 0.29/0.54        ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % combine_common_factor
% 0.29/0.54  thf(fact_2808_combine__common__factor,axiom,
% 0.29/0.54      ! [A: rat,E2: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % combine_common_factor
% 0.29/0.54  thf(fact_2809_combine__common__factor,axiom,
% 0.29/0.54      ! [A: nat,E2: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % combine_common_factor
% 0.29/0.54  thf(fact_2810_combine__common__factor,axiom,
% 0.29/0.54      ! [A: int,E2: int,B: int,C: int] :
% 0.29/0.54        ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 0.29/0.54  
% 0.29/0.54  % combine_common_factor
% 0.29/0.54  thf(fact_2811_distrib__right,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right
% 0.29/0.54  thf(fact_2812_distrib__right,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right
% 0.29/0.54  thf(fact_2813_distrib__right,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right
% 0.29/0.54  thf(fact_2814_distrib__right,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right
% 0.29/0.54  thf(fact_2815_distrib__right,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_right
% 0.29/0.54  thf(fact_2816_distrib__left,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left
% 0.29/0.54  thf(fact_2817_distrib__left,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left
% 0.29/0.54  thf(fact_2818_distrib__left,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left
% 0.29/0.54  thf(fact_2819_distrib__left,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left
% 0.29/0.54  thf(fact_2820_distrib__left,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % distrib_left
% 0.29/0.54  thf(fact_2821_comm__semiring__class_Odistrib,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_semiring_class.distrib
% 0.29/0.54  thf(fact_2822_comm__semiring__class_Odistrib,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_semiring_class.distrib
% 0.29/0.54  thf(fact_2823_comm__semiring__class_Odistrib,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_semiring_class.distrib
% 0.29/0.54  thf(fact_2824_comm__semiring__class_Odistrib,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_semiring_class.distrib
% 0.29/0.54  thf(fact_2825_comm__semiring__class_Odistrib,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % comm_semiring_class.distrib
% 0.29/0.54  thf(fact_2826_ring__class_Oring__distribs_I1_J,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(1)
% 0.29/0.54  thf(fact_2827_ring__class_Oring__distribs_I1_J,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(1)
% 0.29/0.54  thf(fact_2828_ring__class_Oring__distribs_I1_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(1)
% 0.29/0.54  thf(fact_2829_ring__class_Oring__distribs_I1_J,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(1)
% 0.29/0.54  thf(fact_2830_ring__class_Oring__distribs_I2_J,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(2)
% 0.29/0.54  thf(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(2)
% 0.29/0.54  thf(fact_2832_ring__class_Oring__distribs_I2_J,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(2)
% 0.29/0.54  thf(fact_2833_ring__class_Oring__distribs_I2_J,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.29/0.54        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ring_class.ring_distribs(2)
% 0.29/0.54  thf(fact_2834_crossproduct__noteq,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex,D: complex] :
% 0.29/0.54        ( ( ( A != B )
% 0.29/0.54          & ( C != D ) )
% 0.29/0.54        = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
% 0.29/0.54         != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_noteq
% 0.29/0.54  thf(fact_2835_crossproduct__noteq,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ( A != B )
% 0.29/0.54          & ( C != D ) )
% 0.29/0.54        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 0.29/0.54         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_noteq
% 0.29/0.54  thf(fact_2836_crossproduct__noteq,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ( A != B )
% 0.29/0.54          & ( C != D ) )
% 0.29/0.54        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 0.29/0.54         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_noteq
% 0.29/0.54  thf(fact_2837_crossproduct__noteq,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ( A != B )
% 0.29/0.54          & ( C != D ) )
% 0.29/0.54        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 0.29/0.54         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_noteq
% 0.29/0.54  thf(fact_2838_crossproduct__noteq,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ( A != B )
% 0.29/0.54          & ( C != D ) )
% 0.29/0.54        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 0.29/0.54         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_noteq
% 0.29/0.54  thf(fact_2839_crossproduct__eq,axiom,
% 0.29/0.54      ! [W2: complex,Y3: complex,X3: complex,Z2: complex] :
% 0.29/0.54        ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y3 ) @ ( times_times_complex @ X3 @ Z2 ) )
% 0.29/0.54          = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z2 ) @ ( times_times_complex @ X3 @ Y3 ) ) )
% 0.29/0.54        = ( ( W2 = X3 )
% 0.29/0.54          | ( Y3 = Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_eq
% 0.29/0.54  thf(fact_2840_crossproduct__eq,axiom,
% 0.29/0.54      ! [W2: real,Y3: real,X3: real,Z2: real] :
% 0.29/0.54        ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y3 ) @ ( times_times_real @ X3 @ Z2 ) )
% 0.29/0.54          = ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X3 @ Y3 ) ) )
% 0.29/0.54        = ( ( W2 = X3 )
% 0.29/0.54          | ( Y3 = Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_eq
% 0.29/0.54  thf(fact_2841_crossproduct__eq,axiom,
% 0.29/0.54      ! [W2: rat,Y3: rat,X3: rat,Z2: rat] :
% 0.29/0.54        ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y3 ) @ ( times_times_rat @ X3 @ Z2 ) )
% 0.29/0.54          = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X3 @ Y3 ) ) )
% 0.29/0.54        = ( ( W2 = X3 )
% 0.29/0.54          | ( Y3 = Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_eq
% 0.29/0.54  thf(fact_2842_crossproduct__eq,axiom,
% 0.29/0.54      ! [W2: nat,Y3: nat,X3: nat,Z2: nat] :
% 0.29/0.54        ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y3 ) @ ( times_times_nat @ X3 @ Z2 ) )
% 0.29/0.54          = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X3 @ Y3 ) ) )
% 0.29/0.54        = ( ( W2 = X3 )
% 0.29/0.54          | ( Y3 = Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_eq
% 0.29/0.54  thf(fact_2843_crossproduct__eq,axiom,
% 0.29/0.54      ! [W2: int,Y3: int,X3: int,Z2: int] :
% 0.29/0.54        ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y3 ) @ ( times_times_int @ X3 @ Z2 ) )
% 0.29/0.54          = ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X3 @ Y3 ) ) )
% 0.29/0.54        = ( ( W2 = X3 )
% 0.29/0.54          | ( Y3 = Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % crossproduct_eq
% 0.29/0.54  thf(fact_2844_times__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
% 0.29/0.54        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ Y3 @ W2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_times_eq
% 0.29/0.54  thf(fact_2845_times__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: real,Y3: real,Z2: real,W2: real] :
% 0.29/0.54        ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide_divide_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ W2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_times_eq
% 0.29/0.54  thf(fact_2846_times__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
% 0.29/0.54        ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ W2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % times_divide_times_eq
% 0.29/0.54  thf(fact_2847_divide__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
% 0.29/0.54        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W2 ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_times_eq
% 0.29/0.54  thf(fact_2848_divide__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: real,Y3: real,Z2: real,W2: real] :
% 0.29/0.54        ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide_divide_real @ ( times_times_real @ X3 @ W2 ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_times_eq
% 0.29/0.54  thf(fact_2849_divide__divide__times__eq,axiom,
% 0.29/0.54      ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
% 0.29/0.54        ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
% 0.29/0.54        = ( divide_divide_rat @ ( times_times_rat @ X3 @ W2 ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_times_eq
% 0.29/0.54  thf(fact_2850_divide__divide__eq__left_H,axiom,
% 0.29/0.54      ! [A: complex,B: complex,C: complex] :
% 0.29/0.54        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 0.29/0.54        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left'
% 0.29/0.54  thf(fact_2851_divide__divide__eq__left_H,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 0.29/0.54        = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left'
% 0.29/0.54  thf(fact_2852_divide__divide__eq__left_H,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 0.29/0.54        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % divide_divide_eq_left'
% 0.29/0.54  thf(fact_2853_Suc__mult__cancel1,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 0.29/0.54          = ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.29/0.54        = ( M = N2 ) ) ).
% 0.29/0.54  
% 0.29/0.54  % Suc_mult_cancel1
% 0.29/0.54  thf(fact_2854_mult__0,axiom,
% 0.29/0.54      ! [N2: nat] :
% 0.29/0.54        ( ( times_times_nat @ zero_zero_nat @ N2 )
% 0.29/0.54        = zero_zero_nat ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_0
% 0.29/0.54  thf(fact_2855_nat__mult__eq__cancel__disj,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( ( times_times_nat @ K @ M )
% 0.29/0.54          = ( times_times_nat @ K @ N2 ) )
% 0.29/0.54        = ( ( K = zero_zero_nat )
% 0.29/0.54          | ( M = N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_eq_cancel_disj
% 0.29/0.54  thf(fact_2856_mult__le__mono2,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.54       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_mono2
% 0.29/0.54  thf(fact_2857_mult__le__mono1,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.54       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_mono1
% 0.29/0.54  thf(fact_2858_mult__le__mono,axiom,
% 0.29/0.54      ! [I: nat,J: nat,K: nat,L: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ I @ J )
% 0.29/0.54       => ( ( ord_less_eq_nat @ K @ L )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_mono
% 0.29/0.54  thf(fact_2859_le__square,axiom,
% 0.29/0.54      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_square
% 0.29/0.54  thf(fact_2860_le__cube,axiom,
% 0.29/0.54      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % le_cube
% 0.29/0.54  thf(fact_2861_add__mult__distrib2,axiom,
% 0.29/0.54      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.54        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_mult_distrib2
% 0.29/0.54  thf(fact_2862_add__mult__distrib,axiom,
% 0.29/0.54      ! [M: nat,N2: nat,K: nat] :
% 0.29/0.54        ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % add_mult_distrib
% 0.29/0.54  thf(fact_2863_left__add__mult__distrib,axiom,
% 0.29/0.54      ! [I: nat,U: nat,J: nat,K: nat] :
% 0.29/0.54        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 0.29/0.54        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 0.29/0.54  
% 0.29/0.54  % left_add_mult_distrib
% 0.29/0.54  thf(fact_2864_nat__mult__1,axiom,
% 0.29/0.54      ! [N2: nat] :
% 0.29/0.54        ( ( times_times_nat @ one_one_nat @ N2 )
% 0.29/0.54        = N2 ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_1
% 0.29/0.54  thf(fact_2865_nat__mult__1__right,axiom,
% 0.29/0.54      ! [N2: nat] :
% 0.29/0.54        ( ( times_times_nat @ N2 @ one_one_nat )
% 0.29/0.54        = N2 ) ).
% 0.29/0.54  
% 0.29/0.54  % nat_mult_1_right
% 0.29/0.54  thf(fact_2866_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.29/0.54  thf(fact_2867_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.29/0.54  thf(fact_2868_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.29/0.54  thf(fact_2869_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % ordered_comm_semiring_class.comm_mult_left_mono
% 0.29/0.54  thf(fact_2870_zero__le__mult__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_mult_iff
% 0.29/0.54  thf(fact_2871_zero__le__mult__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_mult_iff
% 0.29/0.54  thf(fact_2872_zero__le__mult__iff,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.29/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_mult_iff
% 0.29/0.54  thf(fact_2873_mult__nonneg__nonpos2,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos2
% 0.29/0.54  thf(fact_2874_mult__nonneg__nonpos2,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos2
% 0.29/0.54  thf(fact_2875_mult__nonneg__nonpos2,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos2
% 0.29/0.54  thf(fact_2876_mult__nonneg__nonpos2,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos2
% 0.29/0.54  thf(fact_2877_mult__nonpos__nonneg,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonneg
% 0.29/0.54  thf(fact_2878_mult__nonpos__nonneg,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonneg
% 0.29/0.54  thf(fact_2879_mult__nonpos__nonneg,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonneg
% 0.29/0.54  thf(fact_2880_mult__nonpos__nonneg,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonneg
% 0.29/0.54  thf(fact_2881_mult__nonneg__nonpos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos
% 0.29/0.54  thf(fact_2882_mult__nonneg__nonpos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos
% 0.29/0.54  thf(fact_2883_mult__nonneg__nonpos,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos
% 0.29/0.54  thf(fact_2884_mult__nonneg__nonpos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonpos
% 0.29/0.54  thf(fact_2885_mult__nonneg__nonneg,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.54         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonneg
% 0.29/0.54  thf(fact_2886_mult__nonneg__nonneg,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.54         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonneg
% 0.29/0.54  thf(fact_2887_mult__nonneg__nonneg,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.54         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonneg
% 0.29/0.54  thf(fact_2888_mult__nonneg__nonneg,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.54         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonneg_nonneg
% 0.29/0.54  thf(fact_2889_split__mult__neg__le,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.29/0.54          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 0.29/0.54       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_neg_le
% 0.29/0.54  thf(fact_2890_split__mult__neg__le,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.29/0.54          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 0.29/0.54       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_neg_le
% 0.29/0.54  thf(fact_2891_split__mult__neg__le,axiom,
% 0.29/0.54      ! [A: nat,B: nat] :
% 0.29/0.54        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 0.29/0.54          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 0.29/0.54            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 0.29/0.54       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_neg_le
% 0.29/0.54  thf(fact_2892_split__mult__neg__le,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.29/0.54          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 0.29/0.54       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_neg_le
% 0.29/0.54  thf(fact_2893_mult__le__0__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.29/0.54        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 0.29/0.54          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_0_iff
% 0.29/0.54  thf(fact_2894_mult__le__0__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.54        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 0.29/0.54          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_0_iff
% 0.29/0.54  thf(fact_2895_mult__le__0__iff,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.29/0.54        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 0.29/0.54          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_le_0_iff
% 0.29/0.54  thf(fact_2896_mult__right__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_2897_mult__right__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_2898_mult__right__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_2899_mult__right__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono
% 0.29/0.54  thf(fact_2900_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_2901_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_2902_mult__right__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_right_mono_neg
% 0.29/0.54  thf(fact_2903_mult__left__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_2904_mult__left__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_2905_mult__left__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_2906_mult__left__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono
% 0.29/0.54  thf(fact_2907_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_2908_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_2909_mult__nonpos__nonpos,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_nonpos_nonpos
% 0.29/0.54  thf(fact_2910_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: real,A: real,C: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.54         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_2911_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: rat,A: rat,C: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_2912_mult__left__mono__neg,axiom,
% 0.29/0.54      ! [B: int,A: int,C: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ B @ A )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.54         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_left_mono_neg
% 0.29/0.54  thf(fact_2913_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.29/0.54       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_2914_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.29/0.54            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.29/0.54       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_2915_split__mult__pos__le,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 0.29/0.54          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.29/0.54            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.29/0.54       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % split_mult_pos_le
% 0.29/0.54  thf(fact_2916_zero__le__square,axiom,
% 0.29/0.54      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_2917_zero__le__square,axiom,
% 0.29/0.54      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_2918_zero__le__square,axiom,
% 0.29/0.54      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 0.29/0.54  
% 0.29/0.54  % zero_le_square
% 0.29/0.54  thf(fact_2919_mult__mono_H,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_2920_mult__mono_H,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_2921_mult__mono_H,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_2922_mult__mono_H,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono'
% 0.29/0.54  thf(fact_2923_mult__mono,axiom,
% 0.29/0.54      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.54        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.54           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.54             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_2924_mult__mono,axiom,
% 0.29/0.54      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.54        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.54           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.54             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_2925_mult__mono,axiom,
% 0.29/0.54      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.54        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.54           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.54             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_2926_mult__mono,axiom,
% 0.29/0.54      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.54        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.54       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.54         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.54           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.54             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_mono
% 0.29/0.54  thf(fact_2927_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.54         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_2928_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.54       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.54         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_2929_mult__neg__neg,axiom,
% 0.29/0.54      ! [A: int,B: int] :
% 0.29/0.54        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.54       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.54         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_neg_neg
% 0.29/0.54  thf(fact_2930_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: real] :
% 0.29/0.54        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_2931_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: rat] :
% 0.29/0.54        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_2932_not__square__less__zero,axiom,
% 0.29/0.54      ! [A: int] :
% 0.29/0.54        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 0.29/0.54  
% 0.29/0.54  % not_square_less_zero
% 0.29/0.54  thf(fact_2933_mult__less__0__iff,axiom,
% 0.29/0.54      ! [A: real,B: real] :
% 0.29/0.54        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 0.29/0.54        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.54            & ( ord_less_real @ B @ zero_zero_real ) )
% 0.29/0.54          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.54            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 0.29/0.54  
% 0.29/0.54  % mult_less_0_iff
% 0.29/0.54  thf(fact_2934_mult__less__0__iff,axiom,
% 0.29/0.54      ! [A: rat,B: rat] :
% 0.29/0.54        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 0.29/0.54        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.54            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 0.29/0.54          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.55            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_0_iff
% 0.29/0.55  thf(fact_2935_mult__less__0__iff,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55            & ( ord_less_int @ B @ zero_zero_int ) )
% 0.29/0.55          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.55            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_0_iff
% 0.29/0.55  thf(fact_2936_mult__neg__pos,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_neg_pos
% 0.29/0.55  thf(fact_2937_mult__neg__pos,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_neg_pos
% 0.29/0.55  thf(fact_2938_mult__neg__pos,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ zero_zero_nat )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_neg_pos
% 0.29/0.55  thf(fact_2939_mult__neg__pos,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_neg_pos
% 0.29/0.55  thf(fact_2940_mult__pos__neg,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg
% 0.29/0.55  thf(fact_2941_mult__pos__neg,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg
% 0.29/0.55  thf(fact_2942_mult__pos__neg,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg
% 0.29/0.55  thf(fact_2943_mult__pos__neg,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg
% 0.29/0.55  thf(fact_2944_mult__pos__pos,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_pos
% 0.29/0.55  thf(fact_2945_mult__pos__pos,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.55         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_pos
% 0.29/0.55  thf(fact_2946_mult__pos__pos,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.55         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_pos
% 0.29/0.55  thf(fact_2947_mult__pos__pos,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.55         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_pos
% 0.29/0.55  thf(fact_2948_mult__pos__neg2,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( ord_less_real @ B @ zero_zero_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg2
% 0.29/0.55  thf(fact_2949_mult__pos__neg2,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg2
% 0.29/0.55  thf(fact_2950_mult__pos__neg2,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg2
% 0.29/0.55  thf(fact_2951_mult__pos__neg2,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_pos_neg2
% 0.29/0.55  thf(fact_2952_zero__less__mult__iff,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55            & ( ord_less_real @ zero_zero_real @ B ) )
% 0.29/0.55          | ( ( ord_less_real @ A @ zero_zero_real )
% 0.29/0.55            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_iff
% 0.29/0.55  thf(fact_2953_zero__less__mult__iff,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 0.29/0.55          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.29/0.55            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_iff
% 0.29/0.55  thf(fact_2954_zero__less__mult__iff,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55            & ( ord_less_int @ zero_zero_int @ B ) )
% 0.29/0.55          | ( ( ord_less_int @ A @ zero_zero_int )
% 0.29/0.55            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_iff
% 0.29/0.55  thf(fact_2955_zero__less__mult__pos,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos
% 0.29/0.55  thf(fact_2956_zero__less__mult__pos,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos
% 0.29/0.55  thf(fact_2957_zero__less__mult__pos,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos
% 0.29/0.55  thf(fact_2958_zero__less__mult__pos,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos
% 0.29/0.55  thf(fact_2959_zero__less__mult__pos2,axiom,
% 0.29/0.55      ! [B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos2
% 0.29/0.55  thf(fact_2960_zero__less__mult__pos2,axiom,
% 0.29/0.55      ! [B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos2
% 0.29/0.55  thf(fact_2961_zero__less__mult__pos2,axiom,
% 0.29/0.55      ! [B: nat,A: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos2
% 0.29/0.55  thf(fact_2962_zero__less__mult__pos2,axiom,
% 0.29/0.55      ! [B: int,A: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % zero_less_mult_pos2
% 0.29/0.55  thf(fact_2963_mult__less__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55          = ( ord_less_real @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_neg
% 0.29/0.55  thf(fact_2964_mult__less__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55          = ( ord_less_rat @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_neg
% 0.29/0.55  thf(fact_2965_mult__less__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55          = ( ord_less_int @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_neg
% 0.29/0.55  thf(fact_2966_mult__less__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55          = ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_pos
% 0.29/0.55  thf(fact_2967_mult__less__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55          = ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_pos
% 0.29/0.55  thf(fact_2968_mult__less__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55          = ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_pos
% 0.29/0.55  thf(fact_2969_mult__strict__left__mono__neg,axiom,
% 0.29/0.55      ! [B: real,A: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ B @ A )
% 0.29/0.55       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono_neg
% 0.29/0.55  thf(fact_2970_mult__strict__left__mono__neg,axiom,
% 0.29/0.55      ! [B: rat,A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ B @ A )
% 0.29/0.55       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono_neg
% 0.29/0.55  thf(fact_2971_mult__strict__left__mono__neg,axiom,
% 0.29/0.55      ! [B: int,A: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ B @ A )
% 0.29/0.55       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono_neg
% 0.29/0.55  thf(fact_2972_mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono
% 0.29/0.55  thf(fact_2973_mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono
% 0.29/0.55  thf(fact_2974_mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono
% 0.29/0.55  thf(fact_2975_mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_left_mono
% 0.29/0.55  thf(fact_2976_mult__less__cancel__left__disj,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55            & ( ord_less_real @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_disj
% 0.29/0.55  thf(fact_2977_mult__less__cancel__left__disj,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55            & ( ord_less_rat @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_disj
% 0.29/0.55  thf(fact_2978_mult__less__cancel__left__disj,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55            & ( ord_less_int @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left_disj
% 0.29/0.55  thf(fact_2979_mult__strict__right__mono__neg,axiom,
% 0.29/0.55      ! [B: real,A: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ B @ A )
% 0.29/0.55       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono_neg
% 0.29/0.55  thf(fact_2980_mult__strict__right__mono__neg,axiom,
% 0.29/0.55      ! [B: rat,A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ B @ A )
% 0.29/0.55       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono_neg
% 0.29/0.55  thf(fact_2981_mult__strict__right__mono__neg,axiom,
% 0.29/0.55      ! [B: int,A: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ B @ A )
% 0.29/0.55       => ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono_neg
% 0.29/0.55  thf(fact_2982_mult__strict__right__mono,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono
% 0.29/0.55  thf(fact_2983_mult__strict__right__mono,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono
% 0.29/0.55  thf(fact_2984_mult__strict__right__mono,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono
% 0.29/0.55  thf(fact_2985_mult__strict__right__mono,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_right_mono
% 0.29/0.55  thf(fact_2986_mult__less__cancel__right__disj,axiom,
% 0.29/0.55      ! [A: real,C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55            & ( ord_less_real @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55            & ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right_disj
% 0.29/0.55  thf(fact_2987_mult__less__cancel__right__disj,axiom,
% 0.29/0.55      ! [A: rat,C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55            & ( ord_less_rat @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right_disj
% 0.29/0.55  thf(fact_2988_mult__less__cancel__right__disj,axiom,
% 0.29/0.55      ! [A: int,C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55            & ( ord_less_int @ A @ B ) )
% 0.29/0.55          | ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55            & ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right_disj
% 0.29/0.55  thf(fact_2989_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.55  thf(fact_2990_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.55  thf(fact_2991_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.55  thf(fact_2992_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
% 0.29/0.55  thf(fact_2993_add__scale__eq__noteq,axiom,
% 0.29/0.55      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 0.29/0.55        ( ( R2 != zero_zero_complex )
% 0.29/0.55       => ( ( ( A = B )
% 0.29/0.55            & ( C != D ) )
% 0.29/0.55         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 0.29/0.55           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_scale_eq_noteq
% 0.29/0.55  thf(fact_2994_add__scale__eq__noteq,axiom,
% 0.29/0.55      ! [R2: real,A: real,B: real,C: real,D: real] :
% 0.29/0.55        ( ( R2 != zero_zero_real )
% 0.29/0.55       => ( ( ( A = B )
% 0.29/0.55            & ( C != D ) )
% 0.29/0.55         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 0.29/0.55           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_scale_eq_noteq
% 0.29/0.55  thf(fact_2995_add__scale__eq__noteq,axiom,
% 0.29/0.55      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.55        ( ( R2 != zero_zero_rat )
% 0.29/0.55       => ( ( ( A = B )
% 0.29/0.55            & ( C != D ) )
% 0.29/0.55         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 0.29/0.55           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_scale_eq_noteq
% 0.29/0.55  thf(fact_2996_add__scale__eq__noteq,axiom,
% 0.29/0.55      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.55        ( ( R2 != zero_zero_nat )
% 0.29/0.55       => ( ( ( A = B )
% 0.29/0.55            & ( C != D ) )
% 0.29/0.55         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 0.29/0.55           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_scale_eq_noteq
% 0.29/0.55  thf(fact_2997_add__scale__eq__noteq,axiom,
% 0.29/0.55      ! [R2: int,A: int,B: int,C: int,D: int] :
% 0.29/0.55        ( ( R2 != zero_zero_int )
% 0.29/0.55       => ( ( ( A = B )
% 0.29/0.55            & ( C != D ) )
% 0.29/0.55         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 0.29/0.55           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_scale_eq_noteq
% 0.29/0.55  thf(fact_2998_less__1__mult,axiom,
% 0.29/0.55      ! [M: real,N2: real] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ M )
% 0.29/0.55       => ( ( ord_less_real @ one_one_real @ N2 )
% 0.29/0.55         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_1_mult
% 0.29/0.55  thf(fact_2999_less__1__mult,axiom,
% 0.29/0.55      ! [M: rat,N2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ one_one_rat @ M )
% 0.29/0.55       => ( ( ord_less_rat @ one_one_rat @ N2 )
% 0.29/0.55         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_1_mult
% 0.29/0.55  thf(fact_3000_less__1__mult,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ one_one_nat @ M )
% 0.29/0.55       => ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.29/0.55         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_1_mult
% 0.29/0.55  thf(fact_3001_less__1__mult,axiom,
% 0.29/0.55      ! [M: int,N2: int] :
% 0.29/0.55        ( ( ord_less_int @ one_one_int @ M )
% 0.29/0.55       => ( ( ord_less_int @ one_one_int @ N2 )
% 0.29/0.55         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_1_mult
% 0.29/0.55  thf(fact_3002_frac__eq__eq,axiom,
% 0.29/0.55      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 0.29/0.55        ( ( Y3 != zero_zero_complex )
% 0.29/0.55       => ( ( Z2 != zero_zero_complex )
% 0.29/0.55         => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
% 0.29/0.55              = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.29/0.55            = ( ( times_times_complex @ X3 @ Z2 )
% 0.29/0.55              = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % frac_eq_eq
% 0.29/0.55  thf(fact_3003_frac__eq__eq,axiom,
% 0.29/0.55      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 0.29/0.55        ( ( Y3 != zero_zero_real )
% 0.29/0.55       => ( ( Z2 != zero_zero_real )
% 0.29/0.55         => ( ( ( divide_divide_real @ X3 @ Y3 )
% 0.29/0.55              = ( divide_divide_real @ W2 @ Z2 ) )
% 0.29/0.55            = ( ( times_times_real @ X3 @ Z2 )
% 0.29/0.55              = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % frac_eq_eq
% 0.29/0.55  thf(fact_3004_frac__eq__eq,axiom,
% 0.29/0.55      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 0.29/0.55        ( ( Y3 != zero_zero_rat )
% 0.29/0.55       => ( ( Z2 != zero_zero_rat )
% 0.29/0.55         => ( ( ( divide_divide_rat @ X3 @ Y3 )
% 0.29/0.55              = ( divide_divide_rat @ W2 @ Z2 ) )
% 0.29/0.55            = ( ( times_times_rat @ X3 @ Z2 )
% 0.29/0.55              = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % frac_eq_eq
% 0.29/0.55  thf(fact_3005_divide__eq__eq,axiom,
% 0.29/0.55      ! [B: complex,C: complex,A: complex] :
% 0.29/0.55        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ( C != zero_zero_complex )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_complex @ A @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_complex )
% 0.29/0.55           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq
% 0.29/0.55  thf(fact_3006_divide__eq__eq,axiom,
% 0.29/0.55      ! [B: real,C: real,A: real] :
% 0.29/0.55        ( ( ( divide_divide_real @ B @ C )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ( C != zero_zero_real )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_real @ A @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_real )
% 0.29/0.55           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq
% 0.29/0.55  thf(fact_3007_divide__eq__eq,axiom,
% 0.29/0.55      ! [B: rat,C: rat,A: rat] :
% 0.29/0.55        ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.55          = A )
% 0.29/0.55        = ( ( ( C != zero_zero_rat )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_rat @ A @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_rat )
% 0.29/0.55           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq
% 0.29/0.55  thf(fact_3008_eq__divide__eq,axiom,
% 0.29/0.55      ! [A: complex,B: complex,C: complex] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_complex )
% 0.29/0.55           => ( ( times_times_complex @ A @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_complex )
% 0.29/0.55           => ( A = zero_zero_complex ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq
% 0.29/0.55  thf(fact_3009_eq__divide__eq,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_real )
% 0.29/0.55           => ( ( times_times_real @ A @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_real )
% 0.29/0.55           => ( A = zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq
% 0.29/0.55  thf(fact_3010_eq__divide__eq,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( A
% 0.29/0.55          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_rat )
% 0.29/0.55           => ( ( times_times_rat @ A @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_rat )
% 0.29/0.55           => ( A = zero_zero_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq
% 0.29/0.55  thf(fact_3011_divide__eq__imp,axiom,
% 0.29/0.55      ! [C: complex,B: complex,A: complex] :
% 0.29/0.55        ( ( C != zero_zero_complex )
% 0.29/0.55       => ( ( B
% 0.29/0.55            = ( times_times_complex @ A @ C ) )
% 0.29/0.55         => ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.55            = A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_imp
% 0.29/0.55  thf(fact_3012_divide__eq__imp,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( C != zero_zero_real )
% 0.29/0.55       => ( ( B
% 0.29/0.55            = ( times_times_real @ A @ C ) )
% 0.29/0.55         => ( ( divide_divide_real @ B @ C )
% 0.29/0.55            = A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_imp
% 0.29/0.55  thf(fact_3013_divide__eq__imp,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( C != zero_zero_rat )
% 0.29/0.55       => ( ( B
% 0.29/0.55            = ( times_times_rat @ A @ C ) )
% 0.29/0.55         => ( ( divide_divide_rat @ B @ C )
% 0.29/0.55            = A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_imp
% 0.29/0.55  thf(fact_3014_eq__divide__imp,axiom,
% 0.29/0.55      ! [C: complex,A: complex,B: complex] :
% 0.29/0.55        ( ( C != zero_zero_complex )
% 0.29/0.55       => ( ( ( times_times_complex @ A @ C )
% 0.29/0.55            = B )
% 0.29/0.55         => ( A
% 0.29/0.55            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_imp
% 0.29/0.55  thf(fact_3015_eq__divide__imp,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( C != zero_zero_real )
% 0.29/0.55       => ( ( ( times_times_real @ A @ C )
% 0.29/0.55            = B )
% 0.29/0.55         => ( A
% 0.29/0.55            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_imp
% 0.29/0.55  thf(fact_3016_eq__divide__imp,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( C != zero_zero_rat )
% 0.29/0.55       => ( ( ( times_times_rat @ A @ C )
% 0.29/0.55            = B )
% 0.29/0.55         => ( A
% 0.29/0.55            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_imp
% 0.29/0.55  thf(fact_3017_nonzero__divide__eq__eq,axiom,
% 0.29/0.55      ! [C: complex,B: complex,A: complex] :
% 0.29/0.55        ( ( C != zero_zero_complex )
% 0.29/0.55       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.55            = A )
% 0.29/0.55          = ( B
% 0.29/0.55            = ( times_times_complex @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_divide_eq_eq
% 0.29/0.55  thf(fact_3018_nonzero__divide__eq__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( C != zero_zero_real )
% 0.29/0.55       => ( ( ( divide_divide_real @ B @ C )
% 0.29/0.55            = A )
% 0.29/0.55          = ( B
% 0.29/0.55            = ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_divide_eq_eq
% 0.29/0.55  thf(fact_3019_nonzero__divide__eq__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( C != zero_zero_rat )
% 0.29/0.55       => ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.55            = A )
% 0.29/0.55          = ( B
% 0.29/0.55            = ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_divide_eq_eq
% 0.29/0.55  thf(fact_3020_nonzero__eq__divide__eq,axiom,
% 0.29/0.55      ! [C: complex,A: complex,B: complex] :
% 0.29/0.55        ( ( C != zero_zero_complex )
% 0.29/0.55       => ( ( A
% 0.29/0.55            = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.55          = ( ( times_times_complex @ A @ C )
% 0.29/0.55            = B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_eq_divide_eq
% 0.29/0.55  thf(fact_3021_nonzero__eq__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( C != zero_zero_real )
% 0.29/0.55       => ( ( A
% 0.29/0.55            = ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ( times_times_real @ A @ C )
% 0.29/0.55            = B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_eq_divide_eq
% 0.29/0.55  thf(fact_3022_nonzero__eq__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( C != zero_zero_rat )
% 0.29/0.55       => ( ( A
% 0.29/0.55            = ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ( times_times_rat @ A @ C )
% 0.29/0.55            = B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nonzero_eq_divide_eq
% 0.29/0.55  thf(fact_3023_power__Suc,axiom,
% 0.29/0.55      ! [A: complex,N2: nat] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc
% 0.29/0.55  thf(fact_3024_power__Suc,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc
% 0.29/0.55  thf(fact_3025_power__Suc,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc
% 0.29/0.55  thf(fact_3026_power__Suc,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc
% 0.29/0.55  thf(fact_3027_power__Suc,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc
% 0.29/0.55  thf(fact_3028_power__Suc2,axiom,
% 0.29/0.55      ! [A: complex,N2: nat] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc2
% 0.29/0.55  thf(fact_3029_power__Suc2,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( power_power_real @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc2
% 0.29/0.55  thf(fact_3030_power__Suc2,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc2
% 0.29/0.55  thf(fact_3031_power__Suc2,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc2
% 0.29/0.55  thf(fact_3032_power__Suc2,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( power_power_int @ A @ ( suc @ N2 ) )
% 0.29/0.55        = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc2
% 0.29/0.55  thf(fact_3033_Suc__mult__less__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.29/0.55        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_mult_less_cancel1
% 0.29/0.55  thf(fact_3034_power__add,axiom,
% 0.29/0.55      ! [A: complex,M: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.55        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_add
% 0.29/0.55  thf(fact_3035_power__add,axiom,
% 0.29/0.55      ! [A: real,M: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.55        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_add
% 0.29/0.55  thf(fact_3036_power__add,axiom,
% 0.29/0.55      ! [A: rat,M: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.55        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_add
% 0.29/0.55  thf(fact_3037_power__add,axiom,
% 0.29/0.55      ! [A: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.55        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_add
% 0.29/0.55  thf(fact_3038_power__add,axiom,
% 0.29/0.55      ! [A: int,M: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
% 0.29/0.55        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_add
% 0.29/0.55  thf(fact_3039_mult__less__mono1,axiom,
% 0.29/0.55      ! [I: nat,J: nat,K: nat] :
% 0.29/0.55        ( ( ord_less_nat @ I @ J )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_mono1
% 0.29/0.55  thf(fact_3040_mult__less__mono2,axiom,
% 0.29/0.55      ! [I: nat,J: nat,K: nat] :
% 0.29/0.55        ( ( ord_less_nat @ I @ J )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_mono2
% 0.29/0.55  thf(fact_3041_nat__mult__eq__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( ( times_times_nat @ K @ M )
% 0.29/0.55            = ( times_times_nat @ K @ N2 ) )
% 0.29/0.55          = ( M = N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_eq_cancel1
% 0.29/0.55  thf(fact_3042_nat__mult__less__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.55          = ( ord_less_nat @ M @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_less_cancel1
% 0.29/0.55  thf(fact_3043_Suc__mult__le__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 0.29/0.55        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_mult_le_cancel1
% 0.29/0.55  thf(fact_3044_mult__Suc,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] :
% 0.29/0.55        ( ( times_times_nat @ ( suc @ M ) @ N2 )
% 0.29/0.55        = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_Suc
% 0.29/0.55  thf(fact_3045_mult__eq__self__implies__10,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] :
% 0.29/0.55        ( ( M
% 0.29/0.55          = ( times_times_nat @ M @ N2 ) )
% 0.29/0.55       => ( ( N2 = one_one_nat )
% 0.29/0.55          | ( M = zero_zero_nat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_eq_self_implies_10
% 0.29/0.55  thf(fact_3046_less__mult__imp__div__less,axiom,
% 0.29/0.55      ! [M: nat,I: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
% 0.29/0.55       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_mult_imp_div_less
% 0.29/0.55  thf(fact_3047_times__div__less__eq__dividend,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% 0.29/0.55  
% 0.29/0.55  % times_div_less_eq_dividend
% 0.29/0.55  thf(fact_3048_div__times__less__eq__dividend,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% 0.29/0.55  
% 0.29/0.55  % div_times_less_eq_dividend
% 0.29/0.55  thf(fact_3049_power__odd__eq,axiom,
% 0.29/0.55      ! [A: complex,N2: nat] :
% 0.29/0.55        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.29/0.55        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_odd_eq
% 0.29/0.55  thf(fact_3050_power__odd__eq,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.29/0.55        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_odd_eq
% 0.29/0.55  thf(fact_3051_power__odd__eq,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.29/0.55        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_odd_eq
% 0.29/0.55  thf(fact_3052_power__odd__eq,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.29/0.55        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_odd_eq
% 0.29/0.55  thf(fact_3053_power__odd__eq,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.29/0.55        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_odd_eq
% 0.29/0.55  thf(fact_3054_Suc__double__not__eq__double,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] :
% 0.29/0.55        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.29/0.55       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % Suc_double_not_eq_double
% 0.29/0.55  thf(fact_3055_double__not__eq__Suc__double,axiom,
% 0.29/0.55      ! [M: nat,N2: nat] :
% 0.29/0.55        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.29/0.55       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % double_not_eq_Suc_double
% 0.29/0.55  thf(fact_3056_mult__le__cancel__left,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left
% 0.29/0.55  thf(fact_3057_mult__le__cancel__left,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left
% 0.29/0.55  thf(fact_3058_mult__le__cancel__left,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left
% 0.29/0.55  thf(fact_3059_mult__le__cancel__right,axiom,
% 0.29/0.55      ! [A: real,C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right
% 0.29/0.55  thf(fact_3060_mult__le__cancel__right,axiom,
% 0.29/0.55      ! [A: rat,C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right
% 0.29/0.55  thf(fact_3061_mult__le__cancel__right,axiom,
% 0.29/0.55      ! [A: int,C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right
% 0.29/0.55  thf(fact_3062_mult__left__less__imp__less,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_less_imp_less
% 0.29/0.55  thf(fact_3063_mult__left__less__imp__less,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_less_imp_less
% 0.29/0.55  thf(fact_3064_mult__left__less__imp__less,axiom,
% 0.29/0.55      ! [C: nat,A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_less_imp_less
% 0.29/0.55  thf(fact_3065_mult__left__less__imp__less,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_less_imp_less
% 0.29/0.55  thf(fact_3066_mult__strict__mono,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ C @ D )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono
% 0.29/0.55  thf(fact_3067_mult__strict__mono,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ C @ D )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono
% 0.29/0.55  thf(fact_3068_mult__strict__mono,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ C @ D )
% 0.29/0.55         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.29/0.55           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.55             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono
% 0.29/0.55  thf(fact_3069_mult__strict__mono,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ C @ D )
% 0.29/0.55         => ( ( ord_less_int @ zero_zero_int @ B )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono
% 0.29/0.55  thf(fact_3070_mult__less__cancel__left,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left
% 0.29/0.55  thf(fact_3071_mult__less__cancel__left,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left
% 0.29/0.55  thf(fact_3072_mult__less__cancel__left,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left
% 0.29/0.55  thf(fact_3073_mult__right__less__imp__less,axiom,
% 0.29/0.55      ! [A: real,C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_less_imp_less
% 0.29/0.55  thf(fact_3074_mult__right__less__imp__less,axiom,
% 0.29/0.55      ! [A: rat,C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_less_imp_less
% 0.29/0.55  thf(fact_3075_mult__right__less__imp__less,axiom,
% 0.29/0.55      ! [A: nat,C: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_nat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_less_imp_less
% 0.29/0.55  thf(fact_3076_mult__right__less__imp__less,axiom,
% 0.29/0.55      ! [A: int,C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_less_imp_less
% 0.29/0.55  thf(fact_3077_mult__strict__mono_H,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono'
% 0.29/0.55  thf(fact_3078_mult__strict__mono_H,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono'
% 0.29/0.55  thf(fact_3079_mult__strict__mono_H,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.55           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.55             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono'
% 0.29/0.55  thf(fact_3080_mult__strict__mono_H,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_strict_mono'
% 0.29/0.55  thf(fact_3081_mult__less__cancel__right,axiom,
% 0.29/0.55      ! [A: real,C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right
% 0.29/0.55  thf(fact_3082_mult__less__cancel__right,axiom,
% 0.29/0.55      ! [A: rat,C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right
% 0.29/0.55  thf(fact_3083_mult__less__cancel__right,axiom,
% 0.29/0.55      ! [A: int,C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right
% 0.29/0.55  thf(fact_3084_mult__le__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_neg
% 0.29/0.55  thf(fact_3085_mult__le__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_neg
% 0.29/0.55  thf(fact_3086_mult__le__cancel__left__neg,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_neg
% 0.29/0.55  thf(fact_3087_mult__le__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_pos
% 0.29/0.55  thf(fact_3088_mult__le__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_pos
% 0.29/0.55  thf(fact_3089_mult__le__cancel__left__pos,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left_pos
% 0.29/0.55  thf(fact_3090_mult__left__le__imp__le,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_imp_le
% 0.29/0.55  thf(fact_3091_mult__left__le__imp__le,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_imp_le
% 0.29/0.55  thf(fact_3092_mult__left__le__imp__le,axiom,
% 0.29/0.55      ! [C: nat,A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_imp_le
% 0.29/0.55  thf(fact_3093_mult__left__le__imp__le,axiom,
% 0.29/0.55      ! [C: int,A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_imp_le
% 0.29/0.55  thf(fact_3094_mult__right__le__imp__le,axiom,
% 0.29/0.55      ! [A: real,C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_imp_le
% 0.29/0.55  thf(fact_3095_mult__right__le__imp__le,axiom,
% 0.29/0.55      ! [A: rat,C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_imp_le
% 0.29/0.55  thf(fact_3096_mult__right__le__imp__le,axiom,
% 0.29/0.55      ! [A: nat,C: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_imp_le
% 0.29/0.55  thf(fact_3097_mult__right__le__imp__le,axiom,
% 0.29/0.55      ! [A: int,C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.29/0.55       => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_imp_le
% 0.29/0.55  thf(fact_3098_mult__le__less__imp__less,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ C @ D )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_less_imp_less
% 0.29/0.55  thf(fact_3099_mult__le__less__imp__less,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ C @ D )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_less_imp_less
% 0.29/0.55  thf(fact_3100_mult__le__less__imp__less,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_nat @ C @ D )
% 0.29/0.55         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.29/0.55             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_less_imp_less
% 0.29/0.55  thf(fact_3101_mult__le__less__imp__less,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_int @ C @ D )
% 0.29/0.55         => ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_less_imp_less
% 0.29/0.55  thf(fact_3102_mult__less__le__imp__less,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real,D: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_real @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.55           => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_le_imp_less
% 0.29/0.55  thf(fact_3103_mult__less__le__imp__less,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_rat @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.55           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_le_imp_less
% 0.29/0.55  thf(fact_3104_mult__less__le__imp__less,axiom,
% 0.29/0.55      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.29/0.55        ( ( ord_less_nat @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_nat @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.55           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.29/0.55             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_le_imp_less
% 0.29/0.55  thf(fact_3105_mult__less__le__imp__less,axiom,
% 0.29/0.55      ! [A: int,B: int,C: int,D: int] :
% 0.29/0.55        ( ( ord_less_int @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_int @ C @ D )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.55           => ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_le_imp_less
% 0.29/0.55  thf(fact_3106_mult__left__le,axiom,
% 0.29/0.55      ! [C: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ C @ one_one_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.29/0.55         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le
% 0.29/0.55  thf(fact_3107_mult__left__le,axiom,
% 0.29/0.55      ! [C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.29/0.55         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le
% 0.29/0.55  thf(fact_3108_mult__left__le,axiom,
% 0.29/0.55      ! [C: nat,A: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 0.29/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.29/0.55         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le
% 0.29/0.55  thf(fact_3109_mult__left__le,axiom,
% 0.29/0.55      ! [C: int,A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ C @ one_one_int )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.29/0.55         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le
% 0.29/0.55  thf(fact_3110_mult__le__one,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ one_one_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.29/0.55         => ( ( ord_less_eq_real @ B @ one_one_real )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_one
% 0.29/0.55  thf(fact_3111_mult__le__one,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.29/0.55         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_one
% 0.29/0.55  thf(fact_3112_mult__le__one,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 0.29/0.55       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 0.29/0.55         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 0.29/0.55           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_one
% 0.29/0.55  thf(fact_3113_mult__le__one,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ A @ one_one_int )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.29/0.55         => ( ( ord_less_eq_int @ B @ one_one_int )
% 0.29/0.55           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_one
% 0.29/0.55  thf(fact_3114_mult__right__le__one__le,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_one_le
% 0.29/0.55  thf(fact_3115_mult__right__le__one__le,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_one_le
% 0.29/0.55  thf(fact_3116_mult__right__le__one__le,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.29/0.55           => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_right_le_one_le
% 0.29/0.55  thf(fact_3117_mult__left__le__one__le,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_one_le
% 0.29/0.55  thf(fact_3118_mult__left__le__one__le,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_one_le
% 0.29/0.55  thf(fact_3119_mult__left__le__one__le,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.29/0.55       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.29/0.55         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 0.29/0.55           => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_left_le_one_le
% 0.29/0.55  thf(fact_3120_sum__squares__le__zero__iff,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 0.29/0.55        = ( ( X3 = zero_zero_real )
% 0.29/0.55          & ( Y3 = zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_le_zero_iff
% 0.29/0.55  thf(fact_3121_sum__squares__le__zero__iff,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 0.29/0.55        = ( ( X3 = zero_zero_rat )
% 0.29/0.55          & ( Y3 = zero_zero_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_le_zero_iff
% 0.29/0.55  thf(fact_3122_sum__squares__le__zero__iff,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 0.29/0.55        = ( ( X3 = zero_zero_int )
% 0.29/0.55          & ( Y3 = zero_zero_int ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_le_zero_iff
% 0.29/0.55  thf(fact_3123_sum__squares__ge__zero,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_ge_zero
% 0.29/0.55  thf(fact_3124_sum__squares__ge__zero,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_ge_zero
% 0.29/0.55  thf(fact_3125_sum__squares__ge__zero,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_ge_zero
% 0.29/0.55  thf(fact_3126_sum__squares__gt__zero__iff,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 0.29/0.55        = ( ( X3 != zero_zero_real )
% 0.29/0.55          | ( Y3 != zero_zero_real ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_gt_zero_iff
% 0.29/0.55  thf(fact_3127_sum__squares__gt__zero__iff,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 0.29/0.55        = ( ( X3 != zero_zero_rat )
% 0.29/0.55          | ( Y3 != zero_zero_rat ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_gt_zero_iff
% 0.29/0.55  thf(fact_3128_sum__squares__gt__zero__iff,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 0.29/0.55        = ( ( X3 != zero_zero_int )
% 0.29/0.55          | ( Y3 != zero_zero_int ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % sum_squares_gt_zero_iff
% 0.29/0.55  thf(fact_3129_not__sum__squares__lt__zero,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 0.29/0.55  
% 0.29/0.55  % not_sum_squares_lt_zero
% 0.29/0.55  thf(fact_3130_not__sum__squares__lt__zero,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 0.29/0.55  
% 0.29/0.55  % not_sum_squares_lt_zero
% 0.29/0.55  thf(fact_3131_not__sum__squares__lt__zero,axiom,
% 0.29/0.55      ! [X3: int,Y3: int] :
% 0.29/0.55        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 0.29/0.55  
% 0.29/0.55  % not_sum_squares_lt_zero
% 0.29/0.55  thf(fact_3132_divide__less__eq,axiom,
% 0.29/0.55      ! [B: real,C: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq
% 0.29/0.55  thf(fact_3133_divide__less__eq,axiom,
% 0.29/0.55      ! [B: rat,C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq
% 0.29/0.55  thf(fact_3134_less__divide__eq,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq
% 0.29/0.55  thf(fact_3135_less__divide__eq,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq
% 0.29/0.55  thf(fact_3136_neg__divide__less__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_less_eq
% 0.29/0.55  thf(fact_3137_neg__divide__less__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_less_eq
% 0.29/0.55  thf(fact_3138_neg__less__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_divide_eq
% 0.29/0.55  thf(fact_3139_neg__less__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_less_divide_eq
% 0.29/0.55  thf(fact_3140_pos__divide__less__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_less_eq
% 0.29/0.55  thf(fact_3141_pos__divide__less__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_less_eq
% 0.29/0.55  thf(fact_3142_pos__less__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_less_divide_eq
% 0.29/0.55  thf(fact_3143_pos__less__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_less_divide_eq
% 0.29/0.55  thf(fact_3144_mult__imp__div__pos__less,axiom,
% 0.29/0.55      ! [Y3: real,X3: real,Z2: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.55       => ( ( ord_less_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
% 0.29/0.55         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_less
% 0.29/0.55  thf(fact_3145_mult__imp__div__pos__less,axiom,
% 0.29/0.55      ! [Y3: rat,X3: rat,Z2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55       => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
% 0.29/0.55         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_less
% 0.29/0.55  thf(fact_3146_mult__imp__less__div__pos,axiom,
% 0.29/0.55      ! [Y3: real,Z2: real,X3: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.55       => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
% 0.29/0.55         => ( ord_less_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_less_div_pos
% 0.29/0.55  thf(fact_3147_mult__imp__less__div__pos,axiom,
% 0.29/0.55      ! [Y3: rat,Z2: rat,X3: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55       => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
% 0.29/0.55         => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_less_div_pos
% 0.29/0.55  thf(fact_3148_divide__strict__left__mono,axiom,
% 0.29/0.55      ! [B: real,A: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ B @ A )
% 0.29/0.55       => ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_strict_left_mono
% 0.29/0.55  thf(fact_3149_divide__strict__left__mono,axiom,
% 0.29/0.55      ! [B: rat,A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ B @ A )
% 0.29/0.55       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_strict_left_mono
% 0.29/0.55  thf(fact_3150_divide__strict__left__mono__neg,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_strict_left_mono_neg
% 0.29/0.55  thf(fact_3151_divide__strict__left__mono__neg,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_strict_left_mono_neg
% 0.29/0.55  thf(fact_3152_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: complex,C: complex,W2: num] :
% 0.29/0.55        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.29/0.55          = ( numera6690914467698888265omplex @ W2 ) )
% 0.29/0.55        = ( ( ( C != zero_zero_complex )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_complex )
% 0.29/0.55           => ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.55              = zero_zero_complex ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq_numeral(1)
% 0.29/0.55  thf(fact_3153_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: real,C: real,W2: num] :
% 0.29/0.55        ( ( ( divide_divide_real @ B @ C )
% 0.29/0.55          = ( numeral_numeral_real @ W2 ) )
% 0.29/0.55        = ( ( ( C != zero_zero_real )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_real )
% 0.29/0.55           => ( ( numeral_numeral_real @ W2 )
% 0.29/0.55              = zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq_numeral(1)
% 0.29/0.55  thf(fact_3154_divide__eq__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: rat,C: rat,W2: num] :
% 0.29/0.55        ( ( ( divide_divide_rat @ B @ C )
% 0.29/0.55          = ( numeral_numeral_rat @ W2 ) )
% 0.29/0.55        = ( ( ( C != zero_zero_rat )
% 0.29/0.55           => ( B
% 0.29/0.55              = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ( C = zero_zero_rat )
% 0.29/0.55           => ( ( numeral_numeral_rat @ W2 )
% 0.29/0.55              = zero_zero_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_eq_eq_numeral(1)
% 0.29/0.55  thf(fact_3155_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: complex,C: complex] :
% 0.29/0.55        ( ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.55          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_complex )
% 0.29/0.55           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_complex )
% 0.29/0.55           => ( ( numera6690914467698888265omplex @ W2 )
% 0.29/0.55              = zero_zero_complex ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3156_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: real,C: real] :
% 0.29/0.55        ( ( ( numeral_numeral_real @ W2 )
% 0.29/0.55          = ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_real )
% 0.29/0.55           => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_real )
% 0.29/0.55           => ( ( numeral_numeral_real @ W2 )
% 0.29/0.55              = zero_zero_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3157_eq__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: rat,C: rat] :
% 0.29/0.55        ( ( ( numeral_numeral_rat @ W2 )
% 0.29/0.55          = ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( C != zero_zero_rat )
% 0.29/0.55           => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 0.29/0.55              = B ) )
% 0.29/0.55          & ( ( C = zero_zero_rat )
% 0.29/0.55           => ( ( numeral_numeral_rat @ W2 )
% 0.29/0.55              = zero_zero_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % eq_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3158_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.55      ! [Z2: complex,A: complex,B: complex] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_complex )
% 0.29/0.55         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.29/0.55            = B ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_complex )
% 0.29/0.55         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.29/0.55            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(2)
% 0.29/0.55  thf(fact_3159_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.55      ! [Z2: real,A: real,B: real] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_real )
% 0.29/0.55         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.29/0.55            = B ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_real )
% 0.29/0.55         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.29/0.55            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(2)
% 0.29/0.55  thf(fact_3160_add__divide__eq__if__simps_I2_J,axiom,
% 0.29/0.55      ! [Z2: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_rat )
% 0.29/0.55         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.29/0.55            = B ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_rat )
% 0.29/0.55         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.29/0.55            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(2)
% 0.29/0.55  thf(fact_3161_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.55      ! [Z2: complex,A: complex,B: complex] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_complex )
% 0.29/0.55         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.29/0.55            = A ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_complex )
% 0.29/0.55         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.29/0.55            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(1)
% 0.29/0.55  thf(fact_3162_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.55      ! [Z2: real,A: real,B: real] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_real )
% 0.29/0.55         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.29/0.55            = A ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_real )
% 0.29/0.55         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.29/0.55            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(1)
% 0.29/0.55  thf(fact_3163_add__divide__eq__if__simps_I1_J,axiom,
% 0.29/0.55      ! [Z2: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ( Z2 = zero_zero_rat )
% 0.29/0.55         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.29/0.55            = A ) )
% 0.29/0.55        & ( ( Z2 != zero_zero_rat )
% 0.29/0.55         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.29/0.55            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_if_simps(1)
% 0.29/0.55  thf(fact_3164_add__frac__eq,axiom,
% 0.29/0.55      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 0.29/0.55        ( ( Y3 != zero_zero_complex )
% 0.29/0.55       => ( ( Z2 != zero_zero_complex )
% 0.29/0.55         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.29/0.55            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_eq
% 0.29/0.55  thf(fact_3165_add__frac__eq,axiom,
% 0.29/0.55      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 0.29/0.55        ( ( Y3 != zero_zero_real )
% 0.29/0.55       => ( ( Z2 != zero_zero_real )
% 0.29/0.55         => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.29/0.55            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_eq
% 0.29/0.55  thf(fact_3166_add__frac__eq,axiom,
% 0.29/0.55      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 0.29/0.55        ( ( Y3 != zero_zero_rat )
% 0.29/0.55       => ( ( Z2 != zero_zero_rat )
% 0.29/0.55         => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.29/0.55            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_eq
% 0.29/0.55  thf(fact_3167_add__frac__num,axiom,
% 0.29/0.55      ! [Y3: complex,X3: complex,Z2: complex] :
% 0.29/0.55        ( ( Y3 != zero_zero_complex )
% 0.29/0.55       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z2 )
% 0.29/0.55          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_num
% 0.29/0.55  thf(fact_3168_add__frac__num,axiom,
% 0.29/0.55      ! [Y3: real,X3: real,Z2: real] :
% 0.29/0.55        ( ( Y3 != zero_zero_real )
% 0.29/0.55       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 )
% 0.29/0.55          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_num
% 0.29/0.55  thf(fact_3169_add__frac__num,axiom,
% 0.29/0.55      ! [Y3: rat,X3: rat,Z2: rat] :
% 0.29/0.55        ( ( Y3 != zero_zero_rat )
% 0.29/0.55       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 )
% 0.29/0.55          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_frac_num
% 0.29/0.55  thf(fact_3170_add__num__frac,axiom,
% 0.29/0.55      ! [Y3: complex,Z2: complex,X3: complex] :
% 0.29/0.55        ( ( Y3 != zero_zero_complex )
% 0.29/0.55       => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 0.29/0.55          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_num_frac
% 0.29/0.55  thf(fact_3171_add__num__frac,axiom,
% 0.29/0.55      ! [Y3: real,Z2: real,X3: real] :
% 0.29/0.55        ( ( Y3 != zero_zero_real )
% 0.29/0.55       => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.29/0.55          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_num_frac
% 0.29/0.55  thf(fact_3172_add__num__frac,axiom,
% 0.29/0.55      ! [Y3: rat,Z2: rat,X3: rat] :
% 0.29/0.55        ( ( Y3 != zero_zero_rat )
% 0.29/0.55       => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) )
% 0.29/0.55          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_num_frac
% 0.29/0.55  thf(fact_3173_add__divide__eq__iff,axiom,
% 0.29/0.55      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.29/0.55        ( ( Z2 != zero_zero_complex )
% 0.29/0.55       => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
% 0.29/0.55          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_iff
% 0.29/0.55  thf(fact_3174_add__divide__eq__iff,axiom,
% 0.29/0.55      ! [Z2: real,X3: real,Y3: real] :
% 0.29/0.55        ( ( Z2 != zero_zero_real )
% 0.29/0.55       => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
% 0.29/0.55          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_iff
% 0.29/0.55  thf(fact_3175_add__divide__eq__iff,axiom,
% 0.29/0.55      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.29/0.55        ( ( Z2 != zero_zero_rat )
% 0.29/0.55       => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
% 0.29/0.55          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % add_divide_eq_iff
% 0.29/0.55  thf(fact_3176_divide__add__eq__iff,axiom,
% 0.29/0.55      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.29/0.55        ( ( Z2 != zero_zero_complex )
% 0.29/0.55       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
% 0.29/0.55          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_add_eq_iff
% 0.29/0.55  thf(fact_3177_divide__add__eq__iff,axiom,
% 0.29/0.55      ! [Z2: real,X3: real,Y3: real] :
% 0.29/0.55        ( ( Z2 != zero_zero_real )
% 0.29/0.55       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
% 0.29/0.55          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_add_eq_iff
% 0.29/0.55  thf(fact_3178_divide__add__eq__iff,axiom,
% 0.29/0.55      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.29/0.55        ( ( Z2 != zero_zero_rat )
% 0.29/0.55       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
% 0.29/0.55          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_add_eq_iff
% 0.29/0.55  thf(fact_3179_power__gt1__lemma,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.55       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_gt1_lemma
% 0.29/0.55  thf(fact_3180_power__gt1__lemma,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.55       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_gt1_lemma
% 0.29/0.55  thf(fact_3181_power__gt1__lemma,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.55       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_gt1_lemma
% 0.29/0.55  thf(fact_3182_power__gt1__lemma,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.55       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_gt1_lemma
% 0.29/0.55  thf(fact_3183_power__less__power__Suc,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( ord_less_real @ one_one_real @ A )
% 0.29/0.55       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_less_power_Suc
% 0.29/0.55  thf(fact_3184_power__less__power__Suc,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( ord_less_rat @ one_one_rat @ A )
% 0.29/0.55       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_less_power_Suc
% 0.29/0.55  thf(fact_3185_power__less__power__Suc,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ one_one_nat @ A )
% 0.29/0.55       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_less_power_Suc
% 0.29/0.55  thf(fact_3186_power__less__power__Suc,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( ord_less_int @ one_one_int @ A )
% 0.29/0.55       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_less_power_Suc
% 0.29/0.55  thf(fact_3187_one__less__mult,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % one_less_mult
% 0.29/0.55  thf(fact_3188_n__less__m__mult__n,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % n_less_m_mult_n
% 0.29/0.55  thf(fact_3189_n__less__n__mult__m,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.55       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.29/0.55         => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % n_less_n_mult_m
% 0.29/0.55  thf(fact_3190_nat__mult__le__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.55          = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_le_cancel1
% 0.29/0.55  thf(fact_3191_nat__mult__div__cancel1,axiom,
% 0.29/0.55      ! [K: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.29/0.55       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.29/0.55          = ( divide_divide_nat @ M @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_mult_div_cancel1
% 0.29/0.55  thf(fact_3192_div__less__iff__less__mult,axiom,
% 0.29/0.55      ! [Q4: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.29/0.55       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N2 )
% 0.29/0.55          = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_less_iff_less_mult
% 0.29/0.55  thf(fact_3193_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3194_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3195_mult__le__cancel__left1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left1
% 0.29/0.55  thf(fact_3196_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3197_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3198_mult__le__cancel__left2,axiom,
% 0.29/0.55      ! [C: int,A: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_left2
% 0.29/0.55  thf(fact_3199_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3200_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3201_mult__le__cancel__right1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right1
% 0.29/0.55  thf(fact_3202_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3203_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3204_mult__le__cancel__right2,axiom,
% 0.29/0.55      ! [A: int,C: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_eq_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_le_cancel_right2
% 0.29/0.55  thf(fact_3205_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3206_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3207_mult__less__cancel__left1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left1
% 0.29/0.55  thf(fact_3208_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3209_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3210_mult__less__cancel__left2,axiom,
% 0.29/0.55      ! [C: int,A: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_left2
% 0.29/0.55  thf(fact_3211_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3212_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3213_mult__less__cancel__right1,axiom,
% 0.29/0.55      ! [C: int,B: int] :
% 0.29/0.55        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ B ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right1
% 0.29/0.55  thf(fact_3214_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ A @ one_one_real ) )
% 0.29/0.55          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3215_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ A @ one_one_rat ) )
% 0.29/0.55          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3216_mult__less__cancel__right2,axiom,
% 0.29/0.55      ! [A: int,C: int] :
% 0.29/0.55        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 0.29/0.55        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.29/0.55           => ( ord_less_int @ A @ one_one_int ) )
% 0.29/0.55          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 0.29/0.55           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_less_cancel_right2
% 0.29/0.55  thf(fact_3217_field__le__mult__one__interval,axiom,
% 0.29/0.55      ! [X3: real,Y3: real] :
% 0.29/0.55        ( ! [Z3: real] :
% 0.29/0.55            ( ( ord_less_real @ zero_zero_real @ Z3 )
% 0.29/0.55           => ( ( ord_less_real @ Z3 @ one_one_real )
% 0.29/0.55             => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y3 ) ) )
% 0.29/0.55       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % field_le_mult_one_interval
% 0.29/0.55  thf(fact_3218_field__le__mult__one__interval,axiom,
% 0.29/0.55      ! [X3: rat,Y3: rat] :
% 0.29/0.55        ( ! [Z3: rat] :
% 0.29/0.55            ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 0.29/0.55           => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 0.29/0.55             => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X3 ) @ Y3 ) ) )
% 0.29/0.55       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % field_le_mult_one_interval
% 0.29/0.55  thf(fact_3219_divide__le__eq,axiom,
% 0.29/0.55      ! [B: real,C: real,A: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq
% 0.29/0.55  thf(fact_3220_divide__le__eq,axiom,
% 0.29/0.55      ! [B: rat,C: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq
% 0.29/0.55  thf(fact_3221_le__divide__eq,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq
% 0.29/0.55  thf(fact_3222_le__divide__eq,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq
% 0.29/0.55  thf(fact_3223_divide__left__mono,axiom,
% 0.29/0.55      ! [B: real,A: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ B @ A )
% 0.29/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono
% 0.29/0.55  thf(fact_3224_divide__left__mono,axiom,
% 0.29/0.55      ! [B: rat,A: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ B @ A )
% 0.29/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono
% 0.29/0.55  thf(fact_3225_neg__divide__le__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_le_eq
% 0.29/0.55  thf(fact_3226_neg__divide__le__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_divide_le_eq
% 0.29/0.55  thf(fact_3227_neg__le__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_divide_eq
% 0.29/0.55  thf(fact_3228_neg__le__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % neg_le_divide_eq
% 0.29/0.55  thf(fact_3229_pos__divide__le__eq,axiom,
% 0.29/0.55      ! [C: real,B: real,A: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_le_eq
% 0.29/0.55  thf(fact_3230_pos__divide__le__eq,axiom,
% 0.29/0.55      ! [C: rat,B: rat,A: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 0.29/0.55          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_divide_le_eq
% 0.29/0.55  thf(fact_3231_pos__le__divide__eq,axiom,
% 0.29/0.55      ! [C: real,A: real,B: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_le_divide_eq
% 0.29/0.55  thf(fact_3232_pos__le__divide__eq,axiom,
% 0.29/0.55      ! [C: rat,A: rat,B: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % pos_le_divide_eq
% 0.29/0.55  thf(fact_3233_mult__imp__div__pos__le,axiom,
% 0.29/0.55      ! [Y3: real,X3: real,Z2: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.55       => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
% 0.29/0.55         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_le
% 0.29/0.55  thf(fact_3234_mult__imp__div__pos__le,axiom,
% 0.29/0.55      ! [Y3: rat,X3: rat,Z2: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55       => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
% 0.29/0.55         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_div_pos_le
% 0.29/0.55  thf(fact_3235_mult__imp__le__div__pos,axiom,
% 0.29/0.55      ! [Y3: real,Z2: real,X3: real] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.29/0.55       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
% 0.29/0.55         => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_le_div_pos
% 0.29/0.55  thf(fact_3236_mult__imp__le__div__pos,axiom,
% 0.29/0.55      ! [Y3: rat,Z2: rat,X3: rat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.29/0.55       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
% 0.29/0.55         => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_imp_le_div_pos
% 0.29/0.55  thf(fact_3237_divide__left__mono__neg,axiom,
% 0.29/0.55      ! [A: real,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 0.29/0.55         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono_neg
% 0.29/0.55  thf(fact_3238_divide__left__mono__neg,axiom,
% 0.29/0.55      ! [A: rat,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ A @ B )
% 0.29/0.55       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 0.29/0.55         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 0.29/0.55           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_left_mono_neg
% 0.29/0.55  thf(fact_3239_convex__bound__le,axiom,
% 0.29/0.55      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_eq_real @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.29/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.29/0.55                  = one_one_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3240_convex__bound__le,axiom,
% 0.29/0.55      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_eq_rat @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.29/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.29/0.55                  = one_one_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3241_convex__bound__le,axiom,
% 0.29/0.55      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 0.29/0.55        ( ( ord_less_eq_int @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_eq_int @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.29/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.29/0.55                  = one_one_int )
% 0.29/0.55               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_le
% 0.29/0.55  thf(fact_3242_divide__less__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: real,C: real,W2: num] :
% 0.29/0.55        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq_numeral(1)
% 0.29/0.55  thf(fact_3243_divide__less__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: rat,C: rat,W2: num] :
% 0.29/0.55        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_less_eq_numeral(1)
% 0.29/0.55  thf(fact_3244_less__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3245_less__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3246_power__Suc__less,axiom,
% 0.29/0.55      ! [A: real,N2: nat] :
% 0.29/0.55        ( ( ord_less_real @ zero_zero_real @ A )
% 0.29/0.55       => ( ( ord_less_real @ A @ one_one_real )
% 0.29/0.55         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3247_power__Suc__less,axiom,
% 0.29/0.55      ! [A: rat,N2: nat] :
% 0.29/0.55        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.29/0.55       => ( ( ord_less_rat @ A @ one_one_rat )
% 0.29/0.55         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3248_power__Suc__less,axiom,
% 0.29/0.55      ! [A: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ A )
% 0.29/0.55       => ( ( ord_less_nat @ A @ one_one_nat )
% 0.29/0.55         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3249_power__Suc__less,axiom,
% 0.29/0.55      ! [A: int,N2: nat] :
% 0.29/0.55        ( ( ord_less_int @ zero_zero_int @ A )
% 0.29/0.55       => ( ( ord_less_int @ A @ one_one_int )
% 0.29/0.55         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % power_Suc_less
% 0.29/0.55  thf(fact_3250_mult__2,axiom,
% 0.29/0.55      ! [Z2: complex] :
% 0.29/0.55        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3251_mult__2,axiom,
% 0.29/0.55      ! [Z2: extended_enat] :
% 0.29/0.55        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3252_mult__2,axiom,
% 0.29/0.55      ! [Z2: real] :
% 0.29/0.55        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3253_mult__2,axiom,
% 0.29/0.55      ! [Z2: rat] :
% 0.29/0.55        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3254_mult__2,axiom,
% 0.29/0.55      ! [Z2: nat] :
% 0.29/0.55        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3255_mult__2,axiom,
% 0.29/0.55      ! [Z2: int] :
% 0.29/0.55        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
% 0.29/0.55        = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2
% 0.29/0.55  thf(fact_3256_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: complex] :
% 0.29/0.55        ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3257_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: extended_enat] :
% 0.29/0.55        ( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3258_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: real] :
% 0.29/0.55        ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3259_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: rat] :
% 0.29/0.55        ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3260_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: nat] :
% 0.29/0.55        ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3261_mult__2__right,axiom,
% 0.29/0.55      ! [Z2: int] :
% 0.29/0.55        ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.29/0.55        = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 0.29/0.55  
% 0.29/0.55  % mult_2_right
% 0.29/0.55  thf(fact_3262_left__add__twice,axiom,
% 0.29/0.55      ! [A: complex,B: complex] :
% 0.29/0.55        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 0.29/0.55        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3263_left__add__twice,axiom,
% 0.29/0.55      ! [A: extended_enat,B: extended_enat] :
% 0.29/0.55        ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
% 0.29/0.55        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3264_left__add__twice,axiom,
% 0.29/0.55      ! [A: real,B: real] :
% 0.29/0.55        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.29/0.55        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3265_left__add__twice,axiom,
% 0.29/0.55      ! [A: rat,B: rat] :
% 0.29/0.55        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.29/0.55        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3266_left__add__twice,axiom,
% 0.29/0.55      ! [A: nat,B: nat] :
% 0.29/0.55        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.29/0.55        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3267_left__add__twice,axiom,
% 0.29/0.55      ! [A: int,B: int] :
% 0.29/0.55        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.29/0.55        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 0.29/0.55  
% 0.29/0.55  % left_add_twice
% 0.29/0.55  thf(fact_3268_div__nat__eqI,axiom,
% 0.29/0.55      ! [N2: nat,Q4: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
% 0.29/0.55       => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 0.29/0.55         => ( ( divide_divide_nat @ M @ N2 )
% 0.29/0.55            = Q4 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % div_nat_eqI
% 0.29/0.55  thf(fact_3269_less__eq__div__iff__mult__less__eq,axiom,
% 0.29/0.55      ! [Q4: nat,M: nat,N2: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 0.29/0.55       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q4 ) )
% 0.29/0.55          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N2 ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % less_eq_div_iff_mult_less_eq
% 0.29/0.55  thf(fact_3270_split__div,axiom,
% 0.29/0.55      ! [P: nat > $o,M: nat,N2: nat] :
% 0.29/0.55        ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 0.29/0.55        = ( ( ( N2 = zero_zero_nat )
% 0.29/0.55           => ( P @ zero_zero_nat ) )
% 0.29/0.55          & ( ( N2 != zero_zero_nat )
% 0.29/0.55           => ! [I2: nat,J3: nat] :
% 0.29/0.55                ( ( ord_less_nat @ J3 @ N2 )
% 0.29/0.55               => ( ( M
% 0.29/0.55                    = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
% 0.29/0.55                 => ( P @ I2 ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % split_div
% 0.29/0.55  thf(fact_3271_dividend__less__div__times,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % dividend_less_div_times
% 0.29/0.55  thf(fact_3272_dividend__less__times__div,axiom,
% 0.29/0.55      ! [N2: nat,M: nat] :
% 0.29/0.55        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.29/0.55       => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % dividend_less_times_div
% 0.29/0.55  thf(fact_3273_nat__bit__induct,axiom,
% 0.29/0.55      ! [P: nat > $o,N2: nat] :
% 0.29/0.55        ( ( P @ zero_zero_nat )
% 0.29/0.55       => ( ! [N3: nat] :
% 0.29/0.55              ( ( P @ N3 )
% 0.29/0.55             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.29/0.55               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.29/0.55         => ( ! [N3: nat] :
% 0.29/0.55                ( ( P @ N3 )
% 0.29/0.55               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 0.29/0.55           => ( P @ N2 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % nat_bit_induct
% 0.29/0.55  thf(fact_3274_convex__bound__lt,axiom,
% 0.29/0.55      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 0.29/0.55        ( ( ord_less_real @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_real @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 0.29/0.55           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 0.29/0.55             => ( ( ( plus_plus_real @ U @ V )
% 0.29/0.55                  = one_one_real )
% 0.29/0.55               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3275_convex__bound__lt,axiom,
% 0.29/0.55      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 0.29/0.55        ( ( ord_less_rat @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_rat @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 0.29/0.55           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 0.29/0.55             => ( ( ( plus_plus_rat @ U @ V )
% 0.29/0.55                  = one_one_rat )
% 0.29/0.55               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3276_convex__bound__lt,axiom,
% 0.29/0.55      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 0.29/0.55        ( ( ord_less_int @ X3 @ A )
% 0.29/0.55       => ( ( ord_less_int @ Y3 @ A )
% 0.29/0.55         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 0.29/0.55           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 0.29/0.55             => ( ( ( plus_plus_int @ U @ V )
% 0.29/0.55                  = one_one_int )
% 0.29/0.55               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % convex_bound_lt
% 0.29/0.55  thf(fact_3277_divide__le__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: real,C: real,W2: num] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq_numeral(1)
% 0.29/0.55  thf(fact_3278_divide__le__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [B: rat,C: rat,W2: num] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % divide_le_eq_numeral(1)
% 0.29/0.55  thf(fact_3279_le__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: real,C: real] :
% 0.29/0.55        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.29/0.55           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.29/0.55               => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3280_le__divide__eq__numeral_I1_J,axiom,
% 0.29/0.55      ! [W2: num,B: rat,C: rat] :
% 0.29/0.55        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 0.29/0.55        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 0.29/0.55          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.29/0.55           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 0.29/0.55              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.29/0.55               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % le_divide_eq_numeral(1)
% 0.29/0.55  thf(fact_3281_split__div_H,axiom,
% 0.29/0.55      ! [P: nat > $o,M: nat,N2: nat] :
% 0.29/0.55        ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 0.29/0.55        = ( ( ( N2 = zero_zero_nat )
% 0.29/0.55            & ( P @ zero_zero_nat ) )
% 0.29/0.55          | ? [Q5: nat] :
% 0.29/0.55              ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M )
% 0.29/0.55              & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
% 0.29/0.55              & ( P @ Q5 ) ) ) ) ).
% 0.29/0.55  
% 0.29/0.55  % split_div'
% 0.29/0.55  thf(fact_3282_power2__sum,axiom,
% 0.30/0.55      ! [X3: complex,Y3: complex] :
% 0.30/0.55        ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3283_power2__sum,axiom,
% 0.30/0.55      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.55        ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3284_power2__sum,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] :
% 0.30/0.55        ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3285_power2__sum,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] :
% 0.30/0.55        ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3286_power2__sum,axiom,
% 0.30/0.55      ! [X3: nat,Y3: nat] :
% 0.30/0.55        ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3287_power2__sum,axiom,
% 0.30/0.55      ! [X3: int,Y3: int] :
% 0.30/0.55        ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55        = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % power2_sum
% 0.30/0.55  thf(fact_3288_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_3289_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_3290_zero__le__even__power_H,axiom,
% 0.30/0.55      ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zero_le_even_power'
% 0.30/0.55  thf(fact_3291_triangle__def,axiom,
% 0.30/0.55      ( nat_triangle
% 0.30/0.55      = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % triangle_def
% 0.30/0.55  thf(fact_3292_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: real,N2: nat] :
% 0.30/0.55        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.55       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_0_le_power_imp_0_le
% 0.30/0.55  thf(fact_3293_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: rat,N2: nat] :
% 0.30/0.55        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.55       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_0_le_power_imp_0_le
% 0.30/0.55  thf(fact_3294_odd__0__le__power__imp__0__le,axiom,
% 0.30/0.55      ! [A: int,N2: nat] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.55       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_0_le_power_imp_0_le
% 0.30/0.55  thf(fact_3295_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: real,N2: nat] :
% 0.30/0.55        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.55       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_3296_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: rat,N2: nat] :
% 0.30/0.55        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.55       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_3297_odd__power__less__zero,axiom,
% 0.30/0.55      ! [A: int,N2: nat] :
% 0.30/0.55        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.55       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % odd_power_less_zero
% 0.30/0.55  thf(fact_3298_arith__geo__mean,axiom,
% 0.30/0.55      ! [U: real,X3: real,Y3: real] :
% 0.30/0.55        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.55       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.55         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.55           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % arith_geo_mean
% 0.30/0.55  thf(fact_3299_arith__geo__mean,axiom,
% 0.30/0.55      ! [U: rat,X3: rat,Y3: rat] :
% 0.30/0.55        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.55          = ( times_times_rat @ X3 @ Y3 ) )
% 0.30/0.55       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.55         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.55           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % arith_geo_mean
% 0.30/0.55  thf(fact_3300_sum__squares__bound,axiom,
% 0.30/0.55      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_squares_bound
% 0.30/0.55  thf(fact_3301_sum__squares__bound,axiom,
% 0.30/0.55      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % sum_squares_bound
% 0.30/0.55  thf(fact_3302_set__bit__0,axiom,
% 0.30/0.55      ! [A: nat] :
% 0.30/0.55        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 0.30/0.55        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_0
% 0.30/0.55  thf(fact_3303_set__bit__0,axiom,
% 0.30/0.55      ! [A: code_integer] :
% 0.30/0.55        ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 0.30/0.55        = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_0
% 0.30/0.55  thf(fact_3304_set__bit__0,axiom,
% 0.30/0.55      ! [A: int] :
% 0.30/0.55        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 0.30/0.55        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_0
% 0.30/0.55  thf(fact_3305_pos2,axiom,
% 0.30/0.55      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 0.30/0.55  
% 0.30/0.55  % pos2
% 0.30/0.55  thf(fact_3306_unset__bit__0,axiom,
% 0.30/0.55      ! [A: nat] :
% 0.30/0.55        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 0.30/0.55        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unset_bit_0
% 0.30/0.55  thf(fact_3307_unset__bit__0,axiom,
% 0.30/0.55      ! [A: code_integer] :
% 0.30/0.55        ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 0.30/0.55        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unset_bit_0
% 0.30/0.55  thf(fact_3308_unset__bit__0,axiom,
% 0.30/0.55      ! [A: int] :
% 0.30/0.55        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 0.30/0.55        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unset_bit_0
% 0.30/0.55  thf(fact_3309_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.55      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.55        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.30/0.55       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.55          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.55  thf(fact_3310_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.55      ! [C: nat,A: nat,B: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.55       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.55          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.55  thf(fact_3311_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 0.30/0.55      ! [C: int,A: int,B: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.55       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.55          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_mult2_eq
% 0.30/0.55  thf(fact_3312_discrete,axiom,
% 0.30/0.55      ( ord_less_nat
% 0.30/0.55      = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % discrete
% 0.30/0.55  thf(fact_3313_discrete,axiom,
% 0.30/0.55      ( ord_less_int
% 0.30/0.55      = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % discrete
% 0.30/0.55  thf(fact_3314_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.55      ! [A: code_integer,B: code_integer] :
% 0.30/0.55        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.55       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.55         => ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.55            = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.55  thf(fact_3315_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.55      ! [A: nat,B: nat] :
% 0.30/0.55        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.55       => ( ( ord_less_nat @ A @ B )
% 0.30/0.55         => ( ( divide_divide_nat @ A @ B )
% 0.30/0.55            = zero_zero_nat ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.55  thf(fact_3316_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 0.30/0.55      ! [A: int,B: int] :
% 0.30/0.55        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.55       => ( ( ord_less_int @ A @ B )
% 0.30/0.55         => ( ( divide_divide_int @ A @ B )
% 0.30/0.55            = zero_zero_int ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % unique_euclidean_semiring_numeral_class.div_less
% 0.30/0.55  thf(fact_3317_div__positive,axiom,
% 0.30/0.55      ! [B: code_integer,A: code_integer] :
% 0.30/0.55        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.55       => ( ( ord_le3102999989581377725nteger @ B @ A )
% 0.30/0.55         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_positive
% 0.30/0.55  thf(fact_3318_div__positive,axiom,
% 0.30/0.55      ! [B: nat,A: nat] :
% 0.30/0.55        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.55       => ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.55         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_positive
% 0.30/0.55  thf(fact_3319_div__positive,axiom,
% 0.30/0.55      ! [B: int,A: int] :
% 0.30/0.55        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.55       => ( ( ord_less_eq_int @ B @ A )
% 0.30/0.55         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.55  
% 0.30/0.55  % div_positive
% 0.30/0.55  thf(fact_3320_zle__add1__eq__le,axiom,
% 0.30/0.55      ! [W2: int,Z2: int] :
% 0.30/0.55        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.30/0.55        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.30/0.55  
% 0.30/0.55  % zle_add1_eq_le
% 0.30/0.55  thf(fact_3321_set__bit__negative__int__iff,axiom,
% 0.30/0.55      ! [N2: nat,K: int] :
% 0.30/0.55        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.55        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.55  
% 0.30/0.55  % set_bit_negative_int_iff
% 0.30/0.55  thf(fact_3322_unset__bit__negative__int__iff,axiom,
% 0.30/0.55      ! [N2: nat,K: int] :
% 0.30/0.55        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.56        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unset_bit_negative_int_iff
% 0.30/0.56  thf(fact_3323_div__pos__pos__trivial,axiom,
% 0.30/0.56      ! [K: int,L: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56       => ( ( ord_less_int @ K @ L )
% 0.30/0.56         => ( ( divide_divide_int @ K @ L )
% 0.30/0.56            = zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_pos_pos_trivial
% 0.30/0.56  thf(fact_3324_div__neg__neg__trivial,axiom,
% 0.30/0.56      ! [K: int,L: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_int @ L @ K )
% 0.30/0.56         => ( ( divide_divide_int @ K @ L )
% 0.30/0.56            = zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_neg_neg_trivial
% 0.30/0.56  thf(fact_3325_nonneg1__imp__zdiv__pos__iff,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.30/0.56          = ( ( ord_less_eq_int @ B @ A )
% 0.30/0.56            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % nonneg1_imp_zdiv_pos_iff
% 0.30/0.56  thf(fact_3326_pos__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.30/0.56          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_imp_zdiv_nonneg_iff
% 0.30/0.56  thf(fact_3327_neg__imp__zdiv__nonneg__iff,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 0.30/0.56          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_imp_zdiv_nonneg_iff
% 0.30/0.56  thf(fact_3328_int__one__le__iff__zero__less,axiom,
% 0.30/0.56      ! [Z2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 0.30/0.56        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_one_le_iff_zero_less
% 0.30/0.56  thf(fact_3329_pos__imp__zdiv__pos__iff,axiom,
% 0.30/0.56      ! [K: int,I: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 0.30/0.56          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_imp_zdiv_pos_iff
% 0.30/0.56  thf(fact_3330_div__nonpos__pos__le0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_nonpos_pos_le0
% 0.30/0.56  thf(fact_3331_div__nonneg__neg__le0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.56         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_nonneg_neg_le0
% 0.30/0.56  thf(fact_3332_div__positive__int,axiom,
% 0.30/0.56      ! [L: int,K: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ L @ K )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.56         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_positive_int
% 0.30/0.56  thf(fact_3333_div__int__pos__iff,axiom,
% 0.30/0.56      ! [K: int,L: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 0.30/0.56        = ( ( K = zero_zero_int )
% 0.30/0.56          | ( L = zero_zero_int )
% 0.30/0.56          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 0.30/0.56          | ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.56            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_int_pos_iff
% 0.30/0.56  thf(fact_3334_zless__imp__add1__zle,axiom,
% 0.30/0.56      ! [W2: int,Z2: int] :
% 0.30/0.56        ( ( ord_less_int @ W2 @ Z2 )
% 0.30/0.56       => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zless_imp_add1_zle
% 0.30/0.56  thf(fact_3335_zdiv__mono2__neg,axiom,
% 0.30/0.56      ! [A: int,B6: int,B: int] :
% 0.30/0.56        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.30/0.56         => ( ( ord_less_eq_int @ B6 @ B )
% 0.30/0.56           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2_neg
% 0.30/0.56  thf(fact_3336_zdiv__mono1__neg,axiom,
% 0.30/0.56      ! [A: int,A6: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ A @ A6 )
% 0.30/0.56       => ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.56         => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono1_neg
% 0.30/0.56  thf(fact_3337_int__div__pos__eq,axiom,
% 0.30/0.56      ! [A: int,B: int,Q4: int,R2: int] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.56         => ( ( ord_less_int @ R2 @ B )
% 0.30/0.56           => ( ( divide_divide_int @ A @ B )
% 0.30/0.56              = Q4 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_div_pos_eq
% 0.30/0.56  thf(fact_3338_int__div__neg__eq,axiom,
% 0.30/0.56      ! [A: int,B: int,Q4: int,R2: int] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ B @ R2 )
% 0.30/0.56           => ( ( divide_divide_int @ A @ B )
% 0.30/0.56              = Q4 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_div_neg_eq
% 0.30/0.56  thf(fact_3339_zdiv__eq__0__iff,axiom,
% 0.30/0.56      ! [I: int,K: int] :
% 0.30/0.56        ( ( ( divide_divide_int @ I @ K )
% 0.30/0.56          = zero_zero_int )
% 0.30/0.56        = ( ( K = zero_zero_int )
% 0.30/0.56          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.56            & ( ord_less_int @ I @ K ) )
% 0.30/0.56          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.30/0.56            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_eq_0_iff
% 0.30/0.56  thf(fact_3340_zdiv__mono2,axiom,
% 0.30/0.56      ! [A: int,B6: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.30/0.56         => ( ( ord_less_eq_int @ B6 @ B )
% 0.30/0.56           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2
% 0.30/0.56  thf(fact_3341_zdiv__mono1,axiom,
% 0.30/0.56      ! [A: int,A6: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ A @ A6 )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono1
% 0.30/0.56  thf(fact_3342_split__zdiv,axiom,
% 0.30/0.56      ! [P: int > $o,N2: int,K: int] :
% 0.30/0.56        ( ( P @ ( divide_divide_int @ N2 @ K ) )
% 0.30/0.56        = ( ( ( K = zero_zero_int )
% 0.30/0.56           => ( P @ zero_zero_int ) )
% 0.30/0.56          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56           => ! [I2: int,J3: int] :
% 0.30/0.56                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.56                  & ( ord_less_int @ J3 @ K )
% 0.30/0.56                  & ( N2
% 0.30/0.56                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.56               => ( P @ I2 ) ) )
% 0.30/0.56          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.56           => ! [I2: int,J3: int] :
% 0.30/0.56                ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.56                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.56                  & ( N2
% 0.30/0.56                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.56               => ( P @ I2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % split_zdiv
% 0.30/0.56  thf(fact_3343_le__imp__0__less,axiom,
% 0.30/0.56      ! [Z2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.56       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % le_imp_0_less
% 0.30/0.56  thf(fact_3344_add1__zle__eq,axiom,
% 0.30/0.56      ! [W2: int,Z2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
% 0.30/0.56        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % add1_zle_eq
% 0.30/0.56  thf(fact_3345_odd__less__0__iff,axiom,
% 0.30/0.56      ! [Z2: int] :
% 0.30/0.56        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 0.30/0.56        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_less_0_iff
% 0.30/0.56  thf(fact_3346_pos__zmult__eq__1__iff,axiom,
% 0.30/0.56      ! [M: int,N2: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.56       => ( ( ( times_times_int @ M @ N2 )
% 0.30/0.56            = one_one_int )
% 0.30/0.56          = ( ( M = one_one_int )
% 0.30/0.56            & ( N2 = one_one_int ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_zmult_eq_1_iff
% 0.30/0.56  thf(fact_3347_zless__add1__eq,axiom,
% 0.30/0.56      ! [W2: int,Z2: int] :
% 0.30/0.56        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 0.30/0.56        = ( ( ord_less_int @ W2 @ Z2 )
% 0.30/0.56          | ( W2 = Z2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zless_add1_eq
% 0.30/0.56  thf(fact_3348_int__gr__induct,axiom,
% 0.30/0.56      ! [K: int,I: int,P: int > $o] :
% 0.30/0.56        ( ( ord_less_int @ K @ I )
% 0.30/0.56       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 0.30/0.56         => ( ! [I3: int] :
% 0.30/0.56                ( ( ord_less_int @ K @ I3 )
% 0.30/0.56               => ( ( P @ I3 )
% 0.30/0.56                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.56           => ( P @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_gr_induct
% 0.30/0.56  thf(fact_3349_pos__imp__zdiv__neg__iff,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.30/0.56          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pos_imp_zdiv_neg_iff
% 0.30/0.56  thf(fact_3350_neg__imp__zdiv__neg__iff,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 0.30/0.56          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % neg_imp_zdiv_neg_iff
% 0.30/0.56  thf(fact_3351_int__div__less__self,axiom,
% 0.30/0.56      ! [X3: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.56       => ( ( ord_less_int @ one_one_int @ K )
% 0.30/0.56         => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_div_less_self
% 0.30/0.56  thf(fact_3352_div__neg__pos__less0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_neg_pos_less0
% 0.30/0.56  thf(fact_3353_odd__nonzero,axiom,
% 0.30/0.56      ! [Z2: int] :
% 0.30/0.56        ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
% 0.30/0.56       != zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_nonzero
% 0.30/0.56  thf(fact_3354_times__int__code_I2_J,axiom,
% 0.30/0.56      ! [L: int] :
% 0.30/0.56        ( ( times_times_int @ zero_zero_int @ L )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % times_int_code(2)
% 0.30/0.56  thf(fact_3355_times__int__code_I1_J,axiom,
% 0.30/0.56      ! [K: int] :
% 0.30/0.56        ( ( times_times_int @ K @ zero_zero_int )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % times_int_code(1)
% 0.30/0.56  thf(fact_3356_bot__enat__def,axiom,
% 0.30/0.56      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 0.30/0.56  
% 0.30/0.56  % bot_enat_def
% 0.30/0.56  thf(fact_3357_incr__mult__lemma,axiom,
% 0.30/0.56      ! [D: int,P: int > $o,K: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.56       => ( ! [X4: int] :
% 0.30/0.56              ( ( P @ X4 )
% 0.30/0.56             => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 0.30/0.56         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.56           => ! [X2: int] :
% 0.30/0.56                ( ( P @ X2 )
% 0.30/0.56               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % incr_mult_lemma
% 0.30/0.56  thf(fact_3358_zmult__zless__mono2,axiom,
% 0.30/0.56      ! [I: int,J: int,K: int] :
% 0.30/0.56        ( ( ord_less_int @ I @ J )
% 0.30/0.56       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.56         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zmult_zless_mono2
% 0.30/0.56  thf(fact_3359_less__int__code_I1_J,axiom,
% 0.30/0.56      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % less_int_code(1)
% 0.30/0.56  thf(fact_3360_q__pos__lemma,axiom,
% 0.30/0.56      ! [B6: int,Q6: int,R3: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 0.30/0.56       => ( ( ord_less_int @ R3 @ B6 )
% 0.30/0.56         => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.30/0.56           => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % q_pos_lemma
% 0.30/0.56  thf(fact_3361_zdiv__mono2__lemma,axiom,
% 0.30/0.56      ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
% 0.30/0.56        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 0.30/0.56         => ( ( ord_less_int @ R3 @ B6 )
% 0.30/0.56           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.56             => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.30/0.56               => ( ( ord_less_eq_int @ B6 @ B )
% 0.30/0.56                 => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2_lemma
% 0.30/0.56  thf(fact_3362_zdiv__mono2__neg__lemma,axiom,
% 0.30/0.56      ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
% 0.30/0.56        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 0.30/0.56          = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 0.30/0.56       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ R2 @ B )
% 0.30/0.56           => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 0.30/0.56             => ( ( ord_less_int @ zero_zero_int @ B6 )
% 0.30/0.56               => ( ( ord_less_eq_int @ B6 @ B )
% 0.30/0.56                 => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdiv_mono2_neg_lemma
% 0.30/0.56  thf(fact_3363_unique__quotient__lemma,axiom,
% 0.30/0.56      ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 0.30/0.56         => ( ( ord_less_int @ R3 @ B )
% 0.30/0.56           => ( ( ord_less_int @ R2 @ B )
% 0.30/0.56             => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_quotient_lemma
% 0.30/0.56  thf(fact_3364_unique__quotient__lemma__neg,axiom,
% 0.30/0.56      ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.56       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.56         => ( ( ord_less_int @ B @ R2 )
% 0.30/0.56           => ( ( ord_less_int @ B @ R3 )
% 0.30/0.56             => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_quotient_lemma_neg
% 0.30/0.56  thf(fact_3365_enat__0__less__mult__iff,axiom,
% 0.30/0.56      ! [M: extended_enat,N2: extended_enat] :
% 0.30/0.56        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
% 0.30/0.56        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 0.30/0.56          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % enat_0_less_mult_iff
% 0.30/0.56  thf(fact_3366_realpow__pos__nth2,axiom,
% 0.30/0.56      ! [A: real,N2: nat] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56       => ? [R4: real] :
% 0.30/0.56            ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.30/0.56            & ( ( power_power_real @ R4 @ ( suc @ N2 ) )
% 0.30/0.56              = A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth2
% 0.30/0.56  thf(fact_3367_realpow__pos__nth__unique,axiom,
% 0.30/0.56      ! [N2: nat,A: real] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56         => ? [X4: real] :
% 0.30/0.56              ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.30/0.56              & ( ( power_power_real @ X4 @ N2 )
% 0.30/0.56                = A )
% 0.30/0.56              & ! [Y6: real] :
% 0.30/0.56                  ( ( ( ord_less_real @ zero_zero_real @ Y6 )
% 0.30/0.56                    & ( ( power_power_real @ Y6 @ N2 )
% 0.30/0.56                      = A ) )
% 0.30/0.56                 => ( Y6 = X4 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth_unique
% 0.30/0.56  thf(fact_3368_realpow__pos__nth,axiom,
% 0.30/0.56      ! [N2: nat,A: real] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.56         => ? [R4: real] :
% 0.30/0.56              ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.30/0.56              & ( ( power_power_real @ R4 @ N2 )
% 0.30/0.56                = A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % realpow_pos_nth
% 0.30/0.56  thf(fact_3369_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y3 ) )
% 0.30/0.56          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff2
% 0.30/0.56  thf(fact_3370_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ ( times_times_rat @ Z2 @ Y3 ) )
% 0.30/0.56          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff2
% 0.30/0.56  thf(fact_3371_mult__le__cancel__iff2,axiom,
% 0.30/0.56      ! [Z2: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X3 ) @ ( times_times_int @ Z2 @ Y3 ) )
% 0.30/0.56          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff2
% 0.30/0.56  thf(fact_3372_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff1
% 0.30/0.56  thf(fact_3373_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff1
% 0.30/0.56  thf(fact_3374_mult__le__cancel__iff1,axiom,
% 0.30/0.56      ! [Z2: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.56       => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_le_cancel_iff1
% 0.30/0.56  thf(fact_3375_divides__aux__eq,axiom,
% 0.30/0.56      ! [Q4: nat,R2: nat] :
% 0.30/0.56        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.30/0.56        = ( R2 = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % divides_aux_eq
% 0.30/0.56  thf(fact_3376_divides__aux__eq,axiom,
% 0.30/0.56      ! [Q4: int,R2: int] :
% 0.30/0.56        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.56        = ( R2 = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % divides_aux_eq
% 0.30/0.56  thf(fact_3377_low__def,axiom,
% 0.30/0.56      ( vEBT_VEBT_low
% 0.30/0.56      = ( ^ [X: nat,N: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % low_def
% 0.30/0.56  thf(fact_3378_even__succ__div__exp,axiom,
% 0.30/0.56      ! [A: nat,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_exp
% 0.30/0.56  thf(fact_3379_even__succ__div__exp,axiom,
% 0.30/0.56      ! [A: int,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_exp
% 0.30/0.56  thf(fact_3380_even__succ__div__exp,axiom,
% 0.30/0.56      ! [A: code_integer,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_exp
% 0.30/0.56  thf(fact_3381_set__decode__Suc,axiom,
% 0.30/0.56      ! [N2: nat,X3: nat] :
% 0.30/0.56        ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) )
% 0.30/0.56        = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_decode_Suc
% 0.30/0.56  thf(fact_3382_length__product,axiom,
% 0.30/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3383_length__product,axiom,
% 0.30/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.56        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3384_length__product,axiom,
% 0.30/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.56        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3385_length__product,axiom,
% 0.30/0.56      ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.30/0.56        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3386_length__product,axiom,
% 0.30/0.56      ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3387_length__product,axiom,
% 0.30/0.56      ! [Xs: list_o,Ys: list_o] :
% 0.30/0.56        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3388_length__product,axiom,
% 0.30/0.56      ! [Xs: list_o,Ys: list_nat] :
% 0.30/0.56        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3389_length__product,axiom,
% 0.30/0.56      ! [Xs: list_o,Ys: list_int] :
% 0.30/0.56        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3390_length__product,axiom,
% 0.30/0.56      ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.30/0.56        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3391_length__product,axiom,
% 0.30/0.56      ! [Xs: list_nat,Ys: list_o] :
% 0.30/0.56        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
% 0.30/0.56        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % length_product
% 0.30/0.56  thf(fact_3392_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.30/0.56       => ( ( ord_less_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_less_iff1
% 0.30/0.56  thf(fact_3393_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 0.30/0.56       => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_less_iff1
% 0.30/0.56  thf(fact_3394_mult__less__iff1,axiom,
% 0.30/0.56      ! [Z2: int,X3: int,Y3: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.56       => ( ( ord_less_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
% 0.30/0.56          = ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_less_iff1
% 0.30/0.56  thf(fact_3395_num_Osize__gen_I2_J,axiom,
% 0.30/0.56      ! [X22: num] :
% 0.30/0.56        ( ( size_num @ ( bit0 @ X22 ) )
% 0.30/0.56        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % num.size_gen(2)
% 0.30/0.56  thf(fact_3396_nat__dvd__1__iff__1,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 0.30/0.56        = ( M = one_one_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % nat_dvd_1_iff_1
% 0.30/0.56  thf(fact_3397_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3398_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.56        = ( A = zero_zero_complex ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3399_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.56        = ( A = zero_zero_real ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3400_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.56        = ( A = zero_zero_rat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3401_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.56        = ( A = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3402_dvd__0__left__iff,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.56        = ( A = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left_iff
% 0.30/0.56  thf(fact_3403_dvd__0__right,axiom,
% 0.30/0.56      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3404_dvd__0__right,axiom,
% 0.30/0.56      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3405_dvd__0__right,axiom,
% 0.30/0.56      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3406_dvd__0__right,axiom,
% 0.30/0.56      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3407_dvd__0__right,axiom,
% 0.30/0.56      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3408_dvd__0__right,axiom,
% 0.30/0.56      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_right
% 0.30/0.56  thf(fact_3409_dvd__add__triv__right__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 0.30/0.56        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_right_iff
% 0.30/0.56  thf(fact_3410_dvd__add__triv__right__iff,axiom,
% 0.30/0.56      ! [A: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 0.30/0.56        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_right_iff
% 0.30/0.56  thf(fact_3411_dvd__add__triv__right__iff,axiom,
% 0.30/0.56      ! [A: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 0.30/0.56        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_right_iff
% 0.30/0.56  thf(fact_3412_dvd__add__triv__right__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 0.30/0.56        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_right_iff
% 0.30/0.56  thf(fact_3413_dvd__add__triv__right__iff,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 0.30/0.56        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_right_iff
% 0.30/0.56  thf(fact_3414_dvd__add__triv__left__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.56        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_left_iff
% 0.30/0.56  thf(fact_3415_dvd__add__triv__left__iff,axiom,
% 0.30/0.56      ! [A: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 0.30/0.56        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_left_iff
% 0.30/0.56  thf(fact_3416_dvd__add__triv__left__iff,axiom,
% 0.30/0.56      ! [A: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.56        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_left_iff
% 0.30/0.56  thf(fact_3417_dvd__add__triv__left__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.56        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_left_iff
% 0.30/0.56  thf(fact_3418_dvd__add__triv__left__iff,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 0.30/0.56        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_triv_left_iff
% 0.30/0.56  thf(fact_3419_bits__mod__0,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_0
% 0.30/0.56  thf(fact_3420_bits__mod__0,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_0
% 0.30/0.56  thf(fact_3421_bits__mod__0,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_0
% 0.30/0.56  thf(fact_3422_mod__self,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ A @ A )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_self
% 0.30/0.56  thf(fact_3423_mod__self,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ A @ A )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_self
% 0.30/0.56  thf(fact_3424_mod__self,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ A @ A )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_self
% 0.30/0.56  thf(fact_3425_mod__by__0,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 0.30/0.56        = A ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_0
% 0.30/0.56  thf(fact_3426_mod__by__0,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 0.30/0.56        = A ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_0
% 0.30/0.56  thf(fact_3427_mod__by__0,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.56        = A ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_0
% 0.30/0.56  thf(fact_3428_mod__0,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0
% 0.30/0.56  thf(fact_3429_mod__0,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0
% 0.30/0.56  thf(fact_3430_mod__0,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0
% 0.30/0.56  thf(fact_3431_dvd__1__left,axiom,
% 0.30/0.56      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_1_left
% 0.30/0.56  thf(fact_3432_dvd__1__iff__1,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.56        = ( M
% 0.30/0.56          = ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_1_iff_1
% 0.30/0.56  thf(fact_3433_div__dvd__div,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.56         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 0.30/0.56            = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_dvd_div
% 0.30/0.56  thf(fact_3434_div__dvd__div,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.56         => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 0.30/0.56            = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_dvd_div
% 0.30/0.56  thf(fact_3435_div__dvd__div,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.56         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.56            = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_dvd_div
% 0.30/0.56  thf(fact_3436_not__real__square__gt__zero,axiom,
% 0.30/0.56      ! [X3: real] :
% 0.30/0.56        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
% 0.30/0.56        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_real_square_gt_zero
% 0.30/0.56  thf(fact_3437_mod__add__self2,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self2
% 0.30/0.56  thf(fact_3438_mod__add__self2,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self2
% 0.30/0.56  thf(fact_3439_mod__add__self2,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self2
% 0.30/0.56  thf(fact_3440_mod__add__self1,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self1
% 0.30/0.56  thf(fact_3441_mod__add__self1,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self1
% 0.30/0.56  thf(fact_3442_mod__add__self1,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_self1
% 0.30/0.56  thf(fact_3443_nat__mult__dvd__cancel__disj,axiom,
% 0.30/0.56      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.56        = ( ( K = zero_zero_nat )
% 0.30/0.56          | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % nat_mult_dvd_cancel_disj
% 0.30/0.56  thf(fact_3444_mod__less,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] :
% 0.30/0.56        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.56       => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.56          = M ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_less
% 0.30/0.56  thf(fact_3445_dvd__mult__cancel__left,axiom,
% 0.30/0.56      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.30/0.56        = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.56          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_left
% 0.30/0.56  thf(fact_3446_dvd__mult__cancel__left,axiom,
% 0.30/0.56      ! [C: complex,A: complex,B: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 0.30/0.56        = ( ( C = zero_zero_complex )
% 0.30/0.56          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_left
% 0.30/0.56  thf(fact_3447_dvd__mult__cancel__left,axiom,
% 0.30/0.56      ! [C: real,A: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 0.30/0.56        = ( ( C = zero_zero_real )
% 0.30/0.56          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_left
% 0.30/0.56  thf(fact_3448_dvd__mult__cancel__left,axiom,
% 0.30/0.56      ! [C: rat,A: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 0.30/0.56        = ( ( C = zero_zero_rat )
% 0.30/0.56          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_left
% 0.30/0.56  thf(fact_3449_dvd__mult__cancel__left,axiom,
% 0.30/0.56      ! [C: int,A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 0.30/0.56        = ( ( C = zero_zero_int )
% 0.30/0.56          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_left
% 0.30/0.56  thf(fact_3450_dvd__mult__cancel__right,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.56        = ( ( C = zero_z3403309356797280102nteger )
% 0.30/0.56          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_right
% 0.30/0.56  thf(fact_3451_dvd__mult__cancel__right,axiom,
% 0.30/0.56      ! [A: complex,C: complex,B: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 0.30/0.56        = ( ( C = zero_zero_complex )
% 0.30/0.56          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_right
% 0.30/0.56  thf(fact_3452_dvd__mult__cancel__right,axiom,
% 0.30/0.56      ! [A: real,C: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 0.30/0.56        = ( ( C = zero_zero_real )
% 0.30/0.56          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_right
% 0.30/0.56  thf(fact_3453_dvd__mult__cancel__right,axiom,
% 0.30/0.56      ! [A: rat,C: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 0.30/0.56        = ( ( C = zero_zero_rat )
% 0.30/0.56          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_right
% 0.30/0.56  thf(fact_3454_dvd__mult__cancel__right,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 0.30/0.56        = ( ( C = zero_zero_int )
% 0.30/0.56          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_cancel_right
% 0.30/0.56  thf(fact_3455_dvd__times__left__cancel__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_left_cancel_iff
% 0.30/0.56  thf(fact_3456_dvd__times__left__cancel__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( A != zero_zero_nat )
% 0.30/0.56       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 0.30/0.56          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_left_cancel_iff
% 0.30/0.56  thf(fact_3457_dvd__times__left__cancel__iff,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( A != zero_zero_int )
% 0.30/0.56       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 0.30/0.56          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_left_cancel_iff
% 0.30/0.56  thf(fact_3458_dvd__times__right__cancel__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_right_cancel_iff
% 0.30/0.56  thf(fact_3459_dvd__times__right__cancel__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( A != zero_zero_nat )
% 0.30/0.56       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 0.30/0.56          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_right_cancel_iff
% 0.30/0.56  thf(fact_3460_dvd__times__right__cancel__iff,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( A != zero_zero_int )
% 0.30/0.56       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 0.30/0.56          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_times_right_cancel_iff
% 0.30/0.56  thf(fact_3461_unit__prod,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_prod
% 0.30/0.56  thf(fact_3462_unit__prod,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.56         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_prod
% 0.30/0.56  thf(fact_3463_unit__prod,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.56         => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_prod
% 0.30/0.56  thf(fact_3464_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3465_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: complex,B: complex,C: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3466_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3467_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3468_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3469_dvd__add__times__triv__right__iff,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 0.30/0.56        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_right_iff
% 0.30/0.56  thf(fact_3470_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3471_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: complex,C: complex,B: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_complex @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3472_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: real,C: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3473_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: rat,C: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_rat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3474_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3475_dvd__add__times__triv__left__iff,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 0.30/0.56        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_times_triv_left_iff
% 0.30/0.56  thf(fact_3476_mod__mult__self1__is__0,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1_is_0
% 0.30/0.56  thf(fact_3477_mod__mult__self1__is__0,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1_is_0
% 0.30/0.56  thf(fact_3478_mod__mult__self1__is__0,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1_is_0
% 0.30/0.56  thf(fact_3479_mod__mult__self2__is__0,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2_is_0
% 0.30/0.56  thf(fact_3480_mod__mult__self2__is__0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2_is_0
% 0.30/0.56  thf(fact_3481_mod__mult__self2__is__0,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2_is_0
% 0.30/0.56  thf(fact_3482_bits__mod__by__1,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_by_1
% 0.30/0.56  thf(fact_3483_bits__mod__by__1,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_by_1
% 0.30/0.56  thf(fact_3484_bits__mod__by__1,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_by_1
% 0.30/0.56  thf(fact_3485_mod__by__1,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_1
% 0.30/0.56  thf(fact_3486_mod__by__1,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ A @ one_one_int )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_1
% 0.30/0.56  thf(fact_3487_mod__by__1,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_1
% 0.30/0.56  thf(fact_3488_dvd__div__mult__self,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_mult_self
% 0.30/0.56  thf(fact_3489_dvd__div__mult__self,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_mult_self
% 0.30/0.56  thf(fact_3490_dvd__div__mult__self,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_mult_self
% 0.30/0.56  thf(fact_3491_dvd__mult__div__cancel,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_div_cancel
% 0.30/0.56  thf(fact_3492_dvd__mult__div__cancel,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_div_cancel
% 0.30/0.56  thf(fact_3493_dvd__mult__div__cancel,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_div_cancel
% 0.30/0.56  thf(fact_3494_unit__div,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.56         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div
% 0.30/0.56  thf(fact_3495_unit__div,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.56         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div
% 0.30/0.56  thf(fact_3496_unit__div,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div
% 0.30/0.56  thf(fact_3497_unit__div__1__unit,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_unit
% 0.30/0.56  thf(fact_3498_unit__div__1__unit,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_unit
% 0.30/0.56  thf(fact_3499_unit__div__1__unit,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_unit
% 0.30/0.56  thf(fact_3500_unit__div__1__div__1,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_div_1
% 0.30/0.56  thf(fact_3501_unit__div__1__div__1,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_div_1
% 0.30/0.56  thf(fact_3502_unit__div__1__div__1,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_1_div_1
% 0.30/0.56  thf(fact_3503_div__add,axiom,
% 0.30/0.56      ! [C: nat,A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.56         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.56            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add
% 0.30/0.56  thf(fact_3504_div__add,axiom,
% 0.30/0.56      ! [C: int,A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.56         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.56            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add
% 0.30/0.56  thf(fact_3505_div__add,axiom,
% 0.30/0.56      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.56         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.56            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add
% 0.30/0.56  thf(fact_3506_bits__mod__div__trivial,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_div_trivial
% 0.30/0.56  thf(fact_3507_bits__mod__div__trivial,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_div_trivial
% 0.30/0.56  thf(fact_3508_bits__mod__div__trivial,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % bits_mod_div_trivial
% 0.30/0.56  thf(fact_3509_mod__div__trivial,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_div_trivial
% 0.30/0.56  thf(fact_3510_mod__div__trivial,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 0.30/0.56        = zero_zero_int ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_div_trivial
% 0.30/0.56  thf(fact_3511_mod__div__trivial,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 0.30/0.56        = zero_z3403309356797280102nteger ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_div_trivial
% 0.30/0.56  thf(fact_3512_mod__mult__self4,axiom,
% 0.30/0.56      ! [B: nat,C: nat,A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self4
% 0.30/0.56  thf(fact_3513_mod__mult__self4,axiom,
% 0.30/0.56      ! [B: int,C: int,A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self4
% 0.30/0.56  thf(fact_3514_mod__mult__self4,axiom,
% 0.30/0.56      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self4
% 0.30/0.56  thf(fact_3515_mod__mult__self3,axiom,
% 0.30/0.56      ! [C: nat,B: nat,A: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self3
% 0.30/0.56  thf(fact_3516_mod__mult__self3,axiom,
% 0.30/0.56      ! [C: int,B: int,A: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self3
% 0.30/0.56  thf(fact_3517_mod__mult__self3,axiom,
% 0.30/0.56      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self3
% 0.30/0.56  thf(fact_3518_mod__mult__self2,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2
% 0.30/0.56  thf(fact_3519_mod__mult__self2,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2
% 0.30/0.56  thf(fact_3520_mod__mult__self2,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self2
% 0.30/0.56  thf(fact_3521_mod__mult__self1,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 0.30/0.56        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1
% 0.30/0.56  thf(fact_3522_mod__mult__self1,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 0.30/0.56        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1
% 0.30/0.56  thf(fact_3523_mod__mult__self1,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 0.30/0.56        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_mult_self1
% 0.30/0.56  thf(fact_3524_dvd__imp__mod__0,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( modulo_modulo_nat @ B @ A )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_imp_mod_0
% 0.30/0.56  thf(fact_3525_dvd__imp__mod__0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( modulo_modulo_int @ B @ A )
% 0.30/0.56          = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_imp_mod_0
% 0.30/0.56  thf(fact_3526_dvd__imp__mod__0,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( modulo364778990260209775nteger @ B @ A )
% 0.30/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_imp_mod_0
% 0.30/0.56  thf(fact_3527_mod__by__Suc__0,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_by_Suc_0
% 0.30/0.56  thf(fact_3528_set__decode__zero,axiom,
% 0.30/0.56      ( ( nat_set_decode @ zero_zero_nat )
% 0.30/0.56      = bot_bot_set_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % set_decode_zero
% 0.30/0.56  thf(fact_3529_set__encode__inverse,axiom,
% 0.30/0.56      ! [A2: set_nat] :
% 0.30/0.56        ( ( finite_finite_nat @ A2 )
% 0.30/0.56       => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 0.30/0.56          = A2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_encode_inverse
% 0.30/0.56  thf(fact_3530_unit__div__mult__self,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_mult_self
% 0.30/0.56  thf(fact_3531_unit__div__mult__self,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_mult_self
% 0.30/0.56  thf(fact_3532_unit__div__mult__self,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 0.30/0.56          = B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_div_mult_self
% 0.30/0.56  thf(fact_3533_unit__mult__div__div,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.56       => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 0.30/0.56          = ( divide_divide_nat @ B @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_mult_div_div
% 0.30/0.56  thf(fact_3534_unit__mult__div__div,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.56       => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 0.30/0.56          = ( divide_divide_int @ B @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_mult_div_div
% 0.30/0.56  thf(fact_3535_unit__mult__div__div,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.56       => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 0.30/0.56          = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_mult_div_div
% 0.30/0.56  thf(fact_3536_even__Suc,axiom,
% 0.30/0.56      ! [N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_Suc
% 0.30/0.56  thf(fact_3537_even__Suc__Suc__iff,axiom,
% 0.30/0.56      ! [N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
% 0.30/0.56        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_Suc_Suc_iff
% 0.30/0.56  thf(fact_3538_pow__divides__pow__iff,axiom,
% 0.30/0.56      ! [N2: nat,A: nat,B: nat] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 0.30/0.56          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pow_divides_pow_iff
% 0.30/0.56  thf(fact_3539_pow__divides__pow__iff,axiom,
% 0.30/0.56      ! [N2: nat,A: int,B: int] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 0.30/0.56          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pow_divides_pow_iff
% 0.30/0.56  thf(fact_3540_Suc__mod__mult__self4,axiom,
% 0.30/0.56      ! [N2: nat,K: nat,M: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Suc_mod_mult_self4
% 0.30/0.56  thf(fact_3541_Suc__mod__mult__self3,axiom,
% 0.30/0.56      ! [K: nat,N2: nat,M: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Suc_mod_mult_self3
% 0.30/0.56  thf(fact_3542_Suc__mod__mult__self2,axiom,
% 0.30/0.56      ! [M: nat,N2: nat,K: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Suc_mod_mult_self2
% 0.30/0.56  thf(fact_3543_Suc__mod__mult__self1,axiom,
% 0.30/0.56      ! [M: nat,K: nat,N2: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Suc_mod_mult_self1
% 0.30/0.56  thf(fact_3544_even__add,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.56        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_add
% 0.30/0.56  thf(fact_3545_even__add,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.56        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_add
% 0.30/0.56  thf(fact_3546_even__add,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 0.30/0.56        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_add
% 0.30/0.56  thf(fact_3547_odd__add,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 0.30/0.56        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.56         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_add
% 0.30/0.56  thf(fact_3548_odd__add,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 0.30/0.56        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.56         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_add
% 0.30/0.56  thf(fact_3549_odd__add,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 0.30/0.56        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.56         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_add
% 0.30/0.56  thf(fact_3550_odd__Suc__div__two,axiom,
% 0.30/0.56      ! [N2: nat] :
% 0.30/0.56        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.56       => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_Suc_div_two
% 0.30/0.56  thf(fact_3551_even__Suc__div__two,axiom,
% 0.30/0.56      ! [N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.56       => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_Suc_div_two
% 0.30/0.56  thf(fact_3552_mod2__Suc__Suc,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod2_Suc_Suc
% 0.30/0.56  thf(fact_3553_Suc__times__numeral__mod__eq,axiom,
% 0.30/0.56      ! [K: num,N2: nat] :
% 0.30/0.56        ( ( ( numeral_numeral_nat @ K )
% 0.30/0.56         != one_one_nat )
% 0.30/0.56       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.56          = one_one_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % Suc_times_numeral_mod_eq
% 0.30/0.56  thf(fact_3554_set__decode__0,axiom,
% 0.30/0.56      ! [X3: nat] :
% 0.30/0.56        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % set_decode_0
% 0.30/0.56  thf(fact_3555_zero__le__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: real,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_le_power_eq_numeral
% 0.30/0.56  thf(fact_3556_zero__le__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: rat,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_le_power_eq_numeral
% 0.30/0.56  thf(fact_3557_zero__le__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: int,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_le_power_eq_numeral
% 0.30/0.56  thf(fact_3558_power__less__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: real,W2: num] :
% 0.30/0.56        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq_numeral
% 0.30/0.56  thf(fact_3559_power__less__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: rat,W2: num] :
% 0.30/0.56        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq_numeral
% 0.30/0.56  thf(fact_3560_power__less__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: int,W2: num] :
% 0.30/0.56        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq_numeral
% 0.30/0.56  thf(fact_3561_power__less__zero__eq,axiom,
% 0.30/0.56      ! [A: real,N2: nat] :
% 0.30/0.56        ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.56          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq
% 0.30/0.56  thf(fact_3562_power__less__zero__eq,axiom,
% 0.30/0.56      ! [A: rat,N2: nat] :
% 0.30/0.56        ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.56          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq
% 0.30/0.56  thf(fact_3563_power__less__zero__eq,axiom,
% 0.30/0.56      ! [A: int,N2: nat] :
% 0.30/0.56        ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.56          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_less_zero_eq
% 0.30/0.56  thf(fact_3564_even__plus__one__iff,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_plus_one_iff
% 0.30/0.56  thf(fact_3565_even__plus__one__iff,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_plus_one_iff
% 0.30/0.56  thf(fact_3566_even__plus__one__iff,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_plus_one_iff
% 0.30/0.56  thf(fact_3567_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56         != one_one_nat )
% 0.30/0.56        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_1_eq_0
% 0.30/0.56  thf(fact_3568_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56         != one_one_int )
% 0.30/0.56        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_1_eq_0
% 0.30/0.56  thf(fact_3569_not__mod__2__eq__1__eq__0,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56         != one_one_Code_integer )
% 0.30/0.56        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_1_eq_0
% 0.30/0.56  thf(fact_3570_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56         != zero_zero_nat )
% 0.30/0.56        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = one_one_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_0_eq_1
% 0.30/0.56  thf(fact_3571_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56         != zero_zero_int )
% 0.30/0.56        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = one_one_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_0_eq_1
% 0.30/0.56  thf(fact_3572_not__mod__2__eq__0__eq__1,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56         != zero_z3403309356797280102nteger )
% 0.30/0.56        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = one_one_Code_integer ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod_2_eq_0_eq_1
% 0.30/0.56  thf(fact_3573_not__mod2__eq__Suc__0__eq__0,axiom,
% 0.30/0.56      ! [N2: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56         != ( suc @ zero_zero_nat ) )
% 0.30/0.56        = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % not_mod2_eq_Suc_0_eq_0
% 0.30/0.56  thf(fact_3574_add__self__mod__2,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56        = zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % add_self_mod_2
% 0.30/0.56  thf(fact_3575_zero__less__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: real,W2: num] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( ( numeral_numeral_nat @ W2 )
% 0.30/0.56            = zero_zero_nat )
% 0.30/0.56          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( A != zero_zero_real ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_less_power_eq_numeral
% 0.30/0.56  thf(fact_3576_zero__less__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: rat,W2: num] :
% 0.30/0.56        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( ( numeral_numeral_nat @ W2 )
% 0.30/0.56            = zero_zero_nat )
% 0.30/0.56          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( A != zero_zero_rat ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_less_power_eq_numeral
% 0.30/0.56  thf(fact_3577_zero__less__power__eq__numeral,axiom,
% 0.30/0.56      ! [A: int,W2: num] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 0.30/0.56        = ( ( ( numeral_numeral_nat @ W2 )
% 0.30/0.56            = zero_zero_nat )
% 0.30/0.56          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( A != zero_zero_int ) )
% 0.30/0.56          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zero_less_power_eq_numeral
% 0.30/0.56  thf(fact_3578_even__succ__div__two,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_two
% 0.30/0.56  thf(fact_3579_even__succ__div__two,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_two
% 0.30/0.56  thf(fact_3580_even__succ__div__two,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_two
% 0.30/0.56  thf(fact_3581_odd__succ__div__two,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_succ_div_two
% 0.30/0.56  thf(fact_3582_odd__succ__div__two,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_succ_div_two
% 0.30/0.56  thf(fact_3583_odd__succ__div__two,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_succ_div_two
% 0.30/0.56  thf(fact_3584_even__succ__div__2,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_2
% 0.30/0.56  thf(fact_3585_even__succ__div__2,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_2
% 0.30/0.56  thf(fact_3586_even__succ__div__2,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_div_2
% 0.30/0.56  thf(fact_3587_even__power,axiom,
% 0.30/0.56      ! [A: code_integer,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 0.30/0.56        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_power
% 0.30/0.56  thf(fact_3588_even__power,axiom,
% 0.30/0.56      ! [A: nat,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
% 0.30/0.56        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_power
% 0.30/0.56  thf(fact_3589_even__power,axiom,
% 0.30/0.56      ! [A: int,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
% 0.30/0.56        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_power
% 0.30/0.56  thf(fact_3590_mod2__gr__0,axiom,
% 0.30/0.56      ! [M: nat] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.56        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = one_one_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod2_gr_0
% 0.30/0.56  thf(fact_3591_odd__two__times__div__two__succ,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_two_times_div_two_succ
% 0.30/0.56  thf(fact_3592_odd__two__times__div__two__succ,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_two_times_div_two_succ
% 0.30/0.56  thf(fact_3593_odd__two__times__div__two__succ,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 0.30/0.56          = A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % odd_two_times_div_two_succ
% 0.30/0.56  thf(fact_3594_power__le__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: real,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 0.30/0.56        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.56            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_le_zero_eq_numeral
% 0.30/0.56  thf(fact_3595_power__le__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: rat,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 0.30/0.56        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.56            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_le_zero_eq_numeral
% 0.30/0.56  thf(fact_3596_power__le__zero__eq__numeral,axiom,
% 0.30/0.56      ! [A: int,W2: num] :
% 0.30/0.56        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 0.30/0.56        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.56            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 0.30/0.56              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % power_le_zero_eq_numeral
% 0.30/0.56  thf(fact_3597_even__succ__mod__exp,axiom,
% 0.30/0.56      ! [A: nat,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_mod_exp
% 0.30/0.56  thf(fact_3598_even__succ__mod__exp,axiom,
% 0.30/0.56      ! [A: int,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_mod_exp
% 0.30/0.56  thf(fact_3599_even__succ__mod__exp,axiom,
% 0.30/0.56      ! [A: code_integer,N2: nat] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.56         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.56            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_succ_mod_exp
% 0.30/0.56  thf(fact_3600_real__arch__pow,axiom,
% 0.30/0.56      ! [X3: real,Y3: real] :
% 0.30/0.56        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.56       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % real_arch_pow
% 0.30/0.56  thf(fact_3601_real__arch__pow__inv,axiom,
% 0.30/0.56      ! [Y3: real,X3: real] :
% 0.30/0.56        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.56       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.56         => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y3 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % real_arch_pow_inv
% 0.30/0.56  thf(fact_3602_int__distrib_I2_J,axiom,
% 0.30/0.56      ! [W2: int,Z1: int,Z22: int] :
% 0.30/0.56        ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
% 0.30/0.56        = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(2)
% 0.30/0.56  thf(fact_3603_int__distrib_I1_J,axiom,
% 0.30/0.56      ! [Z1: int,Z22: int,W2: int] :
% 0.30/0.56        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
% 0.30/0.56        = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_distrib(1)
% 0.30/0.56  thf(fact_3604_zdvd__period,axiom,
% 0.30/0.56      ! [A: int,D: int,X3: int,T: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ D )
% 0.30/0.56       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
% 0.30/0.56          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdvd_period
% 0.30/0.56  thf(fact_3605_zdvd__reduce,axiom,
% 0.30/0.56      ! [K: int,N2: int,M: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
% 0.30/0.56        = ( dvd_dvd_int @ K @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdvd_reduce
% 0.30/0.56  thf(fact_3606_less__eq__real__def,axiom,
% 0.30/0.56      ( ord_less_eq_real
% 0.30/0.56      = ( ^ [X: real,Y: real] :
% 0.30/0.56            ( ( ord_less_real @ X @ Y )
% 0.30/0.56            | ( X = Y ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % less_eq_real_def
% 0.30/0.56  thf(fact_3607_mod__eq__0__iff__dvd,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56          = zero_zero_nat )
% 0.30/0.56        = ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_0_iff_dvd
% 0.30/0.56  thf(fact_3608_mod__eq__0__iff__dvd,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56          = zero_zero_int )
% 0.30/0.56        = ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_0_iff_dvd
% 0.30/0.56  thf(fact_3609_mod__eq__0__iff__dvd,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56          = zero_z3403309356797280102nteger )
% 0.30/0.56        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_0_iff_dvd
% 0.30/0.56  thf(fact_3610_dvd__eq__mod__eq__0,axiom,
% 0.30/0.56      ( dvd_dvd_nat
% 0.30/0.56      = ( ^ [A3: nat,B3: nat] :
% 0.30/0.56            ( ( modulo_modulo_nat @ B3 @ A3 )
% 0.30/0.56            = zero_zero_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_eq_mod_eq_0
% 0.30/0.56  thf(fact_3611_dvd__eq__mod__eq__0,axiom,
% 0.30/0.56      ( dvd_dvd_int
% 0.30/0.56      = ( ^ [A3: int,B3: int] :
% 0.30/0.56            ( ( modulo_modulo_int @ B3 @ A3 )
% 0.30/0.56            = zero_zero_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_eq_mod_eq_0
% 0.30/0.56  thf(fact_3612_dvd__eq__mod__eq__0,axiom,
% 0.30/0.56      ( dvd_dvd_Code_integer
% 0.30/0.56      = ( ^ [A3: code_integer,B3: code_integer] :
% 0.30/0.56            ( ( modulo364778990260209775nteger @ B3 @ A3 )
% 0.30/0.56            = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_eq_mod_eq_0
% 0.30/0.56  thf(fact_3613_mod__0__imp__dvd,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56          = zero_zero_nat )
% 0.30/0.56       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0_imp_dvd
% 0.30/0.56  thf(fact_3614_mod__0__imp__dvd,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56          = zero_zero_int )
% 0.30/0.56       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0_imp_dvd
% 0.30/0.56  thf(fact_3615_mod__0__imp__dvd,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56          = zero_z3403309356797280102nteger )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_0_imp_dvd
% 0.30/0.56  thf(fact_3616_dvd__refl,axiom,
% 0.30/0.56      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_refl
% 0.30/0.56  thf(fact_3617_dvd__refl,axiom,
% 0.30/0.56      ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_refl
% 0.30/0.56  thf(fact_3618_dvd__refl,axiom,
% 0.30/0.56      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_refl
% 0.30/0.56  thf(fact_3619_dvd__trans,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.56         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_trans
% 0.30/0.56  thf(fact_3620_dvd__trans,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ B @ C )
% 0.30/0.56         => ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_trans
% 0.30/0.56  thf(fact_3621_dvd__trans,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ B @ C )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_trans
% 0.30/0.56  thf(fact_3622_dvd__mod__iff,axiom,
% 0.30/0.56      ! [C: nat,B: nat,A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56          = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_iff
% 0.30/0.56  thf(fact_3623_dvd__mod__iff,axiom,
% 0.30/0.56      ! [C: int,B: int,A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ C @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56          = ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_iff
% 0.30/0.56  thf(fact_3624_dvd__mod__iff,axiom,
% 0.30/0.56      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_iff
% 0.30/0.56  thf(fact_3625_dvd__mod__imp__dvd,axiom,
% 0.30/0.56      ! [C: nat,A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.56         => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_imp_dvd
% 0.30/0.56  thf(fact_3626_dvd__mod__imp__dvd,axiom,
% 0.30/0.56      ! [C: int,A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.56         => ( dvd_dvd_int @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_imp_dvd
% 0.30/0.56  thf(fact_3627_dvd__mod__imp__dvd,axiom,
% 0.30/0.56      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mod_imp_dvd
% 0.30/0.56  thf(fact_3628_int__ge__induct,axiom,
% 0.30/0.56      ! [K: int,I: int,P: int > $o] :
% 0.30/0.56        ( ( ord_less_eq_int @ K @ I )
% 0.30/0.56       => ( ( P @ K )
% 0.30/0.56         => ( ! [I3: int] :
% 0.30/0.56                ( ( ord_less_eq_int @ K @ I3 )
% 0.30/0.56               => ( ( P @ I3 )
% 0.30/0.56                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.56           => ( P @ I ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % int_ge_induct
% 0.30/0.56  thf(fact_3629_unit__imp__mod__eq__0,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.56       => ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_mod_eq_0
% 0.30/0.56  thf(fact_3630_unit__imp__mod__eq__0,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.56       => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56          = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_mod_eq_0
% 0.30/0.56  thf(fact_3631_unit__imp__mod__eq__0,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.56       => ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_mod_eq_0
% 0.30/0.56  thf(fact_3632_mod__greater__zero__iff__not__dvd,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.56        = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_greater_zero_iff_not_dvd
% 0.30/0.56  thf(fact_3633_mod__add__right__eq,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_right_eq
% 0.30/0.56  thf(fact_3634_mod__add__right__eq,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_right_eq
% 0.30/0.56  thf(fact_3635_mod__add__right__eq,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_right_eq
% 0.30/0.56  thf(fact_3636_mod__add__left__eq,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 0.30/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_left_eq
% 0.30/0.56  thf(fact_3637_mod__add__left__eq,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 0.30/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_left_eq
% 0.30/0.56  thf(fact_3638_mod__add__left__eq,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 0.30/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_left_eq
% 0.30/0.56  thf(fact_3639_mod__add__cong,axiom,
% 0.30/0.56      ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ C )
% 0.30/0.56          = ( modulo_modulo_nat @ A6 @ C ) )
% 0.30/0.56       => ( ( ( modulo_modulo_nat @ B @ C )
% 0.30/0.56            = ( modulo_modulo_nat @ B6 @ C ) )
% 0.30/0.56         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.56            = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_cong
% 0.30/0.56  thf(fact_3640_mod__add__cong,axiom,
% 0.30/0.56      ! [A: int,C: int,A6: int,B: int,B6: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56          = ( modulo_modulo_int @ A6 @ C ) )
% 0.30/0.56       => ( ( ( modulo_modulo_int @ B @ C )
% 0.30/0.56            = ( modulo_modulo_int @ B6 @ C ) )
% 0.30/0.56         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.56            = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_cong
% 0.30/0.56  thf(fact_3641_mod__add__cong,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56          = ( modulo364778990260209775nteger @ A6 @ C ) )
% 0.30/0.56       => ( ( ( modulo364778990260209775nteger @ B @ C )
% 0.30/0.56            = ( modulo364778990260209775nteger @ B6 @ C ) )
% 0.30/0.56         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.56            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_cong
% 0.30/0.56  thf(fact_3642_mod__add__eq,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_eq
% 0.30/0.56  thf(fact_3643_mod__add__eq,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_eq
% 0.30/0.56  thf(fact_3644_mod__add__eq,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 0.30/0.56        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_add_eq
% 0.30/0.56  thf(fact_3645_dvd__0__left,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56       => ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3646_dvd__0__left,axiom,
% 0.30/0.56      ! [A: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 0.30/0.56       => ( A = zero_zero_complex ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3647_dvd__0__left,axiom,
% 0.30/0.56      ! [A: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 0.30/0.56       => ( A = zero_zero_real ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3648_dvd__0__left,axiom,
% 0.30/0.56      ! [A: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 0.30/0.56       => ( A = zero_zero_rat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3649_dvd__0__left,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.56       => ( A = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3650_dvd__0__left,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 0.30/0.56       => ( A = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_0_left
% 0.30/0.56  thf(fact_3651_dvd__field__iff,axiom,
% 0.30/0.56      ( dvd_dvd_complex
% 0.30/0.56      = ( ^ [A3: complex,B3: complex] :
% 0.30/0.56            ( ( A3 = zero_zero_complex )
% 0.30/0.56           => ( B3 = zero_zero_complex ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_field_iff
% 0.30/0.56  thf(fact_3652_dvd__field__iff,axiom,
% 0.30/0.56      ( dvd_dvd_real
% 0.30/0.56      = ( ^ [A3: real,B3: real] :
% 0.30/0.56            ( ( A3 = zero_zero_real )
% 0.30/0.56           => ( B3 = zero_zero_real ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_field_iff
% 0.30/0.56  thf(fact_3653_dvd__field__iff,axiom,
% 0.30/0.56      ( dvd_dvd_rat
% 0.30/0.56      = ( ^ [A3: rat,B3: rat] :
% 0.30/0.56            ( ( A3 = zero_zero_rat )
% 0.30/0.56           => ( B3 = zero_zero_rat ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_field_iff
% 0.30/0.56  thf(fact_3654_mod__Suc__Suc__eq,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_Suc_Suc_eq
% 0.30/0.56  thf(fact_3655_mod__Suc__eq,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] :
% 0.30/0.56        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
% 0.30/0.56        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_Suc_eq
% 0.30/0.56  thf(fact_3656_dvd__productE,axiom,
% 0.30/0.56      ! [P4: nat,A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
% 0.30/0.56       => ~ ! [X4: nat,Y5: nat] :
% 0.30/0.56              ( ( P4
% 0.30/0.56                = ( times_times_nat @ X4 @ Y5 ) )
% 0.30/0.56             => ( ( dvd_dvd_nat @ X4 @ A )
% 0.30/0.56               => ~ ( dvd_dvd_nat @ Y5 @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_productE
% 0.30/0.56  thf(fact_3657_dvd__productE,axiom,
% 0.30/0.56      ! [P4: int,A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
% 0.30/0.56       => ~ ! [X4: int,Y5: int] :
% 0.30/0.56              ( ( P4
% 0.30/0.56                = ( times_times_int @ X4 @ Y5 ) )
% 0.30/0.56             => ( ( dvd_dvd_int @ X4 @ A )
% 0.30/0.56               => ~ ( dvd_dvd_int @ Y5 @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_productE
% 0.30/0.56  thf(fact_3658_division__decomp,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.56       => ? [B8: nat,C5: nat] :
% 0.30/0.56            ( ( A
% 0.30/0.56              = ( times_times_nat @ B8 @ C5 ) )
% 0.30/0.56            & ( dvd_dvd_nat @ B8 @ B )
% 0.30/0.56            & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % division_decomp
% 0.30/0.56  thf(fact_3659_division__decomp,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.56       => ? [B8: int,C5: int] :
% 0.30/0.56            ( ( A
% 0.30/0.56              = ( times_times_int @ B8 @ C5 ) )
% 0.30/0.56            & ( dvd_dvd_int @ B8 @ B )
% 0.30/0.56            & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % division_decomp
% 0.30/0.56  thf(fact_3660_dvdE,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.56       => ~ ! [K2: code_integer] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3661_dvdE,axiom,
% 0.30/0.56      ! [B: complex,A: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ B @ A )
% 0.30/0.56       => ~ ! [K2: complex] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_times_complex @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3662_dvdE,axiom,
% 0.30/0.56      ! [B: real,A: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ B @ A )
% 0.30/0.56       => ~ ! [K2: real] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_times_real @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3663_dvdE,axiom,
% 0.30/0.56      ! [B: rat,A: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.56       => ~ ! [K2: rat] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_times_rat @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3664_dvdE,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.56       => ~ ! [K2: nat] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_times_nat @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3665_dvdE,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ B @ A )
% 0.30/0.56       => ~ ! [K2: int] :
% 0.30/0.56              ( A
% 0.30/0.56             != ( times_times_int @ B @ K2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdE
% 0.30/0.56  thf(fact_3666_dvdI,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,K: code_integer] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_3573771949741848930nteger @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3667_dvdI,axiom,
% 0.30/0.56      ! [A: complex,B: complex,K: complex] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_times_complex @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_complex @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3668_dvdI,axiom,
% 0.30/0.56      ! [A: real,B: real,K: real] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_times_real @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_real @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3669_dvdI,axiom,
% 0.30/0.56      ! [A: rat,B: rat,K: rat] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_times_rat @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_rat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3670_dvdI,axiom,
% 0.30/0.56      ! [A: nat,B: nat,K: nat] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_times_nat @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3671_dvdI,axiom,
% 0.30/0.56      ! [A: int,B: int,K: int] :
% 0.30/0.56        ( ( A
% 0.30/0.56          = ( times_times_int @ B @ K ) )
% 0.30/0.56       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvdI
% 0.30/0.56  thf(fact_3672_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_Code_integer
% 0.30/0.56      = ( ^ [B3: code_integer,A3: code_integer] :
% 0.30/0.56          ? [K3: code_integer] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3673_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_complex
% 0.30/0.56      = ( ^ [B3: complex,A3: complex] :
% 0.30/0.56          ? [K3: complex] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_times_complex @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3674_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_real
% 0.30/0.56      = ( ^ [B3: real,A3: real] :
% 0.30/0.56          ? [K3: real] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_times_real @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3675_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_rat
% 0.30/0.56      = ( ^ [B3: rat,A3: rat] :
% 0.30/0.56          ? [K3: rat] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3676_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_nat
% 0.30/0.56      = ( ^ [B3: nat,A3: nat] :
% 0.30/0.56          ? [K3: nat] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3677_dvd__def,axiom,
% 0.30/0.56      ( dvd_dvd_int
% 0.30/0.56      = ( ^ [B3: int,A3: int] :
% 0.30/0.56          ? [K3: int] :
% 0.30/0.56            ( A3
% 0.30/0.56            = ( times_times_int @ B3 @ K3 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_def
% 0.30/0.56  thf(fact_3678_dvd__mult,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3679_dvd__mult,axiom,
% 0.30/0.56      ! [A: complex,C: complex,B: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ A @ C )
% 0.30/0.56       => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3680_dvd__mult,axiom,
% 0.30/0.56      ! [A: real,C: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ C )
% 0.30/0.56       => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3681_dvd__mult,axiom,
% 0.30/0.56      ! [A: rat,C: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.56       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3682_dvd__mult,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.56       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3683_dvd__mult,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ C )
% 0.30/0.56       => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult
% 0.30/0.56  thf(fact_3684_dvd__mult2,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3685_dvd__mult2,axiom,
% 0.30/0.56      ! [A: complex,B: complex,C: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ A @ B )
% 0.30/0.56       => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3686_dvd__mult2,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ B )
% 0.30/0.56       => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3687_dvd__mult2,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.56       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3688_dvd__mult2,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3689_dvd__mult2,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult2
% 0.30/0.56  thf(fact_3690_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3691_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: complex,B: complex,C: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_complex @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3692_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_real @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3693_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_rat @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3694_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_nat @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3695_dvd__mult__left,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_int @ A @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_left
% 0.30/0.56  thf(fact_3696_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3697_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3698_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3699_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3700_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3701_dvd__triv__left,axiom,
% 0.30/0.56      ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_left
% 0.30/0.56  thf(fact_3702_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3703_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: complex,B: complex,C: complex,D: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_complex @ C @ D )
% 0.30/0.56         => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3704_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_real @ C @ D )
% 0.30/0.56         => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3705_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_rat @ C @ D )
% 0.30/0.56         => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3706_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.56         => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3707_mult__dvd__mono,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.56         => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mult_dvd_mono
% 0.30/0.56  thf(fact_3708_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3709_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: complex,B: complex,C: complex] :
% 0.30/0.56        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_complex @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3710_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_real @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3711_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_rat @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3712_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_nat @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3713_dvd__mult__right,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.56       => ( dvd_dvd_int @ B @ C ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_mult_right
% 0.30/0.56  thf(fact_3714_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3715_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3716_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3717_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3718_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3719_dvd__triv__right,axiom,
% 0.30/0.56      ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_triv_right
% 0.30/0.56  thf(fact_3720_one__dvd,axiom,
% 0.30/0.56      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3721_one__dvd,axiom,
% 0.30/0.56      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3722_one__dvd,axiom,
% 0.30/0.56      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3723_one__dvd,axiom,
% 0.30/0.56      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3724_one__dvd,axiom,
% 0.30/0.56      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3725_one__dvd,axiom,
% 0.30/0.56      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 0.30/0.56  
% 0.30/0.56  % one_dvd
% 0.30/0.56  thf(fact_3726_unit__imp__dvd,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.56       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_dvd
% 0.30/0.56  thf(fact_3727_unit__imp__dvd,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.56       => ( dvd_dvd_nat @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_dvd
% 0.30/0.56  thf(fact_3728_unit__imp__dvd,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.56       => ( dvd_dvd_int @ B @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unit_imp_dvd
% 0.30/0.56  thf(fact_3729_dvd__unit__imp__unit,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_unit_imp_unit
% 0.30/0.56  thf(fact_3730_dvd__unit__imp__unit,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.56         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_unit_imp_unit
% 0.30/0.56  thf(fact_3731_dvd__unit__imp__unit,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.56         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_unit_imp_unit
% 0.30/0.56  thf(fact_3732_dvd__add__right__iff,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_right_iff
% 0.30/0.56  thf(fact_3733_dvd__add__right__iff,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_right_iff
% 0.30/0.56  thf(fact_3734_dvd__add__right__iff,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_right_iff
% 0.30/0.56  thf(fact_3735_dvd__add__right__iff,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_right_iff
% 0.30/0.56  thf(fact_3736_dvd__add__right__iff,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_right_iff
% 0.30/0.56  thf(fact_3737_dvd__add__left__iff,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_left_iff
% 0.30/0.56  thf(fact_3738_dvd__add__left__iff,axiom,
% 0.30/0.56      ! [A: real,C: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ C )
% 0.30/0.56       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_real @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_left_iff
% 0.30/0.56  thf(fact_3739_dvd__add__left__iff,axiom,
% 0.30/0.56      ! [A: rat,C: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.56       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_left_iff
% 0.30/0.56  thf(fact_3740_dvd__add__left__iff,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.56       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_left_iff
% 0.30/0.56  thf(fact_3741_dvd__add__left__iff,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ C )
% 0.30/0.56       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.56          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add_left_iff
% 0.30/0.56  thf(fact_3742_dvd__add,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ A @ C )
% 0.30/0.56         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add
% 0.30/0.56  thf(fact_3743_dvd__add,axiom,
% 0.30/0.56      ! [A: real,B: real,C: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_real @ A @ C )
% 0.30/0.56         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add
% 0.30/0.56  thf(fact_3744_dvd__add,axiom,
% 0.30/0.56      ! [A: rat,B: rat,C: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_rat @ A @ C )
% 0.30/0.56         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add
% 0.30/0.56  thf(fact_3745_dvd__add,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.56         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add
% 0.30/0.56  thf(fact_3746_dvd__add,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ A @ C )
% 0.30/0.56         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_add
% 0.30/0.56  thf(fact_3747_dvd__div__eq__iff,axiom,
% 0.30/0.56      ! [C: nat,A: nat,B: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.56         => ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.56              = ( divide_divide_nat @ B @ C ) )
% 0.30/0.56            = ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_iff
% 0.30/0.56  thf(fact_3748_dvd__div__eq__iff,axiom,
% 0.30/0.56      ! [C: int,A: int,B: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.56         => ( ( ( divide_divide_int @ A @ C )
% 0.30/0.56              = ( divide_divide_int @ B @ C ) )
% 0.30/0.56            = ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_iff
% 0.30/0.56  thf(fact_3749_dvd__div__eq__iff,axiom,
% 0.30/0.56      ! [C: real,A: real,B: real] :
% 0.30/0.56        ( ( dvd_dvd_real @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_real @ C @ B )
% 0.30/0.56         => ( ( ( divide_divide_real @ A @ C )
% 0.30/0.56              = ( divide_divide_real @ B @ C ) )
% 0.30/0.56            = ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_iff
% 0.30/0.56  thf(fact_3750_dvd__div__eq__iff,axiom,
% 0.30/0.56      ! [C: rat,A: rat,B: rat] :
% 0.30/0.56        ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_rat @ C @ B )
% 0.30/0.56         => ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.56              = ( divide_divide_rat @ B @ C ) )
% 0.30/0.56            = ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_iff
% 0.30/0.56  thf(fact_3751_dvd__div__eq__iff,axiom,
% 0.30/0.56      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.56         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.56              = ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.56            = ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_iff
% 0.30/0.56  thf(fact_3752_dvd__div__eq__cancel,axiom,
% 0.30/0.56      ! [A: nat,C: nat,B: nat] :
% 0.30/0.56        ( ( ( divide_divide_nat @ A @ C )
% 0.30/0.56          = ( divide_divide_nat @ B @ C ) )
% 0.30/0.56       => ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.56         => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.56           => ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_cancel
% 0.30/0.56  thf(fact_3753_dvd__div__eq__cancel,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( ( divide_divide_int @ A @ C )
% 0.30/0.56          = ( divide_divide_int @ B @ C ) )
% 0.30/0.56       => ( ( dvd_dvd_int @ C @ A )
% 0.30/0.56         => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.56           => ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_cancel
% 0.30/0.56  thf(fact_3754_dvd__div__eq__cancel,axiom,
% 0.30/0.56      ! [A: real,C: real,B: real] :
% 0.30/0.56        ( ( ( divide_divide_real @ A @ C )
% 0.30/0.56          = ( divide_divide_real @ B @ C ) )
% 0.30/0.56       => ( ( dvd_dvd_real @ C @ A )
% 0.30/0.56         => ( ( dvd_dvd_real @ C @ B )
% 0.30/0.56           => ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_cancel
% 0.30/0.56  thf(fact_3755_dvd__div__eq__cancel,axiom,
% 0.30/0.56      ! [A: rat,C: rat,B: rat] :
% 0.30/0.56        ( ( ( divide_divide_rat @ A @ C )
% 0.30/0.56          = ( divide_divide_rat @ B @ C ) )
% 0.30/0.56       => ( ( dvd_dvd_rat @ C @ A )
% 0.30/0.56         => ( ( dvd_dvd_rat @ C @ B )
% 0.30/0.56           => ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_cancel
% 0.30/0.56  thf(fact_3756_dvd__div__eq__cancel,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ( divide6298287555418463151nteger @ A @ C )
% 0.30/0.56          = ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.56         => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.56           => ( A = B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % dvd_div_eq_cancel
% 0.30/0.56  thf(fact_3757_div__div__div__same,axiom,
% 0.30/0.56      ! [D: nat,B: nat,A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ D @ B )
% 0.30/0.56       => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.56         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 0.30/0.56            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_div_div_same
% 0.30/0.56  thf(fact_3758_div__div__div__same,axiom,
% 0.30/0.56      ! [D: int,B: int,A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ D @ B )
% 0.30/0.56       => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.56         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 0.30/0.56            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_div_div_same
% 0.30/0.56  thf(fact_3759_div__div__div__same,axiom,
% 0.30/0.56      ! [D: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ D @ B )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.56         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 0.30/0.56            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_div_div_same
% 0.30/0.56  thf(fact_3760_mod__less__eq__dividend,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_less_eq_dividend
% 0.30/0.56  thf(fact_3761_gcd__nat_Oextremum__uniqueI,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.56       => ( A = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % gcd_nat.extremum_uniqueI
% 0.30/0.56  thf(fact_3762_gcd__nat_Onot__eq__extremum,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( A != zero_zero_nat )
% 0.30/0.56        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 0.30/0.56          & ( A != zero_zero_nat ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % gcd_nat.not_eq_extremum
% 0.30/0.56  thf(fact_3763_gcd__nat_Oextremum__unique,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.56        = ( A = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % gcd_nat.extremum_unique
% 0.30/0.56  thf(fact_3764_gcd__nat_Oextremum__strict,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 0.30/0.56          & ( zero_zero_nat != A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % gcd_nat.extremum_strict
% 0.30/0.56  thf(fact_3765_gcd__nat_Oextremum,axiom,
% 0.30/0.56      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % gcd_nat.extremum
% 0.30/0.56  thf(fact_3766_zdvd__antisym__nonneg,axiom,
% 0.30/0.56      ! [M: int,N2: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 0.30/0.56       => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 0.30/0.56         => ( ( dvd_dvd_int @ M @ N2 )
% 0.30/0.56           => ( ( dvd_dvd_int @ N2 @ M )
% 0.30/0.56             => ( M = N2 ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdvd_antisym_nonneg
% 0.30/0.56  thf(fact_3767_zdvd__mult__cancel,axiom,
% 0.30/0.56      ! [K: int,M: int,N2: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
% 0.30/0.56       => ( ( K != zero_zero_int )
% 0.30/0.56         => ( dvd_dvd_int @ M @ N2 ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % zdvd_mult_cancel
% 0.30/0.56  thf(fact_3768_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.56      ! [A: nat] :
% 0.30/0.56        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.56        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_iff_mod_2_eq_zero
% 0.30/0.56  thf(fact_3769_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.56      ! [A: int] :
% 0.30/0.56        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.56        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_iff_mod_2_eq_zero
% 0.30/0.56  thf(fact_3770_even__iff__mod__2__eq__zero,axiom,
% 0.30/0.56      ! [A: code_integer] :
% 0.30/0.56        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.56        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % even_iff_mod_2_eq_zero
% 0.30/0.56  thf(fact_3771_subset__decode__imp__le,axiom,
% 0.30/0.56      ! [M: nat,N2: nat] :
% 0.30/0.56        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
% 0.30/0.56       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.56  
% 0.30/0.56  % subset_decode_imp_le
% 0.30/0.56  thf(fact_3772_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.56       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56  thf(fact_3773_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.56       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56  thf(fact_3774_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.56       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
% 0.30/0.56  thf(fact_3775_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56      ! [B: nat,A: nat] :
% 0.30/0.56        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.56       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56  thf(fact_3776_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56      ! [B: int,A: int] :
% 0.30/0.56        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.56       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56  thf(fact_3777_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 0.30/0.56      ! [B: code_integer,A: code_integer] :
% 0.30/0.56        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.56       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 0.30/0.56  
% 0.30/0.56  % unique_euclidean_semiring_numeral_class.pos_mod_bound
% 0.30/0.56  thf(fact_3778_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56      ! [A: nat,B: nat] :
% 0.30/0.56        ( ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.56          = A )
% 0.30/0.56        = ( ( divide_divide_nat @ A @ B )
% 0.30/0.56          = zero_zero_nat ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_self_iff_div_eq_0
% 0.30/0.56  thf(fact_3779_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56      ! [A: int,B: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ B )
% 0.30/0.56          = A )
% 0.30/0.56        = ( ( divide_divide_int @ A @ B )
% 0.30/0.56          = zero_zero_int ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_self_iff_div_eq_0
% 0.30/0.56  thf(fact_3780_mod__eq__self__iff__div__eq__0,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.56          = A )
% 0.30/0.56        = ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.56          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eq_self_iff_div_eq_0
% 0.30/0.56  thf(fact_3781_mod__eqE,axiom,
% 0.30/0.56      ! [A: int,C: int,B: int] :
% 0.30/0.56        ( ( ( modulo_modulo_int @ A @ C )
% 0.30/0.56          = ( modulo_modulo_int @ B @ C ) )
% 0.30/0.56       => ~ ! [D4: int] :
% 0.30/0.56              ( B
% 0.30/0.56             != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eqE
% 0.30/0.56  thf(fact_3782_mod__eqE,axiom,
% 0.30/0.56      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.56        ( ( ( modulo364778990260209775nteger @ A @ C )
% 0.30/0.56          = ( modulo364778990260209775nteger @ B @ C ) )
% 0.30/0.56       => ~ ! [D4: code_integer] :
% 0.30/0.56              ( B
% 0.30/0.56             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % mod_eqE
% 0.30/0.56  thf(fact_3783_div__add1__eq,axiom,
% 0.30/0.56      ! [A: nat,B: nat,C: nat] :
% 0.30/0.56        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.56        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add1_eq
% 0.30/0.56  thf(fact_3784_div__add1__eq,axiom,
% 0.30/0.56      ! [A: int,B: int,C: int] :
% 0.30/0.56        ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.56        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add1_eq
% 0.30/0.56  thf(fact_3785_div__add1__eq,axiom,
% 0.30/0.56      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.56        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.56        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % div_add1_eq
% 0.30/0.56  thf(fact_3786_not__is__unit__0,axiom,
% 0.30/0.56      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 0.30/0.56  
% 0.30/0.56  % not_is_unit_0
% 0.30/0.56  thf(fact_3787_not__is__unit__0,axiom,
% 0.30/0.56      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 0.30/0.56  
% 0.30/0.56  % not_is_unit_0
% 0.30/0.56  thf(fact_3788_not__is__unit__0,axiom,
% 0.30/0.56      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 0.30/0.56  
% 0.30/0.56  % not_is_unit_0
% 0.30/0.56  thf(fact_3789_minf_I10_J,axiom,
% 0.30/0.56      ! [D: code_integer,S: code_integer] :
% 0.30/0.56      ? [Z3: code_integer] :
% 0.30/0.56      ! [X2: code_integer] :
% 0.30/0.56        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(10)
% 0.30/0.56  thf(fact_3790_minf_I10_J,axiom,
% 0.30/0.56      ! [D: real,S: real] :
% 0.30/0.56      ? [Z3: real] :
% 0.30/0.56      ! [X2: real] :
% 0.30/0.56        ( ( ord_less_real @ X2 @ Z3 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(10)
% 0.30/0.56  thf(fact_3791_minf_I10_J,axiom,
% 0.30/0.56      ! [D: rat,S: rat] :
% 0.30/0.56      ? [Z3: rat] :
% 0.30/0.56      ! [X2: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X2 @ Z3 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(10)
% 0.30/0.56  thf(fact_3792_minf_I10_J,axiom,
% 0.30/0.56      ! [D: nat,S: nat] :
% 0.30/0.56      ? [Z3: nat] :
% 0.30/0.56      ! [X2: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X2 @ Z3 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(10)
% 0.30/0.56  thf(fact_3793_minf_I10_J,axiom,
% 0.30/0.56      ! [D: int,S: int] :
% 0.30/0.56      ? [Z3: int] :
% 0.30/0.56      ! [X2: int] :
% 0.30/0.56        ( ( ord_less_int @ X2 @ Z3 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(10)
% 0.30/0.56  thf(fact_3794_minf_I9_J,axiom,
% 0.30/0.56      ! [D: code_integer,S: code_integer] :
% 0.30/0.56      ? [Z3: code_integer] :
% 0.30/0.56      ! [X2: code_integer] :
% 0.30/0.56        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.30/0.56          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(9)
% 0.30/0.56  thf(fact_3795_minf_I9_J,axiom,
% 0.30/0.56      ! [D: real,S: real] :
% 0.30/0.56      ? [Z3: real] :
% 0.30/0.56      ! [X2: real] :
% 0.30/0.56        ( ( ord_less_real @ X2 @ Z3 )
% 0.30/0.56       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.30/0.56          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(9)
% 0.30/0.56  thf(fact_3796_minf_I9_J,axiom,
% 0.30/0.56      ! [D: rat,S: rat] :
% 0.30/0.56      ? [Z3: rat] :
% 0.30/0.56      ! [X2: rat] :
% 0.30/0.56        ( ( ord_less_rat @ X2 @ Z3 )
% 0.30/0.56       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.30/0.56          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(9)
% 0.30/0.56  thf(fact_3797_minf_I9_J,axiom,
% 0.30/0.56      ! [D: nat,S: nat] :
% 0.30/0.56      ? [Z3: nat] :
% 0.30/0.56      ! [X2: nat] :
% 0.30/0.56        ( ( ord_less_nat @ X2 @ Z3 )
% 0.30/0.56       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.30/0.56          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(9)
% 0.30/0.56  thf(fact_3798_minf_I9_J,axiom,
% 0.30/0.56      ! [D: int,S: int] :
% 0.30/0.56      ? [Z3: int] :
% 0.30/0.56      ! [X2: int] :
% 0.30/0.56        ( ( ord_less_int @ X2 @ Z3 )
% 0.30/0.56       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.30/0.56          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % minf(9)
% 0.30/0.56  thf(fact_3799_pinf_I10_J,axiom,
% 0.30/0.56      ! [D: code_integer,S: code_integer] :
% 0.30/0.56      ? [Z3: code_integer] :
% 0.30/0.56      ! [X2: code_integer] :
% 0.30/0.56        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(10)
% 0.30/0.56  thf(fact_3800_pinf_I10_J,axiom,
% 0.30/0.56      ! [D: real,S: real] :
% 0.30/0.56      ? [Z3: real] :
% 0.30/0.56      ! [X2: real] :
% 0.30/0.56        ( ( ord_less_real @ Z3 @ X2 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(10)
% 0.30/0.56  thf(fact_3801_pinf_I10_J,axiom,
% 0.30/0.56      ! [D: rat,S: rat] :
% 0.30/0.56      ? [Z3: rat] :
% 0.30/0.56      ! [X2: rat] :
% 0.30/0.56        ( ( ord_less_rat @ Z3 @ X2 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(10)
% 0.30/0.56  thf(fact_3802_pinf_I10_J,axiom,
% 0.30/0.56      ! [D: nat,S: nat] :
% 0.30/0.56      ? [Z3: nat] :
% 0.30/0.56      ! [X2: nat] :
% 0.30/0.56        ( ( ord_less_nat @ Z3 @ X2 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(10)
% 0.30/0.56  thf(fact_3803_pinf_I10_J,axiom,
% 0.30/0.56      ! [D: int,S: int] :
% 0.30/0.56      ? [Z3: int] :
% 0.30/0.56      ! [X2: int] :
% 0.30/0.56        ( ( ord_less_int @ Z3 @ X2 )
% 0.30/0.56       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 0.30/0.56          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 0.30/0.56  
% 0.30/0.56  % pinf(10)
% 0.30/0.56  thf(fact_3804_pinf_I9_J,axiom,
% 0.30/0.56      ! [D: code_integer,S: code_integer] :
% 0.30/0.56      ? [Z3: code_integer] :
% 0.30/0.56      ! [X2: code_integer] :
% 0.30/0.56        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 0.30/0.56       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pinf(9)
% 0.30/0.57  thf(fact_3805_pinf_I9_J,axiom,
% 0.30/0.57      ! [D: real,S: real] :
% 0.30/0.57      ? [Z3: real] :
% 0.30/0.57      ! [X2: real] :
% 0.30/0.57        ( ( ord_less_real @ Z3 @ X2 )
% 0.30/0.57       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 0.30/0.57          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pinf(9)
% 0.30/0.57  thf(fact_3806_pinf_I9_J,axiom,
% 0.30/0.57      ! [D: rat,S: rat] :
% 0.30/0.57      ? [Z3: rat] :
% 0.30/0.57      ! [X2: rat] :
% 0.30/0.57        ( ( ord_less_rat @ Z3 @ X2 )
% 0.30/0.57       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 0.30/0.57          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pinf(9)
% 0.30/0.57  thf(fact_3807_pinf_I9_J,axiom,
% 0.30/0.57      ! [D: nat,S: nat] :
% 0.30/0.57      ? [Z3: nat] :
% 0.30/0.57      ! [X2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ Z3 @ X2 )
% 0.30/0.57       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 0.30/0.57          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pinf(9)
% 0.30/0.57  thf(fact_3808_pinf_I9_J,axiom,
% 0.30/0.57      ! [D: int,S: int] :
% 0.30/0.57      ? [Z3: int] :
% 0.30/0.57      ! [X2: int] :
% 0.30/0.57        ( ( ord_less_int @ Z3 @ X2 )
% 0.30/0.57       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 0.30/0.57          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pinf(9)
% 0.30/0.57  thf(fact_3809_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: complex,A: complex] :
% 0.30/0.57        ( ( dvd_dvd_complex @ B @ A )
% 0.30/0.57       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.57            = zero_zero_complex )
% 0.30/0.57          = ( A = zero_zero_complex ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3810_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.57       => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.57            = zero_zero_nat )
% 0.30/0.57          = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3811_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ A )
% 0.30/0.57       => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.57            = zero_zero_int )
% 0.30/0.57          = ( A = zero_zero_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3812_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: real,A: real] :
% 0.30/0.57        ( ( dvd_dvd_real @ B @ A )
% 0.30/0.57       => ( ( ( divide_divide_real @ A @ B )
% 0.30/0.57            = zero_zero_real )
% 0.30/0.57          = ( A = zero_zero_real ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3813_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: rat,A: rat] :
% 0.30/0.57        ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.57       => ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.57            = zero_zero_rat )
% 0.30/0.57          = ( A = zero_zero_rat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3814_dvd__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.57       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.57            = zero_z3403309356797280102nteger )
% 0.30/0.57          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_0_iff
% 0.30/0.57  thf(fact_3815_unit__mult__right__cancel,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ( ( ( times_3573771949741848930nteger @ B @ A )
% 0.30/0.57            = ( times_3573771949741848930nteger @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_right_cancel
% 0.30/0.57  thf(fact_3816_unit__mult__right__cancel,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ( ( ( times_times_nat @ B @ A )
% 0.30/0.57            = ( times_times_nat @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_right_cancel
% 0.30/0.57  thf(fact_3817_unit__mult__right__cancel,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ( ( ( times_times_int @ B @ A )
% 0.30/0.57            = ( times_times_int @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_right_cancel
% 0.30/0.57  thf(fact_3818_unit__mult__left__cancel,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ( ( ( times_3573771949741848930nteger @ A @ B )
% 0.30/0.57            = ( times_3573771949741848930nteger @ A @ C ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_left_cancel
% 0.30/0.57  thf(fact_3819_unit__mult__left__cancel,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ( ( ( times_times_nat @ A @ B )
% 0.30/0.57            = ( times_times_nat @ A @ C ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_left_cancel
% 0.30/0.57  thf(fact_3820_unit__mult__left__cancel,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ( ( ( times_times_int @ A @ B )
% 0.30/0.57            = ( times_times_int @ A @ C ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_mult_left_cancel
% 0.30/0.57  thf(fact_3821_mult__unit__dvd__iff_H,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff'
% 0.30/0.57  thf(fact_3822_mult__unit__dvd__iff_H,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff'
% 0.30/0.57  thf(fact_3823_mult__unit__dvd__iff_H,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff'
% 0.30/0.57  thf(fact_3824_dvd__mult__unit__iff_H,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff'
% 0.30/0.57  thf(fact_3825_dvd__mult__unit__iff_H,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.57          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff'
% 0.30/0.57  thf(fact_3826_dvd__mult__unit__iff_H,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff'
% 0.30/0.57  thf(fact_3827_mult__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff
% 0.30/0.57  thf(fact_3828_mult__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff
% 0.30/0.57  thf(fact_3829_mult__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_unit_dvd_iff
% 0.30/0.57  thf(fact_3830_dvd__mult__unit__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff
% 0.30/0.57  thf(fact_3831_dvd__mult__unit__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff
% 0.30/0.57  thf(fact_3832_dvd__mult__unit__iff,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_unit_iff
% 0.30/0.57  thf(fact_3833_is__unit__mult__iff,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 0.30/0.57        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57          & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_mult_iff
% 0.30/0.57  thf(fact_3834_is__unit__mult__iff,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 0.30/0.57        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57          & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_mult_iff
% 0.30/0.57  thf(fact_3835_is__unit__mult__iff,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 0.30/0.57        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57          & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_mult_iff
% 0.30/0.57  thf(fact_3836_mod__Suc,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.57            = N2 )
% 0.30/0.57         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 0.30/0.57            = zero_zero_nat ) )
% 0.30/0.57        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.57           != N2 )
% 0.30/0.57         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 0.30/0.57            = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_Suc
% 0.30/0.57  thf(fact_3837_finite__set__decode,axiom,
% 0.30/0.57      ! [N2: nat] : ( finite_finite_nat @ ( nat_set_decode @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_set_decode
% 0.30/0.57  thf(fact_3838_mod__induct,axiom,
% 0.30/0.57      ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
% 0.30/0.57        ( ( P @ N2 )
% 0.30/0.57       => ( ( ord_less_nat @ N2 @ P4 )
% 0.30/0.57         => ( ( ord_less_nat @ M @ P4 )
% 0.30/0.57           => ( ! [N3: nat] :
% 0.30/0.57                  ( ( ord_less_nat @ N3 @ P4 )
% 0.30/0.57                 => ( ( P @ N3 )
% 0.30/0.57                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 0.30/0.57             => ( P @ M ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_induct
% 0.30/0.57  thf(fact_3839_dvd__div__mult,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.57       => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 0.30/0.57          = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult
% 0.30/0.57  thf(fact_3840_dvd__div__mult,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57       => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 0.30/0.57          = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult
% 0.30/0.57  thf(fact_3841_dvd__div__mult,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 0.30/0.57          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult
% 0.30/0.57  thf(fact_3842_div__mult__swap,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.57       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.57          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_swap
% 0.30/0.57  thf(fact_3843_div__mult__swap,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.57          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_swap
% 0.30/0.57  thf(fact_3844_div__mult__swap,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.57          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_swap
% 0.30/0.57  thf(fact_3845_div__div__eq__right,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.57       => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.57         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.57            = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_div_eq_right
% 0.30/0.57  thf(fact_3846_div__div__eq__right,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57       => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.57         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.57            = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_div_eq_right
% 0.30/0.57  thf(fact_3847_div__div__eq__right,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.57            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_div_eq_right
% 0.30/0.57  thf(fact_3848_dvd__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: nat,C: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 0.30/0.57       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.57          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult2_eq
% 0.30/0.57  thf(fact_3849_dvd__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: int,C: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 0.30/0.57       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult2_eq
% 0.30/0.57  thf(fact_3850_dvd__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 0.30/0.57       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.57          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_mult2_eq
% 0.30/0.57  thf(fact_3851_dvd__mult__imp__div,axiom,
% 0.30/0.57      ! [A: nat,C: nat,B: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 0.30/0.57       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_imp_div
% 0.30/0.57  thf(fact_3852_dvd__mult__imp__div,axiom,
% 0.30/0.57      ! [A: int,C: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 0.30/0.57       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_imp_div
% 0.30/0.57  thf(fact_3853_dvd__mult__imp__div,axiom,
% 0.30/0.57      ! [A: code_integer,C: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 0.30/0.57       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_imp_div
% 0.30/0.57  thf(fact_3854_div__mult__div__if__dvd,axiom,
% 0.30/0.57      ! [B: nat,A: nat,D: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.57       => ( ( dvd_dvd_nat @ D @ C )
% 0.30/0.57         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 0.30/0.57            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_div_if_dvd
% 0.30/0.57  thf(fact_3855_div__mult__div__if__dvd,axiom,
% 0.30/0.57      ! [B: int,A: int,D: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ A )
% 0.30/0.57       => ( ( dvd_dvd_int @ D @ C )
% 0.30/0.57         => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 0.30/0.57            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_div_if_dvd
% 0.30/0.57  thf(fact_3856_div__mult__div__if__dvd,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ D @ C )
% 0.30/0.57         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_div_if_dvd
% 0.30/0.57  thf(fact_3857_gcd__nat__induct,axiom,
% 0.30/0.57      ! [P: nat > nat > $o,M: nat,N2: nat] :
% 0.30/0.57        ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
% 0.30/0.57       => ( ! [M3: nat,N3: nat] :
% 0.30/0.57              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.57             => ( ( P @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
% 0.30/0.57               => ( P @ M3 @ N3 ) ) )
% 0.30/0.57         => ( P @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % gcd_nat_induct
% 0.30/0.57  thf(fact_3858_mod__less__divisor,axiom,
% 0.30/0.57      ! [N2: nat,M: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_less_divisor
% 0.30/0.57  thf(fact_3859_dvd__div__unit__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_unit_iff
% 0.30/0.57  thf(fact_3860_dvd__div__unit__iff,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_unit_iff
% 0.30/0.57  thf(fact_3861_dvd__div__unit__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_unit_iff
% 0.30/0.57  thf(fact_3862_div__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_unit_dvd_iff
% 0.30/0.57  thf(fact_3863_div__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_unit_dvd_iff
% 0.30/0.57  thf(fact_3864_div__unit__dvd__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.57          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_unit_dvd_iff
% 0.30/0.57  thf(fact_3865_unit__div__cancel,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.57            = ( divide_divide_nat @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_cancel
% 0.30/0.57  thf(fact_3866_unit__div__cancel,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.57            = ( divide_divide_int @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_cancel
% 0.30/0.57  thf(fact_3867_unit__div__cancel,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.57            = ( divide6298287555418463151nteger @ C @ A ) )
% 0.30/0.57          = ( B = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_cancel
% 0.30/0.57  thf(fact_3868_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.57       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.57          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_right
% 0.30/0.57  thf(fact_3869_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.57          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_right
% 0.30/0.57  thf(fact_3870_div__plus__div__distrib__dvd__right,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.57          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_right
% 0.30/0.57  thf(fact_3871_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.57      ! [C: nat,A: nat,B: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ A )
% 0.30/0.57       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 0.30/0.57          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_left
% 0.30/0.57  thf(fact_3872_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.57      ! [C: int,A: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.57       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.30/0.57          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_left
% 0.30/0.57  thf(fact_3873_div__plus__div__distrib__dvd__left,axiom,
% 0.30/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.57       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 0.30/0.57          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_plus_div_distrib_dvd_left
% 0.30/0.57  thf(fact_3874_mod__Suc__le__divisor,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_Suc_le_divisor
% 0.30/0.57  thf(fact_3875_mod__eq__0D,axiom,
% 0.30/0.57      ! [M: nat,D: nat] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ M @ D )
% 0.30/0.57          = zero_zero_nat )
% 0.30/0.57       => ? [Q2: nat] :
% 0.30/0.57            ( M
% 0.30/0.57            = ( times_times_nat @ D @ Q2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_eq_0D
% 0.30/0.57  thf(fact_3876_le__imp__power__dvd,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,A: code_integer] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_imp_power_dvd
% 0.30/0.57  thf(fact_3877_le__imp__power__dvd,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_imp_power_dvd
% 0.30/0.57  thf(fact_3878_le__imp__power__dvd,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,A: real] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_imp_power_dvd
% 0.30/0.57  thf(fact_3879_le__imp__power__dvd,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,A: int] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_imp_power_dvd
% 0.30/0.57  thf(fact_3880_le__imp__power__dvd,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,A: complex] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_imp_power_dvd
% 0.30/0.57  thf(fact_3881_power__le__dvd,axiom,
% 0.30/0.57      ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_dvd
% 0.30/0.57  thf(fact_3882_power__le__dvd,axiom,
% 0.30/0.57      ! [A: nat,N2: nat,B: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_dvd
% 0.30/0.57  thf(fact_3883_power__le__dvd,axiom,
% 0.30/0.57      ! [A: real,N2: nat,B: real,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_dvd
% 0.30/0.57  thf(fact_3884_power__le__dvd,axiom,
% 0.30/0.57      ! [A: int,N2: nat,B: int,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_dvd
% 0.30/0.57  thf(fact_3885_power__le__dvd,axiom,
% 0.30/0.57      ! [A: complex,N2: nat,B: complex,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_dvd
% 0.30/0.57  thf(fact_3886_dvd__power__le,axiom,
% 0.30/0.57      ! [X3: code_integer,Y3: code_integer,N2: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N2 ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_le
% 0.30/0.57  thf(fact_3887_dvd__power__le,axiom,
% 0.30/0.57      ! [X3: nat,Y3: nat,N2: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N2 ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_le
% 0.30/0.57  thf(fact_3888_dvd__power__le,axiom,
% 0.30/0.57      ! [X3: real,Y3: real,N2: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_real @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ( dvd_dvd_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_le
% 0.30/0.57  thf(fact_3889_dvd__power__le,axiom,
% 0.30/0.57      ! [X3: int,Y3: int,N2: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ( dvd_dvd_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_le
% 0.30/0.57  thf(fact_3890_dvd__power__le,axiom,
% 0.30/0.57      ! [X3: complex,Y3: complex,N2: nat,M: nat] :
% 0.30/0.57        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_le
% 0.30/0.57  thf(fact_3891_nat__mod__eq__iff,axiom,
% 0.30/0.57      ! [X3: nat,N2: nat,Y3: nat] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ X3 @ N2 )
% 0.30/0.57          = ( modulo_modulo_nat @ Y3 @ N2 ) )
% 0.30/0.57        = ( ? [Q1: nat,Q22: nat] :
% 0.30/0.57              ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N2 @ Q1 ) )
% 0.30/0.57              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_mod_eq_iff
% 0.30/0.57  thf(fact_3892_nat__dvd__not__less,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.57       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.57         => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_dvd_not_less
% 0.30/0.57  thf(fact_3893_dvd__pos__nat,axiom,
% 0.30/0.57      ! [N2: nat,M: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57       => ( ( dvd_dvd_nat @ M @ N2 )
% 0.30/0.57         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_pos_nat
% 0.30/0.57  thf(fact_3894_zdvd__imp__le,axiom,
% 0.30/0.57      ! [Z2: int,N2: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ Z2 @ N2 )
% 0.30/0.57       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.30/0.57         => ( ord_less_eq_int @ Z2 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zdvd_imp_le
% 0.30/0.57  thf(fact_3895_zdvd__not__zless,axiom,
% 0.30/0.57      ! [M: int,N2: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.57       => ( ( ord_less_int @ M @ N2 )
% 0.30/0.57         => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zdvd_not_zless
% 0.30/0.57  thf(fact_3896_bezout__add__nat,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57      ? [D4: nat,X4: nat,Y5: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ D4 @ A )
% 0.30/0.57        & ( dvd_dvd_nat @ D4 @ B )
% 0.30/0.57        & ( ( ( times_times_nat @ A @ X4 )
% 0.30/0.57            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) )
% 0.30/0.57          | ( ( times_times_nat @ B @ X4 )
% 0.30/0.57            = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bezout_add_nat
% 0.30/0.57  thf(fact_3897_bezout__lemma__nat,axiom,
% 0.30/0.57      ! [D: nat,A: nat,B: nat,X3: nat,Y3: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.57       => ( ( dvd_dvd_nat @ D @ B )
% 0.30/0.57         => ( ( ( ( times_times_nat @ A @ X3 )
% 0.30/0.57                = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 0.30/0.57              | ( ( times_times_nat @ B @ X3 )
% 0.30/0.57                = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 0.30/0.57           => ? [X4: nat,Y5: nat] :
% 0.30/0.57                ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.57                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.57                & ( ( ( times_times_nat @ A @ X4 )
% 0.30/0.57                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D ) )
% 0.30/0.57                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 0.30/0.57                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bezout_lemma_nat
% 0.30/0.57  thf(fact_3898_parity__cases,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57           != zero_zero_nat ) )
% 0.30/0.57       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57             != one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % parity_cases
% 0.30/0.57  thf(fact_3899_parity__cases,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57           != zero_zero_int ) )
% 0.30/0.57       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57             != one_one_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % parity_cases
% 0.30/0.57  thf(fact_3900_parity__cases,axiom,
% 0.30/0.57      ! [A: code_integer] :
% 0.30/0.57        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.57           != zero_z3403309356797280102nteger ) )
% 0.30/0.57       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.57             != one_one_Code_integer ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % parity_cases
% 0.30/0.57  thf(fact_3901_mod2__eq__if,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57            = zero_zero_nat ) )
% 0.30/0.57        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57            = one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod2_eq_if
% 0.30/0.57  thf(fact_3902_mod2__eq__if,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57            = zero_zero_int ) )
% 0.30/0.57        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57            = one_one_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod2_eq_if
% 0.30/0.57  thf(fact_3903_mod2__eq__if,axiom,
% 0.30/0.57      ! [A: code_integer] :
% 0.30/0.57        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.57            = zero_z3403309356797280102nteger ) )
% 0.30/0.57        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.57            = one_one_Code_integer ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod2_eq_if
% 0.30/0.57  thf(fact_3904_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.57       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.57         => ( ( modulo364778990260209775nteger @ A @ B )
% 0.30/0.57            = A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.57  thf(fact_3905_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.57       => ( ( ord_less_nat @ A @ B )
% 0.30/0.57         => ( ( modulo_modulo_nat @ A @ B )
% 0.30/0.57            = A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.57  thf(fact_3906_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.57       => ( ( ord_less_int @ A @ B )
% 0.30/0.57         => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.57            = A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_less
% 0.30/0.57  thf(fact_3907_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.57       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.57  thf(fact_3908_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.57       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.57  thf(fact_3909_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.57       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.pos_mod_sign
% 0.30/0.57  thf(fact_3910_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.57      ! [N2: num,Q4: num] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.30/0.57          = zero_zero_nat )
% 0.30/0.57        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.30/0.57          = zero_zero_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(2)
% 0.30/0.57  thf(fact_3911_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.57      ! [N2: num,Q4: num] :
% 0.30/0.57        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.30/0.57          = zero_zero_int )
% 0.30/0.57        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
% 0.30/0.57          = zero_zero_int ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(2)
% 0.30/0.57  thf(fact_3912_cong__exp__iff__simps_I2_J,axiom,
% 0.30/0.57      ! [N2: num,Q4: num] :
% 0.30/0.57        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.30/0.57          = zero_z3403309356797280102nteger )
% 0.30/0.57        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.30/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(2)
% 0.30/0.57  thf(fact_3913_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.57      ! [N2: num] :
% 0.30/0.57        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(1)
% 0.30/0.57  thf(fact_3914_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.57      ! [N2: num] :
% 0.30/0.57        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
% 0.30/0.57        = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(1)
% 0.30/0.57  thf(fact_3915_cong__exp__iff__simps_I1_J,axiom,
% 0.30/0.57      ! [N2: num] :
% 0.30/0.57        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
% 0.30/0.57        = zero_z3403309356797280102nteger ) ).
% 0.30/0.57  
% 0.30/0.57  % cong_exp_iff_simps(1)
% 0.30/0.57  thf(fact_3916_mult__div__mod__eq,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_div_mod_eq
% 0.30/0.57  thf(fact_3917_mult__div__mod__eq,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_div_mod_eq
% 0.30/0.57  thf(fact_3918_mult__div__mod__eq,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mult_div_mod_eq
% 0.30/0.57  thf(fact_3919_mod__mult__div__eq,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_mult_div_eq
% 0.30/0.57  thf(fact_3920_mod__mult__div__eq,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_mult_div_eq
% 0.30/0.57  thf(fact_3921_mod__mult__div__eq,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_mult_div_eq
% 0.30/0.57  thf(fact_3922_mod__div__mult__eq,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_mult_eq
% 0.30/0.57  thf(fact_3923_mod__div__mult__eq,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_mult_eq
% 0.30/0.57  thf(fact_3924_mod__div__mult__eq,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_mult_eq
% 0.30/0.57  thf(fact_3925_div__mult__mod__eq,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_mod_eq
% 0.30/0.57  thf(fact_3926_div__mult__mod__eq,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_mod_eq
% 0.30/0.57  thf(fact_3927_div__mult__mod__eq,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_mod_eq
% 0.30/0.57  thf(fact_3928_mod__div__decomp,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( A
% 0.30/0.57        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_decomp
% 0.30/0.57  thf(fact_3929_mod__div__decomp,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( A
% 0.30/0.57        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_decomp
% 0.30/0.57  thf(fact_3930_mod__div__decomp,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( A
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_div_decomp
% 0.30/0.57  thf(fact_3931_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(1)
% 0.30/0.57  thf(fact_3932_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(1)
% 0.30/0.57  thf(fact_3933_cancel__div__mod__rules_I1_J,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(1)
% 0.30/0.57  thf(fact_3934_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_plus_nat @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(2)
% 0.30/0.57  thf(fact_3935_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_plus_int @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(2)
% 0.30/0.57  thf(fact_3936_cancel__div__mod__rules_I2_J,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_div_mod_rules(2)
% 0.30/0.57  thf(fact_3937_div__mult1__eq,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 0.30/0.57        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult1_eq
% 0.30/0.57  thf(fact_3938_div__mult1__eq,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 0.30/0.57        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult1_eq
% 0.30/0.57  thf(fact_3939_div__mult1__eq,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult1_eq
% 0.30/0.57  thf(fact_3940_unit__dvdE,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57           => ! [C2: code_integer] :
% 0.30/0.57                ( B
% 0.30/0.57               != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_dvdE
% 0.30/0.57  thf(fact_3941_unit__dvdE,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ~ ( ( A != zero_zero_nat )
% 0.30/0.57           => ! [C2: nat] :
% 0.30/0.57                ( B
% 0.30/0.57               != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_dvdE
% 0.30/0.57  thf(fact_3942_unit__dvdE,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ~ ( ( A != zero_zero_int )
% 0.30/0.57           => ! [C2: int] :
% 0.30/0.57                ( B
% 0.30/0.57               != ( times_times_int @ A @ C2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_dvdE
% 0.30/0.57  thf(fact_3943_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: code_integer > $o,L: code_integer] :
% 0.30/0.57        ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: code_integer] :
% 0.30/0.57              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3944_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: complex > $o,L: complex] :
% 0.30/0.57        ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: complex] :
% 0.30/0.57              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3945_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: real > $o,L: real] :
% 0.30/0.57        ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: real] :
% 0.30/0.57              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3946_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: rat > $o,L: rat] :
% 0.30/0.57        ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: rat] :
% 0.30/0.57              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3947_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: nat > $o,L: nat] :
% 0.30/0.57        ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: nat] :
% 0.30/0.57              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3948_unity__coeff__ex,axiom,
% 0.30/0.57      ! [P: int > $o,L: int] :
% 0.30/0.57        ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 0.30/0.57        = ( ? [X: int] :
% 0.30/0.57              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 0.30/0.57              & ( P @ X ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unity_coeff_ex
% 0.30/0.57  thf(fact_3949_dvd__div__eq__mult,axiom,
% 0.30/0.57      ! [A: nat,B: nat,C: nat] :
% 0.30/0.57        ( ( A != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.57         => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.57              = C )
% 0.30/0.57            = ( B
% 0.30/0.57              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_mult
% 0.30/0.57  thf(fact_3950_dvd__div__eq__mult,axiom,
% 0.30/0.57      ! [A: int,B: int,C: int] :
% 0.30/0.57        ( ( A != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ A @ B )
% 0.30/0.57         => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.57              = C )
% 0.30/0.57            = ( B
% 0.30/0.57              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_mult
% 0.30/0.57  thf(fact_3951_dvd__div__eq__mult,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer,C: code_integer] :
% 0.30/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.57         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.57              = C )
% 0.30/0.57            = ( B
% 0.30/0.57              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_eq_mult
% 0.30/0.57  thf(fact_3952_div__dvd__iff__mult,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( B != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.57         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.57            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_dvd_iff_mult
% 0.30/0.57  thf(fact_3953_div__dvd__iff__mult,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( B != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.57         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.57            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_dvd_iff_mult
% 0.30/0.57  thf(fact_3954_div__dvd__iff__mult,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( B != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.57         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.57            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_dvd_iff_mult
% 0.30/0.57  thf(fact_3955_dvd__div__iff__mult,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( C != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ C @ B )
% 0.30/0.57         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.57            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_iff_mult
% 0.30/0.57  thf(fact_3956_dvd__div__iff__mult,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( C != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.57            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_iff_mult
% 0.30/0.57  thf(fact_3957_dvd__div__iff__mult,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.57            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_iff_mult
% 0.30/0.57  thf(fact_3958_dvd__div__div__eq__mult,axiom,
% 0.30/0.57      ! [A: nat,C: nat,B: nat,D: nat] :
% 0.30/0.57        ( ( A != zero_zero_nat )
% 0.30/0.57       => ( ( C != zero_zero_nat )
% 0.30/0.57         => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.57           => ( ( dvd_dvd_nat @ C @ D )
% 0.30/0.57             => ( ( ( divide_divide_nat @ B @ A )
% 0.30/0.57                  = ( divide_divide_nat @ D @ C ) )
% 0.30/0.57                = ( ( times_times_nat @ B @ C )
% 0.30/0.57                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_div_eq_mult
% 0.30/0.57  thf(fact_3959_dvd__div__div__eq__mult,axiom,
% 0.30/0.57      ! [A: int,C: int,B: int,D: int] :
% 0.30/0.57        ( ( A != zero_zero_int )
% 0.30/0.57       => ( ( C != zero_zero_int )
% 0.30/0.57         => ( ( dvd_dvd_int @ A @ B )
% 0.30/0.57           => ( ( dvd_dvd_int @ C @ D )
% 0.30/0.57             => ( ( ( divide_divide_int @ B @ A )
% 0.30/0.57                  = ( divide_divide_int @ D @ C ) )
% 0.30/0.57                = ( ( times_times_int @ B @ C )
% 0.30/0.57                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_div_eq_mult
% 0.30/0.57  thf(fact_3960_dvd__div__div__eq__mult,axiom,
% 0.30/0.57      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.30/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( C != zero_z3403309356797280102nteger )
% 0.30/0.57         => ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.57           => ( ( dvd_dvd_Code_integer @ C @ D )
% 0.30/0.57             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 0.30/0.57                  = ( divide6298287555418463151nteger @ D @ C ) )
% 0.30/0.57                = ( ( times_3573771949741848930nteger @ B @ C )
% 0.30/0.57                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_div_div_eq_mult
% 0.30/0.57  thf(fact_3961_unit__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.57            = zero_zero_nat )
% 0.30/0.57          = ( A = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_eq_0_iff
% 0.30/0.57  thf(fact_3962_unit__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.57            = zero_zero_int )
% 0.30/0.57          = ( A = zero_zero_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_eq_0_iff
% 0.30/0.57  thf(fact_3963_unit__div__eq__0__iff,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.57            = zero_z3403309356797280102nteger )
% 0.30/0.57          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_eq_0_iff
% 0.30/0.57  thf(fact_3964_is__unit__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: nat,C: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.57         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.57            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult2_eq
% 0.30/0.57  thf(fact_3965_is__unit__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: int,C: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.57         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult2_eq
% 0.30/0.57  thf(fact_3966_is__unit__div__mult2__eq,axiom,
% 0.30/0.57      ! [B: code_integer,C: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult2_eq
% 0.30/0.57  thf(fact_3967_unit__div__mult__swap,axiom,
% 0.30/0.57      ! [C: nat,A: nat,B: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.57       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 0.30/0.57          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_mult_swap
% 0.30/0.57  thf(fact_3968_unit__div__mult__swap,axiom,
% 0.30/0.57      ! [C: int,A: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.57       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 0.30/0.57          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_mult_swap
% 0.30/0.57  thf(fact_3969_unit__div__mult__swap,axiom,
% 0.30/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.57       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 0.30/0.57          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_mult_swap
% 0.30/0.57  thf(fact_3970_unit__div__commute,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 0.30/0.57          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_commute
% 0.30/0.57  thf(fact_3971_unit__div__commute,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 0.30/0.57          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_commute
% 0.30/0.57  thf(fact_3972_unit__div__commute,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 0.30/0.57          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_div_commute
% 0.30/0.57  thf(fact_3973_div__mult__unit2,axiom,
% 0.30/0.57      ! [C: nat,B: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.57         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.57            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_unit2
% 0.30/0.57  thf(fact_3974_div__mult__unit2,axiom,
% 0.30/0.57      ! [C: int,B: int,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ one_one_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.57         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_unit2
% 0.30/0.57  thf(fact_3975_div__mult__unit2,axiom,
% 0.30/0.57      ! [C: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mult_unit2
% 0.30/0.57  thf(fact_3976_unit__eq__div2,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( A
% 0.30/0.57            = ( divide_divide_nat @ C @ B ) )
% 0.30/0.57          = ( ( times_times_nat @ A @ B )
% 0.30/0.57            = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div2
% 0.30/0.57  thf(fact_3977_unit__eq__div2,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( A
% 0.30/0.57            = ( divide_divide_int @ C @ B ) )
% 0.30/0.57          = ( ( times_times_int @ A @ B )
% 0.30/0.57            = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div2
% 0.30/0.57  thf(fact_3978_unit__eq__div2,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( A
% 0.30/0.57            = ( divide6298287555418463151nteger @ C @ B ) )
% 0.30/0.57          = ( ( times_3573771949741848930nteger @ A @ B )
% 0.30/0.57            = C ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div2
% 0.30/0.57  thf(fact_3979_unit__eq__div1,axiom,
% 0.30/0.57      ! [B: nat,A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57       => ( ( ( divide_divide_nat @ A @ B )
% 0.30/0.57            = C )
% 0.30/0.57          = ( A
% 0.30/0.57            = ( times_times_nat @ C @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div1
% 0.30/0.57  thf(fact_3980_unit__eq__div1,axiom,
% 0.30/0.57      ! [B: int,A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57       => ( ( ( divide_divide_int @ A @ B )
% 0.30/0.57            = C )
% 0.30/0.57          = ( A
% 0.30/0.57            = ( times_times_int @ C @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div1
% 0.30/0.57  thf(fact_3981_unit__eq__div1,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 0.30/0.57            = C )
% 0.30/0.57          = ( A
% 0.30/0.57            = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unit_eq_div1
% 0.30/0.57  thf(fact_3982_mod__le__divisor,axiom,
% 0.30/0.57      ! [N2: nat,M: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_le_divisor
% 0.30/0.57  thf(fact_3983_is__unit__power__iff,axiom,
% 0.30/0.57      ! [A: code_integer,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
% 0.30/0.57        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_power_iff
% 0.30/0.57  thf(fact_3984_is__unit__power__iff,axiom,
% 0.30/0.57      ! [A: nat,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
% 0.30/0.57        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_power_iff
% 0.30/0.57  thf(fact_3985_is__unit__power__iff,axiom,
% 0.30/0.57      ! [A: int,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
% 0.30/0.57        = ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_power_iff
% 0.30/0.57  thf(fact_3986_div__less__mono,axiom,
% 0.30/0.57      ! [A2: nat,B2: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57         => ( ( ( modulo_modulo_nat @ A2 @ N2 )
% 0.30/0.57              = zero_zero_nat )
% 0.30/0.57           => ( ( ( modulo_modulo_nat @ B2 @ N2 )
% 0.30/0.57                = zero_zero_nat )
% 0.30/0.57             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_less_mono
% 0.30/0.57  thf(fact_3987_mod__eq__nat1E,axiom,
% 0.30/0.57      ! [M: nat,Q4: nat,N2: nat] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.30/0.57          = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 0.30/0.57       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.57         => ~ ! [S2: nat] :
% 0.30/0.57                ( M
% 0.30/0.57               != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_eq_nat1E
% 0.30/0.57  thf(fact_3988_mod__eq__nat2E,axiom,
% 0.30/0.57      ! [M: nat,Q4: nat,N2: nat] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.30/0.57          = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 0.30/0.57       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57         => ~ ! [S2: nat] :
% 0.30/0.57                ( N2
% 0.30/0.57               != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_eq_nat2E
% 0.30/0.57  thf(fact_3989_nat__mod__eq__lemma,axiom,
% 0.30/0.57      ! [X3: nat,N2: nat,Y3: nat] :
% 0.30/0.57        ( ( ( modulo_modulo_nat @ X3 @ N2 )
% 0.30/0.57          = ( modulo_modulo_nat @ Y3 @ N2 ) )
% 0.30/0.57       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.57         => ? [Q2: nat] :
% 0.30/0.57              ( X3
% 0.30/0.57              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_mod_eq_lemma
% 0.30/0.57  thf(fact_3990_dvd__imp__le,axiom,
% 0.30/0.57      ! [K: nat,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ K @ N2 )
% 0.30/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57         => ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_imp_le
% 0.30/0.57  thf(fact_3991_div__mod__decomp,axiom,
% 0.30/0.57      ! [A2: nat,N2: nat] :
% 0.30/0.57        ( A2
% 0.30/0.57        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_mod_decomp
% 0.30/0.57  thf(fact_3992_mod__mult2__eq,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.57        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q4 ) )
% 0.30/0.57        = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_mult2_eq
% 0.30/0.57  thf(fact_3993_dvd__mult__cancel,axiom,
% 0.30/0.57      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.57         => ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_cancel
% 0.30/0.57  thf(fact_3994_nat__mult__dvd__cancel1,axiom,
% 0.30/0.57      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 0.30/0.57          = ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % nat_mult_dvd_cancel1
% 0.30/0.57  thf(fact_3995_bezout__add__strong__nat,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( A != zero_zero_nat )
% 0.30/0.57       => ? [D4: nat,X4: nat,Y5: nat] :
% 0.30/0.57            ( ( dvd_dvd_nat @ D4 @ A )
% 0.30/0.57            & ( dvd_dvd_nat @ D4 @ B )
% 0.30/0.57            & ( ( times_times_nat @ A @ X4 )
% 0.30/0.57              = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bezout_add_strong_nat
% 0.30/0.57  thf(fact_3996_even__zero,axiom,
% 0.30/0.57      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 0.30/0.57  
% 0.30/0.57  % even_zero
% 0.30/0.57  thf(fact_3997_even__zero,axiom,
% 0.30/0.57      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 0.30/0.57  
% 0.30/0.57  % even_zero
% 0.30/0.57  thf(fact_3998_even__zero,axiom,
% 0.30/0.57      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 0.30/0.57  
% 0.30/0.57  % even_zero
% 0.30/0.57  thf(fact_3999_is__unitE,axiom,
% 0.30/0.57      ! [A: nat,C: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 0.30/0.57       => ~ ( ( A != zero_zero_nat )
% 0.30/0.57           => ! [B4: nat] :
% 0.30/0.57                ( ( B4 != zero_zero_nat )
% 0.30/0.57               => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 0.30/0.57                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 0.30/0.57                      = B4 )
% 0.30/0.57                   => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 0.30/0.57                        = A )
% 0.30/0.57                     => ( ( ( times_times_nat @ A @ B4 )
% 0.30/0.57                          = one_one_nat )
% 0.30/0.57                       => ( ( divide_divide_nat @ C @ A )
% 0.30/0.57                         != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unitE
% 0.30/0.57  thf(fact_4000_is__unitE,axiom,
% 0.30/0.57      ! [A: int,C: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ A @ one_one_int )
% 0.30/0.57       => ~ ( ( A != zero_zero_int )
% 0.30/0.57           => ! [B4: int] :
% 0.30/0.57                ( ( B4 != zero_zero_int )
% 0.30/0.57               => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 0.30/0.57                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 0.30/0.57                      = B4 )
% 0.30/0.57                   => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 0.30/0.57                        = A )
% 0.30/0.57                     => ( ( ( times_times_int @ A @ B4 )
% 0.30/0.57                          = one_one_int )
% 0.30/0.57                       => ( ( divide_divide_int @ C @ A )
% 0.30/0.57                         != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unitE
% 0.30/0.57  thf(fact_4001_is__unitE,axiom,
% 0.30/0.57      ! [A: code_integer,C: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 0.30/0.57       => ~ ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57           => ! [B4: code_integer] :
% 0.30/0.57                ( ( B4 != zero_z3403309356797280102nteger )
% 0.30/0.57               => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 0.30/0.57                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 0.30/0.57                      = B4 )
% 0.30/0.57                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 0.30/0.57                        = A )
% 0.30/0.57                     => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 0.30/0.57                          = one_one_Code_integer )
% 0.30/0.57                       => ( ( divide6298287555418463151nteger @ C @ A )
% 0.30/0.57                         != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unitE
% 0.30/0.57  thf(fact_4002_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( A != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 0.30/0.57            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_left
% 0.30/0.57  thf(fact_4003_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( A != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 0.30/0.57            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_left
% 0.30/0.57  thf(fact_4004_is__unit__div__mult__cancel__left,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_left
% 0.30/0.57  thf(fact_4005_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( A != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 0.30/0.57         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 0.30/0.57            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_right
% 0.30/0.57  thf(fact_4006_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( A != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ B @ one_one_int )
% 0.30/0.57         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 0.30/0.57            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_right
% 0.30/0.57  thf(fact_4007_is__unit__div__mult__cancel__right,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % is_unit_div_mult_cancel_right
% 0.30/0.57  thf(fact_4008_odd__even__add,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 0.30/0.57         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % odd_even_add
% 0.30/0.57  thf(fact_4009_odd__even__add,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.57         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % odd_even_add
% 0.30/0.57  thf(fact_4010_odd__even__add,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 0.30/0.57         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % odd_even_add
% 0.30/0.57  thf(fact_4011_dvd__power__iff,axiom,
% 0.30/0.57      ! [X3: code_integer,M: nat,N2: nat] :
% 0.30/0.57        ( ( X3 != zero_z3403309356797280102nteger )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N2 ) )
% 0.30/0.57          = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
% 0.30/0.57            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_iff
% 0.30/0.57  thf(fact_4012_dvd__power__iff,axiom,
% 0.30/0.57      ! [X3: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( X3 != zero_zero_nat )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N2 ) )
% 0.30/0.57          = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
% 0.30/0.57            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_iff
% 0.30/0.57  thf(fact_4013_dvd__power__iff,axiom,
% 0.30/0.57      ! [X3: int,M: nat,N2: nat] :
% 0.30/0.57        ( ( X3 != zero_zero_int )
% 0.30/0.57       => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N2 ) )
% 0.30/0.57          = ( ( dvd_dvd_int @ X3 @ one_one_int )
% 0.30/0.57            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_iff
% 0.30/0.57  thf(fact_4014_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: code_integer] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_Code_integer ) )
% 0.30/0.57       => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4015_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: rat] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_rat ) )
% 0.30/0.57       => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4016_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: nat] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_nat ) )
% 0.30/0.57       => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4017_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: real] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_real ) )
% 0.30/0.57       => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4018_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: int] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_int ) )
% 0.30/0.57       => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4019_dvd__power,axiom,
% 0.30/0.57      ! [N2: nat,X3: complex] :
% 0.30/0.57        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          | ( X3 = one_one_complex ) )
% 0.30/0.57       => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power
% 0.30/0.57  thf(fact_4020_split__mod,axiom,
% 0.30/0.57      ! [P: nat > $o,M: nat,N2: nat] :
% 0.30/0.57        ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
% 0.30/0.57        = ( ( ( N2 = zero_zero_nat )
% 0.30/0.57           => ( P @ M ) )
% 0.30/0.57          & ( ( N2 != zero_zero_nat )
% 0.30/0.57           => ! [I2: nat,J3: nat] :
% 0.30/0.57                ( ( ord_less_nat @ J3 @ N2 )
% 0.30/0.57               => ( ( M
% 0.30/0.57                    = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
% 0.30/0.57                 => ( P @ J3 ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % split_mod
% 0.30/0.57  thf(fact_4021_dvd__mult__cancel1,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
% 0.30/0.57          = ( N2 = one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_cancel1
% 0.30/0.57  thf(fact_4022_dvd__mult__cancel2,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
% 0.30/0.57          = ( N2 = one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_mult_cancel2
% 0.30/0.57  thf(fact_4023_power__dvd__imp__le,axiom,
% 0.30/0.57      ! [I: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
% 0.30/0.57       => ( ( ord_less_nat @ one_one_nat @ I )
% 0.30/0.57         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_dvd_imp_le
% 0.30/0.57  thf(fact_4024_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 0.30/0.57       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 0.30/0.57          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.57  thf(fact_4025_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.57      ! [C: nat,A: nat,B: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 0.30/0.57       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 0.30/0.57          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.57  thf(fact_4026_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 0.30/0.57      ! [C: int,A: int,B: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 0.30/0.57       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 0.30/0.57          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unique_euclidean_semiring_numeral_class.mod_mult2_eq
% 0.30/0.57  thf(fact_4027_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4028_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4029_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4030_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4031_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4032_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_o,Ys: list_o] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4033_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_o,Ys: list_nat] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4034_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_o,Ys: list_int] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4035_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4036_product__nth,axiom,
% 0.30/0.57      ! [N2: nat,Xs: list_nat,Ys: list_o] :
% 0.30/0.57        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 0.30/0.57       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N2 )
% 0.30/0.57          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % product_nth
% 0.30/0.57  thf(fact_4037_power__mono__odd,axiom,
% 0.30/0.57      ! [N2: nat,A: real,B: real] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.57         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_mono_odd
% 0.30/0.57  thf(fact_4038_power__mono__odd,axiom,
% 0.30/0.57      ! [N2: nat,A: rat,B: rat] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.57         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_mono_odd
% 0.30/0.57  thf(fact_4039_power__mono__odd,axiom,
% 0.30/0.57      ! [N2: nat,A: int,B: int] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.57         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_mono_odd
% 0.30/0.57  thf(fact_4040_Suc__times__mod__eq,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 0.30/0.57       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
% 0.30/0.57          = one_one_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Suc_times_mod_eq
% 0.30/0.57  thf(fact_4041_odd__pos,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % odd_pos
% 0.30/0.57  thf(fact_4042_dvd__power__iff__le,axiom,
% 0.30/0.57      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
% 0.30/0.57          = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % dvd_power_iff_le
% 0.30/0.57  thf(fact_4043_even__unset__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_unset_bit_iff
% 0.30/0.57  thf(fact_4044_even__unset__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_unset_bit_iff
% 0.30/0.57  thf(fact_4045_even__unset__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          | ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_unset_bit_iff
% 0.30/0.57  thf(fact_4046_even__set__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_set_bit_iff
% 0.30/0.57  thf(fact_4047_even__set__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_set_bit_iff
% 0.30/0.57  thf(fact_4048_even__set__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57          & ( M != zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_set_bit_iff
% 0.30/0.57  thf(fact_4049_num_Osize__gen_I1_J,axiom,
% 0.30/0.57      ( ( size_num @ one )
% 0.30/0.57      = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % num.size_gen(1)
% 0.30/0.57  thf(fact_4050_divmod__digit__0_I2_J,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.57       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.57            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(2)
% 0.30/0.57  thf(fact_4051_divmod__digit__0_I2_J,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.57       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.57            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(2)
% 0.30/0.57  thf(fact_4052_divmod__digit__0_I2_J,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.57       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 0.30/0.57            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(2)
% 0.30/0.57  thf(fact_4053_bits__stable__imp__add__self,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.57          = A )
% 0.30/0.57       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.57          = zero_zero_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bits_stable_imp_add_self
% 0.30/0.57  thf(fact_4054_bits__stable__imp__add__self,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.57          = A )
% 0.30/0.57       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 0.30/0.57          = zero_zero_int ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bits_stable_imp_add_self
% 0.30/0.57  thf(fact_4055_bits__stable__imp__add__self,axiom,
% 0.30/0.57      ! [A: code_integer] :
% 0.30/0.57        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.57          = A )
% 0.30/0.57       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 0.30/0.57          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.57  
% 0.30/0.57  % bits_stable_imp_add_self
% 0.30/0.57  thf(fact_4056_div__exp__mod__exp__eq,axiom,
% 0.30/0.57      ! [A: nat,N2: nat,M: nat] :
% 0.30/0.57        ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57        = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_exp_mod_exp_eq
% 0.30/0.57  thf(fact_4057_div__exp__mod__exp__eq,axiom,
% 0.30/0.57      ! [A: int,N2: nat,M: nat] :
% 0.30/0.57        ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57        = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_exp_mod_exp_eq
% 0.30/0.57  thf(fact_4058_div__exp__mod__exp__eq,axiom,
% 0.30/0.57      ! [A: code_integer,N2: nat,M: nat] :
% 0.30/0.57        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_exp_mod_exp_eq
% 0.30/0.57  thf(fact_4059_oddE,axiom,
% 0.30/0.57      ! [A: code_integer] :
% 0.30/0.57        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ~ ! [B4: code_integer] :
% 0.30/0.57              ( A
% 0.30/0.57             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % oddE
% 0.30/0.57  thf(fact_4060_oddE,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ~ ! [B4: nat] :
% 0.30/0.57              ( A
% 0.30/0.57             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % oddE
% 0.30/0.57  thf(fact_4061_oddE,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57       => ~ ! [B4: int] :
% 0.30/0.57              ( A
% 0.30/0.57             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % oddE
% 0.30/0.57  thf(fact_4062_zero__le__even__power,axiom,
% 0.30/0.57      ! [N2: nat,A: real] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_even_power
% 0.30/0.57  thf(fact_4063_zero__le__even__power,axiom,
% 0.30/0.57      ! [N2: nat,A: rat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_even_power
% 0.30/0.57  thf(fact_4064_zero__le__even__power,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_even_power
% 0.30/0.57  thf(fact_4065_zero__le__odd__power,axiom,
% 0.30/0.57      ! [N2: nat,A: real] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.30/0.57          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_odd_power
% 0.30/0.57  thf(fact_4066_zero__le__odd__power,axiom,
% 0.30/0.57      ! [N2: nat,A: rat] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.30/0.57          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_odd_power
% 0.30/0.57  thf(fact_4067_zero__le__odd__power,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.30/0.57          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_odd_power
% 0.30/0.57  thf(fact_4068_zero__le__power__eq,axiom,
% 0.30/0.57      ! [A: real,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_power_eq
% 0.30/0.57  thf(fact_4069_zero__le__power__eq,axiom,
% 0.30/0.57      ! [A: rat,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_power_eq
% 0.30/0.57  thf(fact_4070_zero__le__power__eq,axiom,
% 0.30/0.57      ! [A: int,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_le_power_eq
% 0.30/0.57  thf(fact_4071_verit__le__mono__div,axiom,
% 0.30/0.57      ! [A2: nat,B2: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_nat @ A2 @ B2 )
% 0.30/0.57       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57         => ( ord_less_eq_nat
% 0.30/0.57            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
% 0.30/0.57              @ ( if_nat
% 0.30/0.57                @ ( ( modulo_modulo_nat @ B2 @ N2 )
% 0.30/0.57                  = zero_zero_nat )
% 0.30/0.57                @ one_one_nat
% 0.30/0.57                @ zero_zero_nat ) )
% 0.30/0.57            @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % verit_le_mono_div
% 0.30/0.57  thf(fact_4072_even__set__encode__iff,axiom,
% 0.30/0.57      ! [A2: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A2 )
% 0.30/0.57       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 0.30/0.57          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_set_encode_iff
% 0.30/0.57  thf(fact_4073_divmod__digit__0_I1_J,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.57       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(1)
% 0.30/0.57  thf(fact_4074_divmod__digit__0_I1_J,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.57       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(1)
% 0.30/0.57  thf(fact_4075_divmod__digit__0_I1_J,axiom,
% 0.30/0.57      ! [B: code_integer,A: code_integer] :
% 0.30/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.57       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.57         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_0(1)
% 0.30/0.57  thf(fact_4076_zero__less__power__eq,axiom,
% 0.30/0.57      ! [A: real,N2: nat] :
% 0.30/0.57        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 0.30/0.57        = ( ( N2 = zero_zero_nat )
% 0.30/0.57          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( A != zero_zero_real ) )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_power_eq
% 0.30/0.57  thf(fact_4077_zero__less__power__eq,axiom,
% 0.30/0.57      ! [A: rat,N2: nat] :
% 0.30/0.57        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 0.30/0.57        = ( ( N2 = zero_zero_nat )
% 0.30/0.57          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( A != zero_zero_rat ) )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_power_eq
% 0.30/0.57  thf(fact_4078_zero__less__power__eq,axiom,
% 0.30/0.57      ! [A: int,N2: nat] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 0.30/0.57        = ( ( N2 = zero_zero_nat )
% 0.30/0.57          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( A != zero_zero_int ) )
% 0.30/0.57          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_power_eq
% 0.30/0.57  thf(fact_4079_mod__double__modulus,axiom,
% 0.30/0.57      ! [M: code_integer,X3: code_integer] :
% 0.30/0.57        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 0.30/0.57       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.30/0.57         => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( modulo364778990260209775nteger @ X3 @ M ) )
% 0.30/0.57            | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_double_modulus
% 0.30/0.57  thf(fact_4080_mod__double__modulus,axiom,
% 0.30/0.57      ! [M: nat,X3: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.57       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 0.30/0.57         => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( modulo_modulo_nat @ X3 @ M ) )
% 0.30/0.57            | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_double_modulus
% 0.30/0.57  thf(fact_4081_mod__double__modulus,axiom,
% 0.30/0.57      ! [M: int,X3: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ M )
% 0.30/0.57       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.57         => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( modulo_modulo_int @ X3 @ M ) )
% 0.30/0.57            | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 0.30/0.57              = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % mod_double_modulus
% 0.30/0.57  thf(fact_4082_unset__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: nat] :
% 0.30/0.57        ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unset_bit_Suc
% 0.30/0.57  thf(fact_4083_unset__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: code_integer] :
% 0.30/0.57        ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unset_bit_Suc
% 0.30/0.57  thf(fact_4084_unset__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % unset_bit_Suc
% 0.30/0.57  thf(fact_4085_set__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: nat] :
% 0.30/0.57        ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_bit_Suc
% 0.30/0.57  thf(fact_4086_set__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: code_integer] :
% 0.30/0.57        ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_bit_Suc
% 0.30/0.57  thf(fact_4087_set__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_bit_Suc
% 0.30/0.57  thf(fact_4088_power__le__zero__eq,axiom,
% 0.30/0.57      ! [A: real,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 0.30/0.57        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.57            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( A = zero_zero_real ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_zero_eq
% 0.30/0.57  thf(fact_4089_power__le__zero__eq,axiom,
% 0.30/0.57      ! [A: rat,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 0.30/0.57        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.57            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( A = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_zero_eq
% 0.30/0.57  thf(fact_4090_power__le__zero__eq,axiom,
% 0.30/0.57      ! [A: int,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 0.30/0.57        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.57          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.57            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57              & ( A = zero_zero_int ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % power_le_zero_eq
% 0.30/0.57  thf(fact_4091_divmod__digit__1_I1_J,axiom,
% 0.30/0.57      ! [A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.57       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.57         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 0.30/0.57              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_1(1)
% 0.30/0.57  thf(fact_4092_divmod__digit__1_I1_J,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.57       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.57         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57           => ( ( 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 )
% 0.30/0.57              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_1(1)
% 0.30/0.57  thf(fact_4093_divmod__digit__1_I1_J,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.57       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.57         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.57           => ( ( 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 )
% 0.30/0.57              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % divmod_digit_1(1)
% 0.30/0.57  thf(fact_4094_flip__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: nat] :
% 0.30/0.57        ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % flip_bit_Suc
% 0.30/0.57  thf(fact_4095_flip__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: code_integer] :
% 0.30/0.57        ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % flip_bit_Suc
% 0.30/0.57  thf(fact_4096_flip__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % flip_bit_Suc
% 0.30/0.57  thf(fact_4097_pos__eucl__rel__int__mult__2,axiom,
% 0.30/0.57      ! [B: int,A: int,Q4: int,R2: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.57       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.57         => ( 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 ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % pos_eucl_rel_int_mult_2
% 0.30/0.57  thf(fact_4098_signed__take__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: code_integer] :
% 0.30/0.57        ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % signed_take_bit_Suc
% 0.30/0.57  thf(fact_4099_signed__take__bit__Suc,axiom,
% 0.30/0.57      ! [N2: nat,A: int] :
% 0.30/0.57        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
% 0.30/0.57        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % signed_take_bit_Suc
% 0.30/0.57  thf(fact_4100_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.57      ! [A: nat,M: nat,N2: nat] :
% 0.30/0.57        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.57        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.57          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            = zero_zero_nat )
% 0.30/0.57          | ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_mult_exp_div_exp_iff
% 0.30/0.57  thf(fact_4101_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.57      ! [A: int,M: nat,N2: nat] :
% 0.30/0.57        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.57        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.57          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            = zero_zero_int )
% 0.30/0.57          | ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_mult_exp_div_exp_iff
% 0.30/0.57  thf(fact_4102_even__mult__exp__div__exp__iff,axiom,
% 0.30/0.57      ! [A: code_integer,M: nat,N2: nat] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.57        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.57          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.57            = zero_z3403309356797280102nteger )
% 0.30/0.57          | ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_mult_exp_div_exp_iff
% 0.30/0.57  thf(fact_4103_even__flip__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_flip_bit_iff
% 0.30/0.57  thf(fact_4104_even__flip__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: nat] :
% 0.30/0.57        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_flip_bit_iff
% 0.30/0.57  thf(fact_4105_even__flip__bit__iff,axiom,
% 0.30/0.57      ! [M: nat,A: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.30/0.57        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.57         != ( M = zero_zero_nat ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % even_flip_bit_iff
% 0.30/0.57  thf(fact_4106_one__mod__2__pow__eq,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.57        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % one_mod_2_pow_eq
% 0.30/0.57  thf(fact_4107_one__mod__2__pow__eq,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.57        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % one_mod_2_pow_eq
% 0.30/0.57  thf(fact_4108_one__mod__2__pow__eq,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.57        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % one_mod_2_pow_eq
% 0.30/0.57  thf(fact_4109_set__encode__insert,axiom,
% 0.30/0.57      ! [A2: set_nat,N2: nat] :
% 0.30/0.57        ( ( finite_finite_nat @ A2 )
% 0.30/0.57       => ( ~ ( member_nat @ N2 @ A2 )
% 0.30/0.57         => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
% 0.30/0.57            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % set_encode_insert
% 0.30/0.57  thf(fact_4110_insert__absorb2,axiom,
% 0.30/0.57      ! [X3: nat,A2: set_nat] :
% 0.30/0.57        ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.57        = ( insert_nat @ X3 @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_absorb2
% 0.30/0.57  thf(fact_4111_insert__absorb2,axiom,
% 0.30/0.57      ! [X3: int,A2: set_int] :
% 0.30/0.57        ( ( insert_int @ X3 @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.57        = ( insert_int @ X3 @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_absorb2
% 0.30/0.57  thf(fact_4112_insert__absorb2,axiom,
% 0.30/0.57      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.57        ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
% 0.30/0.57        = ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_absorb2
% 0.30/0.57  thf(fact_4113_insert__absorb2,axiom,
% 0.30/0.57      ! [X3: real,A2: set_real] :
% 0.30/0.57        ( ( insert_real @ X3 @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.57        = ( insert_real @ X3 @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_absorb2
% 0.30/0.57  thf(fact_4114_insert__absorb2,axiom,
% 0.30/0.57      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.57        ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
% 0.30/0.57        = ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_absorb2
% 0.30/0.57  thf(fact_4115_insert__iff,axiom,
% 0.30/0.57      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.57        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4116_insert__iff,axiom,
% 0.30/0.57      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.57        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4117_insert__iff,axiom,
% 0.30/0.57      ! [A: real,B: real,A2: set_real] :
% 0.30/0.57        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member_real @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4118_insert__iff,axiom,
% 0.30/0.57      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.30/0.57        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member_set_nat @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4119_insert__iff,axiom,
% 0.30/0.57      ! [A: nat,B: nat,A2: set_nat] :
% 0.30/0.57        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member_nat @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4120_insert__iff,axiom,
% 0.30/0.57      ! [A: int,B: int,A2: set_int] :
% 0.30/0.57        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          | ( member_int @ A @ A2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_iff
% 0.30/0.57  thf(fact_4121_insertCI,axiom,
% 0.30/0.57      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.57        ( ( ~ ( member8757157785044589968at_nat @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4122_insertCI,axiom,
% 0.30/0.57      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.30/0.57        ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4123_insertCI,axiom,
% 0.30/0.57      ! [A: real,B2: set_real,B: real] :
% 0.30/0.57        ( ( ~ ( member_real @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4124_insertCI,axiom,
% 0.30/0.57      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.30/0.57        ( ( ~ ( member_set_nat @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4125_insertCI,axiom,
% 0.30/0.57      ! [A: nat,B2: set_nat,B: nat] :
% 0.30/0.57        ( ( ~ ( member_nat @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4126_insertCI,axiom,
% 0.30/0.57      ! [A: int,B2: set_int,B: int] :
% 0.30/0.57        ( ( ~ ( member_int @ A @ B2 )
% 0.30/0.57         => ( A = B ) )
% 0.30/0.57       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insertCI
% 0.30/0.57  thf(fact_4127_diff__self,axiom,
% 0.30/0.57      ! [A: complex] :
% 0.30/0.57        ( ( minus_minus_complex @ A @ A )
% 0.30/0.57        = zero_zero_complex ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_self
% 0.30/0.57  thf(fact_4128_diff__self,axiom,
% 0.30/0.57      ! [A: real] :
% 0.30/0.57        ( ( minus_minus_real @ A @ A )
% 0.30/0.57        = zero_zero_real ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_self
% 0.30/0.57  thf(fact_4129_diff__self,axiom,
% 0.30/0.57      ! [A: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ A @ A )
% 0.30/0.57        = zero_zero_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_self
% 0.30/0.57  thf(fact_4130_diff__self,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( minus_minus_int @ A @ A )
% 0.30/0.57        = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_self
% 0.30/0.57  thf(fact_4131_diff__0__right,axiom,
% 0.30/0.57      ! [A: complex] :
% 0.30/0.57        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_0_right
% 0.30/0.57  thf(fact_4132_diff__0__right,axiom,
% 0.30/0.57      ! [A: real] :
% 0.30/0.57        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_0_right
% 0.30/0.57  thf(fact_4133_diff__0__right,axiom,
% 0.30/0.57      ! [A: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_0_right
% 0.30/0.57  thf(fact_4134_diff__0__right,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_0_right
% 0.30/0.57  thf(fact_4135_zero__diff,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_diff
% 0.30/0.57  thf(fact_4136_diff__zero,axiom,
% 0.30/0.57      ! [A: complex] :
% 0.30/0.57        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_zero
% 0.30/0.57  thf(fact_4137_diff__zero,axiom,
% 0.30/0.57      ! [A: real] :
% 0.30/0.57        ( ( minus_minus_real @ A @ zero_zero_real )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_zero
% 0.30/0.57  thf(fact_4138_diff__zero,axiom,
% 0.30/0.57      ! [A: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_zero
% 0.30/0.57  thf(fact_4139_diff__zero,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_zero
% 0.30/0.57  thf(fact_4140_diff__zero,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( minus_minus_int @ A @ zero_zero_int )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_zero
% 0.30/0.57  thf(fact_4141_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.57      ! [A: complex] :
% 0.30/0.57        ( ( minus_minus_complex @ A @ A )
% 0.30/0.57        = zero_zero_complex ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.57  thf(fact_4142_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.57      ! [A: real] :
% 0.30/0.57        ( ( minus_minus_real @ A @ A )
% 0.30/0.57        = zero_zero_real ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.57  thf(fact_4143_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.57      ! [A: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ A @ A )
% 0.30/0.57        = zero_zero_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.57  thf(fact_4144_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.57      ! [A: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ A @ A )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.57  thf(fact_4145_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 0.30/0.57      ! [A: int] :
% 0.30/0.57        ( ( minus_minus_int @ A @ A )
% 0.30/0.57        = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % cancel_comm_monoid_add_class.diff_cancel
% 0.30/0.57  thf(fact_4146_add__diff__cancel,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel
% 0.30/0.57  thf(fact_4147_add__diff__cancel,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel
% 0.30/0.57  thf(fact_4148_add__diff__cancel,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel
% 0.30/0.57  thf(fact_4149_diff__add__cancel,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_add_cancel
% 0.30/0.57  thf(fact_4150_diff__add__cancel,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_add_cancel
% 0.30/0.57  thf(fact_4151_diff__add__cancel,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_add_cancel
% 0.30/0.57  thf(fact_4152_add__diff__cancel__left,axiom,
% 0.30/0.57      ! [C: real,A: real,B: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 0.30/0.57        = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left
% 0.30/0.57  thf(fact_4153_add__diff__cancel__left,axiom,
% 0.30/0.57      ! [C: rat,A: rat,B: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 0.30/0.57        = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left
% 0.30/0.57  thf(fact_4154_add__diff__cancel__left,axiom,
% 0.30/0.57      ! [C: nat,A: nat,B: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 0.30/0.57        = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left
% 0.30/0.57  thf(fact_4155_add__diff__cancel__left,axiom,
% 0.30/0.57      ! [C: int,A: int,B: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.30/0.57        = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left
% 0.30/0.57  thf(fact_4156_add__diff__cancel__left_H,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 0.30/0.57        = B ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left'
% 0.30/0.57  thf(fact_4157_add__diff__cancel__left_H,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 0.30/0.57        = B ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left'
% 0.30/0.57  thf(fact_4158_add__diff__cancel__left_H,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 0.30/0.57        = B ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left'
% 0.30/0.57  thf(fact_4159_add__diff__cancel__left_H,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.30/0.57        = B ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_left'
% 0.30/0.57  thf(fact_4160_add__diff__cancel__right,axiom,
% 0.30/0.57      ! [A: real,C: real,B: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 0.30/0.57        = ( minus_minus_real @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right
% 0.30/0.57  thf(fact_4161_add__diff__cancel__right,axiom,
% 0.30/0.57      ! [A: rat,C: rat,B: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.57        = ( minus_minus_rat @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right
% 0.30/0.57  thf(fact_4162_add__diff__cancel__right,axiom,
% 0.30/0.57      ! [A: nat,C: nat,B: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 0.30/0.57        = ( minus_minus_nat @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right
% 0.30/0.57  thf(fact_4163_add__diff__cancel__right,axiom,
% 0.30/0.57      ! [A: int,C: int,B: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.30/0.57        = ( minus_minus_int @ A @ B ) ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right
% 0.30/0.57  thf(fact_4164_add__diff__cancel__right_H,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right'
% 0.30/0.57  thf(fact_4165_add__diff__cancel__right_H,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right'
% 0.30/0.57  thf(fact_4166_add__diff__cancel__right_H,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right'
% 0.30/0.57  thf(fact_4167_add__diff__cancel__right_H,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.30/0.57        = A ) ).
% 0.30/0.57  
% 0.30/0.57  % add_diff_cancel_right'
% 0.30/0.57  thf(fact_4168_singletonI,axiom,
% 0.30/0.57      ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4169_singletonI,axiom,
% 0.30/0.57      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4170_singletonI,axiom,
% 0.30/0.57      ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4171_singletonI,axiom,
% 0.30/0.57      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4172_singletonI,axiom,
% 0.30/0.57      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4173_singletonI,axiom,
% 0.30/0.57      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singletonI
% 0.30/0.57  thf(fact_4174_finite__insert,axiom,
% 0.30/0.57      ! [A: real,A2: set_real] :
% 0.30/0.57        ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
% 0.30/0.57        = ( finite_finite_real @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4175_finite__insert,axiom,
% 0.30/0.57      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.57        ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
% 0.30/0.57        = ( finite4343798906461161616at_nat @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4176_finite__insert,axiom,
% 0.30/0.57      ! [A: nat,A2: set_nat] :
% 0.30/0.57        ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
% 0.30/0.57        = ( finite_finite_nat @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4177_finite__insert,axiom,
% 0.30/0.57      ! [A: int,A2: set_int] :
% 0.30/0.57        ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
% 0.30/0.57        = ( finite_finite_int @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4178_finite__insert,axiom,
% 0.30/0.57      ! [A: complex,A2: set_complex] :
% 0.30/0.57        ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
% 0.30/0.57        = ( finite3207457112153483333omplex @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4179_finite__insert,axiom,
% 0.30/0.57      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.57        ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
% 0.30/0.57        = ( finite6177210948735845034at_nat @ A2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % finite_insert
% 0.30/0.57  thf(fact_4180_insert__subset,axiom,
% 0.30/0.57      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.57        ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.57          & ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4181_insert__subset,axiom,
% 0.30/0.57      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.57        ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.57          & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4182_insert__subset,axiom,
% 0.30/0.57      ! [X3: real,A2: set_real,B2: set_real] :
% 0.30/0.57        ( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member_real @ X3 @ B2 )
% 0.30/0.57          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4183_insert__subset,axiom,
% 0.30/0.57      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.57        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member_set_nat @ X3 @ B2 )
% 0.30/0.57          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4184_insert__subset,axiom,
% 0.30/0.57      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.57        ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member_nat @ X3 @ B2 )
% 0.30/0.57          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4185_insert__subset,axiom,
% 0.30/0.57      ! [X3: int,A2: set_int,B2: set_int] :
% 0.30/0.57        ( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 0.30/0.57        = ( ( member_int @ X3 @ B2 )
% 0.30/0.57          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % insert_subset
% 0.30/0.57  thf(fact_4186_diff__Suc__Suc,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.57        = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_Suc_Suc
% 0.30/0.57  thf(fact_4187_Suc__diff__diff,axiom,
% 0.30/0.57      ! [M: nat,N2: nat,K: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
% 0.30/0.57        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% 0.30/0.57  
% 0.30/0.57  % Suc_diff_diff
% 0.30/0.57  thf(fact_4188_diff__0__eq__0,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_0_eq_0
% 0.30/0.57  thf(fact_4189_diff__self__eq__0,axiom,
% 0.30/0.57      ! [M: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ M @ M )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_self_eq_0
% 0.30/0.57  thf(fact_4190_diff__diff__cancel,axiom,
% 0.30/0.57      ! [I: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ I @ N2 )
% 0.30/0.57       => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
% 0.30/0.57          = I ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_diff_cancel
% 0.30/0.57  thf(fact_4191_diff__diff__left,axiom,
% 0.30/0.57      ! [I: nat,J: nat,K: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.57        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_diff_left
% 0.30/0.57  thf(fact_4192_flip__bit__negative__int__iff,axiom,
% 0.30/0.57      ! [N2: nat,K: int] :
% 0.30/0.57        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.57        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.57  
% 0.30/0.57  % flip_bit_negative_int_iff
% 0.30/0.57  thf(fact_4193_of__bool__less__eq__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.57        = ( P
% 0.30/0.57         => Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_eq_iff
% 0.30/0.57  thf(fact_4194_of__bool__less__eq__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.57        = ( P
% 0.30/0.57         => Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_eq_iff
% 0.30/0.57  thf(fact_4195_of__bool__less__eq__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.57        = ( P
% 0.30/0.57         => Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_eq_iff
% 0.30/0.57  thf(fact_4196_of__bool__eq_I1_J,axiom,
% 0.30/0.57      ( ( zero_n1201886186963655149omplex @ $false )
% 0.30/0.57      = zero_zero_complex ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(1)
% 0.30/0.57  thf(fact_4197_of__bool__eq_I1_J,axiom,
% 0.30/0.57      ( ( zero_n3304061248610475627l_real @ $false )
% 0.30/0.57      = zero_zero_real ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(1)
% 0.30/0.57  thf(fact_4198_of__bool__eq_I1_J,axiom,
% 0.30/0.57      ( ( zero_n2052037380579107095ol_rat @ $false )
% 0.30/0.57      = zero_zero_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(1)
% 0.30/0.57  thf(fact_4199_of__bool__eq_I1_J,axiom,
% 0.30/0.57      ( ( zero_n2687167440665602831ol_nat @ $false )
% 0.30/0.57      = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(1)
% 0.30/0.57  thf(fact_4200_of__bool__eq_I1_J,axiom,
% 0.30/0.57      ( ( zero_n2684676970156552555ol_int @ $false )
% 0.30/0.57      = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(1)
% 0.30/0.57  thf(fact_4201_of__bool__eq__0__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.57          = zero_zero_complex )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_0_iff
% 0.30/0.57  thf(fact_4202_of__bool__eq__0__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.57          = zero_zero_real )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_0_iff
% 0.30/0.57  thf(fact_4203_of__bool__eq__0__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.57          = zero_zero_rat )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_0_iff
% 0.30/0.57  thf(fact_4204_of__bool__eq__0__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.57          = zero_zero_nat )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_0_iff
% 0.30/0.57  thf(fact_4205_of__bool__eq__0__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.57          = zero_zero_int )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_0_iff
% 0.30/0.57  thf(fact_4206_of__bool__less__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 0.30/0.57        = ( ~ P
% 0.30/0.57          & Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_iff
% 0.30/0.57  thf(fact_4207_of__bool__less__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 0.30/0.57        = ( ~ P
% 0.30/0.57          & Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_iff
% 0.30/0.57  thf(fact_4208_of__bool__less__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 0.30/0.57        = ( ~ P
% 0.30/0.57          & Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_iff
% 0.30/0.57  thf(fact_4209_of__bool__less__iff,axiom,
% 0.30/0.57      ! [P: $o,Q: $o] :
% 0.30/0.57        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 0.30/0.57        = ( ~ P
% 0.30/0.57          & Q ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_iff
% 0.30/0.57  thf(fact_4210_of__bool__eq__1__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n1201886186963655149omplex @ P )
% 0.30/0.57          = one_one_complex )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_1_iff
% 0.30/0.57  thf(fact_4211_of__bool__eq__1__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n3304061248610475627l_real @ P )
% 0.30/0.57          = one_one_real )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_1_iff
% 0.30/0.57  thf(fact_4212_of__bool__eq__1__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 0.30/0.57          = one_one_rat )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_1_iff
% 0.30/0.57  thf(fact_4213_of__bool__eq__1__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 0.30/0.57          = one_one_nat )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_1_iff
% 0.30/0.57  thf(fact_4214_of__bool__eq__1__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ( zero_n2684676970156552555ol_int @ P )
% 0.30/0.57          = one_one_int )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq_1_iff
% 0.30/0.57  thf(fact_4215_of__bool__eq_I2_J,axiom,
% 0.30/0.57      ( ( zero_n1201886186963655149omplex @ $true )
% 0.30/0.57      = one_one_complex ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(2)
% 0.30/0.57  thf(fact_4216_of__bool__eq_I2_J,axiom,
% 0.30/0.57      ( ( zero_n3304061248610475627l_real @ $true )
% 0.30/0.57      = one_one_real ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(2)
% 0.30/0.57  thf(fact_4217_of__bool__eq_I2_J,axiom,
% 0.30/0.57      ( ( zero_n2052037380579107095ol_rat @ $true )
% 0.30/0.57      = one_one_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(2)
% 0.30/0.57  thf(fact_4218_of__bool__eq_I2_J,axiom,
% 0.30/0.57      ( ( zero_n2687167440665602831ol_nat @ $true )
% 0.30/0.57      = one_one_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(2)
% 0.30/0.57  thf(fact_4219_of__bool__eq_I2_J,axiom,
% 0.30/0.57      ( ( zero_n2684676970156552555ol_int @ $true )
% 0.30/0.57      = one_one_int ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_eq(2)
% 0.30/0.57  thf(fact_4220_signed__take__bit__of__0,axiom,
% 0.30/0.57      ! [N2: nat] :
% 0.30/0.57        ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
% 0.30/0.57        = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % signed_take_bit_of_0
% 0.30/0.57  thf(fact_4221_diff__ge__0__iff__ge,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.57        = ( ord_less_eq_real @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_ge_0_iff_ge
% 0.30/0.57  thf(fact_4222_diff__ge__0__iff__ge,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.57        = ( ord_less_eq_rat @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_ge_0_iff_ge
% 0.30/0.57  thf(fact_4223_diff__ge__0__iff__ge,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.57        = ( ord_less_eq_int @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_ge_0_iff_ge
% 0.30/0.57  thf(fact_4224_diff__gt__0__iff__gt,axiom,
% 0.30/0.57      ! [A: real,B: real] :
% 0.30/0.57        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.57        = ( ord_less_real @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_gt_0_iff_gt
% 0.30/0.57  thf(fact_4225_diff__gt__0__iff__gt,axiom,
% 0.30/0.57      ! [A: rat,B: rat] :
% 0.30/0.57        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.57        = ( ord_less_rat @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_gt_0_iff_gt
% 0.30/0.57  thf(fact_4226_diff__gt__0__iff__gt,axiom,
% 0.30/0.57      ! [A: int,B: int] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.57        = ( ord_less_int @ B @ A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_gt_0_iff_gt
% 0.30/0.57  thf(fact_4227_le__add__diff__inverse2,axiom,
% 0.30/0.57      ! [B: real,A: real] :
% 0.30/0.57        ( ( ord_less_eq_real @ B @ A )
% 0.30/0.57       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse2
% 0.30/0.57  thf(fact_4228_le__add__diff__inverse2,axiom,
% 0.30/0.57      ! [B: rat,A: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse2
% 0.30/0.57  thf(fact_4229_le__add__diff__inverse2,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse2
% 0.30/0.57  thf(fact_4230_le__add__diff__inverse2,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse2
% 0.30/0.57  thf(fact_4231_le__add__diff__inverse,axiom,
% 0.30/0.57      ! [B: real,A: real] :
% 0.30/0.57        ( ( ord_less_eq_real @ B @ A )
% 0.30/0.57       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse
% 0.30/0.57  thf(fact_4232_le__add__diff__inverse,axiom,
% 0.30/0.57      ! [B: rat,A: rat] :
% 0.30/0.57        ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.57       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse
% 0.30/0.57  thf(fact_4233_le__add__diff__inverse,axiom,
% 0.30/0.57      ! [B: nat,A: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ B @ A )
% 0.30/0.57       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse
% 0.30/0.57  thf(fact_4234_le__add__diff__inverse,axiom,
% 0.30/0.57      ! [B: int,A: int] :
% 0.30/0.57        ( ( ord_less_eq_int @ B @ A )
% 0.30/0.57       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.30/0.57          = A ) ) ).
% 0.30/0.57  
% 0.30/0.57  % le_add_diff_inverse
% 0.30/0.57  thf(fact_4235_diff__numeral__special_I9_J,axiom,
% 0.30/0.57      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 0.30/0.57      = zero_zero_complex ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_numeral_special(9)
% 0.30/0.57  thf(fact_4236_diff__numeral__special_I9_J,axiom,
% 0.30/0.57      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 0.30/0.57      = zero_zero_real ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_numeral_special(9)
% 0.30/0.57  thf(fact_4237_diff__numeral__special_I9_J,axiom,
% 0.30/0.57      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 0.30/0.57      = zero_zero_rat ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_numeral_special(9)
% 0.30/0.57  thf(fact_4238_diff__numeral__special_I9_J,axiom,
% 0.30/0.57      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 0.30/0.57      = zero_zero_int ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_numeral_special(9)
% 0.30/0.57  thf(fact_4239_diff__add__zero,axiom,
% 0.30/0.57      ! [A: nat,B: nat] :
% 0.30/0.57        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.57        = zero_zero_nat ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_add_zero
% 0.30/0.57  thf(fact_4240_div__diff,axiom,
% 0.30/0.57      ! [C: int,A: int,B: int] :
% 0.30/0.57        ( ( dvd_dvd_int @ C @ A )
% 0.30/0.57       => ( ( dvd_dvd_int @ C @ B )
% 0.30/0.57         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.57            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_diff
% 0.30/0.57  thf(fact_4241_div__diff,axiom,
% 0.30/0.57      ! [C: code_integer,A: code_integer,B: code_integer] :
% 0.30/0.57        ( ( dvd_dvd_Code_integer @ C @ A )
% 0.30/0.57       => ( ( dvd_dvd_Code_integer @ C @ B )
% 0.30/0.57         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 0.30/0.57            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % div_diff
% 0.30/0.57  thf(fact_4242_singleton__insert__inj__eq_H,axiom,
% 0.30/0.57      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.57        ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.30/0.57          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq'
% 0.30/0.57  thf(fact_4243_singleton__insert__inj__eq_H,axiom,
% 0.30/0.57      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.30/0.57        ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.30/0.57          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq'
% 0.30/0.57  thf(fact_4244_singleton__insert__inj__eq_H,axiom,
% 0.30/0.57      ! [A: real,A2: set_real,B: real] :
% 0.30/0.57        ( ( ( insert_real @ A @ A2 )
% 0.30/0.57          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq'
% 0.30/0.57  thf(fact_4245_singleton__insert__inj__eq_H,axiom,
% 0.30/0.57      ! [A: nat,A2: set_nat,B: nat] :
% 0.30/0.57        ( ( ( insert_nat @ A @ A2 )
% 0.30/0.57          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq'
% 0.30/0.57  thf(fact_4246_singleton__insert__inj__eq_H,axiom,
% 0.30/0.57      ! [A: int,A2: set_int,B: int] :
% 0.30/0.57        ( ( ( insert_int @ A @ A2 )
% 0.30/0.57          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq'
% 0.30/0.57  thf(fact_4247_singleton__insert__inj__eq,axiom,
% 0.30/0.57      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.57        ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
% 0.30/0.57          = ( insert9069300056098147895at_nat @ A @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq
% 0.30/0.57  thf(fact_4248_singleton__insert__inj__eq,axiom,
% 0.30/0.57      ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.57        ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
% 0.30/0.57          = ( insert8211810215607154385at_nat @ A @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq
% 0.30/0.57  thf(fact_4249_singleton__insert__inj__eq,axiom,
% 0.30/0.57      ! [B: real,A: real,A2: set_real] :
% 0.30/0.57        ( ( ( insert_real @ B @ bot_bot_set_real )
% 0.30/0.57          = ( insert_real @ A @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq
% 0.30/0.57  thf(fact_4250_singleton__insert__inj__eq,axiom,
% 0.30/0.57      ! [B: nat,A: nat,A2: set_nat] :
% 0.30/0.57        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 0.30/0.57          = ( insert_nat @ A @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq
% 0.30/0.57  thf(fact_4251_singleton__insert__inj__eq,axiom,
% 0.30/0.57      ! [B: int,A: int,A2: set_int] :
% 0.30/0.57        ( ( ( insert_int @ B @ bot_bot_set_int )
% 0.30/0.57          = ( insert_int @ A @ A2 ) )
% 0.30/0.57        = ( ( A = B )
% 0.30/0.57          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % singleton_insert_inj_eq
% 0.30/0.57  thf(fact_4252_zero__less__of__bool__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_of_bool_iff
% 0.30/0.57  thf(fact_4253_zero__less__of__bool__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_of_bool_iff
% 0.30/0.57  thf(fact_4254_zero__less__of__bool__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_of_bool_iff
% 0.30/0.57  thf(fact_4255_zero__less__of__bool__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.57        = P ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_of_bool_iff
% 0.30/0.57  thf(fact_4256_zero__less__diff,axiom,
% 0.30/0.57      ! [N2: nat,M: nat] :
% 0.30/0.57        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.57        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % zero_less_diff
% 0.30/0.57  thf(fact_4257_diff__is__0__eq,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.57          = zero_zero_nat )
% 0.30/0.57        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_is_0_eq
% 0.30/0.57  thf(fact_4258_diff__is__0__eq_H,axiom,
% 0.30/0.57      ! [M: nat,N2: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.57       => ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.57          = zero_zero_nat ) ) ).
% 0.30/0.57  
% 0.30/0.57  % diff_is_0_eq'
% 0.30/0.57  thf(fact_4259_of__bool__less__one__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_one_iff
% 0.30/0.57  thf(fact_4260_of__bool__less__one__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_one_iff
% 0.30/0.57  thf(fact_4261_of__bool__less__one__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_one_iff
% 0.30/0.57  thf(fact_4262_of__bool__less__one__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 0.30/0.57        = ( ~ P ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_less_one_iff
% 0.30/0.57  thf(fact_4263_of__bool__not__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( zero_n1201886186963655149omplex @ ~ P )
% 0.30/0.57        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_not_iff
% 0.30/0.57  thf(fact_4264_of__bool__not__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( zero_n3304061248610475627l_real @ ~ P )
% 0.30/0.57        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_not_iff
% 0.30/0.57  thf(fact_4265_of__bool__not__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 0.30/0.57        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_not_iff
% 0.30/0.57  thf(fact_4266_of__bool__not__iff,axiom,
% 0.30/0.57      ! [P: $o] :
% 0.30/0.57        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 0.30/0.57        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 0.30/0.57  
% 0.30/0.57  % of_bool_not_iff
% 0.30/0.57  thf(fact_4267_Nat_Odiff__diff__right,axiom,
% 0.30/0.57      ! [K: nat,J: nat,I: nat] :
% 0.30/0.57        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.57       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.diff_diff_right
% 0.30/0.58  thf(fact_4268_Nat_Oadd__diff__assoc2,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.add_diff_assoc2
% 0.30/0.58  thf(fact_4269_Nat_Oadd__diff__assoc,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.add_diff_assoc
% 0.30/0.58  thf(fact_4270_diff__Suc__1,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
% 0.30/0.58        = N2 ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_Suc_1
% 0.30/0.58  thf(fact_4271_Suc__0__mod__eq,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.58        = ( zero_n2687167440665602831ol_nat
% 0.30/0.58          @ ( N2
% 0.30/0.58           != ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Suc_0_mod_eq
% 0.30/0.58  thf(fact_4272_mod__pos__pos__trivial,axiom,
% 0.30/0.58      ! [K: int,L: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.58       => ( ( ord_less_int @ K @ L )
% 0.30/0.58         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.58            = K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_pos_pos_trivial
% 0.30/0.58  thf(fact_4273_mod__neg__neg__trivial,axiom,
% 0.30/0.58      ! [K: int,L: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.58       => ( ( ord_less_int @ L @ K )
% 0.30/0.58         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.58            = K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_neg_neg_trivial
% 0.30/0.58  thf(fact_4274_signed__take__bit__Suc__1,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
% 0.30/0.58        = one_one_int ) ).
% 0.30/0.58  
% 0.30/0.58  % signed_take_bit_Suc_1
% 0.30/0.58  thf(fact_4275_Suc__pred,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.58       => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.58          = N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Suc_pred
% 0.30/0.58  thf(fact_4276_diff__Suc__diff__eq2,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 0.30/0.58          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_Suc_diff_eq2
% 0.30/0.58  thf(fact_4277_diff__Suc__diff__eq1,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_Suc_diff_eq1
% 0.30/0.58  thf(fact_4278_signed__take__bit__Suc__bit0,axiom,
% 0.30/0.58      ! [N2: nat,K: num] :
% 0.30/0.58        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.58        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % signed_take_bit_Suc_bit0
% 0.30/0.58  thf(fact_4279_Suc__diff__1,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.58       => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 0.30/0.58          = N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Suc_diff_1
% 0.30/0.58  thf(fact_4280_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 0.30/0.58          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4281_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: vEBT_VEBT] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 0.30/0.58          = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4282_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: product_prod_nat_nat] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 0.30/0.58          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4283_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: real] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 0.30/0.58          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4284_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: nat] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 0.30/0.58          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4285_set__replicate,axiom,
% 0.30/0.58      ! [N2: nat,X3: int] :
% 0.30/0.58        ( ( N2 != zero_zero_nat )
% 0.30/0.58       => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 0.30/0.58          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_replicate
% 0.30/0.58  thf(fact_4286_even__diff,axiom,
% 0.30/0.58      ! [A: code_integer,B: code_integer] :
% 0.30/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.58        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % even_diff
% 0.30/0.58  thf(fact_4287_even__diff,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 0.30/0.58        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % even_diff
% 0.30/0.58  thf(fact_4288_of__bool__half__eq__0,axiom,
% 0.30/0.58      ! [B: $o] :
% 0.30/0.58        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.58        = zero_z3403309356797280102nteger ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_half_eq_0
% 0.30/0.58  thf(fact_4289_of__bool__half__eq__0,axiom,
% 0.30/0.58      ! [B: $o] :
% 0.30/0.58        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.58        = zero_zero_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_half_eq_0
% 0.30/0.58  thf(fact_4290_of__bool__half__eq__0,axiom,
% 0.30/0.58      ! [B: $o] :
% 0.30/0.58        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.58        = zero_zero_int ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_half_eq_0
% 0.30/0.58  thf(fact_4291_odd__Suc__minus__one,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.58       => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 0.30/0.58          = N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % odd_Suc_minus_one
% 0.30/0.58  thf(fact_4292_even__diff__nat,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.58        = ( ( ord_less_nat @ M @ N2 )
% 0.30/0.58          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % even_diff_nat
% 0.30/0.58  thf(fact_4293_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
% 0.30/0.58        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.58  
% 0.30/0.58  % semiring_parity_class.even_mask_iff
% 0.30/0.58  thf(fact_4294_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
% 0.30/0.58        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.58  
% 0.30/0.58  % semiring_parity_class.even_mask_iff
% 0.30/0.58  thf(fact_4295_semiring__parity__class_Oeven__mask__iff,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 0.30/0.58        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.58  
% 0.30/0.58  % semiring_parity_class.even_mask_iff
% 0.30/0.58  thf(fact_4296_one__div__2__pow__eq,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % one_div_2_pow_eq
% 0.30/0.58  thf(fact_4297_one__div__2__pow__eq,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % one_div_2_pow_eq
% 0.30/0.58  thf(fact_4298_one__div__2__pow__eq,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % one_div_2_pow_eq
% 0.30/0.58  thf(fact_4299_bits__1__div__exp,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % bits_1_div_exp
% 0.30/0.58  thf(fact_4300_bits__1__div__exp,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % bits_1_div_exp
% 0.30/0.58  thf(fact_4301_bits__1__div__exp,axiom,
% 0.30/0.58      ! [N2: nat] :
% 0.30/0.58        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % bits_1_div_exp
% 0.30/0.58  thf(fact_4302_flip__bit__0,axiom,
% 0.30/0.58      ! [A: code_integer] :
% 0.30/0.58        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 0.30/0.58        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % flip_bit_0
% 0.30/0.58  thf(fact_4303_flip__bit__0,axiom,
% 0.30/0.58      ! [A: nat] :
% 0.30/0.58        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 0.30/0.58        = ( 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 ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % flip_bit_0
% 0.30/0.58  thf(fact_4304_flip__bit__0,axiom,
% 0.30/0.58      ! [A: int] :
% 0.30/0.58        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 0.30/0.58        = ( 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 ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % flip_bit_0
% 0.30/0.58  thf(fact_4305_mod__int__unique,axiom,
% 0.30/0.58      ! [K: int,L: int,Q4: int,R2: int] :
% 0.30/0.58        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.58       => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.58          = R2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_int_unique
% 0.30/0.58  thf(fact_4306_gcd__nat_Oasym,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58       => ~ ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.58            & ( B != A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.asym
% 0.30/0.58  thf(fact_4307_gcd__nat_Orefl,axiom,
% 0.30/0.58      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.refl
% 0.30/0.58  thf(fact_4308_gcd__nat_Otrans,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58       => ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.58         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.trans
% 0.30/0.58  thf(fact_4309_gcd__nat_Oeq__iff,axiom,
% 0.30/0.58      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 0.30/0.58      = ( ^ [A3: nat,B3: nat] :
% 0.30/0.58            ( ( dvd_dvd_nat @ A3 @ B3 )
% 0.30/0.58            & ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.eq_iff
% 0.30/0.58  thf(fact_4310_gcd__nat_Oirrefl,axiom,
% 0.30/0.58      ! [A: nat] :
% 0.30/0.58        ~ ( ( dvd_dvd_nat @ A @ A )
% 0.30/0.58          & ( A != A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.irrefl
% 0.30/0.58  thf(fact_4311_gcd__nat_Oantisym,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58       => ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.58         => ( A = B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.antisym
% 0.30/0.58  thf(fact_4312_gcd__nat_Ostrict__trans,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58       => ( ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.58            & ( B != C ) )
% 0.30/0.58         => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.58            & ( A != C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_trans
% 0.30/0.58  thf(fact_4313_gcd__nat_Ostrict__trans1,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58       => ( ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.58            & ( B != C ) )
% 0.30/0.58         => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.58            & ( A != C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_trans1
% 0.30/0.58  thf(fact_4314_gcd__nat_Ostrict__trans2,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58       => ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.58         => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.58            & ( A != C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_trans2
% 0.30/0.58  thf(fact_4315_gcd__nat_Ostrict__iff__not,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58        = ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_iff_not
% 0.30/0.58  thf(fact_4316_gcd__nat_Oorder__iff__strict,axiom,
% 0.30/0.58      ( dvd_dvd_nat
% 0.30/0.58      = ( ^ [A3: nat,B3: nat] :
% 0.30/0.58            ( ( ( dvd_dvd_nat @ A3 @ B3 )
% 0.30/0.58              & ( A3 != B3 ) )
% 0.30/0.58            | ( A3 = B3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.order_iff_strict
% 0.30/0.58  thf(fact_4317_gcd__nat_Ostrict__iff__order,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58        = ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_iff_order
% 0.30/0.58  thf(fact_4318_gcd__nat_Ostrict__implies__order,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58       => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_implies_order
% 0.30/0.58  thf(fact_4319_gcd__nat_Ostrict__implies__not__eq,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58          & ( A != B ) )
% 0.30/0.58       => ( A != B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.strict_implies_not_eq
% 0.30/0.58  thf(fact_4320_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( A != B )
% 0.30/0.58       => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58         => ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.58            & ( A != B ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % gcd_nat.not_eq_order_implies_strict
% 0.30/0.58  thf(fact_4321_dvd__antisym,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ M @ N2 )
% 0.30/0.58       => ( ( dvd_dvd_nat @ N2 @ M )
% 0.30/0.58         => ( M = N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_antisym
% 0.30/0.58  thf(fact_4322_dvd__diff__nat,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.58       => ( ( dvd_dvd_nat @ K @ N2 )
% 0.30/0.58         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diff_nat
% 0.30/0.58  thf(fact_4323_eucl__rel__int,axiom,
% 0.30/0.58      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eucl_rel_int
% 0.30/0.58  thf(fact_4324_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.58       => ? [B9: set_Pr4329608150637261639at_nat] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert9069300056098147895at_nat @ A @ B9 ) )
% 0.30/0.58            & ~ ( member8757157785044589968at_nat @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4325_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.58       => ? [B9: set_Pr1261947904930325089at_nat] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert8211810215607154385at_nat @ A @ B9 ) )
% 0.30/0.58            & ~ ( member8440522571783428010at_nat @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4326_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: real,A2: set_real] :
% 0.30/0.58        ( ( member_real @ A @ A2 )
% 0.30/0.58       => ? [B9: set_real] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert_real @ A @ B9 ) )
% 0.30/0.58            & ~ ( member_real @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4327_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: set_nat,A2: set_set_nat] :
% 0.30/0.58        ( ( member_set_nat @ A @ A2 )
% 0.30/0.58       => ? [B9: set_set_nat] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert_set_nat @ A @ B9 ) )
% 0.30/0.58            & ~ ( member_set_nat @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4328_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: nat,A2: set_nat] :
% 0.30/0.58        ( ( member_nat @ A @ A2 )
% 0.30/0.58       => ? [B9: set_nat] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert_nat @ A @ B9 ) )
% 0.30/0.58            & ~ ( member_nat @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4329_mk__disjoint__insert,axiom,
% 0.30/0.58      ! [A: int,A2: set_int] :
% 0.30/0.58        ( ( member_int @ A @ A2 )
% 0.30/0.58       => ? [B9: set_int] :
% 0.30/0.58            ( ( A2
% 0.30/0.58              = ( insert_int @ A @ B9 ) )
% 0.30/0.58            & ~ ( member_int @ A @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mk_disjoint_insert
% 0.30/0.58  thf(fact_4330_insert__commute,axiom,
% 0.30/0.58      ! [X3: nat,Y3: nat,A2: set_nat] :
% 0.30/0.58        ( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A2 ) )
% 0.30/0.58        = ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_commute
% 0.30/0.58  thf(fact_4331_insert__commute,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,A2: set_int] :
% 0.30/0.58        ( ( insert_int @ X3 @ ( insert_int @ Y3 @ A2 ) )
% 0.30/0.58        = ( insert_int @ Y3 @ ( insert_int @ X3 @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_commute
% 0.30/0.58  thf(fact_4332_insert__commute,axiom,
% 0.30/0.58      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ A2 ) )
% 0.30/0.58        = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_commute
% 0.30/0.58  thf(fact_4333_insert__commute,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,A2: set_real] :
% 0.30/0.58        ( ( insert_real @ X3 @ ( insert_real @ Y3 @ A2 ) )
% 0.30/0.58        = ( insert_real @ Y3 @ ( insert_real @ X3 @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_commute
% 0.30/0.58  thf(fact_4334_insert__commute,axiom,
% 0.30/0.58      ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y3 @ A2 ) )
% 0.30/0.58        = ( insert9069300056098147895at_nat @ Y3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_commute
% 0.30/0.58  thf(fact_4335_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.58       => ( ~ ( member8757157785044589968at_nat @ B @ B2 )
% 0.30/0.58         => ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.30/0.58              = ( insert9069300056098147895at_nat @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert9069300056098147895at_nat @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member8757157785044589968at_nat @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert9069300056098147895at_nat @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member8757157785044589968at_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4336_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.58       => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
% 0.30/0.58         => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.30/0.58              = ( insert8211810215607154385at_nat @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert8211810215607154385at_nat @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member8440522571783428010at_nat @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert8211810215607154385at_nat @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member8440522571783428010at_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4337_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 0.30/0.58        ( ~ ( member_real @ A @ A2 )
% 0.30/0.58       => ( ~ ( member_real @ B @ B2 )
% 0.30/0.58         => ( ( ( insert_real @ A @ A2 )
% 0.30/0.58              = ( insert_real @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_real] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert_real @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member_real @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert_real @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member_real @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4338_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 0.30/0.58        ( ~ ( member_set_nat @ A @ A2 )
% 0.30/0.58       => ( ~ ( member_set_nat @ B @ B2 )
% 0.30/0.58         => ( ( ( insert_set_nat @ A @ A2 )
% 0.30/0.58              = ( insert_set_nat @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_set_nat] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert_set_nat @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member_set_nat @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert_set_nat @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member_set_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4339_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 0.30/0.58        ( ~ ( member_nat @ A @ A2 )
% 0.30/0.58       => ( ~ ( member_nat @ B @ B2 )
% 0.30/0.58         => ( ( ( insert_nat @ A @ A2 )
% 0.30/0.58              = ( insert_nat @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_nat] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert_nat @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member_nat @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert_nat @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4340_insert__eq__iff,axiom,
% 0.30/0.58      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 0.30/0.58        ( ~ ( member_int @ A @ A2 )
% 0.30/0.58       => ( ~ ( member_int @ B @ B2 )
% 0.30/0.58         => ( ( ( insert_int @ A @ A2 )
% 0.30/0.58              = ( insert_int @ B @ B2 ) )
% 0.30/0.58            = ( ( ( A = B )
% 0.30/0.58               => ( A2 = B2 ) )
% 0.30/0.58              & ( ( A != B )
% 0.30/0.58               => ? [C6: set_int] :
% 0.30/0.58                    ( ( A2
% 0.30/0.58                      = ( insert_int @ B @ C6 ) )
% 0.30/0.58                    & ~ ( member_int @ B @ C6 )
% 0.30/0.58                    & ( B2
% 0.30/0.58                      = ( insert_int @ A @ C6 ) )
% 0.30/0.58                    & ~ ( member_int @ A @ C6 ) ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_eq_iff
% 0.30/0.58  thf(fact_4341_insert__absorb,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.58       => ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4342_insert__absorb,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.58       => ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4343_insert__absorb,axiom,
% 0.30/0.58      ! [A: real,A2: set_real] :
% 0.30/0.58        ( ( member_real @ A @ A2 )
% 0.30/0.58       => ( ( insert_real @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4344_insert__absorb,axiom,
% 0.30/0.58      ! [A: set_nat,A2: set_set_nat] :
% 0.30/0.58        ( ( member_set_nat @ A @ A2 )
% 0.30/0.58       => ( ( insert_set_nat @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4345_insert__absorb,axiom,
% 0.30/0.58      ! [A: nat,A2: set_nat] :
% 0.30/0.58        ( ( member_nat @ A @ A2 )
% 0.30/0.58       => ( ( insert_nat @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4346_insert__absorb,axiom,
% 0.30/0.58      ! [A: int,A2: set_int] :
% 0.30/0.58        ( ( member_int @ A @ A2 )
% 0.30/0.58       => ( ( insert_int @ A @ A2 )
% 0.30/0.58          = A2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_absorb
% 0.30/0.58  thf(fact_4347_insert__ident,axiom,
% 0.30/0.58      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert9069300056098147895at_nat @ X3 @ A2 )
% 0.30/0.58              = ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4348_insert__ident,axiom,
% 0.30/0.58      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert8211810215607154385at_nat @ X3 @ A2 )
% 0.30/0.58              = ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4349_insert__ident,axiom,
% 0.30/0.58      ! [X3: real,A2: set_real,B2: set_real] :
% 0.30/0.58        ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member_real @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert_real @ X3 @ A2 )
% 0.30/0.58              = ( insert_real @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4350_insert__ident,axiom,
% 0.30/0.58      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.58        ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member_set_nat @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert_set_nat @ X3 @ A2 )
% 0.30/0.58              = ( insert_set_nat @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4351_insert__ident,axiom,
% 0.30/0.58      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.58        ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member_nat @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert_nat @ X3 @ A2 )
% 0.30/0.58              = ( insert_nat @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4352_insert__ident,axiom,
% 0.30/0.58      ! [X3: int,A2: set_int,B2: set_int] :
% 0.30/0.58        ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.58       => ( ~ ( member_int @ X3 @ B2 )
% 0.30/0.58         => ( ( ( insert_int @ X3 @ A2 )
% 0.30/0.58              = ( insert_int @ X3 @ B2 ) )
% 0.30/0.58            = ( A2 = B2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_ident
% 0.30/0.58  thf(fact_4353_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_Pr4329608150637261639at_nat] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert9069300056098147895at_nat @ X3 @ B9 ) )
% 0.30/0.58             => ( member8757157785044589968at_nat @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4354_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_Pr1261947904930325089at_nat] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert8211810215607154385at_nat @ X3 @ B9 ) )
% 0.30/0.58             => ( member8440522571783428010at_nat @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4355_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: real,A2: set_real] :
% 0.30/0.58        ( ( member_real @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_real] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert_real @ X3 @ B9 ) )
% 0.30/0.58             => ( member_real @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4356_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: set_nat,A2: set_set_nat] :
% 0.30/0.58        ( ( member_set_nat @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_set_nat] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert_set_nat @ X3 @ B9 ) )
% 0.30/0.58             => ( member_set_nat @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4357_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: nat,A2: set_nat] :
% 0.30/0.58        ( ( member_nat @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_nat] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert_nat @ X3 @ B9 ) )
% 0.30/0.58             => ( member_nat @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4358_Set_Oset__insert,axiom,
% 0.30/0.58      ! [X3: int,A2: set_int] :
% 0.30/0.58        ( ( member_int @ X3 @ A2 )
% 0.30/0.58       => ~ ! [B9: set_int] :
% 0.30/0.58              ( ( A2
% 0.30/0.58                = ( insert_int @ X3 @ B9 ) )
% 0.30/0.58             => ( member_int @ X3 @ B9 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Set.set_insert
% 0.30/0.58  thf(fact_4359_insertI2,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ A @ B2 )
% 0.30/0.58       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4360_insertI2,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ A @ B2 )
% 0.30/0.58       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4361_insertI2,axiom,
% 0.30/0.58      ! [A: real,B2: set_real,B: real] :
% 0.30/0.58        ( ( member_real @ A @ B2 )
% 0.30/0.58       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4362_insertI2,axiom,
% 0.30/0.58      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 0.30/0.58        ( ( member_set_nat @ A @ B2 )
% 0.30/0.58       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4363_insertI2,axiom,
% 0.30/0.58      ! [A: nat,B2: set_nat,B: nat] :
% 0.30/0.58        ( ( member_nat @ A @ B2 )
% 0.30/0.58       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4364_insertI2,axiom,
% 0.30/0.58      ! [A: int,B2: set_int,B: int] :
% 0.30/0.58        ( ( member_int @ A @ B2 )
% 0.30/0.58       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI2
% 0.30/0.58  thf(fact_4365_insertI1,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4366_insertI1,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4367_insertI1,axiom,
% 0.30/0.58      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4368_insertI1,axiom,
% 0.30/0.58      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4369_insertI1,axiom,
% 0.30/0.58      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4370_insertI1,axiom,
% 0.30/0.58      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertI1
% 0.30/0.58  thf(fact_4371_insertE,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4372_insertE,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4373_insertE,axiom,
% 0.30/0.58      ! [A: real,B: real,A2: set_real] :
% 0.30/0.58        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member_real @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4374_insertE,axiom,
% 0.30/0.58      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 0.30/0.58        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member_set_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4375_insertE,axiom,
% 0.30/0.58      ! [A: nat,B: nat,A2: set_nat] :
% 0.30/0.58        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4376_insertE,axiom,
% 0.30/0.58      ! [A: int,B: int,A2: set_int] :
% 0.30/0.58        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 0.30/0.58       => ( ( A != B )
% 0.30/0.58         => ( member_int @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insertE
% 0.30/0.58  thf(fact_4377_of__bool__eq__iff,axiom,
% 0.30/0.58      ! [P4: $o,Q4: $o] :
% 0.30/0.58        ( ( ( zero_n2687167440665602831ol_nat @ P4 )
% 0.30/0.58          = ( zero_n2687167440665602831ol_nat @ Q4 ) )
% 0.30/0.58        = ( P4 = Q4 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_eq_iff
% 0.30/0.58  thf(fact_4378_of__bool__eq__iff,axiom,
% 0.30/0.58      ! [P4: $o,Q4: $o] :
% 0.30/0.58        ( ( ( zero_n2684676970156552555ol_int @ P4 )
% 0.30/0.58          = ( zero_n2684676970156552555ol_int @ Q4 ) )
% 0.30/0.58        = ( P4 = Q4 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_eq_iff
% 0.30/0.58  thf(fact_4379_diff__commute,axiom,
% 0.30/0.58      ! [I: nat,J: nat,K: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 0.30/0.58        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_commute
% 0.30/0.58  thf(fact_4380_diff__right__commute,axiom,
% 0.30/0.58      ! [A: real,C: real,B: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 0.30/0.58        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_commute
% 0.30/0.58  thf(fact_4381_diff__right__commute,axiom,
% 0.30/0.58      ! [A: rat,C: rat,B: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 0.30/0.58        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_commute
% 0.30/0.58  thf(fact_4382_diff__right__commute,axiom,
% 0.30/0.58      ! [A: nat,C: nat,B: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 0.30/0.58        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_commute
% 0.30/0.58  thf(fact_4383_diff__right__commute,axiom,
% 0.30/0.58      ! [A: int,C: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.30/0.58        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_commute
% 0.30/0.58  thf(fact_4384_diff__eq__diff__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.58        ( ( ( minus_minus_real @ A @ B )
% 0.30/0.58          = ( minus_minus_real @ C @ D ) )
% 0.30/0.58       => ( ( A = B )
% 0.30/0.58          = ( C = D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_eq
% 0.30/0.58  thf(fact_4385_diff__eq__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.58        ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.58          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.58       => ( ( A = B )
% 0.30/0.58          = ( C = D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_eq
% 0.30/0.58  thf(fact_4386_diff__eq__diff__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.58        ( ( ( minus_minus_int @ A @ B )
% 0.30/0.58          = ( minus_minus_int @ C @ D ) )
% 0.30/0.58       => ( ( A = B )
% 0.30/0.58          = ( C = D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_eq
% 0.30/0.58  thf(fact_4387_full__exhaustive__int_H_Ocases,axiom,
% 0.30/0.58      ! [X3: produc2285326912895808259nt_int] :
% 0.30/0.58        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D4: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % full_exhaustive_int'.cases
% 0.30/0.58  thf(fact_4388_exhaustive__int_H_Ocases,axiom,
% 0.30/0.58      ! [X3: produc7773217078559923341nt_int] :
% 0.30/0.58        ~ ! [F2: int > option6357759511663192854e_term,D4: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exhaustive_int'.cases
% 0.30/0.58  thf(fact_4389_small__lazy_H_Ocases,axiom,
% 0.30/0.58      ! [X3: product_prod_int_int] :
% 0.30/0.58        ~ ! [D4: int,I3: int] :
% 0.30/0.58            ( X3
% 0.30/0.58           != ( product_Pair_int_int @ D4 @ I3 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % small_lazy'.cases
% 0.30/0.58  thf(fact_4390_unique__quotient,axiom,
% 0.30/0.58      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
% 0.30/0.58        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.58       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
% 0.30/0.58         => ( Q4 = Q6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % unique_quotient
% 0.30/0.58  thf(fact_4391_unique__remainder,axiom,
% 0.30/0.58      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
% 0.30/0.58        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.58       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
% 0.30/0.58         => ( R2 = R3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % unique_remainder
% 0.30/0.58  thf(fact_4392_of__bool__conj,axiom,
% 0.30/0.58      ! [P: $o,Q: $o] :
% 0.30/0.58        ( ( zero_n1201886186963655149omplex
% 0.30/0.58          @ ( P
% 0.30/0.58            & Q ) )
% 0.30/0.58        = ( times_times_complex @ ( zero_n1201886186963655149omplex @ P ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_conj
% 0.30/0.58  thf(fact_4393_of__bool__conj,axiom,
% 0.30/0.58      ! [P: $o,Q: $o] :
% 0.30/0.58        ( ( zero_n3304061248610475627l_real
% 0.30/0.58          @ ( P
% 0.30/0.58            & Q ) )
% 0.30/0.58        = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_conj
% 0.30/0.58  thf(fact_4394_of__bool__conj,axiom,
% 0.30/0.58      ! [P: $o,Q: $o] :
% 0.30/0.58        ( ( zero_n2052037380579107095ol_rat
% 0.30/0.58          @ ( P
% 0.30/0.58            & Q ) )
% 0.30/0.58        = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_conj
% 0.30/0.58  thf(fact_4395_of__bool__conj,axiom,
% 0.30/0.58      ! [P: $o,Q: $o] :
% 0.30/0.58        ( ( zero_n2687167440665602831ol_nat
% 0.30/0.58          @ ( P
% 0.30/0.58            & Q ) )
% 0.30/0.58        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_conj
% 0.30/0.58  thf(fact_4396_of__bool__conj,axiom,
% 0.30/0.58      ! [P: $o,Q: $o] :
% 0.30/0.58        ( ( zero_n2684676970156552555ol_int
% 0.30/0.58          @ ( P
% 0.30/0.58            & Q ) )
% 0.30/0.58        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_conj
% 0.30/0.58  thf(fact_4397_diff__mono,axiom,
% 0.30/0.58      ! [A: real,B: real,D: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_real @ D @ C )
% 0.30/0.58         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_mono
% 0.30/0.58  thf(fact_4398_diff__mono,axiom,
% 0.30/0.58      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_rat @ D @ C )
% 0.30/0.58         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_mono
% 0.30/0.58  thf(fact_4399_diff__mono,axiom,
% 0.30/0.58      ! [A: int,B: int,D: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_int @ D @ C )
% 0.30/0.58         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_mono
% 0.30/0.58  thf(fact_4400_diff__left__mono,axiom,
% 0.30/0.58      ! [B: real,A: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ B @ A )
% 0.30/0.58       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_left_mono
% 0.30/0.58  thf(fact_4401_diff__left__mono,axiom,
% 0.30/0.58      ! [B: rat,A: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ B @ A )
% 0.30/0.58       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_left_mono
% 0.30/0.58  thf(fact_4402_diff__left__mono,axiom,
% 0.30/0.58      ! [B: int,A: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ B @ A )
% 0.30/0.58       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_left_mono
% 0.30/0.58  thf(fact_4403_diff__right__mono,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_mono
% 0.30/0.58  thf(fact_4404_diff__right__mono,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_mono
% 0.30/0.58  thf(fact_4405_diff__right__mono,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_right_mono
% 0.30/0.58  thf(fact_4406_diff__eq__diff__less__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.58        ( ( ( minus_minus_real @ A @ B )
% 0.30/0.58          = ( minus_minus_real @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_eq_real @ A @ B )
% 0.30/0.58          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less_eq
% 0.30/0.58  thf(fact_4407_diff__eq__diff__less__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.58        ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.58          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.58          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less_eq
% 0.30/0.58  thf(fact_4408_diff__eq__diff__less__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.58        ( ( ( minus_minus_int @ A @ B )
% 0.30/0.58          = ( minus_minus_int @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_eq_int @ A @ B )
% 0.30/0.58          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less_eq
% 0.30/0.58  thf(fact_4409_eq__iff__diff__eq__0,axiom,
% 0.30/0.58      ( ( ^ [Y4: complex,Z: complex] : ( Y4 = Z ) )
% 0.30/0.58      = ( ^ [A3: complex,B3: complex] :
% 0.30/0.58            ( ( minus_minus_complex @ A3 @ B3 )
% 0.30/0.58            = zero_zero_complex ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_iff_diff_eq_0
% 0.30/0.58  thf(fact_4410_eq__iff__diff__eq__0,axiom,
% 0.30/0.58      ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 0.30/0.58      = ( ^ [A3: real,B3: real] :
% 0.30/0.58            ( ( minus_minus_real @ A3 @ B3 )
% 0.30/0.58            = zero_zero_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_iff_diff_eq_0
% 0.30/0.58  thf(fact_4411_eq__iff__diff__eq__0,axiom,
% 0.30/0.58      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 0.30/0.58      = ( ^ [A3: rat,B3: rat] :
% 0.30/0.58            ( ( minus_minus_rat @ A3 @ B3 )
% 0.30/0.58            = zero_zero_rat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_iff_diff_eq_0
% 0.30/0.58  thf(fact_4412_eq__iff__diff__eq__0,axiom,
% 0.30/0.58      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 0.30/0.58      = ( ^ [A3: int,B3: int] :
% 0.30/0.58            ( ( minus_minus_int @ A3 @ B3 )
% 0.30/0.58            = zero_zero_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_iff_diff_eq_0
% 0.30/0.58  thf(fact_4413_diff__strict__mono,axiom,
% 0.30/0.58      ! [A: real,B: real,D: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B )
% 0.30/0.58       => ( ( ord_less_real @ D @ C )
% 0.30/0.58         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_mono
% 0.30/0.58  thf(fact_4414_diff__strict__mono,axiom,
% 0.30/0.58      ! [A: rat,B: rat,D: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B )
% 0.30/0.58       => ( ( ord_less_rat @ D @ C )
% 0.30/0.58         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_mono
% 0.30/0.58  thf(fact_4415_diff__strict__mono,axiom,
% 0.30/0.58      ! [A: int,B: int,D: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B )
% 0.30/0.58       => ( ( ord_less_int @ D @ C )
% 0.30/0.58         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_mono
% 0.30/0.58  thf(fact_4416_diff__eq__diff__less,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.58        ( ( ( minus_minus_real @ A @ B )
% 0.30/0.58          = ( minus_minus_real @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_real @ A @ B )
% 0.30/0.58          = ( ord_less_real @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less
% 0.30/0.58  thf(fact_4417_diff__eq__diff__less,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.58        ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.58          = ( minus_minus_rat @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_rat @ A @ B )
% 0.30/0.58          = ( ord_less_rat @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less
% 0.30/0.58  thf(fact_4418_diff__eq__diff__less,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.58        ( ( ( minus_minus_int @ A @ B )
% 0.30/0.58          = ( minus_minus_int @ C @ D ) )
% 0.30/0.58       => ( ( ord_less_int @ A @ B )
% 0.30/0.58          = ( ord_less_int @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_diff_less
% 0.30/0.58  thf(fact_4419_diff__strict__left__mono,axiom,
% 0.30/0.58      ! [B: real,A: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ B @ A )
% 0.30/0.58       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_left_mono
% 0.30/0.58  thf(fact_4420_diff__strict__left__mono,axiom,
% 0.30/0.58      ! [B: rat,A: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ B @ A )
% 0.30/0.58       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_left_mono
% 0.30/0.58  thf(fact_4421_diff__strict__left__mono,axiom,
% 0.30/0.58      ! [B: int,A: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ B @ A )
% 0.30/0.58       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_left_mono
% 0.30/0.58  thf(fact_4422_diff__strict__right__mono,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ B )
% 0.30/0.58       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_right_mono
% 0.30/0.58  thf(fact_4423_diff__strict__right__mono,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ B )
% 0.30/0.58       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_right_mono
% 0.30/0.58  thf(fact_4424_diff__strict__right__mono,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ B )
% 0.30/0.58       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_strict_right_mono
% 0.30/0.58  thf(fact_4425_right__diff__distrib_H,axiom,
% 0.30/0.58      ! [A: complex,B: complex,C: complex] :
% 0.30/0.58        ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 0.30/0.58        = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib'
% 0.30/0.58  thf(fact_4426_right__diff__distrib_H,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.58        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib'
% 0.30/0.58  thf(fact_4427_right__diff__distrib_H,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib'
% 0.30/0.58  thf(fact_4428_right__diff__distrib_H,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib'
% 0.30/0.58  thf(fact_4429_right__diff__distrib_H,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.58        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib'
% 0.30/0.58  thf(fact_4430_left__diff__distrib_H,axiom,
% 0.30/0.58      ! [B: complex,C: complex,A: complex] :
% 0.30/0.58        ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
% 0.30/0.58        = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib'
% 0.30/0.58  thf(fact_4431_left__diff__distrib_H,axiom,
% 0.30/0.58      ! [B: real,C: real,A: real] :
% 0.30/0.58        ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 0.30/0.58        = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib'
% 0.30/0.58  thf(fact_4432_left__diff__distrib_H,axiom,
% 0.30/0.58      ! [B: rat,C: rat,A: rat] :
% 0.30/0.58        ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 0.30/0.58        = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib'
% 0.30/0.58  thf(fact_4433_left__diff__distrib_H,axiom,
% 0.30/0.58      ! [B: nat,C: nat,A: nat] :
% 0.30/0.58        ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 0.30/0.58        = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib'
% 0.30/0.58  thf(fact_4434_left__diff__distrib_H,axiom,
% 0.30/0.58      ! [B: int,C: int,A: int] :
% 0.30/0.58        ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 0.30/0.58        = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib'
% 0.30/0.58  thf(fact_4435_right__diff__distrib,axiom,
% 0.30/0.58      ! [A: complex,B: complex,C: complex] :
% 0.30/0.58        ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 0.30/0.58        = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib
% 0.30/0.58  thf(fact_4436_right__diff__distrib,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.58        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib
% 0.30/0.58  thf(fact_4437_right__diff__distrib,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib
% 0.30/0.58  thf(fact_4438_right__diff__distrib,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.58        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % right_diff_distrib
% 0.30/0.58  thf(fact_4439_left__diff__distrib,axiom,
% 0.30/0.58      ! [A: complex,B: complex,C: complex] :
% 0.30/0.58        ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib
% 0.30/0.58  thf(fact_4440_left__diff__distrib,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib
% 0.30/0.58  thf(fact_4441_left__diff__distrib,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib
% 0.30/0.58  thf(fact_4442_left__diff__distrib,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % left_diff_distrib
% 0.30/0.58  thf(fact_4443_add__diff__add,axiom,
% 0.30/0.58      ! [A: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 0.30/0.58        = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_add
% 0.30/0.58  thf(fact_4444_add__diff__add,axiom,
% 0.30/0.58      ! [A: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 0.30/0.58        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_add
% 0.30/0.58  thf(fact_4445_add__diff__add,axiom,
% 0.30/0.58      ! [A: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 0.30/0.58        = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_add
% 0.30/0.58  thf(fact_4446_group__cancel_Osub1,axiom,
% 0.30/0.58      ! [A2: real,K: real,A: real,B: real] :
% 0.30/0.58        ( ( A2
% 0.30/0.58          = ( plus_plus_real @ K @ A ) )
% 0.30/0.58       => ( ( minus_minus_real @ A2 @ B )
% 0.30/0.58          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % group_cancel.sub1
% 0.30/0.58  thf(fact_4447_group__cancel_Osub1,axiom,
% 0.30/0.58      ! [A2: rat,K: rat,A: rat,B: rat] :
% 0.30/0.58        ( ( A2
% 0.30/0.58          = ( plus_plus_rat @ K @ A ) )
% 0.30/0.58       => ( ( minus_minus_rat @ A2 @ B )
% 0.30/0.58          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % group_cancel.sub1
% 0.30/0.58  thf(fact_4448_group__cancel_Osub1,axiom,
% 0.30/0.58      ! [A2: int,K: int,A: int,B: int] :
% 0.30/0.58        ( ( A2
% 0.30/0.58          = ( plus_plus_int @ K @ A ) )
% 0.30/0.58       => ( ( minus_minus_int @ A2 @ B )
% 0.30/0.58          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % group_cancel.sub1
% 0.30/0.58  thf(fact_4449_diff__eq__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( ( minus_minus_real @ A @ B )
% 0.30/0.58          = C )
% 0.30/0.58        = ( A
% 0.30/0.58          = ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_eq
% 0.30/0.58  thf(fact_4450_diff__eq__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( ( minus_minus_rat @ A @ B )
% 0.30/0.58          = C )
% 0.30/0.58        = ( A
% 0.30/0.58          = ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_eq
% 0.30/0.58  thf(fact_4451_diff__eq__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( ( minus_minus_int @ A @ B )
% 0.30/0.58          = C )
% 0.30/0.58        = ( A
% 0.30/0.58          = ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_eq_eq
% 0.30/0.58  thf(fact_4452_eq__diff__eq,axiom,
% 0.30/0.58      ! [A: real,C: real,B: real] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( minus_minus_real @ C @ B ) )
% 0.30/0.58        = ( ( plus_plus_real @ A @ B )
% 0.30/0.58          = C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_diff_eq
% 0.30/0.58  thf(fact_4453_eq__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,C: rat,B: rat] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( minus_minus_rat @ C @ B ) )
% 0.30/0.58        = ( ( plus_plus_rat @ A @ B )
% 0.30/0.58          = C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_diff_eq
% 0.30/0.58  thf(fact_4454_eq__diff__eq,axiom,
% 0.30/0.58      ! [A: int,C: int,B: int] :
% 0.30/0.58        ( ( A
% 0.30/0.58          = ( minus_minus_int @ C @ B ) )
% 0.30/0.58        = ( ( plus_plus_int @ A @ B )
% 0.30/0.58          = C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_diff_eq
% 0.30/0.58  thf(fact_4455_add__diff__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.58        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_eq
% 0.30/0.58  thf(fact_4456_add__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_eq
% 0.30/0.58  thf(fact_4457_add__diff__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.58        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_eq
% 0.30/0.58  thf(fact_4458_diff__diff__eq2,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 0.30/0.58        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq2
% 0.30/0.58  thf(fact_4459_diff__diff__eq2,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq2
% 0.30/0.58  thf(fact_4460_diff__diff__eq2,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.30/0.58        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq2
% 0.30/0.58  thf(fact_4461_diff__add__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq
% 0.30/0.58  thf(fact_4462_diff__add__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq
% 0.30/0.58  thf(fact_4463_diff__add__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq
% 0.30/0.58  thf(fact_4464_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 0.30/0.58        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq_diff_diff_swap
% 0.30/0.58  thf(fact_4465_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 0.30/0.58        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq_diff_diff_swap
% 0.30/0.58  thf(fact_4466_diff__add__eq__diff__diff__swap,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.30/0.58        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_eq_diff_diff_swap
% 0.30/0.58  thf(fact_4467_add__implies__diff,axiom,
% 0.30/0.58      ! [C: real,B: real,A: real] :
% 0.30/0.58        ( ( ( plus_plus_real @ C @ B )
% 0.30/0.58          = A )
% 0.30/0.58       => ( C
% 0.30/0.58          = ( minus_minus_real @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_implies_diff
% 0.30/0.58  thf(fact_4468_add__implies__diff,axiom,
% 0.30/0.58      ! [C: rat,B: rat,A: rat] :
% 0.30/0.58        ( ( ( plus_plus_rat @ C @ B )
% 0.30/0.58          = A )
% 0.30/0.58       => ( C
% 0.30/0.58          = ( minus_minus_rat @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_implies_diff
% 0.30/0.58  thf(fact_4469_add__implies__diff,axiom,
% 0.30/0.58      ! [C: nat,B: nat,A: nat] :
% 0.30/0.58        ( ( ( plus_plus_nat @ C @ B )
% 0.30/0.58          = A )
% 0.30/0.58       => ( C
% 0.30/0.58          = ( minus_minus_nat @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_implies_diff
% 0.30/0.58  thf(fact_4470_add__implies__diff,axiom,
% 0.30/0.58      ! [C: int,B: int,A: int] :
% 0.30/0.58        ( ( ( plus_plus_int @ C @ B )
% 0.30/0.58          = A )
% 0.30/0.58       => ( C
% 0.30/0.58          = ( minus_minus_int @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_implies_diff
% 0.30/0.58  thf(fact_4471_diff__diff__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq
% 0.30/0.58  thf(fact_4472_diff__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq
% 0.30/0.58  thf(fact_4473_diff__diff__eq,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq
% 0.30/0.58  thf(fact_4474_diff__diff__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_diff_eq
% 0.30/0.58  thf(fact_4475_diff__divide__distrib,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_divide_distrib
% 0.30/0.58  thf(fact_4476_diff__divide__distrib,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_divide_distrib
% 0.30/0.58  thf(fact_4477_singletonD,axiom,
% 0.30/0.58      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4478_singletonD,axiom,
% 0.30/0.58      ! [B: set_nat,A: set_nat] :
% 0.30/0.58        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4479_singletonD,axiom,
% 0.30/0.58      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4480_singletonD,axiom,
% 0.30/0.58      ! [B: real,A: real] :
% 0.30/0.58        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4481_singletonD,axiom,
% 0.30/0.58      ! [B: nat,A: nat] :
% 0.30/0.58        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4482_singletonD,axiom,
% 0.30/0.58      ! [B: int,A: int] :
% 0.30/0.58        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.30/0.58       => ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singletonD
% 0.30/0.58  thf(fact_4483_singleton__iff,axiom,
% 0.30/0.58      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4484_singleton__iff,axiom,
% 0.30/0.58      ! [B: set_nat,A: set_nat] :
% 0.30/0.58        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4485_singleton__iff,axiom,
% 0.30/0.58      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4486_singleton__iff,axiom,
% 0.30/0.58      ! [B: real,A: real] :
% 0.30/0.58        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4487_singleton__iff,axiom,
% 0.30/0.58      ! [B: nat,A: nat] :
% 0.30/0.58        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4488_singleton__iff,axiom,
% 0.30/0.58      ! [B: int,A: int] :
% 0.30/0.58        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.30/0.58        = ( B = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_iff
% 0.30/0.58  thf(fact_4489_doubleton__eq__iff,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58          = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.58        = ( ( ( A = C )
% 0.30/0.58            & ( B = D ) )
% 0.30/0.58          | ( ( A = D )
% 0.30/0.58            & ( B = C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % doubleton_eq_iff
% 0.30/0.58  thf(fact_4490_doubleton__eq__iff,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 0.30/0.58        ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58          = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.58        = ( ( ( A = C )
% 0.30/0.58            & ( B = D ) )
% 0.30/0.58          | ( ( A = D )
% 0.30/0.58            & ( B = C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % doubleton_eq_iff
% 0.30/0.58  thf(fact_4491_doubleton__eq__iff,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.58        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 0.30/0.58          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 0.30/0.58        = ( ( ( A = C )
% 0.30/0.58            & ( B = D ) )
% 0.30/0.58          | ( ( A = D )
% 0.30/0.58            & ( B = C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % doubleton_eq_iff
% 0.30/0.58  thf(fact_4492_doubleton__eq__iff,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.58        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.30/0.58          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 0.30/0.58        = ( ( ( A = C )
% 0.30/0.58            & ( B = D ) )
% 0.30/0.58          | ( ( A = D )
% 0.30/0.58            & ( B = C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % doubleton_eq_iff
% 0.30/0.58  thf(fact_4493_doubleton__eq__iff,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.58        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 0.30/0.58          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 0.30/0.58        = ( ( ( A = C )
% 0.30/0.58            & ( B = D ) )
% 0.30/0.58          | ( ( A = D )
% 0.30/0.58            & ( B = C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % doubleton_eq_iff
% 0.30/0.58  thf(fact_4494_insert__not__empty,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( insert9069300056098147895at_nat @ A @ A2 )
% 0.30/0.58       != bot_bo228742789529271731at_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_not_empty
% 0.30/0.58  thf(fact_4495_insert__not__empty,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( insert8211810215607154385at_nat @ A @ A2 )
% 0.30/0.58       != bot_bo2099793752762293965at_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_not_empty
% 0.30/0.58  thf(fact_4496_insert__not__empty,axiom,
% 0.30/0.58      ! [A: real,A2: set_real] :
% 0.30/0.58        ( ( insert_real @ A @ A2 )
% 0.30/0.58       != bot_bot_set_real ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_not_empty
% 0.30/0.58  thf(fact_4497_insert__not__empty,axiom,
% 0.30/0.58      ! [A: nat,A2: set_nat] :
% 0.30/0.58        ( ( insert_nat @ A @ A2 )
% 0.30/0.58       != bot_bot_set_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_not_empty
% 0.30/0.58  thf(fact_4498_insert__not__empty,axiom,
% 0.30/0.58      ! [A: int,A2: set_int] :
% 0.30/0.58        ( ( insert_int @ A @ A2 )
% 0.30/0.58       != bot_bot_set_int ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_not_empty
% 0.30/0.58  thf(fact_4499_singleton__inject,axiom,
% 0.30/0.58      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
% 0.30/0.58          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58       => ( A = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_inject
% 0.30/0.58  thf(fact_4500_singleton__inject,axiom,
% 0.30/0.58      ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 0.30/0.58        ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
% 0.30/0.58          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58       => ( A = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_inject
% 0.30/0.58  thf(fact_4501_singleton__inject,axiom,
% 0.30/0.58      ! [A: real,B: real] :
% 0.30/0.58        ( ( ( insert_real @ A @ bot_bot_set_real )
% 0.30/0.58          = ( insert_real @ B @ bot_bot_set_real ) )
% 0.30/0.58       => ( A = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_inject
% 0.30/0.58  thf(fact_4502_singleton__inject,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 0.30/0.58          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 0.30/0.58       => ( A = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_inject
% 0.30/0.58  thf(fact_4503_singleton__inject,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( ( insert_int @ A @ bot_bot_set_int )
% 0.30/0.58          = ( insert_int @ B @ bot_bot_set_int ) )
% 0.30/0.58       => ( A = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % singleton_inject
% 0.30/0.58  thf(fact_4504_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_real,A: real] :
% 0.30/0.58        ( ( finite_finite_real @ A2 )
% 0.30/0.58       => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4505_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ A2 )
% 0.30/0.58       => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4506_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_nat,A: nat] :
% 0.30/0.58        ( ( finite_finite_nat @ A2 )
% 0.30/0.58       => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4507_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_int,A: int] :
% 0.30/0.58        ( ( finite_finite_int @ A2 )
% 0.30/0.58       => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4508_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_complex,A: complex] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.58       => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4509_finite_OinsertI,axiom,
% 0.30/0.58      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.58       => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.insertI
% 0.30/0.58  thf(fact_4510_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le1268244103169919719at_nat @ X7 @ A2 )
% 0.30/0.58         => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4511_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le3146513528884898305at_nat @ X7 @ A2 )
% 0.30/0.58         => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4512_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: real,A2: set_real,X7: set_real] :
% 0.30/0.58        ( ( member_real @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_real @ X7 @ A2 )
% 0.30/0.58         => ( ord_less_eq_set_real @ ( insert_real @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4513_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: set_nat,A2: set_set_nat,X7: set_set_nat] :
% 0.30/0.58        ( ( member_set_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le6893508408891458716et_nat @ X7 @ A2 )
% 0.30/0.58         => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4514_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: nat,A2: set_nat,X7: set_nat] :
% 0.30/0.58        ( ( member_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_nat @ X7 @ A2 )
% 0.30/0.58         => ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4515_insert__subsetI,axiom,
% 0.30/0.58      ! [X3: int,A2: set_int,X7: set_int] :
% 0.30/0.58        ( ( member_int @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ X7 @ A2 )
% 0.30/0.58         => ( ord_less_eq_set_int @ ( insert_int @ X3 @ X7 ) @ A2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_subsetI
% 0.30/0.58  thf(fact_4516_insert__mono,axiom,
% 0.30/0.58      ! [C4: set_nat,D6: set_nat,A: nat] :
% 0.30/0.58        ( ( ord_less_eq_set_nat @ C4 @ D6 )
% 0.30/0.58       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_mono
% 0.30/0.58  thf(fact_4517_insert__mono,axiom,
% 0.30/0.58      ! [C4: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.30/0.58        ( ( ord_le3146513528884898305at_nat @ C4 @ D6 )
% 0.30/0.58       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C4 ) @ ( insert8211810215607154385at_nat @ A @ D6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_mono
% 0.30/0.58  thf(fact_4518_insert__mono,axiom,
% 0.30/0.58      ! [C4: set_real,D6: set_real,A: real] :
% 0.30/0.58        ( ( ord_less_eq_set_real @ C4 @ D6 )
% 0.30/0.58       => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_mono
% 0.30/0.58  thf(fact_4519_insert__mono,axiom,
% 0.30/0.58      ! [C4: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ord_le1268244103169919719at_nat @ C4 @ D6 )
% 0.30/0.58       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C4 ) @ ( insert9069300056098147895at_nat @ A @ D6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_mono
% 0.30/0.58  thf(fact_4520_insert__mono,axiom,
% 0.30/0.58      ! [C4: set_int,D6: set_int,A: int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ C4 @ D6 )
% 0.30/0.58       => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D6 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % insert_mono
% 0.30/0.58  thf(fact_4521_subset__insert,axiom,
% 0.30/0.58      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4522_subset__insert,axiom,
% 0.30/0.58      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4523_subset__insert,axiom,
% 0.30/0.58      ! [X3: real,A2: set_real,B2: set_real] :
% 0.30/0.58        ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4524_subset__insert,axiom,
% 0.30/0.58      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.58        ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4525_subset__insert,axiom,
% 0.30/0.58      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.58        ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4526_subset__insert,axiom,
% 0.30/0.58      ! [X3: int,A2: set_int,B2: set_int] :
% 0.30/0.58        ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 0.30/0.58          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insert
% 0.30/0.58  thf(fact_4527_subset__insertI,axiom,
% 0.30/0.58      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI
% 0.30/0.58  thf(fact_4528_subset__insertI,axiom,
% 0.30/0.58      ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI
% 0.30/0.58  thf(fact_4529_subset__insertI,axiom,
% 0.30/0.58      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI
% 0.30/0.58  thf(fact_4530_subset__insertI,axiom,
% 0.30/0.58      ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI
% 0.30/0.58  thf(fact_4531_subset__insertI,axiom,
% 0.30/0.58      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI
% 0.30/0.58  thf(fact_4532_subset__insertI2,axiom,
% 0.30/0.58      ! [A2: set_nat,B2: set_nat,B: nat] :
% 0.30/0.58        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.58       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI2
% 0.30/0.58  thf(fact_4533_subset__insertI2,axiom,
% 0.30/0.58      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.30/0.58        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 0.30/0.58       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI2
% 0.30/0.58  thf(fact_4534_subset__insertI2,axiom,
% 0.30/0.58      ! [A2: set_real,B2: set_real,B: real] :
% 0.30/0.58        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.58       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI2
% 0.30/0.58  thf(fact_4535_subset__insertI2,axiom,
% 0.30/0.58      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
% 0.30/0.58       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI2
% 0.30/0.58  thf(fact_4536_subset__insertI2,axiom,
% 0.30/0.58      ! [A2: set_int,B2: set_int,B: int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.30/0.58       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_insertI2
% 0.30/0.58  thf(fact_4537_dvd__diff,axiom,
% 0.30/0.58      ! [X3: code_integer,Y3: code_integer,Z2: code_integer] :
% 0.30/0.58        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 0.30/0.58       => ( ( dvd_dvd_Code_integer @ X3 @ Z2 )
% 0.30/0.58         => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diff
% 0.30/0.58  thf(fact_4538_dvd__diff,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,Z2: real] :
% 0.30/0.58        ( ( dvd_dvd_real @ X3 @ Y3 )
% 0.30/0.58       => ( ( dvd_dvd_real @ X3 @ Z2 )
% 0.30/0.58         => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diff
% 0.30/0.58  thf(fact_4539_dvd__diff,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,Z2: rat] :
% 0.30/0.58        ( ( dvd_dvd_rat @ X3 @ Y3 )
% 0.30/0.58       => ( ( dvd_dvd_rat @ X3 @ Z2 )
% 0.30/0.58         => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diff
% 0.30/0.58  thf(fact_4540_dvd__diff,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,Z2: int] :
% 0.30/0.58        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.58       => ( ( dvd_dvd_int @ X3 @ Z2 )
% 0.30/0.58         => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diff
% 0.30/0.58  thf(fact_4541_zero__induct__lemma,axiom,
% 0.30/0.58      ! [P: nat > $o,K: nat,I: nat] :
% 0.30/0.58        ( ( P @ K )
% 0.30/0.58       => ( ! [N3: nat] :
% 0.30/0.58              ( ( P @ ( suc @ N3 ) )
% 0.30/0.58             => ( P @ N3 ) )
% 0.30/0.58         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zero_induct_lemma
% 0.30/0.58  thf(fact_4542_minus__nat_Odiff__0,axiom,
% 0.30/0.58      ! [M: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 0.30/0.58        = M ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_nat.diff_0
% 0.30/0.58  thf(fact_4543_diffs0__imp__equal,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( ( minus_minus_nat @ M @ N2 )
% 0.30/0.58          = zero_zero_nat )
% 0.30/0.58       => ( ( ( minus_minus_nat @ N2 @ M )
% 0.30/0.58            = zero_zero_nat )
% 0.30/0.58         => ( M = N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diffs0_imp_equal
% 0.30/0.58  thf(fact_4544_diff__less__mono2,axiom,
% 0.30/0.58      ! [M: nat,N2: nat,L: nat] :
% 0.30/0.58        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.58       => ( ( ord_less_nat @ M @ L )
% 0.30/0.58         => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_mono2
% 0.30/0.58  thf(fact_4545_less__imp__diff__less,axiom,
% 0.30/0.58      ! [J: nat,K: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_nat @ J @ K )
% 0.30/0.58       => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_imp_diff_less
% 0.30/0.58  thf(fact_4546_dvd__minus__self,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.58        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.58          | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_minus_self
% 0.30/0.58  thf(fact_4547_eq__diff__iff,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.58       => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.58         => ( ( ( minus_minus_nat @ M @ K )
% 0.30/0.58              = ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.58            = ( M = N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_diff_iff
% 0.30/0.58  thf(fact_4548_le__diff__iff,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.58       => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.58         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.58            = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_iff
% 0.30/0.58  thf(fact_4549_Nat_Odiff__diff__eq,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.58       => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.58         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.58            = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.diff_diff_eq
% 0.30/0.58  thf(fact_4550_diff__le__mono,axiom,
% 0.30/0.58      ! [M: nat,N2: nat,L: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.58       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_mono
% 0.30/0.58  thf(fact_4551_diff__le__self,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_self
% 0.30/0.58  thf(fact_4552_le__diff__iff_H,axiom,
% 0.30/0.58      ! [A: nat,C: nat,B: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ C )
% 0.30/0.58       => ( ( ord_less_eq_nat @ B @ C )
% 0.30/0.58         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 0.30/0.58            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_iff'
% 0.30/0.58  thf(fact_4553_diff__le__mono2,axiom,
% 0.30/0.58      ! [M: nat,N2: nat,L: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.58       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_mono2
% 0.30/0.58  thf(fact_4554_dvd__diffD,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.58       => ( ( dvd_dvd_nat @ K @ N2 )
% 0.30/0.58         => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.58           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diffD
% 0.30/0.58  thf(fact_4555_dvd__diffD1,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.58       => ( ( dvd_dvd_nat @ K @ M )
% 0.30/0.58         => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.58           => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_diffD1
% 0.30/0.58  thf(fact_4556_less__eq__dvd__minus,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.58       => ( ( dvd_dvd_nat @ M @ N2 )
% 0.30/0.58          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_eq_dvd_minus
% 0.30/0.58  thf(fact_4557_Nat_Odiff__cancel,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 0.30/0.58        = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.diff_cancel
% 0.30/0.58  thf(fact_4558_diff__cancel2,axiom,
% 0.30/0.58      ! [M: nat,K: nat,N2: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.58        = ( minus_minus_nat @ M @ N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_cancel2
% 0.30/0.58  thf(fact_4559_diff__add__inverse,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
% 0.30/0.58        = M ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_inverse
% 0.30/0.58  thf(fact_4560_diff__add__inverse2,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
% 0.30/0.58        = M ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_inverse2
% 0.30/0.58  thf(fact_4561_diff__mult__distrib2,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.58        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_mult_distrib2
% 0.30/0.58  thf(fact_4562_diff__mult__distrib,axiom,
% 0.30/0.58      ! [M: nat,N2: nat,K: nat] :
% 0.30/0.58        ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 0.30/0.58        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_mult_distrib
% 0.30/0.58  thf(fact_4563_bezout1__nat,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58      ? [D4: nat,X4: nat,Y5: nat] :
% 0.30/0.58        ( ( dvd_dvd_nat @ D4 @ A )
% 0.30/0.58        & ( dvd_dvd_nat @ D4 @ B )
% 0.30/0.58        & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
% 0.30/0.58            = D4 )
% 0.30/0.58          | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
% 0.30/0.58            = D4 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % bezout1_nat
% 0.30/0.58  thf(fact_4564_eucl__rel__int__by0,axiom,
% 0.30/0.58      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eucl_rel_int_by0
% 0.30/0.58  thf(fact_4565_div__int__unique,axiom,
% 0.30/0.58      ! [K: int,L: int,Q4: int,R2: int] :
% 0.30/0.58        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.58       => ( ( divide_divide_int @ K @ L )
% 0.30/0.58          = Q4 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % div_int_unique
% 0.30/0.58  thf(fact_4566_zero__less__eq__of__bool,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zero_less_eq_of_bool
% 0.30/0.58  thf(fact_4567_zero__less__eq__of__bool,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zero_less_eq_of_bool
% 0.30/0.58  thf(fact_4568_zero__less__eq__of__bool,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zero_less_eq_of_bool
% 0.30/0.58  thf(fact_4569_zero__less__eq__of__bool,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zero_less_eq_of_bool
% 0.30/0.58  thf(fact_4570_of__bool__less__eq__one,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_less_eq_one
% 0.30/0.58  thf(fact_4571_of__bool__less__eq__one,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_less_eq_one
% 0.30/0.58  thf(fact_4572_of__bool__less__eq__one,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_less_eq_one
% 0.30/0.58  thf(fact_4573_of__bool__less__eq__one,axiom,
% 0.30/0.58      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_less_eq_one
% 0.30/0.58  thf(fact_4574_split__of__bool__asm,axiom,
% 0.30/0.58      ! [P: complex > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.30/0.58        = ( ~ ( ( P4
% 0.30/0.58                & ~ ( P @ one_one_complex ) )
% 0.30/0.58              | ( ~ P4
% 0.30/0.58                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool_asm
% 0.30/0.58  thf(fact_4575_split__of__bool__asm,axiom,
% 0.30/0.58      ! [P: real > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.30/0.58        = ( ~ ( ( P4
% 0.30/0.58                & ~ ( P @ one_one_real ) )
% 0.30/0.58              | ( ~ P4
% 0.30/0.58                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool_asm
% 0.30/0.58  thf(fact_4576_split__of__bool__asm,axiom,
% 0.30/0.58      ! [P: rat > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.30/0.58        = ( ~ ( ( P4
% 0.30/0.58                & ~ ( P @ one_one_rat ) )
% 0.30/0.58              | ( ~ P4
% 0.30/0.58                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool_asm
% 0.30/0.58  thf(fact_4577_split__of__bool__asm,axiom,
% 0.30/0.58      ! [P: nat > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.30/0.58        = ( ~ ( ( P4
% 0.30/0.58                & ~ ( P @ one_one_nat ) )
% 0.30/0.58              | ( ~ P4
% 0.30/0.58                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool_asm
% 0.30/0.58  thf(fact_4578_split__of__bool__asm,axiom,
% 0.30/0.58      ! [P: int > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.30/0.58        = ( ~ ( ( P4
% 0.30/0.58                & ~ ( P @ one_one_int ) )
% 0.30/0.58              | ( ~ P4
% 0.30/0.58                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool_asm
% 0.30/0.58  thf(fact_4579_split__of__bool,axiom,
% 0.30/0.58      ! [P: complex > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 0.30/0.58        = ( ( P4
% 0.30/0.58           => ( P @ one_one_complex ) )
% 0.30/0.58          & ( ~ P4
% 0.30/0.58           => ( P @ zero_zero_complex ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool
% 0.30/0.58  thf(fact_4580_split__of__bool,axiom,
% 0.30/0.58      ! [P: real > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 0.30/0.58        = ( ( P4
% 0.30/0.58           => ( P @ one_one_real ) )
% 0.30/0.58          & ( ~ P4
% 0.30/0.58           => ( P @ zero_zero_real ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool
% 0.30/0.58  thf(fact_4581_split__of__bool,axiom,
% 0.30/0.58      ! [P: rat > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 0.30/0.58        = ( ( P4
% 0.30/0.58           => ( P @ one_one_rat ) )
% 0.30/0.58          & ( ~ P4
% 0.30/0.58           => ( P @ zero_zero_rat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool
% 0.30/0.58  thf(fact_4582_split__of__bool,axiom,
% 0.30/0.58      ! [P: nat > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 0.30/0.58        = ( ( P4
% 0.30/0.58           => ( P @ one_one_nat ) )
% 0.30/0.58          & ( ~ P4
% 0.30/0.58           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool
% 0.30/0.58  thf(fact_4583_split__of__bool,axiom,
% 0.30/0.58      ! [P: int > $o,P4: $o] :
% 0.30/0.58        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 0.30/0.58        = ( ( P4
% 0.30/0.58           => ( P @ one_one_int ) )
% 0.30/0.58          & ( ~ P4
% 0.30/0.58           => ( P @ zero_zero_int ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % split_of_bool
% 0.30/0.58  thf(fact_4584_of__bool__def,axiom,
% 0.30/0.58      ( zero_n1201886186963655149omplex
% 0.30/0.58      = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_def
% 0.30/0.58  thf(fact_4585_of__bool__def,axiom,
% 0.30/0.58      ( zero_n3304061248610475627l_real
% 0.30/0.58      = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_def
% 0.30/0.58  thf(fact_4586_of__bool__def,axiom,
% 0.30/0.58      ( zero_n2052037380579107095ol_rat
% 0.30/0.58      = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_def
% 0.30/0.58  thf(fact_4587_of__bool__def,axiom,
% 0.30/0.58      ( zero_n2687167440665602831ol_nat
% 0.30/0.58      = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_def
% 0.30/0.58  thf(fact_4588_of__bool__def,axiom,
% 0.30/0.58      ( zero_n2684676970156552555ol_int
% 0.30/0.58      = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % of_bool_def
% 0.30/0.58  thf(fact_4589_le__iff__diff__le__0,axiom,
% 0.30/0.58      ( ord_less_eq_real
% 0.30/0.58      = ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_iff_diff_le_0
% 0.30/0.58  thf(fact_4590_le__iff__diff__le__0,axiom,
% 0.30/0.58      ( ord_less_eq_rat
% 0.30/0.58      = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_iff_diff_le_0
% 0.30/0.58  thf(fact_4591_le__iff__diff__le__0,axiom,
% 0.30/0.58      ( ord_less_eq_int
% 0.30/0.58      = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_iff_diff_le_0
% 0.30/0.58  thf(fact_4592_less__iff__diff__less__0,axiom,
% 0.30/0.58      ( ord_less_real
% 0.30/0.58      = ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_iff_diff_less_0
% 0.30/0.58  thf(fact_4593_less__iff__diff__less__0,axiom,
% 0.30/0.58      ( ord_less_rat
% 0.30/0.58      = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_iff_diff_less_0
% 0.30/0.58  thf(fact_4594_less__iff__diff__less__0,axiom,
% 0.30/0.58      ( ord_less_int
% 0.30/0.58      = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_iff_diff_less_0
% 0.30/0.58  thf(fact_4595_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58         => ( ( ( minus_minus_nat @ B @ A )
% 0.30/0.58              = C )
% 0.30/0.58            = ( B
% 0.30/0.58              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 0.30/0.58  thf(fact_4596_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.58          = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.add_diff_inverse
% 0.30/0.58  thf(fact_4597_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.diff_diff_right
% 0.30/0.58  thf(fact_4598_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 0.30/0.58          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
% 0.30/0.58  thf(fact_4599_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
% 0.30/0.58  thf(fact_4600_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 0.30/0.58          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.diff_add_assoc
% 0.30/0.58  thf(fact_4601_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.add_diff_assoc
% 0.30/0.58  thf(fact_4602_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 0.30/0.58          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_cancel_comm_monoid_diff_class.le_diff_conv2
% 0.30/0.58  thf(fact_4603_le__add__diff,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_add_diff
% 0.30/0.58  thf(fact_4604_diff__add,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 0.30/0.58          = B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add
% 0.30/0.58  thf(fact_4605_le__diff__eq,axiom,
% 0.30/0.58      ! [A: real,C: real,B: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.58        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_eq
% 0.30/0.58  thf(fact_4606_le__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,C: rat,B: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.58        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_eq
% 0.30/0.58  thf(fact_4607_le__diff__eq,axiom,
% 0.30/0.58      ! [A: int,C: int,B: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.58        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_eq
% 0.30/0.58  thf(fact_4608_diff__le__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_eq
% 0.30/0.58  thf(fact_4609_diff__le__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_eq
% 0.30/0.58  thf(fact_4610_diff__le__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_le_eq
% 0.30/0.58  thf(fact_4611_add__le__add__imp__diff__le,axiom,
% 0.30/0.58      ! [I: real,K: real,N2: real,J: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 0.30/0.58         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 0.30/0.58           => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 0.30/0.58             => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_add_imp_diff_le
% 0.30/0.58  thf(fact_4612_add__le__add__imp__diff__le,axiom,
% 0.30/0.58      ! [I: rat,K: rat,N2: rat,J: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.58         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 0.30/0.58           => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 0.30/0.58             => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_add_imp_diff_le
% 0.30/0.58  thf(fact_4613_add__le__add__imp__diff__le,axiom,
% 0.30/0.58      ! [I: nat,K: nat,N2: nat,J: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.58         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 0.30/0.58           => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.58             => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_add_imp_diff_le
% 0.30/0.58  thf(fact_4614_add__le__add__imp__diff__le,axiom,
% 0.30/0.58      ! [I: int,K: int,N2: int,J: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 0.30/0.58         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 0.30/0.58           => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 0.30/0.58             => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_add_imp_diff_le
% 0.30/0.58  thf(fact_4615_add__le__imp__le__diff,axiom,
% 0.30/0.58      ! [I: real,K: real,N2: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 0.30/0.58       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_imp_le_diff
% 0.30/0.58  thf(fact_4616_add__le__imp__le__diff,axiom,
% 0.30/0.58      ! [I: rat,K: rat,N2: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 0.30/0.58       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_imp_le_diff
% 0.30/0.58  thf(fact_4617_add__le__imp__le__diff,axiom,
% 0.30/0.58      ! [I: nat,K: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 0.30/0.58       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_imp_le_diff
% 0.30/0.58  thf(fact_4618_add__le__imp__le__diff,axiom,
% 0.30/0.58      ! [I: int,K: int,N2: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 0.30/0.58       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_le_imp_le_diff
% 0.30/0.58  thf(fact_4619_less__diff__eq,axiom,
% 0.30/0.58      ! [A: real,C: real,B: real] :
% 0.30/0.58        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 0.30/0.58        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_eq
% 0.30/0.58  thf(fact_4620_less__diff__eq,axiom,
% 0.30/0.58      ! [A: rat,C: rat,B: rat] :
% 0.30/0.58        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 0.30/0.58        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_eq
% 0.30/0.58  thf(fact_4621_less__diff__eq,axiom,
% 0.30/0.58      ! [A: int,C: int,B: int] :
% 0.30/0.58        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.30/0.58        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_eq
% 0.30/0.58  thf(fact_4622_diff__less__eq,axiom,
% 0.30/0.58      ! [A: real,B: real,C: real] :
% 0.30/0.58        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_eq
% 0.30/0.58  thf(fact_4623_diff__less__eq,axiom,
% 0.30/0.58      ! [A: rat,B: rat,C: rat] :
% 0.30/0.58        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_eq
% 0.30/0.58  thf(fact_4624_diff__less__eq,axiom,
% 0.30/0.58      ! [A: int,B: int,C: int] :
% 0.30/0.58        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.30/0.58        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_eq
% 0.30/0.58  thf(fact_4625_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.58      ! [A: real,B: real] :
% 0.30/0.58        ( ~ ( ord_less_real @ A @ B )
% 0.30/0.58       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 0.30/0.58          = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linordered_semidom_class.add_diff_inverse
% 0.30/0.58  thf(fact_4626_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.58      ! [A: rat,B: rat] :
% 0.30/0.58        ( ~ ( ord_less_rat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.58          = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linordered_semidom_class.add_diff_inverse
% 0.30/0.58  thf(fact_4627_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ A @ B )
% 0.30/0.58       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.58          = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linordered_semidom_class.add_diff_inverse
% 0.30/0.58  thf(fact_4628_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ~ ( ord_less_int @ A @ B )
% 0.30/0.58       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.30/0.58          = A ) ) ).
% 0.30/0.58  
% 0.30/0.58  % linordered_semidom_class.add_diff_inverse
% 0.30/0.58  thf(fact_4629_mult__diff__mult,axiom,
% 0.30/0.58      ! [X3: complex,Y3: complex,A: complex,B: complex] :
% 0.30/0.58        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B ) )
% 0.30/0.58        = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mult_diff_mult
% 0.30/0.58  thf(fact_4630_mult__diff__mult,axiom,
% 0.30/0.58      ! [X3: real,Y3: real,A: real,B: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B ) )
% 0.30/0.58        = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mult_diff_mult
% 0.30/0.58  thf(fact_4631_mult__diff__mult,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat,A: rat,B: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 0.30/0.58        = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mult_diff_mult
% 0.30/0.58  thf(fact_4632_mult__diff__mult,axiom,
% 0.30/0.58      ! [X3: int,Y3: int,A: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B ) )
% 0.30/0.58        = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mult_diff_mult
% 0.30/0.58  thf(fact_4633_square__diff__square__factored,axiom,
% 0.30/0.58      ! [X3: complex,Y3: complex] :
% 0.30/0.58        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
% 0.30/0.58        = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_square_factored
% 0.30/0.58  thf(fact_4634_square__diff__square__factored,axiom,
% 0.30/0.58      ! [X3: real,Y3: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 0.30/0.58        = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_square_factored
% 0.30/0.58  thf(fact_4635_square__diff__square__factored,axiom,
% 0.30/0.58      ! [X3: rat,Y3: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 0.30/0.58        = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_square_factored
% 0.30/0.58  thf(fact_4636_square__diff__square__factored,axiom,
% 0.30/0.58      ! [X3: int,Y3: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 0.30/0.58        = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_square_factored
% 0.30/0.58  thf(fact_4637_eq__add__iff2,axiom,
% 0.30/0.58      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 0.30/0.58        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( C
% 0.30/0.58          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff2
% 0.30/0.58  thf(fact_4638_eq__add__iff2,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( C
% 0.30/0.58          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff2
% 0.30/0.58  thf(fact_4639_eq__add__iff2,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( C
% 0.30/0.58          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff2
% 0.30/0.58  thf(fact_4640_eq__add__iff2,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( C
% 0.30/0.58          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff2
% 0.30/0.58  thf(fact_4641_eq__add__iff1,axiom,
% 0.30/0.58      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 0.30/0.58        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
% 0.30/0.58          = D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff1
% 0.30/0.58  thf(fact_4642_eq__add__iff1,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 0.30/0.58          = D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff1
% 0.30/0.58  thf(fact_4643_eq__add__iff1,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 0.30/0.58          = D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff1
% 0.30/0.58  thf(fact_4644_eq__add__iff1,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 0.30/0.58          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 0.30/0.58          = D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eq_add_iff1
% 0.30/0.58  thf(fact_4645_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ A )
% 0.30/0.58       => ( ( A != bot_bo228742789529271731at_nat )
% 0.30/0.58         => ~ ! [A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                ( ? [A4: produc3843707927480180839at_nat] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert9069300056098147895at_nat @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite4343798906461161616at_nat @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4646_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_complex] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ A )
% 0.30/0.58       => ( ( A != bot_bot_set_complex )
% 0.30/0.58         => ~ ! [A7: set_complex] :
% 0.30/0.58                ( ? [A4: complex] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert_complex @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite3207457112153483333omplex @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4647_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ A )
% 0.30/0.58       => ( ( A != bot_bo2099793752762293965at_nat )
% 0.30/0.58         => ~ ! [A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                ( ? [A4: product_prod_nat_nat] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert8211810215607154385at_nat @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4648_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_real] :
% 0.30/0.58        ( ( finite_finite_real @ A )
% 0.30/0.58       => ( ( A != bot_bot_set_real )
% 0.30/0.58         => ~ ! [A7: set_real] :
% 0.30/0.58                ( ? [A4: real] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert_real @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite_finite_real @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4649_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_nat] :
% 0.30/0.58        ( ( finite_finite_nat @ A )
% 0.30/0.58       => ( ( A != bot_bot_set_nat )
% 0.30/0.58         => ~ ! [A7: set_nat] :
% 0.30/0.58                ( ? [A4: nat] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert_nat @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4650_finite_Ocases,axiom,
% 0.30/0.58      ! [A: set_int] :
% 0.30/0.58        ( ( finite_finite_int @ A )
% 0.30/0.58       => ( ( A != bot_bot_set_int )
% 0.30/0.58         => ~ ! [A7: set_int] :
% 0.30/0.58                ( ? [A4: int] :
% 0.30/0.58                    ( A
% 0.30/0.58                    = ( insert_int @ A4 @ A7 ) )
% 0.30/0.58               => ~ ( finite_finite_int @ A7 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.cases
% 0.30/0.58  thf(fact_4651_finite_Osimps,axiom,
% 0.30/0.58      ( finite4343798906461161616at_nat
% 0.30/0.58      = ( ^ [A3: set_Pr4329608150637261639at_nat] :
% 0.30/0.58            ( ( A3 = bot_bo228742789529271731at_nat )
% 0.30/0.58            | ? [A5: set_Pr4329608150637261639at_nat,B3: produc3843707927480180839at_nat] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert9069300056098147895at_nat @ B3 @ A5 ) )
% 0.30/0.58                & ( finite4343798906461161616at_nat @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4652_finite_Osimps,axiom,
% 0.30/0.58      ( finite3207457112153483333omplex
% 0.30/0.58      = ( ^ [A3: set_complex] :
% 0.30/0.58            ( ( A3 = bot_bot_set_complex )
% 0.30/0.58            | ? [A5: set_complex,B3: complex] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert_complex @ B3 @ A5 ) )
% 0.30/0.58                & ( finite3207457112153483333omplex @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4653_finite_Osimps,axiom,
% 0.30/0.58      ( finite6177210948735845034at_nat
% 0.30/0.58      = ( ^ [A3: set_Pr1261947904930325089at_nat] :
% 0.30/0.58            ( ( A3 = bot_bo2099793752762293965at_nat )
% 0.30/0.58            | ? [A5: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert8211810215607154385at_nat @ B3 @ A5 ) )
% 0.30/0.58                & ( finite6177210948735845034at_nat @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4654_finite_Osimps,axiom,
% 0.30/0.58      ( finite_finite_real
% 0.30/0.58      = ( ^ [A3: set_real] :
% 0.30/0.58            ( ( A3 = bot_bot_set_real )
% 0.30/0.58            | ? [A5: set_real,B3: real] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert_real @ B3 @ A5 ) )
% 0.30/0.58                & ( finite_finite_real @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4655_finite_Osimps,axiom,
% 0.30/0.58      ( finite_finite_nat
% 0.30/0.58      = ( ^ [A3: set_nat] :
% 0.30/0.58            ( ( A3 = bot_bot_set_nat )
% 0.30/0.58            | ? [A5: set_nat,B3: nat] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert_nat @ B3 @ A5 ) )
% 0.30/0.58                & ( finite_finite_nat @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4656_finite_Osimps,axiom,
% 0.30/0.58      ( finite_finite_int
% 0.30/0.58      = ( ^ [A3: set_int] :
% 0.30/0.58            ( ( A3 = bot_bot_set_int )
% 0.30/0.58            | ? [A5: set_int,B3: int] :
% 0.30/0.58                ( ( A3
% 0.30/0.58                  = ( insert_int @ B3 @ A5 ) )
% 0.30/0.58                & ( finite_finite_int @ A5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite.simps
% 0.30/0.58  thf(fact_4657_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ F3 )
% 0.30/0.58       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.58         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                ( ( finite4343798906461161616at_nat @ F4 )
% 0.30/0.58               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4658_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.58        ( ( finite1152437895449049373et_nat @ F3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_set_nat )
% 0.30/0.58         => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.30/0.58                ( ( finite1152437895449049373et_nat @ F4 )
% 0.30/0.58               => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4659_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_complex,P: set_complex > $o] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ F3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex,F4: set_complex] :
% 0.30/0.58                ( ( finite3207457112153483333omplex @ F4 )
% 0.30/0.58               => ( ~ ( member_complex @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4660_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ F3 )
% 0.30/0.58       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.58         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                ( ( finite6177210948735845034at_nat @ F4 )
% 0.30/0.58               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4661_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ F3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real,F4: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ F4 )
% 0.30/0.58               => ( ~ ( member_real @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4662_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ F3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [X4: nat,F4: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ F4 )
% 0.30/0.58               => ( ~ ( member_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4663_finite__induct,axiom,
% 0.30/0.58      ! [F3: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ F3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [X4: int,F4: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ F4 )
% 0.30/0.58               => ( ~ ( member_int @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ F3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_induct
% 0.30/0.58  thf(fact_4664_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bo228742789529271731at_nat )
% 0.30/0.58         => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58           => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bo228742789529271731at_nat )
% 0.30/0.58                   => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4665_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.58        ( ( finite1152437895449049373et_nat @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bot_set_set_nat )
% 0.30/0.58         => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
% 0.30/0.58           => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.30/0.58                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bot_set_set_nat )
% 0.30/0.58                   => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4666_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_complex,P: set_complex > $o] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
% 0.30/0.58           => ( ! [X4: complex,F4: set_complex] :
% 0.30/0.58                  ( ( finite3207457112153483333omplex @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bot_set_complex )
% 0.30/0.58                   => ( ~ ( member_complex @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4667_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bo2099793752762293965at_nat )
% 0.30/0.58         => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58           => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 0.30/0.58                   => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4668_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
% 0.30/0.58           => ( ! [X4: real,F4: set_real] :
% 0.30/0.58                  ( ( finite_finite_real @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bot_set_real )
% 0.30/0.58                   => ( ~ ( member_real @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4669_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bot_set_nat )
% 0.30/0.58         => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
% 0.30/0.58           => ( ! [X4: nat,F4: set_nat] :
% 0.30/0.58                  ( ( finite_finite_nat @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bot_set_nat )
% 0.30/0.58                   => ( ~ ( member_nat @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4670_finite__ne__induct,axiom,
% 0.30/0.58      ! [F3: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ F3 )
% 0.30/0.58       => ( ( F3 != bot_bot_set_int )
% 0.30/0.58         => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
% 0.30/0.58           => ( ! [X4: int,F4: set_int] :
% 0.30/0.58                  ( ( finite_finite_int @ F4 )
% 0.30/0.58                 => ( ( F4 != bot_bot_set_int )
% 0.30/0.58                   => ( ~ ( member_int @ X4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ne_induct
% 0.30/0.58  thf(fact_4671_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.58        ( ! [A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.58            ( ~ ( finite4343798906461161616at_nat @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.58         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                ( ( finite4343798906461161616at_nat @ F4 )
% 0.30/0.58               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4672_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_set_nat > $o,A2: set_set_nat] :
% 0.30/0.58        ( ! [A7: set_set_nat] :
% 0.30/0.58            ( ~ ( finite1152437895449049373et_nat @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bot_set_set_nat )
% 0.30/0.58         => ( ! [X4: set_nat,F4: set_set_nat] :
% 0.30/0.58                ( ( finite1152437895449049373et_nat @ F4 )
% 0.30/0.58               => ( ~ ( member_set_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4673_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_complex > $o,A2: set_complex] :
% 0.30/0.58        ( ! [A7: set_complex] :
% 0.30/0.58            ( ~ ( finite3207457112153483333omplex @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex,F4: set_complex] :
% 0.30/0.58                ( ( finite3207457112153483333omplex @ F4 )
% 0.30/0.58               => ( ~ ( member_complex @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4674_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.58        ( ! [A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.58            ( ~ ( finite6177210948735845034at_nat @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.58         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                ( ( finite6177210948735845034at_nat @ F4 )
% 0.30/0.58               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4675_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_real > $o,A2: set_real] :
% 0.30/0.58        ( ! [A7: set_real] :
% 0.30/0.58            ( ~ ( finite_finite_real @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real,F4: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ F4 )
% 0.30/0.58               => ( ~ ( member_real @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4676_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_nat > $o,A2: set_nat] :
% 0.30/0.58        ( ! [A7: set_nat] :
% 0.30/0.58            ( ~ ( finite_finite_nat @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [X4: nat,F4: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ F4 )
% 0.30/0.58               => ( ~ ( member_nat @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4677_infinite__finite__induct,axiom,
% 0.30/0.58      ! [P: set_int > $o,A2: set_int] :
% 0.30/0.58        ( ! [A7: set_int] :
% 0.30/0.58            ( ~ ( finite_finite_int @ A7 )
% 0.30/0.58           => ( P @ A7 ) )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [X4: int,F4: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ F4 )
% 0.30/0.58               => ( ~ ( member_int @ X4 @ F4 )
% 0.30/0.58                 => ( ( P @ F4 )
% 0.30/0.58                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % infinite_finite_induct
% 0.30/0.58  thf(fact_4678_Euclidean__Division_Opos__mod__bound,axiom,
% 0.30/0.58      ! [L: int,K: int] :
% 0.30/0.58        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.58       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Euclidean_Division.pos_mod_bound
% 0.30/0.58  thf(fact_4679_neg__mod__bound,axiom,
% 0.30/0.58      ! [L: int,K: int] :
% 0.30/0.58        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.58       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % neg_mod_bound
% 0.30/0.58  thf(fact_4680_subset__singleton__iff,axiom,
% 0.30/0.58      ! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58        = ( ( X7 = bot_bo228742789529271731at_nat )
% 0.30/0.58          | ( X7
% 0.30/0.58            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singleton_iff
% 0.30/0.58  thf(fact_4681_subset__singleton__iff,axiom,
% 0.30/0.58      ! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.30/0.58        ( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58        = ( ( X7 = bot_bo2099793752762293965at_nat )
% 0.30/0.58          | ( X7
% 0.30/0.58            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singleton_iff
% 0.30/0.58  thf(fact_4682_subset__singleton__iff,axiom,
% 0.30/0.58      ! [X7: set_real,A: real] :
% 0.30/0.58        ( ( ord_less_eq_set_real @ X7 @ ( insert_real @ A @ bot_bot_set_real ) )
% 0.30/0.58        = ( ( X7 = bot_bot_set_real )
% 0.30/0.58          | ( X7
% 0.30/0.58            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singleton_iff
% 0.30/0.58  thf(fact_4683_subset__singleton__iff,axiom,
% 0.30/0.58      ! [X7: set_nat,A: nat] :
% 0.30/0.58        ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 0.30/0.58        = ( ( X7 = bot_bot_set_nat )
% 0.30/0.58          | ( X7
% 0.30/0.58            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singleton_iff
% 0.30/0.58  thf(fact_4684_subset__singleton__iff,axiom,
% 0.30/0.58      ! [X7: set_int,A: int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
% 0.30/0.58        = ( ( X7 = bot_bot_set_int )
% 0.30/0.58          | ( X7
% 0.30/0.58            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singleton_iff
% 0.30/0.58  thf(fact_4685_subset__singletonD,axiom,
% 0.30/0.58      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
% 0.30/0.58        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 0.30/0.58       => ( ( A2 = bot_bo228742789529271731at_nat )
% 0.30/0.58          | ( A2
% 0.30/0.58            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singletonD
% 0.30/0.58  thf(fact_4686_subset__singletonD,axiom,
% 0.30/0.58      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
% 0.30/0.58        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.58       => ( ( A2 = bot_bo2099793752762293965at_nat )
% 0.30/0.58          | ( A2
% 0.30/0.58            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singletonD
% 0.30/0.58  thf(fact_4687_subset__singletonD,axiom,
% 0.30/0.58      ! [A2: set_real,X3: real] :
% 0.30/0.58        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 0.30/0.58       => ( ( A2 = bot_bot_set_real )
% 0.30/0.58          | ( A2
% 0.30/0.58            = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singletonD
% 0.30/0.58  thf(fact_4688_subset__singletonD,axiom,
% 0.30/0.58      ! [A2: set_nat,X3: nat] :
% 0.30/0.58        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.30/0.58       => ( ( A2 = bot_bot_set_nat )
% 0.30/0.58          | ( A2
% 0.30/0.58            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singletonD
% 0.30/0.58  thf(fact_4689_subset__singletonD,axiom,
% 0.30/0.58      ! [A2: set_int,X3: int] :
% 0.30/0.58        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 0.30/0.58       => ( ( A2 = bot_bot_set_int )
% 0.30/0.58          | ( A2
% 0.30/0.58            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % subset_singletonD
% 0.30/0.58  thf(fact_4690_dvd__minus__mod,axiom,
% 0.30/0.58      ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_minus_mod
% 0.30/0.58  thf(fact_4691_dvd__minus__mod,axiom,
% 0.30/0.58      ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_minus_mod
% 0.30/0.58  thf(fact_4692_dvd__minus__mod,axiom,
% 0.30/0.58      ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_minus_mod
% 0.30/0.58  thf(fact_4693_diff__less__Suc,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_Suc
% 0.30/0.58  thf(fact_4694_Suc__diff__Suc,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_nat @ N2 @ M )
% 0.30/0.58       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.58          = ( minus_minus_nat @ M @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Suc_diff_Suc
% 0.30/0.58  thf(fact_4695_diff__less,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.58       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.58         => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less
% 0.30/0.58  thf(fact_4696_Suc__diff__le,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.58       => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 0.30/0.58          = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Suc_diff_le
% 0.30/0.58  thf(fact_4697_diff__less__mono,axiom,
% 0.30/0.58      ! [A: nat,B: nat,C: nat] :
% 0.30/0.58        ( ( ord_less_nat @ A @ B )
% 0.30/0.58       => ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.58         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_less_mono
% 0.30/0.58  thf(fact_4698_less__diff__iff,axiom,
% 0.30/0.58      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.58       => ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.58         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 0.30/0.58            = ( ord_less_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_iff
% 0.30/0.58  thf(fact_4699_diff__add__0,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.58        = zero_zero_nat ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_add_0
% 0.30/0.58  thf(fact_4700_add__diff__inverse__nat,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.58       => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.58          = M ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_diff_inverse_nat
% 0.30/0.58  thf(fact_4701_less__diff__conv,axiom,
% 0.30/0.58      ! [I: nat,J: nat,K: nat] :
% 0.30/0.58        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.58        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_conv
% 0.30/0.58  thf(fact_4702_Nat_Ole__imp__diff__is__add,axiom,
% 0.30/0.58      ! [I: nat,J: nat,K: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.58       => ( ( ( minus_minus_nat @ J @ I )
% 0.30/0.58            = K )
% 0.30/0.58          = ( J
% 0.30/0.58            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.le_imp_diff_is_add
% 0.30/0.58  thf(fact_4703_Nat_Odiff__add__assoc2,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 0.30/0.58          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.diff_add_assoc2
% 0.30/0.58  thf(fact_4704_Nat_Odiff__add__assoc,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 0.30/0.58          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.diff_add_assoc
% 0.30/0.58  thf(fact_4705_Nat_Ole__diff__conv2,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 0.30/0.58          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Nat.le_diff_conv2
% 0.30/0.58  thf(fact_4706_le__diff__conv,axiom,
% 0.30/0.58      ! [J: nat,K: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.58        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_diff_conv
% 0.30/0.58  thf(fact_4707_diff__Suc__eq__diff__pred,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 0.30/0.58        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_Suc_eq_diff_pred
% 0.30/0.58  thf(fact_4708_mod__geq,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.58       => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.58          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_geq
% 0.30/0.58  thf(fact_4709_mod__if,axiom,
% 0.30/0.58      ( modulo_modulo_nat
% 0.30/0.58      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_if
% 0.30/0.58  thf(fact_4710_le__mod__geq,axiom,
% 0.30/0.58      ! [N2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.58       => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.58          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % le_mod_geq
% 0.30/0.58  thf(fact_4711_mod__eq__dvd__iff__nat,axiom,
% 0.30/0.58      ! [N2: nat,M: nat,Q4: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.58       => ( ( ( modulo_modulo_nat @ M @ Q4 )
% 0.30/0.58            = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 0.30/0.58          = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_eq_dvd_iff_nat
% 0.30/0.58  thf(fact_4712_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff2
% 0.30/0.58  thf(fact_4713_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff2
% 0.30/0.58  thf(fact_4714_ordered__ring__class_Ole__add__iff2,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff2
% 0.30/0.58  thf(fact_4715_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff1
% 0.30/0.58  thf(fact_4716_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff1
% 0.30/0.58  thf(fact_4717_ordered__ring__class_Ole__add__iff1,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % ordered_ring_class.le_add_iff1
% 0.30/0.58  thf(fact_4718_less__add__iff2,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff2
% 0.30/0.58  thf(fact_4719_less__add__iff2,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff2
% 0.30/0.58  thf(fact_4720_less__add__iff2,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff2
% 0.30/0.58  thf(fact_4721_less__add__iff1,axiom,
% 0.30/0.58      ! [A: real,E2: real,C: real,B: real,D: real] :
% 0.30/0.58        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff1
% 0.30/0.58  thf(fact_4722_less__add__iff1,axiom,
% 0.30/0.58      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 0.30/0.58        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff1
% 0.30/0.58  thf(fact_4723_less__add__iff1,axiom,
% 0.30/0.58      ! [A: int,E2: int,C: int,B: int,D: int] :
% 0.30/0.58        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 0.30/0.58        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_add_iff1
% 0.30/0.58  thf(fact_4724_divide__diff__eq__iff,axiom,
% 0.30/0.58      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.30/0.58        ( ( Z2 != zero_zero_complex )
% 0.30/0.58       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
% 0.30/0.58          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % divide_diff_eq_iff
% 0.30/0.58  thf(fact_4725_divide__diff__eq__iff,axiom,
% 0.30/0.58      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.58        ( ( Z2 != zero_zero_real )
% 0.30/0.58       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
% 0.30/0.58          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % divide_diff_eq_iff
% 0.30/0.58  thf(fact_4726_divide__diff__eq__iff,axiom,
% 0.30/0.58      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.58        ( ( Z2 != zero_zero_rat )
% 0.30/0.58       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
% 0.30/0.58          = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % divide_diff_eq_iff
% 0.30/0.58  thf(fact_4727_diff__divide__eq__iff,axiom,
% 0.30/0.58      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.30/0.58        ( ( Z2 != zero_zero_complex )
% 0.30/0.58       => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
% 0.30/0.58          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_divide_eq_iff
% 0.30/0.58  thf(fact_4728_diff__divide__eq__iff,axiom,
% 0.30/0.58      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.58        ( ( Z2 != zero_zero_real )
% 0.30/0.58       => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
% 0.30/0.58          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_divide_eq_iff
% 0.30/0.58  thf(fact_4729_diff__divide__eq__iff,axiom,
% 0.30/0.58      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.58        ( ( Z2 != zero_zero_rat )
% 0.30/0.58       => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
% 0.30/0.58          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_divide_eq_iff
% 0.30/0.58  thf(fact_4730_diff__frac__eq,axiom,
% 0.30/0.58      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 0.30/0.58        ( ( Y3 != zero_zero_complex )
% 0.30/0.58       => ( ( Z2 != zero_zero_complex )
% 0.30/0.58         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 0.30/0.58            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_frac_eq
% 0.30/0.58  thf(fact_4731_diff__frac__eq,axiom,
% 0.30/0.58      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 0.30/0.58        ( ( Y3 != zero_zero_real )
% 0.30/0.58       => ( ( Z2 != zero_zero_real )
% 0.30/0.58         => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.30/0.58            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_frac_eq
% 0.30/0.58  thf(fact_4732_diff__frac__eq,axiom,
% 0.30/0.58      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 0.30/0.58        ( ( Y3 != zero_zero_rat )
% 0.30/0.58       => ( ( Z2 != zero_zero_rat )
% 0.30/0.58         => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.30/0.58            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_frac_eq
% 0.30/0.58  thf(fact_4733_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.58      ! [Z2: complex,A: complex,B: complex] :
% 0.30/0.58        ( ( ( Z2 = zero_zero_complex )
% 0.30/0.58         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.30/0.58            = A ) )
% 0.30/0.58        & ( ( Z2 != zero_zero_complex )
% 0.30/0.58         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 0.30/0.58            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_divide_eq_if_simps(4)
% 0.30/0.58  thf(fact_4734_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.58      ! [Z2: real,A: real,B: real] :
% 0.30/0.58        ( ( ( Z2 = zero_zero_real )
% 0.30/0.58         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.30/0.58            = A ) )
% 0.30/0.58        & ( ( Z2 != zero_zero_real )
% 0.30/0.58         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 0.30/0.58            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_divide_eq_if_simps(4)
% 0.30/0.58  thf(fact_4735_add__divide__eq__if__simps_I4_J,axiom,
% 0.30/0.58      ! [Z2: rat,A: rat,B: rat] :
% 0.30/0.58        ( ( ( Z2 = zero_zero_rat )
% 0.30/0.58         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.30/0.58            = A ) )
% 0.30/0.58        & ( ( Z2 != zero_zero_rat )
% 0.30/0.58         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 0.30/0.58            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % add_divide_eq_if_simps(4)
% 0.30/0.58  thf(fact_4736_square__diff__one__factored,axiom,
% 0.30/0.58      ! [X3: complex] :
% 0.30/0.58        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
% 0.30/0.58        = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_one_factored
% 0.30/0.58  thf(fact_4737_square__diff__one__factored,axiom,
% 0.30/0.58      ! [X3: real] :
% 0.30/0.58        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
% 0.30/0.58        = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_one_factored
% 0.30/0.58  thf(fact_4738_square__diff__one__factored,axiom,
% 0.30/0.58      ! [X3: rat] :
% 0.30/0.58        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
% 0.30/0.58        = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_one_factored
% 0.30/0.58  thf(fact_4739_square__diff__one__factored,axiom,
% 0.30/0.58      ! [X3: int] :
% 0.30/0.58        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
% 0.30/0.58        = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % square_diff_one_factored
% 0.30/0.58  thf(fact_4740_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex,S5: set_complex] :
% 0.30/0.58                ( ( finite3207457112153483333omplex @ S5 )
% 0.30/0.58               => ( ! [Y6: complex] :
% 0.30/0.58                      ( ( member_complex @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4741_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_real,P: set_real > $o,F: real > rat] :
% 0.30/0.58        ( ( finite_finite_real @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real,S5: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ S5 )
% 0.30/0.58               => ( ! [Y6: real] :
% 0.30/0.58                      ( ( member_real @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4742_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
% 0.30/0.58        ( ( finite_finite_nat @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [X4: nat,S5: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ S5 )
% 0.30/0.58               => ( ! [Y6: nat] :
% 0.30/0.58                      ( ( member_nat @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4743_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_int,P: set_int > $o,F: int > rat] :
% 0.30/0.58        ( ( finite_finite_int @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [X4: int,S5: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ S5 )
% 0.30/0.58               => ( ! [Y6: int] :
% 0.30/0.58                      ( ( member_int @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4744_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex,S5: set_complex] :
% 0.30/0.58                ( ( finite3207457112153483333omplex @ S5 )
% 0.30/0.58               => ( ! [Y6: complex] :
% 0.30/0.58                      ( ( member_complex @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4745_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_real,P: set_real > $o,F: real > num] :
% 0.30/0.58        ( ( finite_finite_real @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real,S5: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ S5 )
% 0.30/0.58               => ( ! [Y6: real] :
% 0.30/0.58                      ( ( member_real @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4746_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
% 0.30/0.58        ( ( finite_finite_nat @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [X4: nat,S5: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ S5 )
% 0.30/0.58               => ( ! [Y6: nat] :
% 0.30/0.58                      ( ( member_nat @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4747_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_int,P: set_int > $o,F: int > num] :
% 0.30/0.58        ( ( finite_finite_int @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [X4: int,S5: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ S5 )
% 0.30/0.58               => ( ! [Y6: int] :
% 0.30/0.58                      ( ( member_int @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4748_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_complex,P: set_complex > $o,F: complex > nat] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_complex )
% 0.30/0.58         => ( ! [X4: complex,S5: set_complex] :
% 0.30/0.58                ( ( finite3207457112153483333omplex @ S5 )
% 0.30/0.58               => ( ! [Y6: complex] :
% 0.30/0.58                      ( ( member_complex @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4749_finite__ranking__induct,axiom,
% 0.30/0.58      ! [S3: set_real,P: set_real > $o,F: real > nat] :
% 0.30/0.58        ( ( finite_finite_real @ S3 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [X4: real,S5: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ S5 )
% 0.30/0.58               => ( ! [Y6: real] :
% 0.30/0.58                      ( ( member_real @ Y6 @ S5 )
% 0.30/0.58                     => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 0.30/0.58                 => ( ( P @ S5 )
% 0.30/0.58                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 0.30/0.58           => ( P @ S3 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_ranking_induct
% 0.30/0.58  thf(fact_4750_finite__linorder__min__induct,axiom,
% 0.30/0.58      ! [A2: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [B4: real,A7: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ A7 )
% 0.30/0.58               => ( ! [X2: real] :
% 0.30/0.58                      ( ( member_real @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_real @ B4 @ X2 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_min_induct
% 0.30/0.58  thf(fact_4751_finite__linorder__min__induct,axiom,
% 0.30/0.58      ! [A2: set_rat,P: set_rat > $o] :
% 0.30/0.58        ( ( finite_finite_rat @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_rat )
% 0.30/0.58         => ( ! [B4: rat,A7: set_rat] :
% 0.30/0.58                ( ( finite_finite_rat @ A7 )
% 0.30/0.58               => ( ! [X2: rat] :
% 0.30/0.58                      ( ( member_rat @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_rat @ B4 @ X2 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_min_induct
% 0.30/0.58  thf(fact_4752_finite__linorder__min__induct,axiom,
% 0.30/0.58      ! [A2: set_num,P: set_num > $o] :
% 0.30/0.58        ( ( finite_finite_num @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_num )
% 0.30/0.58         => ( ! [B4: num,A7: set_num] :
% 0.30/0.58                ( ( finite_finite_num @ A7 )
% 0.30/0.58               => ( ! [X2: num] :
% 0.30/0.58                      ( ( member_num @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_num @ B4 @ X2 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_min_induct
% 0.30/0.58  thf(fact_4753_finite__linorder__min__induct,axiom,
% 0.30/0.58      ! [A2: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [B4: nat,A7: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ A7 )
% 0.30/0.58               => ( ! [X2: nat] :
% 0.30/0.58                      ( ( member_nat @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_nat @ B4 @ X2 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_min_induct
% 0.30/0.58  thf(fact_4754_finite__linorder__min__induct,axiom,
% 0.30/0.58      ! [A2: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [B4: int,A7: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ A7 )
% 0.30/0.58               => ( ! [X2: int] :
% 0.30/0.58                      ( ( member_int @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_int @ B4 @ X2 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_min_induct
% 0.30/0.58  thf(fact_4755_finite__linorder__max__induct,axiom,
% 0.30/0.58      ! [A2: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_real )
% 0.30/0.58         => ( ! [B4: real,A7: set_real] :
% 0.30/0.58                ( ( finite_finite_real @ A7 )
% 0.30/0.58               => ( ! [X2: real] :
% 0.30/0.58                      ( ( member_real @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_real @ X2 @ B4 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_max_induct
% 0.30/0.58  thf(fact_4756_finite__linorder__max__induct,axiom,
% 0.30/0.58      ! [A2: set_rat,P: set_rat > $o] :
% 0.30/0.58        ( ( finite_finite_rat @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_rat )
% 0.30/0.58         => ( ! [B4: rat,A7: set_rat] :
% 0.30/0.58                ( ( finite_finite_rat @ A7 )
% 0.30/0.58               => ( ! [X2: rat] :
% 0.30/0.58                      ( ( member_rat @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_rat @ X2 @ B4 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_max_induct
% 0.30/0.58  thf(fact_4757_finite__linorder__max__induct,axiom,
% 0.30/0.58      ! [A2: set_num,P: set_num > $o] :
% 0.30/0.58        ( ( finite_finite_num @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_num )
% 0.30/0.58         => ( ! [B4: num,A7: set_num] :
% 0.30/0.58                ( ( finite_finite_num @ A7 )
% 0.30/0.58               => ( ! [X2: num] :
% 0.30/0.58                      ( ( member_num @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_num @ X2 @ B4 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_max_induct
% 0.30/0.58  thf(fact_4758_finite__linorder__max__induct,axiom,
% 0.30/0.58      ! [A2: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_nat )
% 0.30/0.58         => ( ! [B4: nat,A7: set_nat] :
% 0.30/0.58                ( ( finite_finite_nat @ A7 )
% 0.30/0.58               => ( ! [X2: nat] :
% 0.30/0.58                      ( ( member_nat @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_nat @ X2 @ B4 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_max_induct
% 0.30/0.58  thf(fact_4759_finite__linorder__max__induct,axiom,
% 0.30/0.58      ! [A2: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ A2 )
% 0.30/0.58       => ( ( P @ bot_bot_set_int )
% 0.30/0.58         => ( ! [B4: int,A7: set_int] :
% 0.30/0.58                ( ( finite_finite_int @ A7 )
% 0.30/0.58               => ( ! [X2: int] :
% 0.30/0.58                      ( ( member_int @ X2 @ A7 )
% 0.30/0.58                     => ( ord_less_int @ X2 @ B4 ) )
% 0.30/0.58                 => ( ( P @ A7 )
% 0.30/0.58                   => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
% 0.30/0.58           => ( P @ A2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_linorder_max_induct
% 0.30/0.58  thf(fact_4760_inf__period_I4_J,axiom,
% 0.30/0.58      ! [D: code_integer,D6: code_integer,T: code_integer] :
% 0.30/0.58        ( ( dvd_dvd_Code_integer @ D @ D6 )
% 0.30/0.58       => ! [X2: code_integer,K4: code_integer] :
% 0.30/0.58            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) ) )
% 0.30/0.58            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(4)
% 0.30/0.58  thf(fact_4761_inf__period_I4_J,axiom,
% 0.30/0.58      ! [D: complex,D6: complex,T: complex] :
% 0.30/0.58        ( ( dvd_dvd_complex @ D @ D6 )
% 0.30/0.58       => ! [X2: complex,K4: complex] :
% 0.30/0.58            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) ) )
% 0.30/0.58            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(4)
% 0.30/0.58  thf(fact_4762_inf__period_I4_J,axiom,
% 0.30/0.58      ! [D: real,D6: real,T: real] :
% 0.30/0.58        ( ( dvd_dvd_real @ D @ D6 )
% 0.30/0.58       => ! [X2: real,K4: real] :
% 0.30/0.58            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
% 0.30/0.58            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(4)
% 0.30/0.58  thf(fact_4763_inf__period_I4_J,axiom,
% 0.30/0.58      ! [D: rat,D6: rat,T: rat] :
% 0.30/0.58        ( ( dvd_dvd_rat @ D @ D6 )
% 0.30/0.58       => ! [X2: rat,K4: rat] :
% 0.30/0.58            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
% 0.30/0.58            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(4)
% 0.30/0.58  thf(fact_4764_inf__period_I4_J,axiom,
% 0.30/0.58      ! [D: int,D6: int,T: int] :
% 0.30/0.58        ( ( dvd_dvd_int @ D @ D6 )
% 0.30/0.58       => ! [X2: int,K4: int] :
% 0.30/0.58            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
% 0.30/0.58            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(4)
% 0.30/0.58  thf(fact_4765_inf__period_I3_J,axiom,
% 0.30/0.58      ! [D: code_integer,D6: code_integer,T: code_integer] :
% 0.30/0.58        ( ( dvd_dvd_Code_integer @ D @ D6 )
% 0.30/0.58       => ! [X2: code_integer,K4: code_integer] :
% 0.30/0.58            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) )
% 0.30/0.58            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(3)
% 0.30/0.58  thf(fact_4766_inf__period_I3_J,axiom,
% 0.30/0.58      ! [D: complex,D6: complex,T: complex] :
% 0.30/0.58        ( ( dvd_dvd_complex @ D @ D6 )
% 0.30/0.58       => ! [X2: complex,K4: complex] :
% 0.30/0.58            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) )
% 0.30/0.58            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(3)
% 0.30/0.58  thf(fact_4767_inf__period_I3_J,axiom,
% 0.30/0.58      ! [D: real,D6: real,T: real] :
% 0.30/0.58        ( ( dvd_dvd_real @ D @ D6 )
% 0.30/0.58       => ! [X2: real,K4: real] :
% 0.30/0.58            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
% 0.30/0.58            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(3)
% 0.30/0.58  thf(fact_4768_inf__period_I3_J,axiom,
% 0.30/0.58      ! [D: rat,D6: rat,T: rat] :
% 0.30/0.58        ( ( dvd_dvd_rat @ D @ D6 )
% 0.30/0.58       => ! [X2: rat,K4: rat] :
% 0.30/0.58            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
% 0.30/0.58            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(3)
% 0.30/0.58  thf(fact_4769_inf__period_I3_J,axiom,
% 0.30/0.58      ! [D: int,D6: int,T: int] :
% 0.30/0.58        ( ( dvd_dvd_int @ D @ D6 )
% 0.30/0.58       => ! [X2: int,K4: int] :
% 0.30/0.58            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 0.30/0.58            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % inf_period(3)
% 0.30/0.58  thf(fact_4770_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ F3 )
% 0.30/0.58       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.58           => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.30/0.58                 => ( ( member8757157785044589968at_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4771_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.58        ( ( finite1152437895449049373et_nat @ F3 )
% 0.30/0.58       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_set_nat )
% 0.30/0.58           => ( ! [A4: set_nat,F4: set_set_nat] :
% 0.30/0.58                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.30/0.58                 => ( ( member_set_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member_set_nat @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4772_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ F3 )
% 0.30/0.58       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_complex )
% 0.30/0.58           => ( ! [A4: complex,F4: set_complex] :
% 0.30/0.58                  ( ( finite3207457112153483333omplex @ F4 )
% 0.30/0.58                 => ( ( member_complex @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member_complex @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4773_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ F3 )
% 0.30/0.58       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.58           => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.30/0.58                 => ( ( member8440522571783428010at_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4774_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_real )
% 0.30/0.58           => ( ! [A4: real,F4: set_real] :
% 0.30/0.58                  ( ( finite_finite_real @ F4 )
% 0.30/0.58                 => ( ( member_real @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member_real @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4775_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_nat )
% 0.30/0.58           => ( ! [A4: nat,F4: set_nat] :
% 0.30/0.58                  ( ( finite_finite_nat @ F4 )
% 0.30/0.58                 => ( ( member_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member_nat @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4776_finite__subset__induct,axiom,
% 0.30/0.58      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_int )
% 0.30/0.58           => ( ! [A4: int,F4: set_int] :
% 0.30/0.58                  ( ( finite_finite_int @ F4 )
% 0.30/0.58                 => ( ( member_int @ A4 @ A2 )
% 0.30/0.58                   => ( ~ ( member_int @ A4 @ F4 )
% 0.30/0.58                     => ( ( P @ F4 )
% 0.30/0.58                       => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct
% 0.30/0.58  thf(fact_4777_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.58        ( ( finite4343798906461161616at_nat @ F3 )
% 0.30/0.58       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.58           => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 0.30/0.58                  ( ( finite4343798906461161616at_nat @ F4 )
% 0.30/0.58                 => ( ( member8757157785044589968at_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4778_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.58        ( ( finite1152437895449049373et_nat @ F3 )
% 0.30/0.58       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_set_nat )
% 0.30/0.58           => ( ! [A4: set_nat,F4: set_set_nat] :
% 0.30/0.58                  ( ( finite1152437895449049373et_nat @ F4 )
% 0.30/0.58                 => ( ( member_set_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member_set_nat @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4779_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 0.30/0.58        ( ( finite3207457112153483333omplex @ F3 )
% 0.30/0.58       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_complex )
% 0.30/0.58           => ( ! [A4: complex,F4: set_complex] :
% 0.30/0.58                  ( ( finite3207457112153483333omplex @ F4 )
% 0.30/0.58                 => ( ( member_complex @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member_complex @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4780_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.58        ( ( finite6177210948735845034at_nat @ F3 )
% 0.30/0.58       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.58           => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 0.30/0.58                  ( ( finite6177210948735845034at_nat @ F4 )
% 0.30/0.58                 => ( ( member8440522571783428010at_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4781_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 0.30/0.58        ( ( finite_finite_real @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_real )
% 0.30/0.58           => ( ! [A4: real,F4: set_real] :
% 0.30/0.58                  ( ( finite_finite_real @ F4 )
% 0.30/0.58                 => ( ( member_real @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_less_eq_set_real @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member_real @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4782_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 0.30/0.58        ( ( finite_finite_nat @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_nat )
% 0.30/0.58           => ( ! [A4: nat,F4: set_nat] :
% 0.30/0.58                  ( ( finite_finite_nat @ F4 )
% 0.30/0.58                 => ( ( member_nat @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_less_eq_set_nat @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member_nat @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4783_finite__subset__induct_H,axiom,
% 0.30/0.58      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 0.30/0.58        ( ( finite_finite_int @ F3 )
% 0.30/0.58       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 0.30/0.58         => ( ( P @ bot_bot_set_int )
% 0.30/0.58           => ( ! [A4: int,F4: set_int] :
% 0.30/0.58                  ( ( finite_finite_int @ F4 )
% 0.30/0.58                 => ( ( member_int @ A4 @ A2 )
% 0.30/0.58                   => ( ( ord_less_eq_set_int @ F4 @ A2 )
% 0.30/0.58                     => ( ~ ( member_int @ A4 @ F4 )
% 0.30/0.58                       => ( ( P @ F4 )
% 0.30/0.58                         => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) ) )
% 0.30/0.58             => ( P @ F3 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % finite_subset_induct'
% 0.30/0.58  thf(fact_4784_minus__mult__div__eq__mod,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 0.30/0.58        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mult_div_eq_mod
% 0.30/0.58  thf(fact_4785_minus__mult__div__eq__mod,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 0.30/0.58        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mult_div_eq_mod
% 0.30/0.58  thf(fact_4786_minus__mult__div__eq__mod,axiom,
% 0.30/0.58      ! [A: code_integer,B: code_integer] :
% 0.30/0.58        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 0.30/0.58        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mult_div_eq_mod
% 0.30/0.58  thf(fact_4787_minus__mod__eq__mult__div,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.58        = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_mult_div
% 0.30/0.58  thf(fact_4788_minus__mod__eq__mult__div,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.58        = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_mult_div
% 0.30/0.58  thf(fact_4789_minus__mod__eq__mult__div,axiom,
% 0.30/0.58      ! [A: code_integer,B: code_integer] :
% 0.30/0.58        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.58        = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_mult_div
% 0.30/0.58  thf(fact_4790_minus__mod__eq__div__mult,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 0.30/0.58        = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_div_mult
% 0.30/0.58  thf(fact_4791_minus__mod__eq__div__mult,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.58        = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_div_mult
% 0.30/0.58  thf(fact_4792_minus__mod__eq__div__mult,axiom,
% 0.30/0.58      ! [A: code_integer,B: code_integer] :
% 0.30/0.58        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 0.30/0.58        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_mod_eq_div_mult
% 0.30/0.58  thf(fact_4793_minus__div__mult__eq__mod,axiom,
% 0.30/0.58      ! [A: nat,B: nat] :
% 0.30/0.58        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 0.30/0.58        = ( modulo_modulo_nat @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_div_mult_eq_mod
% 0.30/0.58  thf(fact_4794_minus__div__mult__eq__mod,axiom,
% 0.30/0.58      ! [A: int,B: int] :
% 0.30/0.58        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 0.30/0.58        = ( modulo_modulo_int @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_div_mult_eq_mod
% 0.30/0.58  thf(fact_4795_minus__div__mult__eq__mod,axiom,
% 0.30/0.58      ! [A: code_integer,B: code_integer] :
% 0.30/0.58        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 0.30/0.58        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 0.30/0.58  
% 0.30/0.58  % minus_div_mult_eq_mod
% 0.30/0.58  thf(fact_4796_eucl__rel__int__dividesI,axiom,
% 0.30/0.58      ! [L: int,K: int,Q4: int] :
% 0.30/0.58        ( ( L != zero_zero_int )
% 0.30/0.58       => ( ( K
% 0.30/0.58            = ( times_times_int @ Q4 @ L ) )
% 0.30/0.58         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % eucl_rel_int_dividesI
% 0.30/0.58  thf(fact_4797_Euclidean__Division_Opos__mod__sign,axiom,
% 0.30/0.58      ! [L: int,K: int] :
% 0.30/0.58        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.58       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % Euclidean_Division.pos_mod_sign
% 0.30/0.58  thf(fact_4798_neg__mod__sign,axiom,
% 0.30/0.58      ! [L: int,K: int] :
% 0.30/0.58        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.58       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 0.30/0.58  
% 0.30/0.58  % neg_mod_sign
% 0.30/0.58  thf(fact_4799_neg__mod__conj,axiom,
% 0.30/0.58      ! [B: int,A: int] :
% 0.30/0.58        ( ( ord_less_int @ B @ zero_zero_int )
% 0.30/0.58       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 0.30/0.58          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % neg_mod_conj
% 0.30/0.58  thf(fact_4800_pos__mod__conj,axiom,
% 0.30/0.58      ! [B: int,A: int] :
% 0.30/0.58        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.58       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 0.30/0.58          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % pos_mod_conj
% 0.30/0.58  thf(fact_4801_zmod__trivial__iff,axiom,
% 0.30/0.58      ! [I: int,K: int] :
% 0.30/0.58        ( ( ( modulo_modulo_int @ I @ K )
% 0.30/0.58          = I )
% 0.30/0.58        = ( ( K = zero_zero_int )
% 0.30/0.58          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.58            & ( ord_less_int @ I @ K ) )
% 0.30/0.58          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 0.30/0.58            & ( ord_less_int @ K @ I ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zmod_trivial_iff
% 0.30/0.58  thf(fact_4802_mod__int__pos__iff,axiom,
% 0.30/0.58      ! [K: int,L: int] :
% 0.30/0.58        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 0.30/0.58        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.58          | ( ( L = zero_zero_int )
% 0.30/0.58            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 0.30/0.58          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_int_pos_iff
% 0.30/0.58  thf(fact_4803_diff__Suc__less,axiom,
% 0.30/0.58      ! [N2: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.58       => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% 0.30/0.58  
% 0.30/0.58  % diff_Suc_less
% 0.30/0.58  thf(fact_4804_zdiv__mono__strict,axiom,
% 0.30/0.58      ! [A2: int,B2: int,N2: int] :
% 0.30/0.58        ( ( ord_less_int @ A2 @ B2 )
% 0.30/0.58       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.30/0.58         => ( ( ( modulo_modulo_int @ A2 @ N2 )
% 0.30/0.58              = zero_zero_int )
% 0.30/0.58           => ( ( ( modulo_modulo_int @ B2 @ N2 )
% 0.30/0.58                = zero_zero_int )
% 0.30/0.58             => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % zdiv_mono_strict
% 0.30/0.58  thf(fact_4805_nat__diff__split__asm,axiom,
% 0.30/0.58      ! [P: nat > $o,A: nat,B: nat] :
% 0.30/0.58        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.58        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 0.30/0.58                & ~ ( P @ zero_zero_nat ) )
% 0.30/0.58              | ? [D5: nat] :
% 0.30/0.58                  ( ( A
% 0.30/0.58                    = ( plus_plus_nat @ B @ D5 ) )
% 0.30/0.58                  & ~ ( P @ D5 ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_diff_split_asm
% 0.30/0.58  thf(fact_4806_nat__diff__split,axiom,
% 0.30/0.58      ! [P: nat > $o,A: nat,B: nat] :
% 0.30/0.58        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.58        = ( ( ( ord_less_nat @ A @ B )
% 0.30/0.58           => ( P @ zero_zero_nat ) )
% 0.30/0.58          & ! [D5: nat] :
% 0.30/0.58              ( ( A
% 0.30/0.58                = ( plus_plus_nat @ B @ D5 ) )
% 0.30/0.58             => ( P @ D5 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_diff_split
% 0.30/0.58  thf(fact_4807_less__diff__conv2,axiom,
% 0.30/0.58      ! [K: nat,J: nat,I: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ K @ J )
% 0.30/0.58       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 0.30/0.58          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % less_diff_conv2
% 0.30/0.58  thf(fact_4808_nat__diff__add__eq2,axiom,
% 0.30/0.58      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_diff_add_eq2
% 0.30/0.58  thf(fact_4809_nat__diff__add__eq1,axiom,
% 0.30/0.58      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.58       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_diff_add_eq1
% 0.30/0.58  thf(fact_4810_nat__le__add__iff2,axiom,
% 0.30/0.58      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.58       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_le_add_iff2
% 0.30/0.58  thf(fact_4811_nat__le__add__iff1,axiom,
% 0.30/0.58      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.58       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_le_add_iff1
% 0.30/0.58  thf(fact_4812_nat__eq__add__iff2,axiom,
% 0.30/0.58      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.58       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.30/0.58            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( M
% 0.30/0.58            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_eq_add_iff2
% 0.30/0.58  thf(fact_4813_nat__eq__add__iff1,axiom,
% 0.30/0.58      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.58       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 0.30/0.58            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.58          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 0.30/0.58            = N2 ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % nat_eq_add_iff1
% 0.30/0.58  thf(fact_4814_dvd__minus__add,axiom,
% 0.30/0.58      ! [Q4: nat,N2: nat,R2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_eq_nat @ Q4 @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
% 0.30/0.58         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q4 ) )
% 0.30/0.58            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % dvd_minus_add
% 0.30/0.58  thf(fact_4815_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ ( insert8211810215607154385at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4816_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_real,I: nat,X3: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ ( insert_real @ X3 @ ( set_real2 @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4817_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_P5464809261938338413at_nat,I: nat,X3: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X3 ) ) @ ( insert9069300056098147895at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4818_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ ( insert_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4819_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_nat,I: nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ ( insert_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4820_set__update__subset__insert,axiom,
% 0.30/0.58      ! [Xs: list_int,I: nat,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ ( insert_int @ X3 @ ( set_int2 @ Xs ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % set_update_subset_insert
% 0.30/0.58  thf(fact_4821_mod__nat__eqI,axiom,
% 0.30/0.58      ! [R2: nat,N2: nat,M: nat] :
% 0.30/0.58        ( ( ord_less_nat @ R2 @ N2 )
% 0.30/0.58       => ( ( ord_less_eq_nat @ R2 @ M )
% 0.30/0.58         => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
% 0.30/0.58           => ( ( modulo_modulo_nat @ M @ N2 )
% 0.30/0.58              = R2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % mod_nat_eqI
% 0.30/0.58  thf(fact_4822_exp__div__exp__eq,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( times_3573771949741848930nteger
% 0.30/0.58          @ ( zero_n356916108424825756nteger
% 0.30/0.58            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 0.30/0.58               != zero_z3403309356797280102nteger )
% 0.30/0.58              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.30/0.58          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exp_div_exp_eq
% 0.30/0.58  thf(fact_4823_exp__div__exp__eq,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( times_times_nat
% 0.30/0.58          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.58            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 0.30/0.58               != zero_zero_nat )
% 0.30/0.58              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.30/0.58          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exp_div_exp_eq
% 0.30/0.58  thf(fact_4824_exp__div__exp__eq,axiom,
% 0.30/0.58      ! [M: nat,N2: nat] :
% 0.30/0.58        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.58        = ( times_times_int
% 0.30/0.58          @ ( zero_n2684676970156552555ol_int
% 0.30/0.58            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 0.30/0.58               != zero_zero_int )
% 0.30/0.58              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 0.30/0.58          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % exp_div_exp_eq
% 0.30/0.58  thf(fact_4825_frac__le__eq,axiom,
% 0.30/0.58      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 0.30/0.58        ( ( Y3 != zero_zero_real )
% 0.30/0.58       => ( ( Z2 != zero_zero_real )
% 0.30/0.58         => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.30/0.58            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % frac_le_eq
% 0.30/0.58  thf(fact_4826_frac__le__eq,axiom,
% 0.30/0.58      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 0.30/0.58        ( ( Y3 != zero_zero_rat )
% 0.30/0.58       => ( ( Z2 != zero_zero_rat )
% 0.30/0.58         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.30/0.58            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.58  
% 0.30/0.58  % frac_le_eq
% 0.30/0.58  thf(fact_4827_frac__less__eq,axiom,
% 0.30/0.58      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 0.30/0.58        ( ( Y3 != zero_zero_real )
% 0.30/0.58       => ( ( Z2 != zero_zero_real )
% 0.30/0.59         => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 0.30/0.59            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % frac_less_eq
% 0.30/0.59  thf(fact_4828_frac__less__eq,axiom,
% 0.30/0.59      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 0.30/0.59        ( ( Y3 != zero_zero_rat )
% 0.30/0.59       => ( ( Z2 != zero_zero_rat )
% 0.30/0.59         => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 0.30/0.59            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % frac_less_eq
% 0.30/0.59  thf(fact_4829_signed__take__bit__int__less__exp,axiom,
% 0.30/0.59      ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_int_less_exp
% 0.30/0.59  thf(fact_4830_power__diff,axiom,
% 0.30/0.59      ! [A: complex,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_complex )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4831_power__diff,axiom,
% 0.30/0.59      ! [A: nat,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_nat )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4832_power__diff,axiom,
% 0.30/0.59      ! [A: int,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_int )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4833_power__diff,axiom,
% 0.30/0.59      ! [A: real,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_real )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4834_power__diff,axiom,
% 0.30/0.59      ! [A: rat,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_rat )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4835_power__diff,axiom,
% 0.30/0.59      ! [A: code_integer,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.59            = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff
% 0.30/0.59  thf(fact_4836_mod__pos__neg__trivial,axiom,
% 0.30/0.59      ! [K: int,L: int] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.59       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.59         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.59            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_pos_neg_trivial
% 0.30/0.59  thf(fact_4837_Suc__diff__eq__diff__pred,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 0.30/0.59          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Suc_diff_eq_diff_pred
% 0.30/0.59  thf(fact_4838_Suc__pred_H,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( N2
% 0.30/0.59          = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Suc_pred'
% 0.30/0.59  thf(fact_4839_div__geq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ~ ( ord_less_nat @ M @ N2 )
% 0.30/0.59         => ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.59            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_geq
% 0.30/0.59  thf(fact_4840_div__if,axiom,
% 0.30/0.59      ( divide_divide_nat
% 0.30/0.59      = ( ^ [M2: nat,N: nat] :
% 0.30/0.59            ( if_nat
% 0.30/0.59            @ ( ( ord_less_nat @ M2 @ N )
% 0.30/0.59              | ( N = zero_zero_nat ) )
% 0.30/0.59            @ zero_zero_nat
% 0.30/0.59            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % div_if
% 0.30/0.59  thf(fact_4841_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 0.30/0.59        ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4842_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: vEBT_VEBT] :
% 0.30/0.59        ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4843_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: product_prod_nat_nat] :
% 0.30/0.59        ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4844_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: real] :
% 0.30/0.59        ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert_real @ X3 @ bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4845_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: nat] :
% 0.30/0.59        ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4846_set__replicate__Suc,axiom,
% 0.30/0.59      ! [N2: nat,X3: int] :
% 0.30/0.59        ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X3 ) )
% 0.30/0.59        = ( insert_int @ X3 @ bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_Suc
% 0.30/0.59  thf(fact_4847_add__eq__if,axiom,
% 0.30/0.59      ( plus_plus_nat
% 0.30/0.59      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_if
% 0.30/0.59  thf(fact_4848_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 0.30/0.59            = bot_bo228742789529271731at_nat ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 0.30/0.59            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4849_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: vEBT_VEBT] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 0.30/0.59            = bot_bo8194388402131092736T_VEBT ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 0.30/0.59            = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4850_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: product_prod_nat_nat] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 0.30/0.59            = bot_bo2099793752762293965at_nat ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 0.30/0.59            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4851_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: real] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 0.30/0.59            = bot_bot_set_real ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 0.30/0.59            = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4852_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: nat] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 0.30/0.59            = bot_bot_set_nat ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 0.30/0.59            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4853_set__replicate__conv__if,axiom,
% 0.30/0.59      ! [N2: nat,X3: int] :
% 0.30/0.59        ( ( ( N2 = zero_zero_nat )
% 0.30/0.59         => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 0.30/0.59            = bot_bot_set_int ) )
% 0.30/0.59        & ( ( N2 != zero_zero_nat )
% 0.30/0.59         => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 0.30/0.59            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_replicate_conv_if
% 0.30/0.59  thf(fact_4854_nat__less__add__iff2,axiom,
% 0.30/0.59      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.59       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.59          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nat_less_add_iff2
% 0.30/0.59  thf(fact_4855_nat__less__add__iff1,axiom,
% 0.30/0.59      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.59       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 0.30/0.59          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nat_less_add_iff1
% 0.30/0.59  thf(fact_4856_mult__eq__if,axiom,
% 0.30/0.59      ( times_times_nat
% 0.30/0.59      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_eq_if
% 0.30/0.59  thf(fact_4857_scaling__mono,axiom,
% 0.30/0.59      ! [U: real,V: real,R2: real,S: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ U @ V )
% 0.30/0.59       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/0.59         => ( ( ord_less_eq_real @ R2 @ S )
% 0.30/0.59           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % scaling_mono
% 0.30/0.59  thf(fact_4858_scaling__mono,axiom,
% 0.30/0.59      ! [U: rat,V: rat,R2: rat,S: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ U @ V )
% 0.30/0.59       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 0.30/0.59         => ( ( ord_less_eq_rat @ R2 @ S )
% 0.30/0.59           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % scaling_mono
% 0.30/0.59  thf(fact_4859_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] :
% 0.30/0.59        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.59         != zero_zero_nat )
% 0.30/0.59       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.59         != zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.59  thf(fact_4860_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] :
% 0.30/0.59        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.59         != zero_zero_int )
% 0.30/0.59       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 0.30/0.59         != zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % exp_not_zero_imp_exp_diff_not_zero
% 0.30/0.59  thf(fact_4861_signed__take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.59      ! [K: int,N2: nat] :
% 0.30/0.59        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 0.30/0.59        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_int_greater_eq_self_iff
% 0.30/0.59  thf(fact_4862_signed__take__bit__int__less__self__iff,axiom,
% 0.30/0.59      ! [N2: nat,K: int] :
% 0.30/0.59        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 0.30/0.59        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_int_less_self_iff
% 0.30/0.59  thf(fact_4863_power__diff__power__eq,axiom,
% 0.30/0.59      ! [A: nat,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_nat )
% 0.30/0.59       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.30/0.59              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 0.30/0.59          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 0.30/0.59              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff_power_eq
% 0.30/0.59  thf(fact_4864_power__diff__power__eq,axiom,
% 0.30/0.59      ! [A: int,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_zero_int )
% 0.30/0.59       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.30/0.59              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 0.30/0.59          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 0.30/0.59              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff_power_eq
% 0.30/0.59  thf(fact_4865_power__diff__power__eq,axiom,
% 0.30/0.59      ! [A: code_integer,N2: nat,M: nat] :
% 0.30/0.59        ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.59       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 0.30/0.59              = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 0.30/0.59          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 0.30/0.59              = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_diff_power_eq
% 0.30/0.59  thf(fact_4866_split__zmod,axiom,
% 0.30/0.59      ! [P: int > $o,N2: int,K: int] :
% 0.30/0.59        ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
% 0.30/0.59        = ( ( ( K = zero_zero_int )
% 0.30/0.59           => ( P @ N2 ) )
% 0.30/0.59          & ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.59           => ! [I2: int,J3: int] :
% 0.30/0.59                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.59                  & ( ord_less_int @ J3 @ K )
% 0.30/0.59                  & ( N2
% 0.30/0.59                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.59               => ( P @ J3 ) ) )
% 0.30/0.59          & ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.59           => ! [I2: int,J3: int] :
% 0.30/0.59                ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.59                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.59                  & ( N2
% 0.30/0.59                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.59               => ( P @ J3 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % split_zmod
% 0.30/0.59  thf(fact_4867_int__mod__neg__eq,axiom,
% 0.30/0.59      ! [A: int,B: int,Q4: int,R2: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.59       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 0.30/0.59         => ( ( ord_less_int @ B @ R2 )
% 0.30/0.59           => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.59              = R2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_mod_neg_eq
% 0.30/0.59  thf(fact_4868_int__mod__pos__eq,axiom,
% 0.30/0.59      ! [A: int,B: int,Q4: int,R2: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 0.30/0.59       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.59         => ( ( ord_less_int @ R2 @ B )
% 0.30/0.59           => ( ( modulo_modulo_int @ A @ B )
% 0.30/0.59              = R2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_mod_pos_eq
% 0.30/0.59  thf(fact_4869_power__eq__if,axiom,
% 0.30/0.59      ( power_power_complex
% 0.30/0.59      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_eq_if
% 0.30/0.59  thf(fact_4870_power__eq__if,axiom,
% 0.30/0.59      ( power_power_real
% 0.30/0.59      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_eq_if
% 0.30/0.59  thf(fact_4871_power__eq__if,axiom,
% 0.30/0.59      ( power_power_rat
% 0.30/0.59      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_eq_if
% 0.30/0.59  thf(fact_4872_power__eq__if,axiom,
% 0.30/0.59      ( power_power_nat
% 0.30/0.59      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_eq_if
% 0.30/0.59  thf(fact_4873_power__eq__if,axiom,
% 0.30/0.59      ( power_power_int
% 0.30/0.59      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_eq_if
% 0.30/0.59  thf(fact_4874_power__minus__mult,axiom,
% 0.30/0.59      ! [N2: nat,A: complex] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.30/0.59          = ( power_power_complex @ A @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_minus_mult
% 0.30/0.59  thf(fact_4875_power__minus__mult,axiom,
% 0.30/0.59      ! [N2: nat,A: real] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.30/0.59          = ( power_power_real @ A @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_minus_mult
% 0.30/0.59  thf(fact_4876_power__minus__mult,axiom,
% 0.30/0.59      ! [N2: nat,A: rat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.30/0.59          = ( power_power_rat @ A @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_minus_mult
% 0.30/0.59  thf(fact_4877_power__minus__mult,axiom,
% 0.30/0.59      ! [N2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.30/0.59          = ( power_power_nat @ A @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_minus_mult
% 0.30/0.59  thf(fact_4878_power__minus__mult,axiom,
% 0.30/0.59      ! [N2: nat,A: int] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 0.30/0.59          = ( power_power_int @ A @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power_minus_mult
% 0.30/0.59  thf(fact_4879_diff__le__diff__pow,axiom,
% 0.30/0.59      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.59       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_le_diff_pow
% 0.30/0.59  thf(fact_4880_le__div__geq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( divide_divide_nat @ M @ N2 )
% 0.30/0.59            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_div_geq
% 0.30/0.59  thf(fact_4881_bits__induct,axiom,
% 0.30/0.59      ! [P: code_integer > $o,A: code_integer] :
% 0.30/0.59        ( ! [A4: code_integer] :
% 0.30/0.59            ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59              = A4 )
% 0.30/0.59           => ( P @ A4 ) )
% 0.30/0.59       => ( ! [A4: code_integer,B4: $o] :
% 0.30/0.59              ( ( P @ A4 )
% 0.30/0.59             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.59                  = A4 )
% 0.30/0.59               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59         => ( P @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bits_induct
% 0.30/0.59  thf(fact_4882_bits__induct,axiom,
% 0.30/0.59      ! [P: nat > $o,A: nat] :
% 0.30/0.59        ( ! [A4: nat] :
% 0.30/0.59            ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59              = A4 )
% 0.30/0.59           => ( P @ A4 ) )
% 0.30/0.59       => ( ! [A4: nat,B4: $o] :
% 0.30/0.59              ( ( P @ A4 )
% 0.30/0.59             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59                  = A4 )
% 0.30/0.59               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59         => ( P @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bits_induct
% 0.30/0.59  thf(fact_4883_bits__induct,axiom,
% 0.30/0.59      ! [P: int > $o,A: int] :
% 0.30/0.59        ( ! [A4: int] :
% 0.30/0.59            ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59              = A4 )
% 0.30/0.59           => ( P @ A4 ) )
% 0.30/0.59       => ( ! [A4: int,B4: $o] :
% 0.30/0.59              ( ( P @ A4 )
% 0.30/0.59             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59                  = A4 )
% 0.30/0.59               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 0.30/0.59         => ( P @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % bits_induct
% 0.30/0.59  thf(fact_4884_verit__le__mono__div__int,axiom,
% 0.30/0.59      ! [A2: int,B2: int,N2: int] :
% 0.30/0.59        ( ( ord_less_int @ A2 @ B2 )
% 0.30/0.59       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 0.30/0.59         => ( ord_less_eq_int
% 0.30/0.59            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
% 0.30/0.59              @ ( if_int
% 0.30/0.59                @ ( ( modulo_modulo_int @ B2 @ N2 )
% 0.30/0.59                  = zero_zero_int )
% 0.30/0.59                @ one_one_int
% 0.30/0.59                @ zero_zero_int ) )
% 0.30/0.59            @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_le_mono_div_int
% 0.30/0.59  thf(fact_4885_split__pos__lemma,axiom,
% 0.30/0.59      ! [K: int,P: int > int > $o,N2: int] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.59       => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 0.30/0.59          = ( ! [I2: int,J3: int] :
% 0.30/0.59                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 0.30/0.59                  & ( ord_less_int @ J3 @ K )
% 0.30/0.59                  & ( N2
% 0.30/0.59                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.59               => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % split_pos_lemma
% 0.30/0.59  thf(fact_4886_split__neg__lemma,axiom,
% 0.30/0.59      ! [K: int,P: int > int > $o,N2: int] :
% 0.30/0.59        ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.59       => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 0.30/0.59          = ( ! [I2: int,J3: int] :
% 0.30/0.59                ( ( ( ord_less_int @ K @ J3 )
% 0.30/0.59                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 0.30/0.59                  & ( N2
% 0.30/0.59                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 0.30/0.59               => ( P @ I2 @ J3 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % split_neg_lemma
% 0.30/0.59  thf(fact_4887_int__power__div__base,axiom,
% 0.30/0.59      ! [M: nat,K: int] :
% 0.30/0.59        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.59       => ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.59         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 0.30/0.59            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_power_div_base
% 0.30/0.59  thf(fact_4888_set__decode__plus__power__2,axiom,
% 0.30/0.59      ! [N2: nat,Z2: nat] :
% 0.30/0.59        ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z2 ) )
% 0.30/0.59       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z2 ) )
% 0.30/0.59          = ( insert_nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % set_decode_plus_power_2
% 0.30/0.59  thf(fact_4889_exp__mod__exp,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % exp_mod_exp
% 0.30/0.59  thf(fact_4890_exp__mod__exp,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % exp_mod_exp
% 0.30/0.59  thf(fact_4891_exp__mod__exp,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % exp_mod_exp
% 0.30/0.59  thf(fact_4892_power2__diff,axiom,
% 0.30/0.59      ! [X3: complex,Y3: complex] :
% 0.30/0.59        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power2_diff
% 0.30/0.59  thf(fact_4893_power2__diff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power2_diff
% 0.30/0.59  thf(fact_4894_power2__diff,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat] :
% 0.30/0.59        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power2_diff
% 0.30/0.59  thf(fact_4895_power2__diff,axiom,
% 0.30/0.59      ! [X3: int,Y3: int] :
% 0.30/0.59        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % power2_diff
% 0.30/0.59  thf(fact_4896_mult__exp__mod__exp__eq,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_exp_mod_exp_eq
% 0.30/0.59  thf(fact_4897_mult__exp__mod__exp__eq,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,A: int] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_exp_mod_exp_eq
% 0.30/0.59  thf(fact_4898_mult__exp__mod__exp__eq,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,A: code_integer] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_exp_mod_exp_eq
% 0.30/0.59  thf(fact_4899_even__mod__4__div__2,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.59          = ( suc @ zero_zero_nat ) )
% 0.30/0.59       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mod_4_div_2
% 0.30/0.59  thf(fact_4900_eucl__rel__int__iff,axiom,
% 0.30/0.59      ! [K: int,L: int,Q4: int,R2: int] :
% 0.30/0.59        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59        = ( ( K
% 0.30/0.59            = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 0.30/0.59          & ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.59           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 0.30/0.59              & ( ord_less_int @ R2 @ L ) ) )
% 0.30/0.59          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.59           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.59               => ( ( ord_less_int @ L @ R2 )
% 0.30/0.59                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 0.30/0.59              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.59               => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eucl_rel_int_iff
% 0.30/0.59  thf(fact_4901_divmod__digit__1_I2_J,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.59       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.59         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.59           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.59              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_digit_1(2)
% 0.30/0.59  thf(fact_4902_divmod__digit__1_I2_J,axiom,
% 0.30/0.59      ! [A: nat,B: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 0.30/0.59         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.59           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.59              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_digit_1(2)
% 0.30/0.59  thf(fact_4903_divmod__digit__1_I2_J,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.59       => ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.59         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 0.30/0.59           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 0.30/0.59              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_digit_1(2)
% 0.30/0.59  thf(fact_4904_even__mask__div__iff_H,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff'
% 0.30/0.59  thf(fact_4905_even__mask__div__iff_H,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff'
% 0.30/0.59  thf(fact_4906_even__mask__div__iff_H,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff'
% 0.30/0.59  thf(fact_4907_even__mask__div__iff,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.59            = zero_zero_nat )
% 0.30/0.59          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff
% 0.30/0.59  thf(fact_4908_even__mask__div__iff,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.59            = zero_zero_int )
% 0.30/0.59          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff
% 0.30/0.59  thf(fact_4909_even__mask__div__iff,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( 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 ) ) @ N2 ) ) )
% 0.30/0.59        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.59            = zero_z3403309356797280102nteger )
% 0.30/0.59          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_mask_div_iff
% 0.30/0.59  thf(fact_4910_divmod__step__eq,axiom,
% 0.30/0.59      ! [L: num,R2: nat,Q4: nat] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.30/0.59         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.30/0.59            = ( 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 ) ) ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 0.30/0.59         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 0.30/0.59            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_step_eq
% 0.30/0.59  thf(fact_4911_divmod__step__eq,axiom,
% 0.30/0.59      ! [L: num,R2: int,Q4: int] :
% 0.30/0.59        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.30/0.59         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59            = ( 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 ) ) ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 0.30/0.59         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_step_eq
% 0.30/0.59  thf(fact_4912_divmod__step__eq,axiom,
% 0.30/0.59      ! [L: num,R2: code_integer,Q4: code_integer] :
% 0.30/0.59        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.30/0.59         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 0.30/0.59            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 0.30/0.59        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 0.30/0.59         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 0.30/0.59            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divmod_step_eq
% 0.30/0.59  thf(fact_4913_inrange,axiom,
% 0.30/0.59      ! [T: vEBT_VEBT,N2: nat] :
% 0.30/0.59        ( ( vEBT_invar_vebt @ T @ N2 )
% 0.30/0.59       => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % inrange
% 0.30/0.59  thf(fact_4914_Divides_Oadjust__div__eq,axiom,
% 0.30/0.59      ! [Q4: int,R2: int] :
% 0.30/0.59        ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59        = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Divides.adjust_div_eq
% 0.30/0.59  thf(fact_4915_signed__take__bit__rec,axiom,
% 0.30/0.59      ( bit_ri6519982836138164636nteger
% 0.30/0.59      = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_rec
% 0.30/0.59  thf(fact_4916_signed__take__bit__rec,axiom,
% 0.30/0.59      ( bit_ri631733984087533419it_int
% 0.30/0.59      = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_rec
% 0.30/0.59  thf(fact_4917_neg__eucl__rel__int__mult__2,axiom,
% 0.30/0.59      ! [B: int,A: int,Q4: int,R2: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 0.30/0.59       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59         => ( 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 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eucl_rel_int_mult_2
% 0.30/0.59  thf(fact_4918_diff__shunt__var,axiom,
% 0.30/0.59      ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ( minus_1356011639430497352at_nat @ X3 @ Y3 )
% 0.30/0.59          = bot_bo2099793752762293965at_nat )
% 0.30/0.59        = ( ord_le3146513528884898305at_nat @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_shunt_var
% 0.30/0.59  thf(fact_4919_diff__shunt__var,axiom,
% 0.30/0.59      ! [X3: set_real,Y3: set_real] :
% 0.30/0.59        ( ( ( minus_minus_set_real @ X3 @ Y3 )
% 0.30/0.59          = bot_bot_set_real )
% 0.30/0.59        = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_shunt_var
% 0.30/0.59  thf(fact_4920_diff__shunt__var,axiom,
% 0.30/0.59      ! [X3: set_nat,Y3: set_nat] :
% 0.30/0.59        ( ( ( minus_minus_set_nat @ X3 @ Y3 )
% 0.30/0.59          = bot_bot_set_nat )
% 0.30/0.59        = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_shunt_var
% 0.30/0.59  thf(fact_4921_diff__shunt__var,axiom,
% 0.30/0.59      ! [X3: set_int,Y3: set_int] :
% 0.30/0.59        ( ( ( minus_minus_set_int @ X3 @ Y3 )
% 0.30/0.59          = bot_bot_set_int )
% 0.30/0.59        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_shunt_var
% 0.30/0.59  thf(fact_4922_take__bit__rec,axiom,
% 0.30/0.59      ( bit_se1745604003318907178nteger
% 0.30/0.59      = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_rec
% 0.30/0.59  thf(fact_4923_take__bit__rec,axiom,
% 0.30/0.59      ( bit_se2925701944663578781it_nat
% 0.30/0.59      = ( ^ [N: nat,A3: nat] : ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_rec
% 0.30/0.59  thf(fact_4924_take__bit__rec,axiom,
% 0.30/0.59      ( bit_se2923211474154528505it_int
% 0.30/0.59      = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_rec
% 0.30/0.59  thf(fact_4925_add_Oinverse__inverse,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = A ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_inverse
% 0.30/0.59  thf(fact_4926_add_Oinverse__inverse,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = A ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_inverse
% 0.30/0.59  thf(fact_4927_add_Oinverse__inverse,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = A ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_inverse
% 0.30/0.59  thf(fact_4928_add_Oinverse__inverse,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = A ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_inverse
% 0.30/0.59  thf(fact_4929_neg__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( uminus_uminus_int @ A )
% 0.30/0.59          = ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( A = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_iff_equal
% 0.30/0.59  thf(fact_4930_neg__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ A )
% 0.30/0.59          = ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( A = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_iff_equal
% 0.30/0.59  thf(fact_4931_neg__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( A = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_iff_equal
% 0.30/0.59  thf(fact_4932_neg__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( A = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_iff_equal
% 0.30/0.59  thf(fact_4933_Diff__cancel,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
% 0.30/0.59        = bot_bo2099793752762293965at_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_cancel
% 0.30/0.59  thf(fact_4934_Diff__cancel,axiom,
% 0.30/0.59      ! [A2: set_real] :
% 0.30/0.59        ( ( minus_minus_set_real @ A2 @ A2 )
% 0.30/0.59        = bot_bot_set_real ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_cancel
% 0.30/0.59  thf(fact_4935_Diff__cancel,axiom,
% 0.30/0.59      ! [A2: set_int] :
% 0.30/0.59        ( ( minus_minus_set_int @ A2 @ A2 )
% 0.30/0.59        = bot_bot_set_int ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_cancel
% 0.30/0.59  thf(fact_4936_Diff__cancel,axiom,
% 0.30/0.59      ! [A2: set_nat] :
% 0.30/0.59        ( ( minus_minus_set_nat @ A2 @ A2 )
% 0.30/0.59        = bot_bot_set_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_cancel
% 0.30/0.59  thf(fact_4937_empty__Diff,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
% 0.30/0.59        = bot_bo2099793752762293965at_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % empty_Diff
% 0.30/0.59  thf(fact_4938_empty__Diff,axiom,
% 0.30/0.59      ! [A2: set_real] :
% 0.30/0.59        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 0.30/0.59        = bot_bot_set_real ) ).
% 0.30/0.59  
% 0.30/0.59  % empty_Diff
% 0.30/0.59  thf(fact_4939_empty__Diff,axiom,
% 0.30/0.59      ! [A2: set_int] :
% 0.30/0.59        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 0.30/0.59        = bot_bot_set_int ) ).
% 0.30/0.59  
% 0.30/0.59  % empty_Diff
% 0.30/0.59  thf(fact_4940_empty__Diff,axiom,
% 0.30/0.59      ! [A2: set_nat] :
% 0.30/0.59        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 0.30/0.59        = bot_bot_set_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % empty_Diff
% 0.30/0.59  thf(fact_4941_Diff__empty,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 0.30/0.59        = A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_empty
% 0.30/0.59  thf(fact_4942_Diff__empty,axiom,
% 0.30/0.59      ! [A2: set_real] :
% 0.30/0.59        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 0.30/0.59        = A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_empty
% 0.30/0.59  thf(fact_4943_Diff__empty,axiom,
% 0.30/0.59      ! [A2: set_int] :
% 0.30/0.59        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 0.30/0.59        = A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_empty
% 0.30/0.59  thf(fact_4944_Diff__empty,axiom,
% 0.30/0.59      ! [A2: set_nat] :
% 0.30/0.59        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 0.30/0.59        = A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_empty
% 0.30/0.59  thf(fact_4945_finite__Diff2,axiom,
% 0.30/0.59      ! [B2: set_int,A2: set_int] :
% 0.30/0.59        ( ( finite_finite_int @ B2 )
% 0.30/0.59       => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.59          = ( finite_finite_int @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff2
% 0.30/0.59  thf(fact_4946_finite__Diff2,axiom,
% 0.30/0.59      ! [B2: set_complex,A2: set_complex] :
% 0.30/0.59        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.59       => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.59          = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff2
% 0.30/0.59  thf(fact_4947_finite__Diff2,axiom,
% 0.30/0.59      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( finite6177210948735845034at_nat @ B2 )
% 0.30/0.59       => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.59          = ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff2
% 0.30/0.59  thf(fact_4948_finite__Diff2,axiom,
% 0.30/0.59      ! [B2: set_nat,A2: set_nat] :
% 0.30/0.59        ( ( finite_finite_nat @ B2 )
% 0.30/0.59       => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.59          = ( finite_finite_nat @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff2
% 0.30/0.59  thf(fact_4949_finite__Diff,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] :
% 0.30/0.59        ( ( finite_finite_int @ A2 )
% 0.30/0.59       => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff
% 0.30/0.59  thf(fact_4950_finite__Diff,axiom,
% 0.30/0.59      ! [A2: set_complex,B2: set_complex] :
% 0.30/0.59        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.59       => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff
% 0.30/0.59  thf(fact_4951_finite__Diff,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.59       => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff
% 0.30/0.59  thf(fact_4952_finite__Diff,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat] :
% 0.30/0.59        ( ( finite_finite_nat @ A2 )
% 0.30/0.59       => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff
% 0.30/0.59  thf(fact_4953_Compl__subset__Compl__iff,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 0.30/0.59        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_subset_Compl_iff
% 0.30/0.59  thf(fact_4954_Compl__anti__mono,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.30/0.59       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_anti_mono
% 0.30/0.59  thf(fact_4955_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4956_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4957_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: real,A2: set_real,B2: set_real] :
% 0.30/0.59        ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4958_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.59        ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4959_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: int,A2: set_int,B2: set_int] :
% 0.30/0.59        ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4960_Diff__insert0,axiom,
% 0.30/0.59      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.59        ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 0.30/0.59          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert0
% 0.30/0.59  thf(fact_4961_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4962_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4963_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: real,B2: set_real,A2: set_real] :
% 0.30/0.59        ( ( member_real @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4964_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.30/0.59        ( ( member_set_nat @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4965_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: int,B2: set_int,A2: set_int] :
% 0.30/0.59        ( ( member_int @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4966_insert__Diff1,axiom,
% 0.30/0.59      ! [X3: nat,B2: set_nat,A2: set_nat] :
% 0.30/0.59        ( ( member_nat @ X3 @ B2 )
% 0.30/0.59       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff1
% 0.30/0.59  thf(fact_4967_compl__le__compl__iff,axiom,
% 0.30/0.59      ! [X3: set_int,Y3: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 0.30/0.59        = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % compl_le_compl_iff
% 0.30/0.59  thf(fact_4968_neg__le__iff__le,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_iff_le
% 0.30/0.59  thf(fact_4969_neg__le__iff__le,axiom,
% 0.30/0.59      ! [B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_iff_le
% 0.30/0.59  thf(fact_4970_neg__le__iff__le,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_iff_le
% 0.30/0.59  thf(fact_4971_neg__le__iff__le,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_iff_le
% 0.30/0.59  thf(fact_4972_neg__equal__zero,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ( uminus_uminus_int @ A )
% 0.30/0.59          = A )
% 0.30/0.59        = ( A = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_zero
% 0.30/0.59  thf(fact_4973_neg__equal__zero,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ A )
% 0.30/0.59          = A )
% 0.30/0.59        = ( A = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_zero
% 0.30/0.59  thf(fact_4974_neg__equal__zero,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = A )
% 0.30/0.59        = ( A = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_zero
% 0.30/0.59  thf(fact_4975_neg__equal__zero,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = A )
% 0.30/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_zero
% 0.30/0.59  thf(fact_4976_equal__neg__zero,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( A = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equal_neg_zero
% 0.30/0.59  thf(fact_4977_equal__neg__zero,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( A = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equal_neg_zero
% 0.30/0.59  thf(fact_4978_equal__neg__zero,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( A = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equal_neg_zero
% 0.30/0.59  thf(fact_4979_equal__neg__zero,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equal_neg_zero
% 0.30/0.59  thf(fact_4980_neg__equal__0__iff__equal,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.59          = zero_zero_complex )
% 0.30/0.59        = ( A = zero_zero_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_0_iff_equal
% 0.30/0.59  thf(fact_4981_neg__equal__0__iff__equal,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ( uminus_uminus_int @ A )
% 0.30/0.59          = zero_zero_int )
% 0.30/0.59        = ( A = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_0_iff_equal
% 0.30/0.59  thf(fact_4982_neg__equal__0__iff__equal,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ A )
% 0.30/0.59          = zero_zero_real )
% 0.30/0.59        = ( A = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_0_iff_equal
% 0.30/0.59  thf(fact_4983_neg__equal__0__iff__equal,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = zero_zero_rat )
% 0.30/0.59        = ( A = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_0_iff_equal
% 0.30/0.59  thf(fact_4984_neg__equal__0__iff__equal,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = zero_z3403309356797280102nteger )
% 0.30/0.59        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_equal_0_iff_equal
% 0.30/0.59  thf(fact_4985_neg__0__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( zero_zero_complex
% 0.30/0.59          = ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.59        = ( zero_zero_complex = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_equal_iff_equal
% 0.30/0.59  thf(fact_4986_neg__0__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( zero_zero_int
% 0.30/0.59          = ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( zero_zero_int = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_equal_iff_equal
% 0.30/0.59  thf(fact_4987_neg__0__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( zero_zero_real
% 0.30/0.59          = ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( zero_zero_real = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_equal_iff_equal
% 0.30/0.59  thf(fact_4988_neg__0__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( zero_zero_rat
% 0.30/0.59          = ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( zero_zero_rat = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_equal_iff_equal
% 0.30/0.59  thf(fact_4989_neg__0__equal__iff__equal,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( zero_z3403309356797280102nteger
% 0.30/0.59          = ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( zero_z3403309356797280102nteger = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_equal_iff_equal
% 0.30/0.59  thf(fact_4990_add_Oinverse__neutral,axiom,
% 0.30/0.59      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 0.30/0.59      = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_neutral
% 0.30/0.59  thf(fact_4991_add_Oinverse__neutral,axiom,
% 0.30/0.59      ( ( uminus_uminus_int @ zero_zero_int )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_neutral
% 0.30/0.59  thf(fact_4992_add_Oinverse__neutral,axiom,
% 0.30/0.59      ( ( uminus_uminus_real @ zero_zero_real )
% 0.30/0.59      = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_neutral
% 0.30/0.59  thf(fact_4993_add_Oinverse__neutral,axiom,
% 0.30/0.59      ( ( uminus_uminus_rat @ zero_zero_rat )
% 0.30/0.59      = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_neutral
% 0.30/0.59  thf(fact_4994_add_Oinverse__neutral,axiom,
% 0.30/0.59      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 0.30/0.59      = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_neutral
% 0.30/0.59  thf(fact_4995_neg__less__iff__less,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_int @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_iff_less
% 0.30/0.59  thf(fact_4996_neg__less__iff__less,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_iff_less
% 0.30/0.59  thf(fact_4997_neg__less__iff__less,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_iff_less
% 0.30/0.59  thf(fact_4998_neg__less__iff__less,axiom,
% 0.30/0.59      ! [B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_iff_less
% 0.30/0.59  thf(fact_4999_mult__minus__left,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.59        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_left
% 0.30/0.59  thf(fact_5000_mult__minus__left,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59        = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_left
% 0.30/0.59  thf(fact_5001_mult__minus__left,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59        = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_left
% 0.30/0.59  thf(fact_5002_mult__minus__left,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_left
% 0.30/0.59  thf(fact_5003_mult__minus__left,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_left
% 0.30/0.59  thf(fact_5004_minus__mult__minus,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.59        = ( times_times_complex @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_minus
% 0.30/0.59  thf(fact_5005_minus__mult__minus,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( times_times_int @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_minus
% 0.30/0.59  thf(fact_5006_minus__mult__minus,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( times_times_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_minus
% 0.30/0.59  thf(fact_5007_minus__mult__minus,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( times_times_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_minus
% 0.30/0.59  thf(fact_5008_minus__mult__minus,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_minus
% 0.30/0.59  thf(fact_5009_mult__minus__right,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.59        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_right
% 0.30/0.59  thf(fact_5010_mult__minus__right,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_right
% 0.30/0.59  thf(fact_5011_mult__minus__right,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_right
% 0.30/0.59  thf(fact_5012_mult__minus__right,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_right
% 0.30/0.59  thf(fact_5013_mult__minus__right,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % mult_minus_right
% 0.30/0.59  thf(fact_5014_minus__add__distrib,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.59        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_distrib
% 0.30/0.59  thf(fact_5015_minus__add__distrib,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.59        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_distrib
% 0.30/0.59  thf(fact_5016_minus__add__distrib,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.59        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_distrib
% 0.30/0.59  thf(fact_5017_minus__add__distrib,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_distrib
% 0.30/0.59  thf(fact_5018_minus__add__cancel,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_cancel
% 0.30/0.59  thf(fact_5019_minus__add__cancel,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_cancel
% 0.30/0.59  thf(fact_5020_minus__add__cancel,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_cancel
% 0.30/0.59  thf(fact_5021_minus__add__cancel,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_add_cancel
% 0.30/0.59  thf(fact_5022_add__minus__cancel,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % add_minus_cancel
% 0.30/0.59  thf(fact_5023_add__minus__cancel,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % add_minus_cancel
% 0.30/0.59  thf(fact_5024_add__minus__cancel,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % add_minus_cancel
% 0.30/0.59  thf(fact_5025_add__minus__cancel,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 0.30/0.59        = B ) ).
% 0.30/0.59  
% 0.30/0.59  % add_minus_cancel
% 0.30/0.59  thf(fact_5026_minus__diff__eq,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.59        = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_eq
% 0.30/0.59  thf(fact_5027_minus__diff__eq,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.59        = ( minus_minus_real @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_eq
% 0.30/0.59  thf(fact_5028_minus__diff__eq,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.59        = ( minus_minus_rat @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_eq
% 0.30/0.59  thf(fact_5029_minus__diff__eq,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.59        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_eq
% 0.30/0.59  thf(fact_5030_dvd__minus__iff,axiom,
% 0.30/0.59      ! [X3: int,Y3: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 0.30/0.59        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_minus_iff
% 0.30/0.59  thf(fact_5031_dvd__minus__iff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
% 0.30/0.59        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_minus_iff
% 0.30/0.59  thf(fact_5032_dvd__minus__iff,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
% 0.30/0.59        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_minus_iff
% 0.30/0.59  thf(fact_5033_dvd__minus__iff,axiom,
% 0.30/0.59      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_minus_iff
% 0.30/0.59  thf(fact_5034_minus__dvd__iff,axiom,
% 0.30/0.59      ! [X3: int,Y3: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 0.30/0.59        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_dvd_iff
% 0.30/0.59  thf(fact_5035_minus__dvd__iff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
% 0.30/0.59        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_dvd_iff
% 0.30/0.59  thf(fact_5036_minus__dvd__iff,axiom,
% 0.30/0.59      ! [X3: rat,Y3: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
% 0.30/0.59        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_dvd_iff
% 0.30/0.59  thf(fact_5037_minus__dvd__iff,axiom,
% 0.30/0.59      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
% 0.30/0.59        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_dvd_iff
% 0.30/0.59  thf(fact_5038_Diff__eq__empty__iff,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
% 0.30/0.59          = bot_bo2099793752762293965at_nat )
% 0.30/0.59        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_eq_empty_iff
% 0.30/0.59  thf(fact_5039_Diff__eq__empty__iff,axiom,
% 0.30/0.59      ! [A2: set_real,B2: set_real] :
% 0.30/0.59        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 0.30/0.59          = bot_bot_set_real )
% 0.30/0.59        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_eq_empty_iff
% 0.30/0.59  thf(fact_5040_Diff__eq__empty__iff,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat] :
% 0.30/0.59        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 0.30/0.59          = bot_bot_set_nat )
% 0.30/0.59        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_eq_empty_iff
% 0.30/0.59  thf(fact_5041_Diff__eq__empty__iff,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] :
% 0.30/0.59        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 0.30/0.59          = bot_bot_set_int )
% 0.30/0.59        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_eq_empty_iff
% 0.30/0.59  thf(fact_5042_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: set_nat,L: set_nat,U: set_nat] :
% 0.30/0.59        ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_set_nat @ L @ I )
% 0.30/0.59          & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5043_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: set_int,L: set_int,U: set_int] :
% 0.30/0.59        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_set_int @ L @ I )
% 0.30/0.59          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5044_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: rat,L: rat,U: rat] :
% 0.30/0.59        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_rat @ L @ I )
% 0.30/0.59          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5045_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: num,L: num,U: num] :
% 0.30/0.59        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_num @ L @ I )
% 0.30/0.59          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5046_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: nat,L: nat,U: nat] :
% 0.30/0.59        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_nat @ L @ I )
% 0.30/0.59          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5047_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: int,L: int,U: int] :
% 0.30/0.59        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_int @ L @ I )
% 0.30/0.59          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5048_atLeastAtMost__iff,axiom,
% 0.30/0.59      ! [I: real,L: real,U: real] :
% 0.30/0.59        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 0.30/0.59        = ( ( ord_less_eq_real @ L @ I )
% 0.30/0.59          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_iff
% 0.30/0.59  thf(fact_5049_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 0.30/0.59        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 0.30/0.59          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5050_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 0.30/0.59        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 0.30/0.59          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5051_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: num,H2: num,L3: num,H3: num] :
% 0.30/0.59        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 0.30/0.59          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5052_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 0.30/0.59        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 0.30/0.59          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5053_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: int,H2: int,L3: int,H3: int] :
% 0.30/0.59        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 0.30/0.59          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5054_Icc__eq__Icc,axiom,
% 0.30/0.59      ! [L: real,H2: real,L3: real,H3: real] :
% 0.30/0.59        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 0.30/0.59          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 0.30/0.59        = ( ( ( L = L3 )
% 0.30/0.59            & ( H2 = H3 ) )
% 0.30/0.59          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.30/0.59            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Icc_eq_Icc
% 0.30/0.59  thf(fact_5055_insert__Diff__single,axiom,
% 0.30/0.59      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.59        = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_single
% 0.30/0.59  thf(fact_5056_insert__Diff__single,axiom,
% 0.30/0.59      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.59        = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_single
% 0.30/0.59  thf(fact_5057_insert__Diff__single,axiom,
% 0.30/0.59      ! [A: real,A2: set_real] :
% 0.30/0.59        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.59        = ( insert_real @ A @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_single
% 0.30/0.59  thf(fact_5058_insert__Diff__single,axiom,
% 0.30/0.59      ! [A: int,A2: set_int] :
% 0.30/0.59        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.59        = ( insert_int @ A @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_single
% 0.30/0.59  thf(fact_5059_insert__Diff__single,axiom,
% 0.30/0.59      ! [A: nat,A2: set_nat] :
% 0.30/0.59        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.59        = ( insert_nat @ A @ A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_single
% 0.30/0.59  thf(fact_5060_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_real,A: real,B2: set_real] :
% 0.30/0.59        ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
% 0.30/0.59        = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5061_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
% 0.30/0.59        = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5062_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_int,A: int,B2: set_int] :
% 0.30/0.59        ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
% 0.30/0.59        = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5063_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_complex,A: complex,B2: set_complex] :
% 0.30/0.59        ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
% 0.30/0.59        = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5064_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
% 0.30/0.59        = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5065_finite__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.30/0.59        ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
% 0.30/0.59        = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_Diff_insert
% 0.30/0.59  thf(fact_5066_take__bit__of__0,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
% 0.30/0.59        = zero_zero_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_0
% 0.30/0.59  thf(fact_5067_take__bit__of__0,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_0
% 0.30/0.59  thf(fact_5068_finite__atLeastAtMost,axiom,
% 0.30/0.59      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_atLeastAtMost
% 0.30/0.59  thf(fact_5069_neg__less__eq__nonneg,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.59        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_eq_nonneg
% 0.30/0.59  thf(fact_5070_neg__less__eq__nonneg,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_eq_nonneg
% 0.30/0.59  thf(fact_5071_neg__less__eq__nonneg,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.59        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_eq_nonneg
% 0.30/0.59  thf(fact_5072_neg__less__eq__nonneg,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.59        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_eq_nonneg
% 0.30/0.59  thf(fact_5073_less__eq__neg__nonpos,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_eq_neg_nonpos
% 0.30/0.59  thf(fact_5074_less__eq__neg__nonpos,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_eq_neg_nonpos
% 0.30/0.59  thf(fact_5075_less__eq__neg__nonpos,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_eq_neg_nonpos
% 0.30/0.59  thf(fact_5076_less__eq__neg__nonpos,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_eq_neg_nonpos
% 0.30/0.59  thf(fact_5077_neg__le__0__iff__le,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.59        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_0_iff_le
% 0.30/0.59  thf(fact_5078_neg__le__0__iff__le,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_0_iff_le
% 0.30/0.59  thf(fact_5079_neg__le__0__iff__le,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.59        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_0_iff_le
% 0.30/0.59  thf(fact_5080_neg__le__0__iff__le,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.59        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_le_0_iff_le
% 0.30/0.59  thf(fact_5081_neg__0__le__iff__le,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_le_iff_le
% 0.30/0.59  thf(fact_5082_neg__0__le__iff__le,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_le_iff_le
% 0.30/0.59  thf(fact_5083_neg__0__le__iff__le,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_le_iff_le
% 0.30/0.59  thf(fact_5084_neg__0__le__iff__le,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_le_iff_le
% 0.30/0.59  thf(fact_5085_less__neg__neg,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_neg_neg
% 0.30/0.59  thf(fact_5086_less__neg__neg,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_neg_neg
% 0.30/0.59  thf(fact_5087_less__neg__neg,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_neg_neg
% 0.30/0.59  thf(fact_5088_less__neg__neg,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_neg_neg
% 0.30/0.59  thf(fact_5089_neg__less__pos,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.59        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_pos
% 0.30/0.59  thf(fact_5090_neg__less__pos,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.59        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_pos
% 0.30/0.59  thf(fact_5091_neg__less__pos,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.59        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_pos
% 0.30/0.59  thf(fact_5092_neg__less__pos,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_pos
% 0.30/0.59  thf(fact_5093_neg__0__less__iff__less,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_less_iff_less
% 0.30/0.59  thf(fact_5094_neg__0__less__iff__less,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_less_iff_less
% 0.30/0.59  thf(fact_5095_neg__0__less__iff__less,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_less_iff_less
% 0.30/0.59  thf(fact_5096_neg__0__less__iff__less,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_0_less_iff_less
% 0.30/0.59  thf(fact_5097_neg__less__0__iff__less,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 0.30/0.59        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_0_iff_less
% 0.30/0.59  thf(fact_5098_neg__less__0__iff__less,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 0.30/0.59        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_0_iff_less
% 0.30/0.59  thf(fact_5099_neg__less__0__iff__less,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 0.30/0.59        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_0_iff_less
% 0.30/0.59  thf(fact_5100_neg__less__0__iff__less,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_less_0_iff_less
% 0.30/0.59  thf(fact_5101_add_Oright__inverse,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 0.30/0.59        = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % add.right_inverse
% 0.30/0.59  thf(fact_5102_add_Oright__inverse,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % add.right_inverse
% 0.30/0.59  thf(fact_5103_add_Oright__inverse,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 0.30/0.59        = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % add.right_inverse
% 0.30/0.59  thf(fact_5104_add_Oright__inverse,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 0.30/0.59        = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % add.right_inverse
% 0.30/0.59  thf(fact_5105_add_Oright__inverse,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.59        = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % add.right_inverse
% 0.30/0.59  thf(fact_5106_ab__left__minus,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.59        = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_left_minus
% 0.30/0.59  thf(fact_5107_ab__left__minus,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_left_minus
% 0.30/0.59  thf(fact_5108_ab__left__minus,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.59        = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_left_minus
% 0.30/0.59  thf(fact_5109_ab__left__minus,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.59        = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_left_minus
% 0.30/0.59  thf(fact_5110_ab__left__minus,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.59        = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_left_minus
% 0.30/0.59  thf(fact_5111_verit__minus__simplify_I3_J,axiom,
% 0.30/0.59      ! [B: complex] :
% 0.30/0.59        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 0.30/0.59        = ( uminus1482373934393186551omplex @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_minus_simplify(3)
% 0.30/0.59  thf(fact_5112_verit__minus__simplify_I3_J,axiom,
% 0.30/0.59      ! [B: int] :
% 0.30/0.59        ( ( minus_minus_int @ zero_zero_int @ B )
% 0.30/0.59        = ( uminus_uminus_int @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_minus_simplify(3)
% 0.30/0.59  thf(fact_5113_verit__minus__simplify_I3_J,axiom,
% 0.30/0.59      ! [B: real] :
% 0.30/0.59        ( ( minus_minus_real @ zero_zero_real @ B )
% 0.30/0.59        = ( uminus_uminus_real @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_minus_simplify(3)
% 0.30/0.59  thf(fact_5114_verit__minus__simplify_I3_J,axiom,
% 0.30/0.59      ! [B: rat] :
% 0.30/0.59        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 0.30/0.59        = ( uminus_uminus_rat @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_minus_simplify(3)
% 0.30/0.59  thf(fact_5115_verit__minus__simplify_I3_J,axiom,
% 0.30/0.59      ! [B: code_integer] :
% 0.30/0.59        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_minus_simplify(3)
% 0.30/0.59  thf(fact_5116_diff__0,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 0.30/0.59        = ( uminus1482373934393186551omplex @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_0
% 0.30/0.59  thf(fact_5117_diff__0,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( minus_minus_int @ zero_zero_int @ A )
% 0.30/0.59        = ( uminus_uminus_int @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_0
% 0.30/0.59  thf(fact_5118_diff__0,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( minus_minus_real @ zero_zero_real @ A )
% 0.30/0.59        = ( uminus_uminus_real @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_0
% 0.30/0.59  thf(fact_5119_diff__0,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 0.30/0.59        = ( uminus_uminus_rat @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_0
% 0.30/0.59  thf(fact_5120_diff__0,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_0
% 0.30/0.59  thf(fact_5121_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.59        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_simps(3)
% 0.30/0.59  thf(fact_5122_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.30/0.59        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_simps(3)
% 0.30/0.59  thf(fact_5123_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.30/0.59        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_simps(3)
% 0.30/0.59  thf(fact_5124_add__neg__numeral__simps_I3_J,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_simps(3)
% 0.30/0.59  thf(fact_5125_uminus__add__conv__diff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59        = ( minus_minus_int @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % uminus_add_conv_diff
% 0.30/0.59  thf(fact_5126_uminus__add__conv__diff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59        = ( minus_minus_real @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % uminus_add_conv_diff
% 0.30/0.59  thf(fact_5127_uminus__add__conv__diff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59        = ( minus_minus_rat @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % uminus_add_conv_diff
% 0.30/0.59  thf(fact_5128_uminus__add__conv__diff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % uminus_add_conv_diff
% 0.30/0.59  thf(fact_5129_diff__minus__eq__add,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( plus_plus_int @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_minus_eq_add
% 0.30/0.59  thf(fact_5130_diff__minus__eq__add,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( plus_plus_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_minus_eq_add
% 0.30/0.59  thf(fact_5131_diff__minus__eq__add,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( plus_plus_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_minus_eq_add
% 0.30/0.59  thf(fact_5132_diff__minus__eq__add,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_minus_eq_add
% 0.30/0.59  thf(fact_5133_divide__minus1,axiom,
% 0.30/0.59      ! [X3: complex] :
% 0.30/0.59        ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.59        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_minus1
% 0.30/0.59  thf(fact_5134_divide__minus1,axiom,
% 0.30/0.59      ! [X3: real] :
% 0.30/0.59        ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59        = ( uminus_uminus_real @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_minus1
% 0.30/0.59  thf(fact_5135_divide__minus1,axiom,
% 0.30/0.59      ! [X3: rat] :
% 0.30/0.59        ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59        = ( uminus_uminus_rat @ X3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_minus1
% 0.30/0.59  thf(fact_5136_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: set_int,B: set_int] :
% 0.30/0.59        ( ( ( set_or370866239135849197et_int @ A @ B )
% 0.30/0.59          = bot_bot_set_set_int )
% 0.30/0.59        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5137_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( set_or633870826150836451st_rat @ A @ B )
% 0.30/0.59          = bot_bot_set_rat )
% 0.30/0.59        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5138_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: num,B: num] :
% 0.30/0.59        ( ( ( set_or7049704709247886629st_num @ A @ B )
% 0.30/0.59          = bot_bot_set_num )
% 0.30/0.59        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5139_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: nat,B: nat] :
% 0.30/0.59        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.59          = bot_bot_set_nat )
% 0.30/0.59        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5140_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.59          = bot_bot_set_int )
% 0.30/0.59        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5141_atLeastatMost__empty__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.59          = bot_bot_set_real )
% 0.30/0.59        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff
% 0.30/0.59  thf(fact_5142_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: set_int,B: set_int] :
% 0.30/0.59        ( ( bot_bot_set_set_int
% 0.30/0.59          = ( set_or370866239135849197et_int @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5143_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( bot_bot_set_rat
% 0.30/0.59          = ( set_or633870826150836451st_rat @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5144_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: num,B: num] :
% 0.30/0.59        ( ( bot_bot_set_num
% 0.30/0.59          = ( set_or7049704709247886629st_num @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5145_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: nat,B: nat] :
% 0.30/0.59        ( ( bot_bot_set_nat
% 0.30/0.59          = ( set_or1269000886237332187st_nat @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5146_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( bot_bot_set_int
% 0.30/0.59          = ( set_or1266510415728281911st_int @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5147_atLeastatMost__empty__iff2,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( bot_bot_set_real
% 0.30/0.59          = ( set_or1222579329274155063t_real @ A @ B ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty_iff2
% 0.30/0.59  thf(fact_5148_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.30/0.59        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.59            & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5149_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.59        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.59            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5150_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: num,B: num,C: num,D: num] :
% 0.30/0.59        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.59            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5151_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.59            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5152_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.59            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5153_atLeastatMost__subset__iff,axiom,
% 0.30/0.59      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.59        = ( ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.59          | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.59            & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_subset_iff
% 0.30/0.59  thf(fact_5154_atLeastatMost__empty,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ B @ A )
% 0.30/0.59       => ( ( set_or633870826150836451st_rat @ A @ B )
% 0.30/0.59          = bot_bot_set_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty
% 0.30/0.59  thf(fact_5155_atLeastatMost__empty,axiom,
% 0.30/0.59      ! [B: num,A: num] :
% 0.30/0.59        ( ( ord_less_num @ B @ A )
% 0.30/0.59       => ( ( set_or7049704709247886629st_num @ A @ B )
% 0.30/0.59          = bot_bot_set_num ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty
% 0.30/0.59  thf(fact_5156_atLeastatMost__empty,axiom,
% 0.30/0.59      ! [B: nat,A: nat] :
% 0.30/0.59        ( ( ord_less_nat @ B @ A )
% 0.30/0.59       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.59          = bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty
% 0.30/0.59  thf(fact_5157_atLeastatMost__empty,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( ord_less_int @ B @ A )
% 0.30/0.59       => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.59          = bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty
% 0.30/0.59  thf(fact_5158_atLeastatMost__empty,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( ord_less_real @ B @ A )
% 0.30/0.59       => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.59          = bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_empty
% 0.30/0.59  thf(fact_5159_infinite__Icc__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 0.30/0.59        = ( ord_less_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_Icc_iff
% 0.30/0.59  thf(fact_5160_infinite__Icc__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 0.30/0.59        = ( ord_less_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_Icc_iff
% 0.30/0.59  thf(fact_5161_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 0.30/0.59        ( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.59        = ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5162_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_set_nat,B: set_nat] :
% 0.30/0.59        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 0.30/0.59        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5163_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 0.30/0.59        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.59        = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5164_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_real,B: real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 0.30/0.59        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5165_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_nat,B: nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 0.30/0.59        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5166_subset__Compl__singleton,axiom,
% 0.30/0.59      ! [A2: set_int,B: int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 0.30/0.59        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_singleton
% 0.30/0.59  thf(fact_5167_take__bit__0,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 0.30/0.59        = zero_zero_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_0
% 0.30/0.59  thf(fact_5168_take__bit__0,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_0
% 0.30/0.59  thf(fact_5169_atLeastAtMost__singleton,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( set_or1269000886237332187st_nat @ A @ A )
% 0.30/0.59        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton
% 0.30/0.59  thf(fact_5170_atLeastAtMost__singleton,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( set_or1266510415728281911st_int @ A @ A )
% 0.30/0.59        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton
% 0.30/0.59  thf(fact_5171_atLeastAtMost__singleton,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( set_or1222579329274155063t_real @ A @ A )
% 0.30/0.59        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton
% 0.30/0.59  thf(fact_5172_atLeastAtMost__singleton__iff,axiom,
% 0.30/0.59      ! [A: nat,B: nat,C: nat] :
% 0.30/0.59        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.59          = ( insert_nat @ C @ bot_bot_set_nat ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          & ( B = C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton_iff
% 0.30/0.59  thf(fact_5173_atLeastAtMost__singleton__iff,axiom,
% 0.30/0.59      ! [A: int,B: int,C: int] :
% 0.30/0.59        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.59          = ( insert_int @ C @ bot_bot_set_int ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          & ( B = C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton_iff
% 0.30/0.59  thf(fact_5174_atLeastAtMost__singleton__iff,axiom,
% 0.30/0.59      ! [A: real,B: real,C: real] :
% 0.30/0.59        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.59          = ( insert_real @ C @ bot_bot_set_real ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          & ( B = C ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton_iff
% 0.30/0.59  thf(fact_5175_take__bit__Suc__1,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
% 0.30/0.59        = one_one_nat ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_1
% 0.30/0.59  thf(fact_5176_take__bit__Suc__1,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
% 0.30/0.59        = one_one_int ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_1
% 0.30/0.59  thf(fact_5177_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.59      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 0.30/0.59      = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(8)
% 0.30/0.59  thf(fact_5178_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.59      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(8)
% 0.30/0.59  thf(fact_5179_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.59      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 0.30/0.59      = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(8)
% 0.30/0.59  thf(fact_5180_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.59      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 0.30/0.59      = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(8)
% 0.30/0.59  thf(fact_5181_add__neg__numeral__special_I8_J,axiom,
% 0.30/0.59      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 0.30/0.59      = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(8)
% 0.30/0.59  thf(fact_5182_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.59      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.59      = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(7)
% 0.30/0.59  thf(fact_5183_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.59      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(7)
% 0.30/0.59  thf(fact_5184_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.59      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59      = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(7)
% 0.30/0.59  thf(fact_5185_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.59      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59      = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(7)
% 0.30/0.59  thf(fact_5186_add__neg__numeral__special_I7_J,axiom,
% 0.30/0.59      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.59      = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(7)
% 0.30/0.59  thf(fact_5187_diff__numeral__special_I12_J,axiom,
% 0.30/0.59      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.59      = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_numeral_special(12)
% 0.30/0.59  thf(fact_5188_diff__numeral__special_I12_J,axiom,
% 0.30/0.59      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_numeral_special(12)
% 0.30/0.59  thf(fact_5189_diff__numeral__special_I12_J,axiom,
% 0.30/0.59      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59      = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_numeral_special(12)
% 0.30/0.59  thf(fact_5190_diff__numeral__special_I12_J,axiom,
% 0.30/0.59      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59      = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_numeral_special(12)
% 0.30/0.59  thf(fact_5191_diff__numeral__special_I12_J,axiom,
% 0.30/0.59      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.59      = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_numeral_special(12)
% 0.30/0.59  thf(fact_5192_mod__minus1__right,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_minus1_right
% 0.30/0.59  thf(fact_5193_mod__minus1__right,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.59        = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % mod_minus1_right
% 0.30/0.59  thf(fact_5194_take__bit__of__1__eq__0__iff,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
% 0.30/0.59          = zero_zero_nat )
% 0.30/0.59        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_1_eq_0_iff
% 0.30/0.59  thf(fact_5195_take__bit__of__1__eq__0__iff,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
% 0.30/0.59          = zero_zero_int )
% 0.30/0.59        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_1_eq_0_iff
% 0.30/0.59  thf(fact_5196_semiring__norm_I168_J,axiom,
% 0.30/0.59      ! [V: num,W2: num,Y3: int] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
% 0.30/0.59        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_norm(168)
% 0.30/0.59  thf(fact_5197_semiring__norm_I168_J,axiom,
% 0.30/0.59      ! [V: num,W2: num,Y3: real] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y3 ) )
% 0.30/0.59        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_norm(168)
% 0.30/0.59  thf(fact_5198_semiring__norm_I168_J,axiom,
% 0.30/0.59      ! [V: num,W2: num,Y3: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
% 0.30/0.59        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_norm(168)
% 0.30/0.59  thf(fact_5199_semiring__norm_I168_J,axiom,
% 0.30/0.59      ! [V: num,W2: num,Y3: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % semiring_norm(168)
% 0.30/0.59  thf(fact_5200_neg__numeral__le__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_iff
% 0.30/0.59  thf(fact_5201_neg__numeral__le__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_iff
% 0.30/0.59  thf(fact_5202_neg__numeral__le__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_iff
% 0.30/0.59  thf(fact_5203_neg__numeral__le__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.59        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_iff
% 0.30/0.59  thf(fact_5204_neg__numeral__less__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.59        = ( ord_less_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_iff
% 0.30/0.59  thf(fact_5205_neg__numeral__less__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.30/0.59        = ( ord_less_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_iff
% 0.30/0.59  thf(fact_5206_neg__numeral__less__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.30/0.59        = ( ord_less_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_iff
% 0.30/0.59  thf(fact_5207_neg__numeral__less__iff,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.30/0.59        = ( ord_less_num @ N2 @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_iff
% 0.30/0.59  thf(fact_5208_zle__diff1__eq,axiom,
% 0.30/0.59      ! [W2: int,Z2: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 0.30/0.59        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zle_diff1_eq
% 0.30/0.59  thf(fact_5209_take__bit__of__Suc__0,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.59        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_Suc_0
% 0.30/0.59  thf(fact_5210_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_le_neg_numeral_iff
% 0.30/0.59  thf(fact_5211_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_le_neg_numeral_iff
% 0.30/0.59  thf(fact_5212_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_le_neg_numeral_iff
% 0.30/0.59  thf(fact_5213_not__neg__one__le__neg__numeral__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_le_neg_numeral_iff
% 0.30/0.59  thf(fact_5214_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: real,W2: num,A: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 0.30/0.59        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_le_eq_numeral1(2)
% 0.30/0.59  thf(fact_5215_divide__le__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: rat,W2: num,A: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 0.30/0.59        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_le_eq_numeral1(2)
% 0.30/0.59  thf(fact_5216_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: real,B: real,W2: num] :
% 0.30/0.59        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.30/0.59        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5217_le__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: rat,B: rat,W2: num] :
% 0.30/0.59        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.30/0.59        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5218_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: complex,W2: num,A: complex] :
% 0.30/0.59        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.59             != zero_zero_complex )
% 0.30/0.59           => ( B
% 0.30/0.59              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 0.30/0.59          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.59              = zero_zero_complex )
% 0.30/0.59           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_eq_numeral1(2)
% 0.30/0.59  thf(fact_5219_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: real,W2: num,A: real] :
% 0.30/0.59        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.59             != zero_zero_real )
% 0.30/0.59           => ( B
% 0.30/0.59              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 0.30/0.59          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.59              = zero_zero_real )
% 0.30/0.59           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_eq_numeral1(2)
% 0.30/0.59  thf(fact_5220_divide__eq__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: rat,W2: num,A: rat] :
% 0.30/0.59        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.59             != zero_zero_rat )
% 0.30/0.59           => ( B
% 0.30/0.59              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 0.30/0.59          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.59              = zero_zero_rat )
% 0.30/0.59           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_eq_numeral1(2)
% 0.30/0.59  thf(fact_5221_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: complex,B: complex,W2: num] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 0.30/0.59        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.59             != zero_zero_complex )
% 0.30/0.59           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.30/0.59              = B ) )
% 0.30/0.59          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.59              = zero_zero_complex )
% 0.30/0.59           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5222_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: real,B: real,W2: num] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.30/0.59        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.59             != zero_zero_real )
% 0.30/0.59           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.30/0.59              = B ) )
% 0.30/0.59          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.59              = zero_zero_real )
% 0.30/0.59           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5223_eq__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: rat,B: rat,W2: num] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.30/0.59        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.59             != zero_zero_rat )
% 0.30/0.59           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.30/0.59              = B ) )
% 0.30/0.59          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.59              = zero_zero_rat )
% 0.30/0.59           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5224_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_neg_one_iff
% 0.30/0.59  thf(fact_5225_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_neg_one_iff
% 0.30/0.59  thf(fact_5226_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_neg_one_iff
% 0.30/0.59  thf(fact_5227_neg__numeral__less__neg__one__iff,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.59        = ( M != one ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_neg_one_iff
% 0.30/0.59  thf(fact_5228_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: real,B: real,W2: num] :
% 0.30/0.59        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 0.30/0.59        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5229_less__divide__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [A: rat,B: rat,W2: num] :
% 0.30/0.59        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 0.30/0.59        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_divide_eq_numeral1(2)
% 0.30/0.59  thf(fact_5230_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: real,W2: num,A: real] :
% 0.30/0.59        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 0.30/0.59        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_less_eq_numeral1(2)
% 0.30/0.59  thf(fact_5231_divide__less__eq__numeral1_I2_J,axiom,
% 0.30/0.59      ! [B: rat,W2: num,A: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 0.30/0.59        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_less_eq_numeral1(2)
% 0.30/0.59  thf(fact_5232_take__bit__of__1,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
% 0.30/0.59        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_1
% 0.30/0.59  thf(fact_5233_take__bit__of__1,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
% 0.30/0.59        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_1
% 0.30/0.59  thf(fact_5234_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.59      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.59      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(9)
% 0.30/0.59  thf(fact_5235_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.59      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(9)
% 0.30/0.59  thf(fact_5236_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.59      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(9)
% 0.30/0.59  thf(fact_5237_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.59      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(9)
% 0.30/0.59  thf(fact_5238_add__neg__numeral__special_I9_J,axiom,
% 0.30/0.59      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.59      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_neg_numeral_special(9)
% 0.30/0.59  thf(fact_5239_even__take__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
% 0.30/0.59        = ( ( N2 = zero_zero_nat )
% 0.30/0.59          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_take_bit_eq
% 0.30/0.59  thf(fact_5240_even__take__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,A: nat] :
% 0.30/0.59        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
% 0.30/0.59        = ( ( N2 = zero_zero_nat )
% 0.30/0.59          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_take_bit_eq
% 0.30/0.59  thf(fact_5241_even__take__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
% 0.30/0.59        = ( ( N2 = zero_zero_nat )
% 0.30/0.59          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % even_take_bit_eq
% 0.30/0.59  thf(fact_5242_signed__take__bit__Suc__minus__bit0,axiom,
% 0.30/0.59      ! [N2: nat,K: num] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.59        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_Suc_minus_bit0
% 0.30/0.59  thf(fact_5243_take__bit__Suc__0,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 0.30/0.59        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_0
% 0.30/0.59  thf(fact_5244_take__bit__Suc__0,axiom,
% 0.30/0.59      ! [A: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 0.30/0.59        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_0
% 0.30/0.59  thf(fact_5245_take__bit__Suc__0,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 0.30/0.59        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_0
% 0.30/0.59  thf(fact_5246_signed__take__bit__0,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 0.30/0.59        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_0
% 0.30/0.59  thf(fact_5247_signed__take__bit__0,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 0.30/0.59        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_0
% 0.30/0.59  thf(fact_5248_take__bit__of__exp,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_exp
% 0.30/0.59  thf(fact_5249_take__bit__of__exp,axiom,
% 0.30/0.59      ! [M: nat,N2: nat] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_exp
% 0.30/0.59  thf(fact_5250_take__bit__of__2,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_2
% 0.30/0.59  thf(fact_5251_take__bit__of__2,axiom,
% 0.30/0.59      ! [N2: nat] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.59        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_of_2
% 0.30/0.59  thf(fact_5252_minus__int__code_I2_J,axiom,
% 0.30/0.59      ! [L: int] :
% 0.30/0.59        ( ( minus_minus_int @ zero_zero_int @ L )
% 0.30/0.59        = ( uminus_uminus_int @ L ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_int_code(2)
% 0.30/0.59  thf(fact_5253_compl__mono,axiom,
% 0.30/0.59      ! [X3: set_int,Y3: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 0.30/0.59       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % compl_mono
% 0.30/0.59  thf(fact_5254_compl__le__swap1,axiom,
% 0.30/0.59      ! [Y3: set_int,X3: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
% 0.30/0.59       => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % compl_le_swap1
% 0.30/0.59  thf(fact_5255_compl__le__swap2,axiom,
% 0.30/0.59      ! [Y3: set_int,X3: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
% 0.30/0.59       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % compl_le_swap2
% 0.30/0.59  thf(fact_5256_equation__minus__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equation_minus_iff
% 0.30/0.59  thf(fact_5257_equation__minus__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equation_minus_iff
% 0.30/0.59  thf(fact_5258_equation__minus__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equation_minus_iff
% 0.30/0.59  thf(fact_5259_equation__minus__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % equation_minus_iff
% 0.30/0.59  thf(fact_5260_minus__equation__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( uminus_uminus_int @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( uminus_uminus_int @ B )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_equation_iff
% 0.30/0.59  thf(fact_5261_minus__equation__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( uminus_uminus_real @ B )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_equation_iff
% 0.30/0.59  thf(fact_5262_minus__equation__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( uminus_uminus_rat @ B )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_equation_iff
% 0.30/0.59  thf(fact_5263_minus__equation__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( uminus1351360451143612070nteger @ B )
% 0.30/0.59          = A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_equation_iff
% 0.30/0.59  thf(fact_5264_Compl__insert,axiom,
% 0.30/0.59      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
% 0.30/0.59        = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_insert
% 0.30/0.59  thf(fact_5265_Compl__insert,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
% 0.30/0.59        = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_insert
% 0.30/0.59  thf(fact_5266_Compl__insert,axiom,
% 0.30/0.59      ! [X3: real,A2: set_real] :
% 0.30/0.59        ( ( uminus612125837232591019t_real @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.59        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_insert
% 0.30/0.59  thf(fact_5267_Compl__insert,axiom,
% 0.30/0.59      ! [X3: int,A2: set_int] :
% 0.30/0.59        ( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.59        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_insert
% 0.30/0.59  thf(fact_5268_Compl__insert,axiom,
% 0.30/0.59      ! [X3: nat,A2: set_nat] :
% 0.30/0.59        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.59        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Compl_insert
% 0.30/0.59  thf(fact_5269_take__bit__add,axiom,
% 0.30/0.59      ! [N2: nat,A: nat,B: nat] :
% 0.30/0.59        ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
% 0.30/0.59        = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_add
% 0.30/0.59  thf(fact_5270_take__bit__add,axiom,
% 0.30/0.59      ! [N2: nat,A: int,B: int] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
% 0.30/0.59        = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_add
% 0.30/0.59  thf(fact_5271_take__bit__tightened,axiom,
% 0.30/0.59      ! [N2: nat,A: nat,B: nat,M: nat] :
% 0.30/0.59        ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 0.30/0.59          = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
% 0.30/0.59       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 0.30/0.59            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_tightened
% 0.30/0.59  thf(fact_5272_take__bit__tightened,axiom,
% 0.30/0.59      ! [N2: nat,A: int,B: int,M: nat] :
% 0.30/0.59        ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 0.30/0.59          = ( bit_se2923211474154528505it_int @ N2 @ B ) )
% 0.30/0.59       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 0.30/0.59            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_tightened
% 0.30/0.59  thf(fact_5273_take__bit__nat__less__eq__self,axiom,
% 0.30/0.59      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_nat_less_eq_self
% 0.30/0.59  thf(fact_5274_take__bit__tightened__less__eq__nat,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q4 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_tightened_less_eq_nat
% 0.30/0.59  thf(fact_5275_le__imp__neg__le,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ A @ B )
% 0.30/0.59       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_imp_neg_le
% 0.30/0.59  thf(fact_5276_le__imp__neg__le,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.59       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_imp_neg_le
% 0.30/0.59  thf(fact_5277_le__imp__neg__le,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.59       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_imp_neg_le
% 0.30/0.59  thf(fact_5278_le__imp__neg__le,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ A @ B )
% 0.30/0.59       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_imp_neg_le
% 0.30/0.59  thf(fact_5279_minus__le__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_le_iff
% 0.30/0.59  thf(fact_5280_minus__le__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_le_iff
% 0.30/0.59  thf(fact_5281_minus__le__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_le_iff
% 0.30/0.59  thf(fact_5282_minus__le__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_le_iff
% 0.30/0.59  thf(fact_5283_le__minus__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_iff
% 0.30/0.59  thf(fact_5284_le__minus__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_iff
% 0.30/0.59  thf(fact_5285_le__minus__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_iff
% 0.30/0.59  thf(fact_5286_le__minus__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_iff
% 0.30/0.59  thf(fact_5287_minus__less__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_less_iff
% 0.30/0.59  thf(fact_5288_minus__less__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_less_iff
% 0.30/0.59  thf(fact_5289_minus__less__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_less_iff
% 0.30/0.59  thf(fact_5290_minus__less__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_less_iff
% 0.30/0.59  thf(fact_5291_less__minus__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_iff
% 0.30/0.59  thf(fact_5292_less__minus__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_iff
% 0.30/0.59  thf(fact_5293_less__minus__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_iff
% 0.30/0.59  thf(fact_5294_less__minus__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_iff
% 0.30/0.59  thf(fact_5295_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ord_less_int @ A @ B )
% 0.30/0.59       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_negate_coefficient(2)
% 0.30/0.59  thf(fact_5296_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ B )
% 0.30/0.59       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_negate_coefficient(2)
% 0.30/0.59  thf(fact_5297_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ B )
% 0.30/0.59       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_negate_coefficient(2)
% 0.30/0.59  thf(fact_5298_verit__negate__coefficient_I2_J,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.59       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % verit_negate_coefficient(2)
% 0.30/0.59  thf(fact_5299_square__eq__iff,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( ( times_times_complex @ A @ A )
% 0.30/0.59          = ( times_times_complex @ B @ B ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          | ( A
% 0.30/0.59            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_iff
% 0.30/0.59  thf(fact_5300_square__eq__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( times_times_int @ A @ A )
% 0.30/0.59          = ( times_times_int @ B @ B ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          | ( A
% 0.30/0.59            = ( uminus_uminus_int @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_iff
% 0.30/0.59  thf(fact_5301_square__eq__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( times_times_real @ A @ A )
% 0.30/0.59          = ( times_times_real @ B @ B ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          | ( A
% 0.30/0.59            = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_iff
% 0.30/0.59  thf(fact_5302_square__eq__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( times_times_rat @ A @ A )
% 0.30/0.59          = ( times_times_rat @ B @ B ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          | ( A
% 0.30/0.59            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_iff
% 0.30/0.59  thf(fact_5303_square__eq__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( times_3573771949741848930nteger @ A @ A )
% 0.30/0.59          = ( times_3573771949741848930nteger @ B @ B ) )
% 0.30/0.59        = ( ( A = B )
% 0.30/0.59          | ( A
% 0.30/0.59            = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_iff
% 0.30/0.59  thf(fact_5304_minus__mult__commute,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 0.30/0.59        = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_commute
% 0.30/0.59  thf(fact_5305_minus__mult__commute,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59        = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_commute
% 0.30/0.59  thf(fact_5306_minus__mult__commute,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59        = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_commute
% 0.30/0.59  thf(fact_5307_minus__mult__commute,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59        = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_commute
% 0.30/0.59  thf(fact_5308_minus__mult__commute,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59        = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_mult_commute
% 0.30/0.59  thf(fact_5309_is__num__normalize_I8_J,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.59        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % is_num_normalize(8)
% 0.30/0.59  thf(fact_5310_is__num__normalize_I8_J,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.59        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % is_num_normalize(8)
% 0.30/0.59  thf(fact_5311_is__num__normalize_I8_J,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.59        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % is_num_normalize(8)
% 0.30/0.59  thf(fact_5312_is__num__normalize_I8_J,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % is_num_normalize(8)
% 0.30/0.59  thf(fact_5313_add_Oinverse__distrib__swap,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.30/0.59        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_distrib_swap
% 0.30/0.59  thf(fact_5314_add_Oinverse__distrib__swap,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 0.30/0.59        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_distrib_swap
% 0.30/0.59  thf(fact_5315_add_Oinverse__distrib__swap,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 0.30/0.59        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_distrib_swap
% 0.30/0.59  thf(fact_5316_add_Oinverse__distrib__swap,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 0.30/0.59        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_distrib_swap
% 0.30/0.59  thf(fact_5317_group__cancel_Oneg1,axiom,
% 0.30/0.59      ! [A2: int,K: int,A: int] :
% 0.30/0.59        ( ( A2
% 0.30/0.59          = ( plus_plus_int @ K @ A ) )
% 0.30/0.59       => ( ( uminus_uminus_int @ A2 )
% 0.30/0.59          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.neg1
% 0.30/0.59  thf(fact_5318_group__cancel_Oneg1,axiom,
% 0.30/0.59      ! [A2: real,K: real,A: real] :
% 0.30/0.59        ( ( A2
% 0.30/0.59          = ( plus_plus_real @ K @ A ) )
% 0.30/0.59       => ( ( uminus_uminus_real @ A2 )
% 0.30/0.59          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.neg1
% 0.30/0.59  thf(fact_5319_group__cancel_Oneg1,axiom,
% 0.30/0.59      ! [A2: rat,K: rat,A: rat] :
% 0.30/0.59        ( ( A2
% 0.30/0.59          = ( plus_plus_rat @ K @ A ) )
% 0.30/0.59       => ( ( uminus_uminus_rat @ A2 )
% 0.30/0.59          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.neg1
% 0.30/0.59  thf(fact_5320_group__cancel_Oneg1,axiom,
% 0.30/0.59      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 0.30/0.59        ( ( A2
% 0.30/0.59          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 0.30/0.59       => ( ( uminus1351360451143612070nteger @ A2 )
% 0.30/0.59          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.neg1
% 0.30/0.59  thf(fact_5321_minus__diff__commute,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.30/0.59        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_commute
% 0.30/0.59  thf(fact_5322_minus__diff__commute,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 0.30/0.59        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_commute
% 0.30/0.59  thf(fact_5323_minus__diff__commute,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 0.30/0.59        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_commute
% 0.30/0.59  thf(fact_5324_minus__diff__commute,axiom,
% 0.30/0.59      ! [B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 0.30/0.59        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_diff_commute
% 0.30/0.59  thf(fact_5325_minus__divide__right,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.59        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_right
% 0.30/0.59  thf(fact_5326_minus__divide__right,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.59        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_right
% 0.30/0.59  thf(fact_5327_minus__divide__divide,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( divide_divide_real @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_divide
% 0.30/0.59  thf(fact_5328_minus__divide__divide,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( divide_divide_rat @ A @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_divide
% 0.30/0.59  thf(fact_5329_minus__divide__left,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.59        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_left
% 0.30/0.59  thf(fact_5330_minus__divide__left,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.59        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_left
% 0.30/0.59  thf(fact_5331_Diff__infinite__finite,axiom,
% 0.30/0.59      ! [T3: set_int,S3: set_int] :
% 0.30/0.59        ( ( finite_finite_int @ T3 )
% 0.30/0.59       => ( ~ ( finite_finite_int @ S3 )
% 0.30/0.59         => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_infinite_finite
% 0.30/0.59  thf(fact_5332_Diff__infinite__finite,axiom,
% 0.30/0.59      ! [T3: set_complex,S3: set_complex] :
% 0.30/0.59        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.59       => ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.30/0.59         => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_infinite_finite
% 0.30/0.59  thf(fact_5333_Diff__infinite__finite,axiom,
% 0.30/0.59      ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( finite6177210948735845034at_nat @ T3 )
% 0.30/0.59       => ( ~ ( finite6177210948735845034at_nat @ S3 )
% 0.30/0.59         => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ T3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_infinite_finite
% 0.30/0.59  thf(fact_5334_Diff__infinite__finite,axiom,
% 0.30/0.59      ! [T3: set_nat,S3: set_nat] :
% 0.30/0.59        ( ( finite_finite_nat @ T3 )
% 0.30/0.59       => ( ~ ( finite_finite_nat @ S3 )
% 0.30/0.59         => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_infinite_finite
% 0.30/0.59  thf(fact_5335_double__diff,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat,C4: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 0.30/0.59         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.59            = A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % double_diff
% 0.30/0.59  thf(fact_5336_double__diff,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int,C4: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 0.30/0.59       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 0.30/0.59         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 0.30/0.59            = A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % double_diff
% 0.30/0.59  thf(fact_5337_Diff__subset,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_subset
% 0.30/0.59  thf(fact_5338_Diff__subset,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_subset
% 0.30/0.59  thf(fact_5339_Diff__mono,axiom,
% 0.30/0.59      ! [A2: set_nat,C4: set_nat,D6: set_nat,B2: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.59       => ( ( ord_less_eq_set_nat @ D6 @ B2 )
% 0.30/0.59         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D6 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_mono
% 0.30/0.59  thf(fact_5340_Diff__mono,axiom,
% 0.30/0.59      ! [A2: set_int,C4: set_int,D6: set_int,B2: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ C4 )
% 0.30/0.59       => ( ( ord_less_eq_set_int @ D6 @ B2 )
% 0.30/0.59         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C4 @ D6 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_mono
% 0.30/0.59  thf(fact_5341_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert9069300056098147895at_nat @ X3 @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5342_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert8211810215607154385at_nat @ X3 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5343_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: real,B2: set_real,A2: set_real] :
% 0.30/0.59        ( ( ( member_real @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member_real @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert_real @ X3 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5344_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 0.30/0.59        ( ( ( member_set_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member_set_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert_set_nat @ X3 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5345_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: int,B2: set_int,A2: set_int] :
% 0.30/0.59        ( ( ( member_int @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member_int @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert_int @ X3 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5346_insert__Diff__if,axiom,
% 0.30/0.59      ! [X3: nat,B2: set_nat,A2: set_nat] :
% 0.30/0.59        ( ( ( member_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 0.30/0.59        & ( ~ ( member_nat @ X3 @ B2 )
% 0.30/0.59         => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 0.30/0.59            = ( insert_nat @ X3 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff_if
% 0.30/0.59  thf(fact_5347_minus__int__code_I1_J,axiom,
% 0.30/0.59      ! [K: int] :
% 0.30/0.59        ( ( minus_minus_int @ K @ zero_zero_int )
% 0.30/0.59        = K ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_int_code(1)
% 0.30/0.59  thf(fact_5348_uminus__int__code_I1_J,axiom,
% 0.30/0.59      ( ( uminus_uminus_int @ zero_zero_int )
% 0.30/0.59      = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % uminus_int_code(1)
% 0.30/0.59  thf(fact_5349_int__distrib_I4_J,axiom,
% 0.30/0.59      ! [W2: int,Z1: int,Z22: int] :
% 0.30/0.59        ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
% 0.30/0.59        = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_distrib(4)
% 0.30/0.59  thf(fact_5350_int__distrib_I3_J,axiom,
% 0.30/0.59      ! [Z1: int,Z22: int,W2: int] :
% 0.30/0.59        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
% 0.30/0.59        = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_distrib(3)
% 0.30/0.59  thf(fact_5351_psubset__imp__ex__mem,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 0.30/0.59       => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % psubset_imp_ex_mem
% 0.30/0.59  thf(fact_5352_psubset__imp__ex__mem,axiom,
% 0.30/0.59      ! [A2: set_real,B2: set_real] :
% 0.30/0.59        ( ( ord_less_set_real @ A2 @ B2 )
% 0.30/0.59       => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % psubset_imp_ex_mem
% 0.30/0.59  thf(fact_5353_psubset__imp__ex__mem,axiom,
% 0.30/0.59      ! [A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.59        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 0.30/0.59       => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % psubset_imp_ex_mem
% 0.30/0.59  thf(fact_5354_psubset__imp__ex__mem,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int] :
% 0.30/0.59        ( ( ord_less_set_int @ A2 @ B2 )
% 0.30/0.59       => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % psubset_imp_ex_mem
% 0.30/0.59  thf(fact_5355_psubset__imp__ex__mem,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat] :
% 0.30/0.59        ( ( ord_less_set_nat @ A2 @ B2 )
% 0.30/0.59       => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % psubset_imp_ex_mem
% 0.30/0.59  thf(fact_5356_zdvd__zdiffD,axiom,
% 0.30/0.59      ! [K: int,M: int,N2: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
% 0.30/0.59       => ( ( dvd_dvd_int @ K @ N2 )
% 0.30/0.59         => ( dvd_dvd_int @ K @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zdvd_zdiffD
% 0.30/0.59  thf(fact_5357_zmod__minus1,axiom,
% 0.30/0.59      ! [B: int] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.59       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.30/0.59          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zmod_minus1
% 0.30/0.59  thf(fact_5358_take__bit__minus__small__eq,axiom,
% 0.30/0.59      ! [K: int,N2: nat] :
% 0.30/0.59        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.59       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
% 0.30/0.59            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_minus_small_eq
% 0.30/0.59  thf(fact_5359_infinite__Icc,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ord_less_rat @ A @ B )
% 0.30/0.59       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_Icc
% 0.30/0.59  thf(fact_5360_infinite__Icc,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ord_less_real @ A @ B )
% 0.30/0.59       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % infinite_Icc
% 0.30/0.59  thf(fact_5361_atLeastAtMost__singleton_H,axiom,
% 0.30/0.59      ! [A: nat,B: nat] :
% 0.30/0.59        ( ( A = B )
% 0.30/0.59       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 0.30/0.59          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton'
% 0.30/0.59  thf(fact_5362_atLeastAtMost__singleton_H,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( A = B )
% 0.30/0.59       => ( ( set_or1266510415728281911st_int @ A @ B )
% 0.30/0.59          = ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton'
% 0.30/0.59  thf(fact_5363_atLeastAtMost__singleton_H,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( A = B )
% 0.30/0.59       => ( ( set_or1222579329274155063t_real @ A @ B )
% 0.30/0.59          = ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastAtMost_singleton'
% 0.30/0.59  thf(fact_5364_take__bit__tightened__less__eq__int,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,K: int] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.59       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_tightened_less_eq_int
% 0.30/0.59  thf(fact_5365_take__bit__int__greater__self__iff,axiom,
% 0.30/0.59      ! [K: int,N2: nat] :
% 0.30/0.59        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 0.30/0.59        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_int_greater_self_iff
% 0.30/0.59  thf(fact_5366_not__take__bit__negative,axiom,
% 0.30/0.59      ! [N2: nat,K: int] :
% 0.30/0.59        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % not_take_bit_negative
% 0.30/0.59  thf(fact_5367_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,A: int,B: int] :
% 0.30/0.59        ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
% 0.30/0.59          = ( bit_ri631733984087533419it_int @ N2 @ B ) )
% 0.30/0.59        = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
% 0.30/0.59          = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_eq_iff_take_bit_eq
% 0.30/0.59  thf(fact_5368_signed__take__bit__take__bit,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,A: int] :
% 0.30/0.59        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
% 0.30/0.59        = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_take_bit
% 0.30/0.59  thf(fact_5369_ex__nat__less,axiom,
% 0.30/0.59      ! [N2: nat,P: nat > $o] :
% 0.30/0.59        ( ( ? [M2: nat] :
% 0.30/0.59              ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.30/0.59              & ( P @ M2 ) ) )
% 0.30/0.59        = ( ? [X: nat] :
% 0.30/0.59              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.59              & ( P @ X ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % ex_nat_less
% 0.30/0.59  thf(fact_5370_all__nat__less,axiom,
% 0.30/0.59      ! [N2: nat,P: nat > $o] :
% 0.30/0.59        ( ( ! [M2: nat] :
% 0.30/0.59              ( ( ord_less_eq_nat @ M2 @ N2 )
% 0.30/0.59             => ( P @ M2 ) ) )
% 0.30/0.59        = ( ! [X: nat] :
% 0.30/0.59              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.59             => ( P @ X ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % all_nat_less
% 0.30/0.59  thf(fact_5371_neg__numeral__le__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_numeral
% 0.30/0.59  thf(fact_5372_neg__numeral__le__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_numeral
% 0.30/0.59  thf(fact_5373_neg__numeral__le__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_numeral
% 0.30/0.59  thf(fact_5374_neg__numeral__le__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_numeral
% 0.30/0.59  thf(fact_5375_not__numeral__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_numeral
% 0.30/0.59  thf(fact_5376_not__numeral__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_numeral
% 0.30/0.59  thf(fact_5377_not__numeral__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_numeral
% 0.30/0.59  thf(fact_5378_not__numeral__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_numeral
% 0.30/0.59  thf(fact_5379_zero__neq__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ( zero_zero_complex
% 0.30/0.59       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_numeral
% 0.30/0.59  thf(fact_5380_zero__neq__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ( zero_zero_int
% 0.30/0.59       != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_numeral
% 0.30/0.59  thf(fact_5381_zero__neq__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ( zero_zero_real
% 0.30/0.59       != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_numeral
% 0.30/0.59  thf(fact_5382_zero__neq__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ( zero_zero_rat
% 0.30/0.59       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_numeral
% 0.30/0.59  thf(fact_5383_zero__neq__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ( zero_z3403309356797280102nteger
% 0.30/0.59       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_numeral
% 0.30/0.59  thf(fact_5384_not__numeral__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_numeral
% 0.30/0.59  thf(fact_5385_not__numeral__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_numeral
% 0.30/0.59  thf(fact_5386_not__numeral__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_numeral
% 0.30/0.59  thf(fact_5387_not__numeral__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] :
% 0.30/0.59        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_numeral
% 0.30/0.59  thf(fact_5388_neg__numeral__less__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_numeral
% 0.30/0.59  thf(fact_5389_neg__numeral__less__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_numeral
% 0.30/0.59  thf(fact_5390_neg__numeral__less__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_numeral
% 0.30/0.59  thf(fact_5391_neg__numeral__less__numeral,axiom,
% 0.30/0.59      ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_numeral
% 0.30/0.59  thf(fact_5392_le__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(2)
% 0.30/0.59  thf(fact_5393_le__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(2)
% 0.30/0.59  thf(fact_5394_le__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(2)
% 0.30/0.59  thf(fact_5395_le__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(2)
% 0.30/0.59  thf(fact_5396_le__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(4)
% 0.30/0.59  thf(fact_5397_le__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(4)
% 0.30/0.59  thf(fact_5398_le__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(4)
% 0.30/0.59  thf(fact_5399_le__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(4)
% 0.30/0.59  thf(fact_5400_zero__neq__neg__one,axiom,
% 0.30/0.59      ( zero_zero_complex
% 0.30/0.59     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_one
% 0.30/0.59  thf(fact_5401_zero__neq__neg__one,axiom,
% 0.30/0.59      ( zero_zero_int
% 0.30/0.59     != ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_one
% 0.30/0.59  thf(fact_5402_zero__neq__neg__one,axiom,
% 0.30/0.59      ( zero_zero_real
% 0.30/0.59     != ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_one
% 0.30/0.59  thf(fact_5403_zero__neq__neg__one,axiom,
% 0.30/0.59      ( zero_zero_rat
% 0.30/0.59     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_one
% 0.30/0.59  thf(fact_5404_zero__neq__neg__one,axiom,
% 0.30/0.59      ( zero_z3403309356797280102nteger
% 0.30/0.59     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zero_neq_neg_one
% 0.30/0.59  thf(fact_5405_add__eq__0__iff,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.59          = zero_zero_complex )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_0_iff
% 0.30/0.59  thf(fact_5406_add__eq__0__iff,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( plus_plus_int @ A @ B )
% 0.30/0.59          = zero_zero_int )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_0_iff
% 0.30/0.59  thf(fact_5407_add__eq__0__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( plus_plus_real @ A @ B )
% 0.30/0.59          = zero_zero_real )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_0_iff
% 0.30/0.59  thf(fact_5408_add__eq__0__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.59          = zero_zero_rat )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_0_iff
% 0.30/0.59  thf(fact_5409_add__eq__0__iff,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.59          = zero_z3403309356797280102nteger )
% 0.30/0.59        = ( B
% 0.30/0.59          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add_eq_0_iff
% 0.30/0.59  thf(fact_5410_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.59      ! [A: complex] :
% 0.30/0.59        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 0.30/0.59        = zero_zero_complex ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_left_minus
% 0.30/0.59  thf(fact_5411_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.59      ! [A: int] :
% 0.30/0.59        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 0.30/0.59        = zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_left_minus
% 0.30/0.59  thf(fact_5412_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.59      ! [A: real] :
% 0.30/0.59        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 0.30/0.59        = zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_left_minus
% 0.30/0.59  thf(fact_5413_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.59      ! [A: rat] :
% 0.30/0.59        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 0.30/0.59        = zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_left_minus
% 0.30/0.59  thf(fact_5414_ab__group__add__class_Oab__left__minus,axiom,
% 0.30/0.59      ! [A: code_integer] :
% 0.30/0.59        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 0.30/0.59        = zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_left_minus
% 0.30/0.59  thf(fact_5415_add_Oinverse__unique,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( ( plus_plus_complex @ A @ B )
% 0.30/0.59          = zero_zero_complex )
% 0.30/0.59       => ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.59          = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_unique
% 0.30/0.59  thf(fact_5416_add_Oinverse__unique,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( plus_plus_int @ A @ B )
% 0.30/0.59          = zero_zero_int )
% 0.30/0.59       => ( ( uminus_uminus_int @ A )
% 0.30/0.59          = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_unique
% 0.30/0.59  thf(fact_5417_add_Oinverse__unique,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( plus_plus_real @ A @ B )
% 0.30/0.59          = zero_zero_real )
% 0.30/0.59       => ( ( uminus_uminus_real @ A )
% 0.30/0.59          = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_unique
% 0.30/0.59  thf(fact_5418_add_Oinverse__unique,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( plus_plus_rat @ A @ B )
% 0.30/0.59          = zero_zero_rat )
% 0.30/0.59       => ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_unique
% 0.30/0.59  thf(fact_5419_add_Oinverse__unique,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.59          = zero_z3403309356797280102nteger )
% 0.30/0.59       => ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = B ) ) ).
% 0.30/0.59  
% 0.30/0.59  % add.inverse_unique
% 0.30/0.59  thf(fact_5420_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.59        = ( ( plus_plus_complex @ A @ B )
% 0.30/0.59          = zero_zero_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_neg_iff_add_eq_0
% 0.30/0.59  thf(fact_5421_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_int @ B ) )
% 0.30/0.59        = ( ( plus_plus_int @ A @ B )
% 0.30/0.59          = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_neg_iff_add_eq_0
% 0.30/0.59  thf(fact_5422_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_real @ B ) )
% 0.30/0.59        = ( ( plus_plus_real @ A @ B )
% 0.30/0.59          = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_neg_iff_add_eq_0
% 0.30/0.59  thf(fact_5423_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_rat @ B ) )
% 0.30/0.59        = ( ( plus_plus_rat @ A @ B )
% 0.30/0.59          = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_neg_iff_add_eq_0
% 0.30/0.59  thf(fact_5424_eq__neg__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.59          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_neg_iff_add_eq_0
% 0.30/0.59  thf(fact_5425_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( plus_plus_complex @ A @ B )
% 0.30/0.59          = zero_zero_complex ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eq_iff_add_eq_0
% 0.30/0.59  thf(fact_5426_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: int,B: int] :
% 0.30/0.59        ( ( ( uminus_uminus_int @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( plus_plus_int @ A @ B )
% 0.30/0.59          = zero_zero_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eq_iff_add_eq_0
% 0.30/0.59  thf(fact_5427_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( plus_plus_real @ A @ B )
% 0.30/0.59          = zero_zero_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eq_iff_add_eq_0
% 0.30/0.59  thf(fact_5428_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( plus_plus_rat @ A @ B )
% 0.30/0.59          = zero_zero_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eq_iff_add_eq_0
% 0.30/0.59  thf(fact_5429_neg__eq__iff__add__eq__0,axiom,
% 0.30/0.59      ! [A: code_integer,B: code_integer] :
% 0.30/0.59        ( ( ( uminus1351360451143612070nteger @ A )
% 0.30/0.59          = B )
% 0.30/0.59        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 0.30/0.59          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_eq_iff_add_eq_0
% 0.30/0.59  thf(fact_5430_less__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(4)
% 0.30/0.59  thf(fact_5431_less__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(4)
% 0.30/0.59  thf(fact_5432_less__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(4)
% 0.30/0.59  thf(fact_5433_less__minus__one__simps_I4_J,axiom,
% 0.30/0.59      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(4)
% 0.30/0.59  thf(fact_5434_less__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(2)
% 0.30/0.59  thf(fact_5435_less__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(2)
% 0.30/0.59  thf(fact_5436_less__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(2)
% 0.30/0.59  thf(fact_5437_less__minus__one__simps_I2_J,axiom,
% 0.30/0.59      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(2)
% 0.30/0.59  thf(fact_5438_nonzero__minus__divide__right,axiom,
% 0.30/0.59      ! [B: complex,A: complex] :
% 0.30/0.59        ( ( B != zero_zero_complex )
% 0.30/0.59       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.59          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_right
% 0.30/0.59  thf(fact_5439_nonzero__minus__divide__right,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( B != zero_zero_real )
% 0.30/0.59       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.59          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_right
% 0.30/0.59  thf(fact_5440_nonzero__minus__divide__right,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( B != zero_zero_rat )
% 0.30/0.59       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.59          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_right
% 0.30/0.59  thf(fact_5441_nonzero__minus__divide__divide,axiom,
% 0.30/0.59      ! [B: complex,A: complex] :
% 0.30/0.59        ( ( B != zero_zero_complex )
% 0.30/0.59       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 0.30/0.59          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_divide
% 0.30/0.59  thf(fact_5442_nonzero__minus__divide__divide,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( B != zero_zero_real )
% 0.30/0.59       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 0.30/0.59          = ( divide_divide_real @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_divide
% 0.30/0.59  thf(fact_5443_nonzero__minus__divide__divide,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( B != zero_zero_rat )
% 0.30/0.59       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59          = ( divide_divide_rat @ A @ B ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_minus_divide_divide
% 0.30/0.59  thf(fact_5444_square__eq__1__iff,axiom,
% 0.30/0.59      ! [X3: complex] :
% 0.30/0.59        ( ( ( times_times_complex @ X3 @ X3 )
% 0.30/0.59          = one_one_complex )
% 0.30/0.59        = ( ( X3 = one_one_complex )
% 0.30/0.59          | ( X3
% 0.30/0.59            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_1_iff
% 0.30/0.59  thf(fact_5445_square__eq__1__iff,axiom,
% 0.30/0.59      ! [X3: int] :
% 0.30/0.59        ( ( ( times_times_int @ X3 @ X3 )
% 0.30/0.59          = one_one_int )
% 0.30/0.59        = ( ( X3 = one_one_int )
% 0.30/0.59          | ( X3
% 0.30/0.59            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_1_iff
% 0.30/0.59  thf(fact_5446_square__eq__1__iff,axiom,
% 0.30/0.59      ! [X3: real] :
% 0.30/0.59        ( ( ( times_times_real @ X3 @ X3 )
% 0.30/0.59          = one_one_real )
% 0.30/0.59        = ( ( X3 = one_one_real )
% 0.30/0.59          | ( X3
% 0.30/0.59            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_1_iff
% 0.30/0.59  thf(fact_5447_square__eq__1__iff,axiom,
% 0.30/0.59      ! [X3: rat] :
% 0.30/0.59        ( ( ( times_times_rat @ X3 @ X3 )
% 0.30/0.59          = one_one_rat )
% 0.30/0.59        = ( ( X3 = one_one_rat )
% 0.30/0.59          | ( X3
% 0.30/0.59            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_1_iff
% 0.30/0.59  thf(fact_5448_square__eq__1__iff,axiom,
% 0.30/0.59      ! [X3: code_integer] :
% 0.30/0.59        ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
% 0.30/0.59          = one_one_Code_integer )
% 0.30/0.59        = ( ( X3 = one_one_Code_integer )
% 0.30/0.59          | ( X3
% 0.30/0.59            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % square_eq_1_iff
% 0.30/0.59  thf(fact_5449_group__cancel_Osub2,axiom,
% 0.30/0.59      ! [B2: int,K: int,B: int,A: int] :
% 0.30/0.59        ( ( B2
% 0.30/0.59          = ( plus_plus_int @ K @ B ) )
% 0.30/0.59       => ( ( minus_minus_int @ A @ B2 )
% 0.30/0.59          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.sub2
% 0.30/0.59  thf(fact_5450_group__cancel_Osub2,axiom,
% 0.30/0.59      ! [B2: real,K: real,B: real,A: real] :
% 0.30/0.59        ( ( B2
% 0.30/0.59          = ( plus_plus_real @ K @ B ) )
% 0.30/0.59       => ( ( minus_minus_real @ A @ B2 )
% 0.30/0.59          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.sub2
% 0.30/0.59  thf(fact_5451_group__cancel_Osub2,axiom,
% 0.30/0.59      ! [B2: rat,K: rat,B: rat,A: rat] :
% 0.30/0.59        ( ( B2
% 0.30/0.59          = ( plus_plus_rat @ K @ B ) )
% 0.30/0.59       => ( ( minus_minus_rat @ A @ B2 )
% 0.30/0.59          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.sub2
% 0.30/0.59  thf(fact_5452_group__cancel_Osub2,axiom,
% 0.30/0.59      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( B2
% 0.30/0.59          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 0.30/0.59       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 0.30/0.59          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % group_cancel.sub2
% 0.30/0.59  thf(fact_5453_diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_int
% 0.30/0.59      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_conv_add_uminus
% 0.30/0.59  thf(fact_5454_diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_real
% 0.30/0.59      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_conv_add_uminus
% 0.30/0.59  thf(fact_5455_diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_rat
% 0.30/0.59      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_conv_add_uminus
% 0.30/0.59  thf(fact_5456_diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_8373710615458151222nteger
% 0.30/0.59      = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % diff_conv_add_uminus
% 0.30/0.59  thf(fact_5457_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_int
% 0.30/0.59      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.59  thf(fact_5458_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_real
% 0.30/0.59      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.59  thf(fact_5459_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_minus_rat
% 0.30/0.59      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.59  thf(fact_5460_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.30/0.59      ( minus_8373710615458151222nteger
% 0.30/0.59      = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % ab_group_add_class.ab_diff_conv_add_uminus
% 0.30/0.59  thf(fact_5461_take__bit__unset__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: nat] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_unset_bit_eq
% 0.30/0.59  thf(fact_5462_take__bit__unset__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: int] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_unset_bit_eq
% 0.30/0.59  thf(fact_5463_take__bit__set__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: nat] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_set_bit_eq
% 0.30/0.59  thf(fact_5464_take__bit__set__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: int] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_set_bit_eq
% 0.30/0.59  thf(fact_5465_take__bit__flip__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: nat] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 0.30/0.59            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_flip_bit_eq
% 0.30/0.59  thf(fact_5466_take__bit__flip__bit__eq,axiom,
% 0.30/0.59      ! [N2: nat,M: nat,A: int] :
% 0.30/0.59        ( ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 0.30/0.59        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.59         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 0.30/0.59            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_flip_bit_eq
% 0.30/0.59  thf(fact_5467_dvd__neg__div,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.59          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_neg_div
% 0.30/0.59  thf(fact_5468_dvd__neg__div,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ B @ A )
% 0.30/0.59       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.59          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_neg_div
% 0.30/0.59  thf(fact_5469_dvd__neg__div,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.59       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.59          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_neg_div
% 0.30/0.59  thf(fact_5470_dvd__neg__div,axiom,
% 0.30/0.59      ! [B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.59          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_neg_div
% 0.30/0.59  thf(fact_5471_dvd__div__neg,axiom,
% 0.30/0.59      ! [B: int,A: int] :
% 0.30/0.59        ( ( dvd_dvd_int @ B @ A )
% 0.30/0.59       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 0.30/0.59          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_neg
% 0.30/0.59  thf(fact_5472_dvd__div__neg,axiom,
% 0.30/0.59      ! [B: real,A: real] :
% 0.30/0.59        ( ( dvd_dvd_real @ B @ A )
% 0.30/0.59       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 0.30/0.59          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_neg
% 0.30/0.59  thf(fact_5473_dvd__div__neg,axiom,
% 0.30/0.59      ! [B: rat,A: rat] :
% 0.30/0.59        ( ( dvd_dvd_rat @ B @ A )
% 0.30/0.59       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 0.30/0.59          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_neg
% 0.30/0.59  thf(fact_5474_dvd__div__neg,axiom,
% 0.30/0.59      ! [B: code_integer,A: code_integer] :
% 0.30/0.59        ( ( dvd_dvd_Code_integer @ B @ A )
% 0.30/0.59       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 0.30/0.59          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % dvd_div_neg
% 0.30/0.59  thf(fact_5475_subset__Compl__self__eq,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.30/0.59        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_self_eq
% 0.30/0.59  thf(fact_5476_subset__Compl__self__eq,axiom,
% 0.30/0.59      ! [A2: set_real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 0.30/0.59        = ( A2 = bot_bot_set_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_self_eq
% 0.30/0.59  thf(fact_5477_subset__Compl__self__eq,axiom,
% 0.30/0.59      ! [A2: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.30/0.59        = ( A2 = bot_bot_set_nat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_self_eq
% 0.30/0.59  thf(fact_5478_subset__Compl__self__eq,axiom,
% 0.30/0.59      ! [A2: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.30/0.59        = ( A2 = bot_bot_set_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Compl_self_eq
% 0.30/0.59  thf(fact_5479_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5480_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: set_nat,A2: set_set_nat] :
% 0.30/0.59        ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5481_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5482_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: real,A2: set_real] :
% 0.30/0.59        ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5483_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: int,A2: set_int] :
% 0.30/0.59        ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5484_Diff__insert__absorb,axiom,
% 0.30/0.59      ! [X3: nat,A2: set_nat] :
% 0.30/0.59        ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.59       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert_absorb
% 0.30/0.59  thf(fact_5485_Diff__insert2,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert2
% 0.30/0.59  thf(fact_5486_Diff__insert2,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert2
% 0.30/0.59  thf(fact_5487_Diff__insert2,axiom,
% 0.30/0.59      ! [A2: set_real,A: real,B2: set_real] :
% 0.30/0.59        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert2
% 0.30/0.59  thf(fact_5488_Diff__insert2,axiom,
% 0.30/0.59      ! [A2: set_int,A: int,B2: set_int] :
% 0.30/0.59        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert2
% 0.30/0.59  thf(fact_5489_Diff__insert2,axiom,
% 0.30/0.59      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.30/0.59        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert2
% 0.30/0.59  thf(fact_5490_insert__Diff,axiom,
% 0.30/0.59      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.59       => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5491_insert__Diff,axiom,
% 0.30/0.59      ! [A: set_nat,A2: set_set_nat] :
% 0.30/0.59        ( ( member_set_nat @ A @ A2 )
% 0.30/0.59       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5492_insert__Diff,axiom,
% 0.30/0.59      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.59       => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5493_insert__Diff,axiom,
% 0.30/0.59      ! [A: real,A2: set_real] :
% 0.30/0.59        ( ( member_real @ A @ A2 )
% 0.30/0.59       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5494_insert__Diff,axiom,
% 0.30/0.59      ! [A: int,A2: set_int] :
% 0.30/0.59        ( ( member_int @ A @ A2 )
% 0.30/0.59       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5495_insert__Diff,axiom,
% 0.30/0.59      ! [A: nat,A2: set_nat] :
% 0.30/0.59        ( ( member_nat @ A @ A2 )
% 0.30/0.59       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.59          = A2 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % insert_Diff
% 0.30/0.59  thf(fact_5496_Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert
% 0.30/0.59  thf(fact_5497_Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert
% 0.30/0.59  thf(fact_5498_Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_real,A: real,B2: set_real] :
% 0.30/0.59        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert
% 0.30/0.59  thf(fact_5499_Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_int,A: int,B2: set_int] :
% 0.30/0.59        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert
% 0.30/0.59  thf(fact_5500_Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_nat,A: nat,B2: set_nat] :
% 0.30/0.59        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 0.30/0.59        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % Diff_insert
% 0.30/0.59  thf(fact_5501_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat] :
% 0.30/0.59        ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ ( insert9069300056098147895at_nat @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member8757157785044589968at_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5502_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
% 0.30/0.59        ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member8440522571783428010at_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5503_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_real,B2: set_real,X3: real,C4: set_real] :
% 0.30/0.59        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member_real @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5504_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat,C4: set_set_nat] :
% 0.30/0.59        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member_set_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5505_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_nat,B2: set_nat,X3: nat,C4: set_nat] :
% 0.30/0.59        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member_nat @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5506_subset__Diff__insert,axiom,
% 0.30/0.59      ! [A2: set_int,B2: set_int,X3: int,C4: set_int] :
% 0.30/0.59        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X3 @ C4 ) ) )
% 0.30/0.59        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C4 ) )
% 0.30/0.59          & ~ ( member_int @ X3 @ A2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % subset_Diff_insert
% 0.30/0.59  thf(fact_5507_real__0__less__add__iff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.59        = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 0.30/0.59  
% 0.30/0.59  % real_0_less_add_iff
% 0.30/0.59  thf(fact_5508_real__add__less__0__iff,axiom,
% 0.30/0.59      ! [X3: real,Y3: real] :
% 0.30/0.59        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 0.30/0.59        = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % real_add_less_0_iff
% 0.30/0.59  thf(fact_5509_zminus1__lemma,axiom,
% 0.30/0.59      ! [A: int,B: int,Q4: int,R2: int] :
% 0.30/0.59        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 0.30/0.59       => ( ( B != zero_zero_int )
% 0.30/0.59         => ( 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 ) ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zminus1_lemma
% 0.30/0.59  thf(fact_5510_int__le__induct,axiom,
% 0.30/0.59      ! [I: int,K: int,P: int > $o] :
% 0.30/0.59        ( ( ord_less_eq_int @ I @ K )
% 0.30/0.59       => ( ( P @ K )
% 0.30/0.59         => ( ! [I3: int] :
% 0.30/0.59                ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.59               => ( ( P @ I3 )
% 0.30/0.59                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.59           => ( P @ I ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_le_induct
% 0.30/0.59  thf(fact_5511_int__less__induct,axiom,
% 0.30/0.59      ! [I: int,K: int,P: int > $o] :
% 0.30/0.59        ( ( ord_less_int @ I @ K )
% 0.30/0.59       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 0.30/0.59         => ( ! [I3: int] :
% 0.30/0.59                ( ( ord_less_int @ I3 @ K )
% 0.30/0.59               => ( ( P @ I3 )
% 0.30/0.59                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.59           => ( P @ I ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % int_less_induct
% 0.30/0.59  thf(fact_5512_pos__zmult__eq__1__iff__lemma,axiom,
% 0.30/0.59      ! [M: int,N2: int] :
% 0.30/0.59        ( ( ( times_times_int @ M @ N2 )
% 0.30/0.59          = one_one_int )
% 0.30/0.59       => ( ( M = one_one_int )
% 0.30/0.59          | ( M
% 0.30/0.59            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % pos_zmult_eq_1_iff_lemma
% 0.30/0.59  thf(fact_5513_zmult__eq__1__iff,axiom,
% 0.30/0.59      ! [M: int,N2: int] :
% 0.30/0.59        ( ( ( times_times_int @ M @ N2 )
% 0.30/0.59          = one_one_int )
% 0.30/0.59        = ( ( ( M = one_one_int )
% 0.30/0.59            & ( N2 = one_one_int ) )
% 0.30/0.59          | ( ( M
% 0.30/0.59              = ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.59            & ( N2
% 0.30/0.59              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % zmult_eq_1_iff
% 0.30/0.59  thf(fact_5514_signed__take__bit__eq__take__bit__shift,axiom,
% 0.30/0.59      ( bit_ri631733984087533419it_int
% 0.30/0.59      = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % signed_take_bit_eq_take_bit_shift
% 0.30/0.59  thf(fact_5515_take__bit__Suc__minus__bit0,axiom,
% 0.30/0.59      ! [N2: nat,K: num] :
% 0.30/0.59        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.59        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_Suc_minus_bit0
% 0.30/0.59  thf(fact_5516_take__bit__int__less__eq,axiom,
% 0.30/0.59      ! [N2: nat,K: int] :
% 0.30/0.59        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 0.30/0.59       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.59         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_int_less_eq
% 0.30/0.59  thf(fact_5517_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 0.30/0.59        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_set_int @ C @ A )
% 0.30/0.59              & ( ord_less_eq_set_int @ B @ D )
% 0.30/0.59              & ( ( ord_less_set_int @ C @ A )
% 0.30/0.59                | ( ord_less_set_int @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5518_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat,C: rat,D: rat] :
% 0.30/0.59        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_rat @ C @ A )
% 0.30/0.59              & ( ord_less_eq_rat @ B @ D )
% 0.30/0.59              & ( ( ord_less_rat @ C @ A )
% 0.30/0.59                | ( ord_less_rat @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5519_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: num,B: num,C: num,D: num] :
% 0.30/0.59        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_num @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_num @ C @ A )
% 0.30/0.59              & ( ord_less_eq_num @ B @ D )
% 0.30/0.59              & ( ( ord_less_num @ C @ A )
% 0.30/0.59                | ( ord_less_num @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5520_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: nat,B: nat,C: nat,D: nat] :
% 0.30/0.59        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_nat @ C @ A )
% 0.30/0.59              & ( ord_less_eq_nat @ B @ D )
% 0.30/0.59              & ( ( ord_less_nat @ C @ A )
% 0.30/0.59                | ( ord_less_nat @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5521_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: int,B: int,C: int,D: int] :
% 0.30/0.59        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_int @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_int @ C @ A )
% 0.30/0.59              & ( ord_less_eq_int @ B @ D )
% 0.30/0.59              & ( ( ord_less_int @ C @ A )
% 0.30/0.59                | ( ord_less_int @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5522_atLeastatMost__psubset__iff,axiom,
% 0.30/0.59      ! [A: real,B: real,C: real,D: real] :
% 0.30/0.59        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 0.30/0.59        = ( ( ~ ( ord_less_eq_real @ A @ B )
% 0.30/0.59            | ( ( ord_less_eq_real @ C @ A )
% 0.30/0.59              & ( ord_less_eq_real @ B @ D )
% 0.30/0.59              & ( ( ord_less_real @ C @ A )
% 0.30/0.59                | ( ord_less_real @ B @ D ) ) ) )
% 0.30/0.59          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % atLeastatMost_psubset_iff
% 0.30/0.59  thf(fact_5523_take__bit__signed__take__bit,axiom,
% 0.30/0.59      ! [M: nat,N2: nat,A: int] :
% 0.30/0.59        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.59       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
% 0.30/0.59          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % take_bit_signed_take_bit
% 0.30/0.59  thf(fact_5524_not__zero__le__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_le_neg_numeral
% 0.30/0.59  thf(fact_5525_not__zero__le__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_le_neg_numeral
% 0.30/0.59  thf(fact_5526_not__zero__le__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_le_neg_numeral
% 0.30/0.59  thf(fact_5527_not__zero__le__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_le_neg_numeral
% 0.30/0.59  thf(fact_5528_neg__numeral__le__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_zero
% 0.30/0.59  thf(fact_5529_neg__numeral__le__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_zero
% 0.30/0.59  thf(fact_5530_neg__numeral__le__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_zero
% 0.30/0.59  thf(fact_5531_neg__numeral__le__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_zero
% 0.30/0.59  thf(fact_5532_neg__numeral__less__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_zero
% 0.30/0.59  thf(fact_5533_neg__numeral__less__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_zero
% 0.30/0.59  thf(fact_5534_neg__numeral__less__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_zero
% 0.30/0.59  thf(fact_5535_neg__numeral__less__zero,axiom,
% 0.30/0.59      ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_zero
% 0.30/0.59  thf(fact_5536_not__zero__less__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_less_neg_numeral
% 0.30/0.59  thf(fact_5537_not__zero__less__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_less_neg_numeral
% 0.30/0.59  thf(fact_5538_not__zero__less__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_less_neg_numeral
% 0.30/0.59  thf(fact_5539_not__zero__less__neg__numeral,axiom,
% 0.30/0.59      ! [N2: num] :
% 0.30/0.59        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_zero_less_neg_numeral
% 0.30/0.59  thf(fact_5540_le__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(1)
% 0.30/0.59  thf(fact_5541_le__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(1)
% 0.30/0.59  thf(fact_5542_le__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(1)
% 0.30/0.59  thf(fact_5543_le__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(1)
% 0.30/0.59  thf(fact_5544_le__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(3)
% 0.30/0.59  thf(fact_5545_le__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(3)
% 0.30/0.59  thf(fact_5546_le__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(3)
% 0.30/0.59  thf(fact_5547_le__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % le_minus_one_simps(3)
% 0.30/0.59  thf(fact_5548_less__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(3)
% 0.30/0.59  thf(fact_5549_less__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(3)
% 0.30/0.59  thf(fact_5550_less__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(3)
% 0.30/0.59  thf(fact_5551_less__minus__one__simps_I3_J,axiom,
% 0.30/0.59      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(3)
% 0.30/0.59  thf(fact_5552_less__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(1)
% 0.30/0.59  thf(fact_5553_less__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(1)
% 0.30/0.59  thf(fact_5554_less__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(1)
% 0.30/0.59  thf(fact_5555_less__minus__one__simps_I1_J,axiom,
% 0.30/0.59      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 0.30/0.59  
% 0.30/0.59  % less_minus_one_simps(1)
% 0.30/0.59  thf(fact_5556_neg__numeral__le__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_one
% 0.30/0.59  thf(fact_5557_neg__numeral__le__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_one
% 0.30/0.59  thf(fact_5558_neg__numeral__le__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_one
% 0.30/0.59  thf(fact_5559_neg__numeral__le__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_one
% 0.30/0.59  thf(fact_5560_neg__one__le__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_le_numeral
% 0.30/0.59  thf(fact_5561_neg__one__le__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_le_numeral
% 0.30/0.59  thf(fact_5562_neg__one__le__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_le_numeral
% 0.30/0.59  thf(fact_5563_neg__one__le__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_le_numeral
% 0.30/0.59  thf(fact_5564_neg__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_neg_one
% 0.30/0.59  thf(fact_5565_neg__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_neg_one
% 0.30/0.59  thf(fact_5566_neg__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_neg_one
% 0.30/0.59  thf(fact_5567_neg__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_le_neg_one
% 0.30/0.59  thf(fact_5568_not__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_one
% 0.30/0.59  thf(fact_5569_not__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_one
% 0.30/0.59  thf(fact_5570_not__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_one
% 0.30/0.59  thf(fact_5571_not__numeral__le__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_le_neg_one
% 0.30/0.59  thf(fact_5572_not__one__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_le_neg_numeral
% 0.30/0.59  thf(fact_5573_not__one__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_le_neg_numeral
% 0.30/0.59  thf(fact_5574_not__one__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_le_neg_numeral
% 0.30/0.59  thf(fact_5575_not__one__le__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_le_neg_numeral
% 0.30/0.59  thf(fact_5576_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_less_neg_numeral
% 0.30/0.59  thf(fact_5577_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_less_neg_numeral
% 0.30/0.59  thf(fact_5578_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_less_neg_numeral
% 0.30/0.59  thf(fact_5579_not__neg__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_neg_one_less_neg_numeral
% 0.30/0.59  thf(fact_5580_not__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_less_neg_numeral
% 0.30/0.59  thf(fact_5581_not__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_less_neg_numeral
% 0.30/0.59  thf(fact_5582_not__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_less_neg_numeral
% 0.30/0.59  thf(fact_5583_not__one__less__neg__numeral,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_one_less_neg_numeral
% 0.30/0.59  thf(fact_5584_not__numeral__less__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_one
% 0.30/0.59  thf(fact_5585_not__numeral__less__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_one
% 0.30/0.59  thf(fact_5586_not__numeral__less__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_one
% 0.30/0.59  thf(fact_5587_not__numeral__less__neg__one,axiom,
% 0.30/0.59      ! [M: num] :
% 0.30/0.59        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.59  
% 0.30/0.59  % not_numeral_less_neg_one
% 0.30/0.59  thf(fact_5588_neg__one__less__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_less_numeral
% 0.30/0.59  thf(fact_5589_neg__one__less__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_less_numeral
% 0.30/0.59  thf(fact_5590_neg__one__less__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_less_numeral
% 0.30/0.59  thf(fact_5591_neg__one__less__numeral,axiom,
% 0.30/0.59      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_one_less_numeral
% 0.30/0.59  thf(fact_5592_neg__numeral__less__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_one
% 0.30/0.59  thf(fact_5593_neg__numeral__less__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_one
% 0.30/0.59  thf(fact_5594_neg__numeral__less__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_one
% 0.30/0.59  thf(fact_5595_neg__numeral__less__one,axiom,
% 0.30/0.59      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 0.30/0.59  
% 0.30/0.59  % neg_numeral_less_one
% 0.30/0.59  thf(fact_5596_eq__minus__divide__eq,axiom,
% 0.30/0.59      ! [A: complex,B: complex,C: complex] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 0.30/0.59        = ( ( ( C != zero_zero_complex )
% 0.30/0.59           => ( ( times_times_complex @ A @ C )
% 0.30/0.59              = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.59          & ( ( C = zero_zero_complex )
% 0.30/0.59           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_minus_divide_eq
% 0.30/0.59  thf(fact_5597_eq__minus__divide__eq,axiom,
% 0.30/0.59      ! [A: real,B: real,C: real] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.59        = ( ( ( C != zero_zero_real )
% 0.30/0.59           => ( ( times_times_real @ A @ C )
% 0.30/0.59              = ( uminus_uminus_real @ B ) ) )
% 0.30/0.59          & ( ( C = zero_zero_real )
% 0.30/0.59           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_minus_divide_eq
% 0.30/0.59  thf(fact_5598_eq__minus__divide__eq,axiom,
% 0.30/0.59      ! [A: rat,B: rat,C: rat] :
% 0.30/0.59        ( ( A
% 0.30/0.59          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.59        = ( ( ( C != zero_zero_rat )
% 0.30/0.59           => ( ( times_times_rat @ A @ C )
% 0.30/0.59              = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.59          & ( ( C = zero_zero_rat )
% 0.30/0.59           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % eq_minus_divide_eq
% 0.30/0.59  thf(fact_5599_minus__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: complex,C: complex,A: complex] :
% 0.30/0.59        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( C != zero_zero_complex )
% 0.30/0.59           => ( ( uminus1482373934393186551omplex @ B )
% 0.30/0.59              = ( times_times_complex @ A @ C ) ) )
% 0.30/0.59          & ( ( C = zero_zero_complex )
% 0.30/0.59           => ( A = zero_zero_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_eq_eq
% 0.30/0.59  thf(fact_5600_minus__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: real,C: real,A: real] :
% 0.30/0.59        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( C != zero_zero_real )
% 0.30/0.59           => ( ( uminus_uminus_real @ B )
% 0.30/0.59              = ( times_times_real @ A @ C ) ) )
% 0.30/0.59          & ( ( C = zero_zero_real )
% 0.30/0.59           => ( A = zero_zero_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_eq_eq
% 0.30/0.59  thf(fact_5601_minus__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: rat,C: rat,A: rat] :
% 0.30/0.59        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.59          = A )
% 0.30/0.59        = ( ( ( C != zero_zero_rat )
% 0.30/0.59           => ( ( uminus_uminus_rat @ B )
% 0.30/0.59              = ( times_times_rat @ A @ C ) ) )
% 0.30/0.59          & ( ( C = zero_zero_rat )
% 0.30/0.59           => ( A = zero_zero_rat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % minus_divide_eq_eq
% 0.30/0.59  thf(fact_5602_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: complex,A: complex,C: complex] :
% 0.30/0.59        ( ( B != zero_zero_complex )
% 0.30/0.59       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.59            = C )
% 0.30/0.59          = ( ( uminus1482373934393186551omplex @ A )
% 0.30/0.59            = ( times_times_complex @ C @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq
% 0.30/0.59  thf(fact_5603_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: real,A: real,C: real] :
% 0.30/0.59        ( ( B != zero_zero_real )
% 0.30/0.59       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.59            = C )
% 0.30/0.59          = ( ( uminus_uminus_real @ A )
% 0.30/0.59            = ( times_times_real @ C @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq
% 0.30/0.59  thf(fact_5604_nonzero__neg__divide__eq__eq,axiom,
% 0.30/0.59      ! [B: rat,A: rat,C: rat] :
% 0.30/0.59        ( ( B != zero_zero_rat )
% 0.30/0.59       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.59            = C )
% 0.30/0.59          = ( ( uminus_uminus_rat @ A )
% 0.30/0.59            = ( times_times_rat @ C @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq
% 0.30/0.59  thf(fact_5605_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.59      ! [B: complex,C: complex,A: complex] :
% 0.30/0.59        ( ( B != zero_zero_complex )
% 0.30/0.59       => ( ( C
% 0.30/0.59            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 0.30/0.59          = ( ( times_times_complex @ C @ B )
% 0.30/0.59            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq2
% 0.30/0.59  thf(fact_5606_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.59      ! [B: real,C: real,A: real] :
% 0.30/0.59        ( ( B != zero_zero_real )
% 0.30/0.59       => ( ( C
% 0.30/0.59            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 0.30/0.59          = ( ( times_times_real @ C @ B )
% 0.30/0.59            = ( uminus_uminus_real @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq2
% 0.30/0.59  thf(fact_5607_nonzero__neg__divide__eq__eq2,axiom,
% 0.30/0.59      ! [B: rat,C: rat,A: rat] :
% 0.30/0.59        ( ( B != zero_zero_rat )
% 0.30/0.59       => ( ( C
% 0.30/0.59            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 0.30/0.59          = ( ( times_times_rat @ C @ B )
% 0.30/0.59            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % nonzero_neg_divide_eq_eq2
% 0.30/0.59  thf(fact_5608_divide__eq__minus__1__iff,axiom,
% 0.30/0.59      ! [A: complex,B: complex] :
% 0.30/0.59        ( ( ( divide1717551699836669952omplex @ A @ B )
% 0.30/0.59          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.59        = ( ( B != zero_zero_complex )
% 0.30/0.59          & ( A
% 0.30/0.59            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_minus_1_iff
% 0.30/0.59  thf(fact_5609_divide__eq__minus__1__iff,axiom,
% 0.30/0.59      ! [A: real,B: real] :
% 0.30/0.59        ( ( ( divide_divide_real @ A @ B )
% 0.30/0.59          = ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.59        = ( ( B != zero_zero_real )
% 0.30/0.59          & ( A
% 0.30/0.59            = ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_minus_1_iff
% 0.30/0.59  thf(fact_5610_divide__eq__minus__1__iff,axiom,
% 0.30/0.59      ! [A: rat,B: rat] :
% 0.30/0.59        ( ( ( divide_divide_rat @ A @ B )
% 0.30/0.59          = ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.59        = ( ( B != zero_zero_rat )
% 0.30/0.59          & ( A
% 0.30/0.59            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % divide_eq_minus_1_iff
% 0.30/0.59  thf(fact_5611_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.59        ( ( finite4343798906461161616at_nat @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.59                ( ( finite4343798906461161616at_nat @ A7 )
% 0.30/0.59               => ( ( member8757157785044589968at_nat @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5612_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.59        ( ( finite1152437895449049373et_nat @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: set_nat,A7: set_set_nat] :
% 0.30/0.59                ( ( finite1152437895449049373et_nat @ A7 )
% 0.30/0.59               => ( ( member_set_nat @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5613_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_complex,P: set_complex > $o] :
% 0.30/0.59        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: complex,A7: set_complex] :
% 0.30/0.59                ( ( finite3207457112153483333omplex @ A7 )
% 0.30/0.59               => ( ( member_complex @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bot_set_complex ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5614_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.59        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.59                ( ( finite6177210948735845034at_nat @ A7 )
% 0.30/0.59               => ( ( member8440522571783428010at_nat @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5615_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_real,P: set_real > $o] :
% 0.30/0.59        ( ( finite_finite_real @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: real,A7: set_real] :
% 0.30/0.59                ( ( finite_finite_real @ A7 )
% 0.30/0.59               => ( ( member_real @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5616_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_int,P: set_int > $o] :
% 0.30/0.59        ( ( finite_finite_int @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: int,A7: set_int] :
% 0.30/0.59                ( ( finite_finite_int @ A7 )
% 0.30/0.59               => ( ( member_int @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5617_finite__empty__induct,axiom,
% 0.30/0.59      ! [A2: set_nat,P: set_nat > $o] :
% 0.30/0.59        ( ( finite_finite_nat @ A2 )
% 0.30/0.59       => ( ( P @ A2 )
% 0.30/0.59         => ( ! [A4: nat,A7: set_nat] :
% 0.30/0.59                ( ( finite_finite_nat @ A7 )
% 0.30/0.59               => ( ( member_nat @ A4 @ A7 )
% 0.30/0.59                 => ( ( P @ A7 )
% 0.30/0.59                   => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
% 0.30/0.59           => ( P @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.59  
% 0.30/0.59  % finite_empty_induct
% 0.30/0.59  thf(fact_5618_infinite__coinduct,axiom,
% 0.30/0.59      ! [X7: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: produc3843707927480180839at_nat] :
% 0.30/0.60                  ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.60                    | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5619_infinite__coinduct,axiom,
% 0.30/0.60      ! [X7: set_complex > $o,A2: set_complex] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_complex] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: complex] :
% 0.30/0.60                  ( ( member_complex @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 0.30/0.60                    | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5620_infinite__coinduct,axiom,
% 0.30/0.60      ! [X7: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: product_prod_nat_nat] :
% 0.30/0.60                  ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.60                    | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5621_infinite__coinduct,axiom,
% 0.30/0.60      ! [X7: set_real > $o,A2: set_real] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_real] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: real] :
% 0.30/0.60                  ( ( member_real @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 0.30/0.60                    | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite_finite_real @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5622_infinite__coinduct,axiom,
% 0.30/0.60      ! [X7: set_int > $o,A2: set_int] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_int] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: int] :
% 0.30/0.60                  ( ( member_int @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 0.30/0.60                    | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite_finite_int @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5623_infinite__coinduct,axiom,
% 0.30/0.60      ! [X7: set_nat > $o,A2: set_nat] :
% 0.30/0.60        ( ( X7 @ A2 )
% 0.30/0.60       => ( ! [A7: set_nat] :
% 0.30/0.60              ( ( X7 @ A7 )
% 0.30/0.60             => ? [X2: nat] :
% 0.30/0.60                  ( ( member_nat @ X2 @ A7 )
% 0.30/0.60                  & ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 0.30/0.60                    | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
% 0.30/0.60         => ~ ( finite_finite_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_coinduct
% 0.30/0.60  thf(fact_5624_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 0.30/0.60        ( ~ ( finite4343798906461161616at_nat @ S3 )
% 0.30/0.60       => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5625_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_complex,A: complex] :
% 0.30/0.60        ( ~ ( finite3207457112153483333omplex @ S3 )
% 0.30/0.60       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5626_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 0.30/0.60        ( ~ ( finite6177210948735845034at_nat @ S3 )
% 0.30/0.60       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5627_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_real,A: real] :
% 0.30/0.60        ( ~ ( finite_finite_real @ S3 )
% 0.30/0.60       => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5628_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_int,A: int] :
% 0.30/0.60        ( ~ ( finite_finite_int @ S3 )
% 0.30/0.60       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5629_infinite__remove,axiom,
% 0.30/0.60      ! [S3: set_nat,A: nat] :
% 0.30/0.60        ( ~ ( finite_finite_nat @ S3 )
% 0.30/0.60       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % infinite_remove
% 0.30/0.60  thf(fact_5630_Diff__single__insert,axiom,
% 0.30/0.60      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.60        ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 )
% 0.30/0.60       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_single_insert
% 0.30/0.60  thf(fact_5631_Diff__single__insert,axiom,
% 0.30/0.60      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 0.30/0.60       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_single_insert
% 0.30/0.60  thf(fact_5632_Diff__single__insert,axiom,
% 0.30/0.60      ! [A2: set_real,X3: real,B2: set_real] :
% 0.30/0.60        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 )
% 0.30/0.60       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_single_insert
% 0.30/0.60  thf(fact_5633_Diff__single__insert,axiom,
% 0.30/0.60      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 0.30/0.60        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 )
% 0.30/0.60       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_single_insert
% 0.30/0.60  thf(fact_5634_Diff__single__insert,axiom,
% 0.30/0.60      ! [A2: set_int,X3: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 )
% 0.30/0.60       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_single_insert
% 0.30/0.60  thf(fact_5635_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.60        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5636_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_set_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5637_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5638_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_real,X3: real,B2: set_real] :
% 0.30/0.60        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_real @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5639_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 0.30/0.60        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5640_subset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_int,X3: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_int @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
% 0.30/0.60          & ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.60           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_insert_iff
% 0.30/0.60  thf(fact_5641_take__bit__Suc__minus__1__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_minus_1_eq
% 0.30/0.60  thf(fact_5642_take__bit__Suc__minus__1__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_minus_1_eq
% 0.30/0.60  thf(fact_5643_atLeast0__atMost__Suc,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.30/0.60        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % atLeast0_atMost_Suc
% 0.30/0.60  thf(fact_5644_mod__pos__geq,axiom,
% 0.30/0.60      ! [L: int,K: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.60       => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.60         => ( ( modulo_modulo_int @ K @ L )
% 0.30/0.60            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_pos_geq
% 0.30/0.60  thf(fact_5645_Icc__eq__insert__lb__nat,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.60       => ( ( set_or1269000886237332187st_nat @ M @ N2 )
% 0.30/0.60          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Icc_eq_insert_lb_nat
% 0.30/0.60  thf(fact_5646_atLeastAtMostSuc__conv,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.60       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
% 0.30/0.60          = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % atLeastAtMostSuc_conv
% 0.30/0.60  thf(fact_5647_atLeastAtMost__insertL,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.60       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.30/0.60          = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % atLeastAtMost_insertL
% 0.30/0.60  thf(fact_5648_plusinfinity,axiom,
% 0.30/0.60      ! [D: int,P5: int > $o,P: int > $o] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.60       => ( ! [X4: int,K2: int] :
% 0.30/0.60              ( ( P5 @ X4 )
% 0.30/0.60              = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.30/0.60         => ( ? [Z4: int] :
% 0.30/0.60              ! [X4: int] :
% 0.30/0.60                ( ( ord_less_int @ Z4 @ X4 )
% 0.30/0.60               => ( ( P @ X4 )
% 0.30/0.60                  = ( P5 @ X4 ) ) )
% 0.30/0.60           => ( ? [X_12: int] : ( P5 @ X_12 )
% 0.30/0.60             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % plusinfinity
% 0.30/0.60  thf(fact_5649_minusinfinity,axiom,
% 0.30/0.60      ! [D: int,P1: int > $o,P: int > $o] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.60       => ( ! [X4: int,K2: int] :
% 0.30/0.60              ( ( P1 @ X4 )
% 0.30/0.60              = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.30/0.60         => ( ? [Z4: int] :
% 0.30/0.60              ! [X4: int] :
% 0.30/0.60                ( ( ord_less_int @ X4 @ Z4 )
% 0.30/0.60               => ( ( P @ X4 )
% 0.30/0.60                  = ( P1 @ X4 ) ) )
% 0.30/0.60           => ( ? [X_12: int] : ( P1 @ X_12 )
% 0.30/0.60             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minusinfinity
% 0.30/0.60  thf(fact_5650_int__induct,axiom,
% 0.30/0.60      ! [P: int > $o,K: int,I: int] :
% 0.30/0.60        ( ( P @ K )
% 0.30/0.60       => ( ! [I3: int] :
% 0.30/0.60              ( ( ord_less_eq_int @ K @ I3 )
% 0.30/0.60             => ( ( P @ I3 )
% 0.30/0.60               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.60         => ( ! [I3: int] :
% 0.30/0.60                ( ( ord_less_eq_int @ I3 @ K )
% 0.30/0.60               => ( ( P @ I3 )
% 0.30/0.60                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 0.30/0.60           => ( P @ I ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % int_induct
% 0.30/0.60  thf(fact_5651_subset__eq__atLeast0__atMost__finite,axiom,
% 0.30/0.60      ! [N7: set_nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.60       => ( finite_finite_nat @ N7 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % subset_eq_atLeast0_atMost_finite
% 0.30/0.60  thf(fact_5652_take__bit__int__less__exp,axiom,
% 0.30/0.60      ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_less_exp
% 0.30/0.60  thf(fact_5653_pos__minus__divide__less__eq,axiom,
% 0.30/0.60      ! [C: real,B: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_minus_divide_less_eq
% 0.30/0.60  thf(fact_5654_pos__minus__divide__less__eq,axiom,
% 0.30/0.60      ! [C: rat,B: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_minus_divide_less_eq
% 0.30/0.60  thf(fact_5655_pos__less__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: real,A: real,B: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_less_minus_divide_eq
% 0.30/0.60  thf(fact_5656_pos__less__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_less_minus_divide_eq
% 0.30/0.60  thf(fact_5657_neg__minus__divide__less__eq,axiom,
% 0.30/0.60      ! [C: real,B: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_minus_divide_less_eq
% 0.30/0.60  thf(fact_5658_neg__minus__divide__less__eq,axiom,
% 0.30/0.60      ! [C: rat,B: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_minus_divide_less_eq
% 0.30/0.60  thf(fact_5659_neg__less__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: real,A: real,B: real] :
% 0.30/0.60        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_minus_divide_eq
% 0.30/0.60  thf(fact_5660_neg__less__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_less_minus_divide_eq
% 0.30/0.60  thf(fact_5661_minus__divide__less__eq,axiom,
% 0.30/0.60      ! [B: real,C: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_less_eq
% 0.30/0.60  thf(fact_5662_minus__divide__less__eq,axiom,
% 0.30/0.60      ! [B: rat,C: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_less_eq
% 0.30/0.60  thf(fact_5663_less__minus__divide__eq,axiom,
% 0.30/0.60      ! [A: real,B: real,C: real] :
% 0.30/0.60        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_minus_divide_eq
% 0.30/0.60  thf(fact_5664_less__minus__divide__eq,axiom,
% 0.30/0.60      ! [A: rat,B: rat,C: rat] :
% 0.30/0.60        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_minus_divide_eq
% 0.30/0.60  thf(fact_5665_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: complex,C: complex,W2: num] :
% 0.30/0.60        ( ( ( divide1717551699836669952omplex @ B @ C )
% 0.30/0.60          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 0.30/0.60        = ( ( ( C != zero_zero_complex )
% 0.30/0.60           => ( B
% 0.30/0.60              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ( C = zero_zero_complex )
% 0.30/0.60           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.60              = zero_zero_complex ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_eq_eq_numeral(2)
% 0.30/0.60  thf(fact_5666_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: real,C: real,W2: num] :
% 0.30/0.60        ( ( ( divide_divide_real @ B @ C )
% 0.30/0.60          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.30/0.60        = ( ( ( C != zero_zero_real )
% 0.30/0.60           => ( B
% 0.30/0.60              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ( C = zero_zero_real )
% 0.30/0.60           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.60              = zero_zero_real ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_eq_eq_numeral(2)
% 0.30/0.60  thf(fact_5667_divide__eq__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: rat,C: rat,W2: num] :
% 0.30/0.60        ( ( ( divide_divide_rat @ B @ C )
% 0.30/0.60          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.30/0.60        = ( ( ( C != zero_zero_rat )
% 0.30/0.60           => ( B
% 0.30/0.60              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ( C = zero_zero_rat )
% 0.30/0.60           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.60              = zero_zero_rat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_eq_eq_numeral(2)
% 0.30/0.60  thf(fact_5668_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: complex,C: complex] :
% 0.30/0.60        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.60          = ( divide1717551699836669952omplex @ B @ C ) )
% 0.30/0.60        = ( ( ( C != zero_zero_complex )
% 0.30/0.60           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 0.30/0.60              = B ) )
% 0.30/0.60          & ( ( C = zero_zero_complex )
% 0.30/0.60           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 0.30/0.60              = zero_zero_complex ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eq_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5669_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: real,C: real] :
% 0.30/0.60        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.60          = ( divide_divide_real @ B @ C ) )
% 0.30/0.60        = ( ( ( C != zero_zero_real )
% 0.30/0.60           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 0.30/0.60              = B ) )
% 0.30/0.60          & ( ( C = zero_zero_real )
% 0.30/0.60           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.60              = zero_zero_real ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eq_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5670_eq__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: rat,C: rat] :
% 0.30/0.60        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.60          = ( divide_divide_rat @ B @ C ) )
% 0.30/0.60        = ( ( ( C != zero_zero_rat )
% 0.30/0.60           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 0.30/0.60              = B ) )
% 0.30/0.60          & ( ( C = zero_zero_rat )
% 0.30/0.60           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 0.30/0.60              = zero_zero_rat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eq_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5671_minus__divide__add__eq__iff,axiom,
% 0.30/0.60      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.30/0.60        ( ( Z2 != zero_zero_complex )
% 0.30/0.60       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_add_eq_iff
% 0.30/0.60  thf(fact_5672_minus__divide__add__eq__iff,axiom,
% 0.30/0.60      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.60        ( ( Z2 != zero_zero_real )
% 0.30/0.60       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_add_eq_iff
% 0.30/0.60  thf(fact_5673_minus__divide__add__eq__iff,axiom,
% 0.30/0.60      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.60        ( ( Z2 != zero_zero_rat )
% 0.30/0.60       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_add_eq_iff
% 0.30/0.60  thf(fact_5674_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60      ! [Z2: complex,A: complex,B: complex] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_complex )
% 0.30/0.60         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.30/0.60            = B ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_complex )
% 0.30/0.60         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(3)
% 0.30/0.60  thf(fact_5675_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60      ! [Z2: real,A: real,B: real] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_real )
% 0.30/0.60         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.30/0.60            = B ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_real )
% 0.30/0.60         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(3)
% 0.30/0.60  thf(fact_5676_add__divide__eq__if__simps_I3_J,axiom,
% 0.30/0.60      ! [Z2: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_rat )
% 0.30/0.60         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.30/0.60            = B ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_rat )
% 0.30/0.60         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(3)
% 0.30/0.60  thf(fact_5677_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.60      ! [Z2: complex,A: complex,B: complex] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_complex )
% 0.30/0.60         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_complex )
% 0.30/0.60         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(6)
% 0.30/0.60  thf(fact_5678_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.60      ! [Z2: real,A: real,B: real] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_real )
% 0.30/0.60         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( uminus_uminus_real @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_real )
% 0.30/0.60         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(6)
% 0.30/0.60  thf(fact_5679_add__divide__eq__if__simps_I6_J,axiom,
% 0.30/0.60      ! [Z2: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_rat )
% 0.30/0.60         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_rat )
% 0.30/0.60         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 0.30/0.60            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(6)
% 0.30/0.60  thf(fact_5680_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.60      ! [Z2: complex,A: complex,B: complex] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_complex )
% 0.30/0.60         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.30/0.60            = ( uminus1482373934393186551omplex @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_complex )
% 0.30/0.60         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 0.30/0.60            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(5)
% 0.30/0.60  thf(fact_5681_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.60      ! [Z2: real,A: real,B: real] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_real )
% 0.30/0.60         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.30/0.60            = ( uminus_uminus_real @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_real )
% 0.30/0.60         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 0.30/0.60            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(5)
% 0.30/0.60  thf(fact_5682_add__divide__eq__if__simps_I5_J,axiom,
% 0.30/0.60      ! [Z2: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ( Z2 = zero_zero_rat )
% 0.30/0.60         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.30/0.60            = ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60        & ( ( Z2 != zero_zero_rat )
% 0.30/0.60         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 0.30/0.60            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_divide_eq_if_simps(5)
% 0.30/0.60  thf(fact_5683_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60      ! [Z2: complex,X3: complex,Y3: complex] :
% 0.30/0.60        ( ( Z2 != zero_zero_complex )
% 0.30/0.60       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_diff_eq_iff
% 0.30/0.60  thf(fact_5684_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60      ! [Z2: real,X3: real,Y3: real] :
% 0.30/0.60        ( ( Z2 != zero_zero_real )
% 0.30/0.60       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_diff_eq_iff
% 0.30/0.60  thf(fact_5685_minus__divide__diff__eq__iff,axiom,
% 0.30/0.60      ! [Z2: rat,X3: rat,Y3: rat] :
% 0.30/0.60        ( ( Z2 != zero_zero_rat )
% 0.30/0.60       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
% 0.30/0.60          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_diff_eq_iff
% 0.30/0.60  thf(fact_5686_remove__induct,axiom,
% 0.30/0.60      ! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.60        ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.60       => ( ( ~ ( finite4343798906461161616at_nat @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.60                ( ( finite4343798906461161616at_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bo228742789529271731at_nat )
% 0.30/0.60                 => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: produc3843707927480180839at_nat] :
% 0.30/0.60                          ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5687_remove__induct,axiom,
% 0.30/0.60      ! [P: set_set_nat > $o,B2: set_set_nat] :
% 0.30/0.60        ( ( P @ bot_bot_set_set_nat )
% 0.30/0.60       => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_set_nat] :
% 0.30/0.60                ( ( finite1152437895449049373et_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_set_nat )
% 0.30/0.60                 => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: set_nat] :
% 0.30/0.60                          ( ( member_set_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5688_remove__induct,axiom,
% 0.30/0.60      ! [P: set_complex > $o,B2: set_complex] :
% 0.30/0.60        ( ( P @ bot_bot_set_complex )
% 0.30/0.60       => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_complex] :
% 0.30/0.60                ( ( finite3207457112153483333omplex @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_complex )
% 0.30/0.60                 => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: complex] :
% 0.30/0.60                          ( ( member_complex @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5689_remove__induct,axiom,
% 0.30/0.60      ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.60       => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.60                ( ( finite6177210948735845034at_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bo2099793752762293965at_nat )
% 0.30/0.60                 => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: product_prod_nat_nat] :
% 0.30/0.60                          ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5690_remove__induct,axiom,
% 0.30/0.60      ! [P: set_real > $o,B2: set_real] :
% 0.30/0.60        ( ( P @ bot_bot_set_real )
% 0.30/0.60       => ( ( ~ ( finite_finite_real @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_real] :
% 0.30/0.60                ( ( finite_finite_real @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_real )
% 0.30/0.60                 => ( ( ord_less_eq_set_real @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: real] :
% 0.30/0.60                          ( ( member_real @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5691_remove__induct,axiom,
% 0.30/0.60      ! [P: set_nat > $o,B2: set_nat] :
% 0.30/0.60        ( ( P @ bot_bot_set_nat )
% 0.30/0.60       => ( ( ~ ( finite_finite_nat @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_nat] :
% 0.30/0.60                ( ( finite_finite_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_nat )
% 0.30/0.60                 => ( ( ord_less_eq_set_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: nat] :
% 0.30/0.60                          ( ( member_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5692_remove__induct,axiom,
% 0.30/0.60      ! [P: set_int > $o,B2: set_int] :
% 0.30/0.60        ( ( P @ bot_bot_set_int )
% 0.30/0.60       => ( ( ~ ( finite_finite_int @ B2 )
% 0.30/0.60           => ( P @ B2 ) )
% 0.30/0.60         => ( ! [A7: set_int] :
% 0.30/0.60                ( ( finite_finite_int @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_int )
% 0.30/0.60                 => ( ( ord_less_eq_set_int @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: int] :
% 0.30/0.60                          ( ( member_int @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % remove_induct
% 0.30/0.60  thf(fact_5693_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.60        ( ( finite4343798906461161616at_nat @ B2 )
% 0.30/0.60       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.60         => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 0.30/0.60                ( ( finite4343798906461161616at_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bo228742789529271731at_nat )
% 0.30/0.60                 => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: produc3843707927480180839at_nat] :
% 0.30/0.60                          ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5694_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_set_nat,P: set_set_nat > $o] :
% 0.30/0.60        ( ( finite1152437895449049373et_nat @ B2 )
% 0.30/0.60       => ( ( P @ bot_bot_set_set_nat )
% 0.30/0.60         => ( ! [A7: set_set_nat] :
% 0.30/0.60                ( ( finite1152437895449049373et_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_set_nat )
% 0.30/0.60                 => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: set_nat] :
% 0.30/0.60                          ( ( member_set_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5695_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_complex,P: set_complex > $o] :
% 0.30/0.60        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.60       => ( ( P @ bot_bot_set_complex )
% 0.30/0.60         => ( ! [A7: set_complex] :
% 0.30/0.60                ( ( finite3207457112153483333omplex @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_complex )
% 0.30/0.60                 => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: complex] :
% 0.30/0.60                          ( ( member_complex @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5696_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.60        ( ( finite6177210948735845034at_nat @ B2 )
% 0.30/0.60       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.60         => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 0.30/0.60                ( ( finite6177210948735845034at_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bo2099793752762293965at_nat )
% 0.30/0.60                 => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: product_prod_nat_nat] :
% 0.30/0.60                          ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5697_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_real,P: set_real > $o] :
% 0.30/0.60        ( ( finite_finite_real @ B2 )
% 0.30/0.60       => ( ( P @ bot_bot_set_real )
% 0.30/0.60         => ( ! [A7: set_real] :
% 0.30/0.60                ( ( finite_finite_real @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_real )
% 0.30/0.60                 => ( ( ord_less_eq_set_real @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: real] :
% 0.30/0.60                          ( ( member_real @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5698_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_nat,P: set_nat > $o] :
% 0.30/0.60        ( ( finite_finite_nat @ B2 )
% 0.30/0.60       => ( ( P @ bot_bot_set_nat )
% 0.30/0.60         => ( ! [A7: set_nat] :
% 0.30/0.60                ( ( finite_finite_nat @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_nat )
% 0.30/0.60                 => ( ( ord_less_eq_set_nat @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: nat] :
% 0.30/0.60                          ( ( member_nat @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5699_finite__remove__induct,axiom,
% 0.30/0.60      ! [B2: set_int,P: set_int > $o] :
% 0.30/0.60        ( ( finite_finite_int @ B2 )
% 0.30/0.60       => ( ( P @ bot_bot_set_int )
% 0.30/0.60         => ( ! [A7: set_int] :
% 0.30/0.60                ( ( finite_finite_int @ A7 )
% 0.30/0.60               => ( ( A7 != bot_bot_set_int )
% 0.30/0.60                 => ( ( ord_less_eq_set_int @ A7 @ B2 )
% 0.30/0.60                   => ( ! [X2: int] :
% 0.30/0.60                          ( ( member_int @ X2 @ A7 )
% 0.30/0.60                         => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 0.30/0.60                     => ( P @ A7 ) ) ) ) )
% 0.30/0.60           => ( P @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_remove_induct
% 0.30/0.60  thf(fact_5700_signed__take__bit__int__greater__self__iff,axiom,
% 0.30/0.60      ! [K: int,N2: nat] :
% 0.30/0.60        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 0.30/0.60        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_greater_self_iff
% 0.30/0.60  thf(fact_5701_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 0.30/0.60        ( ( finite4343798906461161616at_nat @ S3 )
% 0.30/0.60       => ( ( P @ bot_bo228742789529271731at_nat )
% 0.30/0.60         => ( ! [T4: set_Pr4329608150637261639at_nat] :
% 0.30/0.60                ( ( ord_le2604355607129572851at_nat @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: produc3843707927480180839at_nat] :
% 0.30/0.60                      ( ( member8757157785044589968at_nat @ X2 @ ( minus_3314409938677909166at_nat @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert9069300056098147895at_nat @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5702_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_complex,P: set_complex > $o] :
% 0.30/0.60        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.60       => ( ( P @ bot_bot_set_complex )
% 0.30/0.60         => ( ! [T4: set_complex] :
% 0.30/0.60                ( ( ord_less_set_complex @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: complex] :
% 0.30/0.60                      ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert_complex @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5703_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 0.30/0.60        ( ( finite6177210948735845034at_nat @ S3 )
% 0.30/0.60       => ( ( P @ bot_bo2099793752762293965at_nat )
% 0.30/0.60         => ( ! [T4: set_Pr1261947904930325089at_nat] :
% 0.30/0.60                ( ( ord_le7866589430770878221at_nat @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: product_prod_nat_nat] :
% 0.30/0.60                      ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert8211810215607154385at_nat @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5704_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_real,P: set_real > $o] :
% 0.30/0.60        ( ( finite_finite_real @ S3 )
% 0.30/0.60       => ( ( P @ bot_bot_set_real )
% 0.30/0.60         => ( ! [T4: set_real] :
% 0.30/0.60                ( ( ord_less_set_real @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: real] :
% 0.30/0.60                      ( ( member_real @ X2 @ ( minus_minus_set_real @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert_real @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5705_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_int,P: set_int > $o] :
% 0.30/0.60        ( ( finite_finite_int @ S3 )
% 0.30/0.60       => ( ( P @ bot_bot_set_int )
% 0.30/0.60         => ( ! [T4: set_int] :
% 0.30/0.60                ( ( ord_less_set_int @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: int] :
% 0.30/0.60                      ( ( member_int @ X2 @ ( minus_minus_set_int @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert_int @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5706_finite__induct__select,axiom,
% 0.30/0.60      ! [S3: set_nat,P: set_nat > $o] :
% 0.30/0.60        ( ( finite_finite_nat @ S3 )
% 0.30/0.60       => ( ( P @ bot_bot_set_nat )
% 0.30/0.60         => ( ! [T4: set_nat] :
% 0.30/0.60                ( ( ord_less_set_nat @ T4 @ S3 )
% 0.30/0.60               => ( ( P @ T4 )
% 0.30/0.60                 => ? [X2: nat] :
% 0.30/0.60                      ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S3 @ T4 ) )
% 0.30/0.60                      & ( P @ ( insert_nat @ X2 @ T4 ) ) ) ) )
% 0.30/0.60           => ( P @ S3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_induct_select
% 0.30/0.60  thf(fact_5707_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 0.30/0.60        ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.60           => ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5708_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_set_nat @ X3 @ B2 )
% 0.30/0.60           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member_set_nat @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member_set_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member_set_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5709_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.60           => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5710_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_real,X3: real,B2: set_real] :
% 0.30/0.60        ( ( ord_less_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_real @ X3 @ B2 )
% 0.30/0.60           => ( ord_less_set_real @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member_real @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member_real @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5711_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 0.30/0.60        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_nat @ X3 @ B2 )
% 0.30/0.60           => ( ord_less_set_nat @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member_nat @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5712_psubset__insert__iff,axiom,
% 0.30/0.60      ! [A2: set_int,X3: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 0.30/0.60        = ( ( ( member_int @ X3 @ B2 )
% 0.30/0.60           => ( ord_less_set_int @ A2 @ B2 ) )
% 0.30/0.60          & ( ~ ( member_int @ X3 @ B2 )
% 0.30/0.60           => ( ( ( member_int @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
% 0.30/0.60              & ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.60               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % psubset_insert_iff
% 0.30/0.60  thf(fact_5713_decr__mult__lemma,axiom,
% 0.30/0.60      ! [D: int,P: int > $o,K: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.60       => ( ! [X4: int] :
% 0.30/0.60              ( ( P @ X4 )
% 0.30/0.60             => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 0.30/0.60         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.60           => ! [X2: int] :
% 0.30/0.60                ( ( P @ X2 )
% 0.30/0.60               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % decr_mult_lemma
% 0.30/0.60  thf(fact_5714_verit__less__mono__div__int2,axiom,
% 0.30/0.60      ! [A2: int,B2: int,N2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ A2 @ B2 )
% 0.30/0.60       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
% 0.30/0.60         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % verit_less_mono_div_int2
% 0.30/0.60  thf(fact_5715_div__eq__minus1,axiom,
% 0.30/0.60      ! [B: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.60       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 0.30/0.60          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_eq_minus1
% 0.30/0.60  thf(fact_5716_take__bit__Suc__bit0,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 0.30/0.60        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_bit0
% 0.30/0.60  thf(fact_5717_take__bit__Suc__bit0,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 0.30/0.60        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_bit0
% 0.30/0.60  thf(fact_5718_take__bit__nat__eq__self,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.60       => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 0.30/0.60          = M ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_nat_eq_self
% 0.30/0.60  thf(fact_5719_take__bit__nat__less__exp,axiom,
% 0.30/0.60      ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_nat_less_exp
% 0.30/0.60  thf(fact_5720_take__bit__nat__eq__self__iff,axiom,
% 0.30/0.60      ! [N2: nat,M: nat] :
% 0.30/0.60        ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 0.30/0.60          = M )
% 0.30/0.60        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_nat_eq_self_iff
% 0.30/0.60  thf(fact_5721_take__bit__int__less__self__iff,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_less_self_iff
% 0.30/0.60  thf(fact_5722_take__bit__int__greater__eq__self__iff,axiom,
% 0.30/0.60      ! [K: int,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 0.30/0.60        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_greater_eq_self_iff
% 0.30/0.60  thf(fact_5723_le__minus__divide__eq,axiom,
% 0.30/0.60      ! [A: real,B: real,C: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_minus_divide_eq
% 0.30/0.60  thf(fact_5724_le__minus__divide__eq,axiom,
% 0.30/0.60      ! [A: rat,B: rat,C: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_minus_divide_eq
% 0.30/0.60  thf(fact_5725_minus__divide__le__eq,axiom,
% 0.30/0.60      ! [B: real,C: real,A: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_le_eq
% 0.30/0.60  thf(fact_5726_minus__divide__le__eq,axiom,
% 0.30/0.60      ! [B: rat,C: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % minus_divide_le_eq
% 0.30/0.60  thf(fact_5727_neg__le__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: real,A: real,B: real] :
% 0.30/0.60        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_minus_divide_eq
% 0.30/0.60  thf(fact_5728_neg__le__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_le_minus_divide_eq
% 0.30/0.60  thf(fact_5729_neg__minus__divide__le__eq,axiom,
% 0.30/0.60      ! [C: real,B: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_minus_divide_le_eq
% 0.30/0.60  thf(fact_5730_neg__minus__divide__le__eq,axiom,
% 0.30/0.60      ! [C: rat,B: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_minus_divide_le_eq
% 0.30/0.60  thf(fact_5731_pos__le__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: real,A: real,B: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_le_minus_divide_eq
% 0.30/0.60  thf(fact_5732_pos__le__minus__divide__eq,axiom,
% 0.30/0.60      ! [C: rat,A: rat,B: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 0.30/0.60          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_le_minus_divide_eq
% 0.30/0.60  thf(fact_5733_pos__minus__divide__le__eq,axiom,
% 0.30/0.60      ! [C: real,B: real,A: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_minus_divide_le_eq
% 0.30/0.60  thf(fact_5734_pos__minus__divide__le__eq,axiom,
% 0.30/0.60      ! [C: rat,B: rat,A: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 0.30/0.60          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % pos_minus_divide_le_eq
% 0.30/0.60  thf(fact_5735_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: real,C: real,W2: num] :
% 0.30/0.60        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_less_eq_numeral(2)
% 0.30/0.60  thf(fact_5736_divide__less__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: rat,C: rat,W2: num] :
% 0.30/0.60        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_less_eq_numeral(2)
% 0.30/0.60  thf(fact_5737_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: real,C: real] :
% 0.30/0.60        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5738_less__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: rat,C: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5739_signed__take__bit__int__less__eq,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 0.30/0.60       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_less_eq
% 0.30/0.60  thf(fact_5740_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.60      ! [K: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.60       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.60          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.60  thf(fact_5741_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.60      ! [K: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.60       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.60  thf(fact_5742_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.60      ! [K: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.60       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.60          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.60  thf(fact_5743_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.60      ! [K: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.60       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.60          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.60  thf(fact_5744_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 0.30/0.60      ! [K: nat,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.60       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
% 0.30/0.60          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_one_power_add_eq_neg_one_power_diff
% 0.30/0.60  thf(fact_5745_signed__take__bit__int__eq__self,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 0.30/0.60       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.60         => ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 0.30/0.60            = K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_eq_self
% 0.30/0.60  thf(fact_5746_signed__take__bit__int__eq__self__iff,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 0.30/0.60          = K )
% 0.30/0.60        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 0.30/0.60          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_eq_self_iff
% 0.30/0.60  thf(fact_5747_take__bit__eq__0__iff,axiom,
% 0.30/0.60      ! [N2: nat,A: code_integer] :
% 0.30/0.60        ( ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 0.30/0.60          = zero_z3403309356797280102nteger )
% 0.30/0.60        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_eq_0_iff
% 0.30/0.60  thf(fact_5748_take__bit__eq__0__iff,axiom,
% 0.30/0.60      ! [N2: nat,A: nat] :
% 0.30/0.60        ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_eq_0_iff
% 0.30/0.60  thf(fact_5749_take__bit__eq__0__iff,axiom,
% 0.30/0.60      ! [N2: nat,A: int] :
% 0.30/0.60        ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_eq_0_iff
% 0.30/0.60  thf(fact_5750_take__bit__nat__less__self__iff,axiom,
% 0.30/0.60      ! [N2: nat,M: nat] :
% 0.30/0.60        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
% 0.30/0.60        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_nat_less_self_iff
% 0.30/0.60  thf(fact_5751_take__bit__int__eq__self,axiom,
% 0.30/0.60      ! [K: int,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.60       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.60         => ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.30/0.60            = K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_eq_self
% 0.30/0.60  thf(fact_5752_take__bit__int__eq__self__iff,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 0.30/0.60          = K )
% 0.30/0.60        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.60          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_eq_self_iff
% 0.30/0.60  thf(fact_5753_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: real,C: real,W2: num] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_le_eq_numeral(2)
% 0.30/0.60  thf(fact_5754_divide__le__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [B: rat,C: rat,W2: num] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divide_le_eq_numeral(2)
% 0.30/0.60  thf(fact_5755_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: real,C: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 0.30/0.60        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.60           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 0.30/0.60               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5756_le__divide__eq__numeral_I2_J,axiom,
% 0.30/0.60      ! [W2: num,B: rat,C: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 0.30/0.60        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 0.30/0.60          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 0.30/0.60           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 0.30/0.60              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 0.30/0.60               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_divide_eq_numeral(2)
% 0.30/0.60  thf(fact_5757_square__le__1,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.60         => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % square_le_1
% 0.30/0.60  thf(fact_5758_square__le__1,axiom,
% 0.30/0.60      ! [X3: code_integer] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
% 0.30/0.60       => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
% 0.30/0.60         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % square_le_1
% 0.30/0.60  thf(fact_5759_square__le__1,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
% 0.30/0.60         => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % square_le_1
% 0.30/0.60  thf(fact_5760_square__le__1,axiom,
% 0.30/0.60      ! [X3: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_int @ X3 @ one_one_int )
% 0.30/0.60         => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % square_le_1
% 0.30/0.60  thf(fact_5761_div__pos__geq,axiom,
% 0.30/0.60      ! [L: int,K: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ L )
% 0.30/0.60       => ( ( ord_less_eq_int @ L @ K )
% 0.30/0.60         => ( ( divide_divide_int @ K @ L )
% 0.30/0.60            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_pos_geq
% 0.30/0.60  thf(fact_5762_div__pos__neg__trivial,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.60       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 0.30/0.60         => ( ( divide_divide_int @ K @ L )
% 0.30/0.60            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_pos_neg_trivial
% 0.30/0.60  thf(fact_5763_take__bit__int__greater__eq,axiom,
% 0.30/0.60      ! [K: int,N2: nat] :
% 0.30/0.60        ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.60       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_int_greater_eq
% 0.30/0.60  thf(fact_5764_power__minus1__odd,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_minus1_odd
% 0.30/0.60  thf(fact_5765_power__minus1__odd,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_minus1_odd
% 0.30/0.60  thf(fact_5766_power__minus1__odd,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_minus1_odd
% 0.30/0.60  thf(fact_5767_power__minus1__odd,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_minus1_odd
% 0.30/0.60  thf(fact_5768_power__minus1__odd,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60  
% 0.30/0.60  % power_minus1_odd
% 0.30/0.60  thf(fact_5769_take__bit__Suc,axiom,
% 0.30/0.60      ! [N2: nat,A: code_integer] :
% 0.30/0.60        ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
% 0.30/0.60        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc
% 0.30/0.60  thf(fact_5770_take__bit__Suc,axiom,
% 0.30/0.60      ! [N2: nat,A: nat] :
% 0.30/0.60        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
% 0.30/0.60        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc
% 0.30/0.60  thf(fact_5771_take__bit__Suc,axiom,
% 0.30/0.60      ! [N2: nat,A: int] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc
% 0.30/0.60  thf(fact_5772_signed__take__bit__int__greater__eq,axiom,
% 0.30/0.60      ! [K: int,N2: nat] :
% 0.30/0.60        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_int_greater_eq
% 0.30/0.60  thf(fact_5773_stable__imp__take__bit__eq,axiom,
% 0.30/0.60      ! [A: code_integer,N2: nat] :
% 0.30/0.60        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.60          = A )
% 0.30/0.60       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 0.30/0.60              = zero_z3403309356797280102nteger ) )
% 0.30/0.60          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 0.30/0.60              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % stable_imp_take_bit_eq
% 0.30/0.60  thf(fact_5774_stable__imp__take__bit__eq,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60          = A )
% 0.30/0.60       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 0.30/0.60              = zero_zero_nat ) )
% 0.30/0.60          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 0.30/0.60              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % stable_imp_take_bit_eq
% 0.30/0.60  thf(fact_5775_stable__imp__take__bit__eq,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.60          = A )
% 0.30/0.60       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 0.30/0.60              = zero_zero_int ) )
% 0.30/0.60          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60           => ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 0.30/0.60              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % stable_imp_take_bit_eq
% 0.30/0.60  thf(fact_5776_signed__take__bit__Suc__minus__bit1,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_Suc_minus_bit1
% 0.30/0.60  thf(fact_5777_concat__bit__Suc,axiom,
% 0.30/0.60      ! [N2: nat,K: int,L: int] :
% 0.30/0.60        ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
% 0.30/0.60        = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % concat_bit_Suc
% 0.30/0.60  thf(fact_5778_Bolzano,axiom,
% 0.30/0.60      ! [A: real,B: real,P: real > real > $o] :
% 0.30/0.60        ( ( ord_less_eq_real @ A @ B )
% 0.30/0.60       => ( ! [A4: real,B4: real,C2: real] :
% 0.30/0.60              ( ( P @ A4 @ B4 )
% 0.30/0.60             => ( ( P @ B4 @ C2 )
% 0.30/0.60               => ( ( ord_less_eq_real @ A4 @ B4 )
% 0.30/0.60                 => ( ( ord_less_eq_real @ B4 @ C2 )
% 0.30/0.60                   => ( P @ A4 @ C2 ) ) ) ) )
% 0.30/0.60         => ( ! [X4: real] :
% 0.30/0.60                ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.60               => ( ( ord_less_eq_real @ X4 @ B )
% 0.30/0.60                 => ? [D3: real] :
% 0.30/0.60                      ( ( ord_less_real @ zero_zero_real @ D3 )
% 0.30/0.60                      & ! [A4: real,B4: real] :
% 0.30/0.60                          ( ( ( ord_less_eq_real @ A4 @ X4 )
% 0.30/0.60                            & ( ord_less_eq_real @ X4 @ B4 )
% 0.30/0.60                            & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
% 0.30/0.60                         => ( P @ A4 @ B4 ) ) ) ) )
% 0.30/0.60           => ( P @ A @ B ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Bolzano
% 0.30/0.60  thf(fact_5779_signed__take__bit__Suc__bit1,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_Suc_bit1
% 0.30/0.60  thf(fact_5780_and__int_Oelims,axiom,
% 0.30/0.60      ! [X3: int,Xa2: int,Y3: int] :
% 0.30/0.60        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 0.30/0.60          = Y3 )
% 0.30/0.60       => ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60           => ( Y3
% 0.30/0.60              = ( uminus_uminus_int
% 0.30/0.60                @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.60          & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60           => ( Y3
% 0.30/0.60              = ( plus_plus_int
% 0.30/0.60                @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.60                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_int.elims
% 0.30/0.60  thf(fact_5781_and__int_Osimps,axiom,
% 0.30/0.60      ( bit_se725231765392027082nd_int
% 0.30/0.60      = ( ^ [K3: int,L2: int] :
% 0.30/0.60            ( if_int
% 0.30/0.60            @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60            @ ( uminus_uminus_int
% 0.30/0.60              @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.30/0.60                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 0.30/0.60            @ ( plus_plus_int
% 0.30/0.60              @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 0.30/0.60                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 0.30/0.60              @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_int.simps
% 0.30/0.60  thf(fact_5782_Diff__idemp,axiom,
% 0.30/0.60      ! [A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 0.30/0.60        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_idemp
% 0.30/0.60  thf(fact_5783_Diff__iff,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.60        = ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.60          & ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_iff
% 0.30/0.60  thf(fact_5784_Diff__iff,axiom,
% 0.30/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.30/0.60        = ( ( member_real @ C @ A2 )
% 0.30/0.60          & ~ ( member_real @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_iff
% 0.30/0.60  thf(fact_5785_Diff__iff,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.30/0.60        = ( ( member_set_nat @ C @ A2 )
% 0.30/0.60          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_iff
% 0.30/0.60  thf(fact_5786_Diff__iff,axiom,
% 0.30/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.60        = ( ( member_int @ C @ A2 )
% 0.30/0.60          & ~ ( member_int @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_iff
% 0.30/0.60  thf(fact_5787_Diff__iff,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.60        = ( ( member_nat @ C @ A2 )
% 0.30/0.60          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Diff_iff
% 0.30/0.60  thf(fact_5788_DiffI,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.60       => ( ~ ( member8440522571783428010at_nat @ C @ B2 )
% 0.30/0.60         => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffI
% 0.30/0.60  thf(fact_5789_DiffI,axiom,
% 0.30/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ A2 )
% 0.30/0.60       => ( ~ ( member_real @ C @ B2 )
% 0.30/0.60         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffI
% 0.30/0.60  thf(fact_5790_DiffI,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ A2 )
% 0.30/0.60       => ( ~ ( member_set_nat @ C @ B2 )
% 0.30/0.60         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffI
% 0.30/0.60  thf(fact_5791_DiffI,axiom,
% 0.30/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ A2 )
% 0.30/0.60       => ( ~ ( member_int @ C @ B2 )
% 0.30/0.60         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffI
% 0.30/0.60  thf(fact_5792_DiffI,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ A2 )
% 0.30/0.60       => ( ~ ( member_nat @ C @ B2 )
% 0.30/0.60         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffI
% 0.30/0.60  thf(fact_5793_ComplI,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ~ ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.60       => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplI
% 0.30/0.60  thf(fact_5794_ComplI,axiom,
% 0.30/0.60      ! [C: real,A2: set_real] :
% 0.30/0.60        ( ~ ( member_real @ C @ A2 )
% 0.30/0.60       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplI
% 0.30/0.60  thf(fact_5795_ComplI,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat] :
% 0.30/0.60        ( ~ ( member_set_nat @ C @ A2 )
% 0.30/0.60       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplI
% 0.30/0.60  thf(fact_5796_ComplI,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat] :
% 0.30/0.60        ( ~ ( member_nat @ C @ A2 )
% 0.30/0.60       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplI
% 0.30/0.60  thf(fact_5797_ComplI,axiom,
% 0.30/0.60      ! [C: int,A2: set_int] :
% 0.30/0.60        ( ~ ( member_int @ C @ A2 )
% 0.30/0.60       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplI
% 0.30/0.60  thf(fact_5798_Compl__iff,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.30/0.60        = ( ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_iff
% 0.30/0.60  thf(fact_5799_Compl__iff,axiom,
% 0.30/0.60      ! [C: real,A2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.30/0.60        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_iff
% 0.30/0.60  thf(fact_5800_Compl__iff,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.30/0.60        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_iff
% 0.30/0.60  thf(fact_5801_Compl__iff,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.30/0.60        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_iff
% 0.30/0.60  thf(fact_5802_Compl__iff,axiom,
% 0.30/0.60      ! [C: int,A2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.30/0.60        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Compl_iff
% 0.30/0.60  thf(fact_5803_and__zero__eq,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % and_zero_eq
% 0.30/0.60  thf(fact_5804_and__zero__eq,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_zero_eq
% 0.30/0.60  thf(fact_5805_zero__and__eq,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_and_eq
% 0.30/0.60  thf(fact_5806_zero__and__eq,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_and_eq
% 0.30/0.60  thf(fact_5807_bit_Oconj__zero__left,axiom,
% 0.30/0.60      ! [X3: int] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X3 )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % bit.conj_zero_left
% 0.30/0.60  thf(fact_5808_bit_Oconj__zero__right,axiom,
% 0.30/0.60      ! [X3: int] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ X3 @ zero_zero_int )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % bit.conj_zero_right
% 0.30/0.60  thf(fact_5809_concat__bit__0,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 0.30/0.60        = L ) ).
% 0.30/0.60  
% 0.30/0.60  % concat_bit_0
% 0.30/0.60  thf(fact_5810_semiring__norm_I80_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_norm(80)
% 0.30/0.60  thf(fact_5811_and__negative__int__iff,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 0.30/0.60        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.60          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_negative_int_iff
% 0.30/0.60  thf(fact_5812_semiring__norm_I81_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_norm(81)
% 0.30/0.60  thf(fact_5813_semiring__norm_I77_J,axiom,
% 0.30/0.60      ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_norm(77)
% 0.30/0.60  thf(fact_5814_concat__bit__negative__iff,axiom,
% 0.30/0.60      ! [N2: nat,K: int,L: int] :
% 0.30/0.60        ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % concat_bit_negative_iff
% 0.30/0.60  thf(fact_5815_semiring__norm_I79_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_norm(79)
% 0.30/0.60  thf(fact_5816_semiring__norm_I74_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_norm(74)
% 0.30/0.60  thf(fact_5817_and__numerals_I5_J,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ one_one_int )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(5)
% 0.30/0.60  thf(fact_5818_and__numerals_I5_J,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ one_one_nat )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(5)
% 0.30/0.60  thf(fact_5819_and__numerals_I1_J,axiom,
% 0.30/0.60      ! [Y3: num] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(1)
% 0.30/0.60  thf(fact_5820_and__numerals_I1_J,axiom,
% 0.30/0.60      ! [Y3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(1)
% 0.30/0.60  thf(fact_5821_div__Suc__eq__div__add3,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 0.30/0.60        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % div_Suc_eq_div_add3
% 0.30/0.60  thf(fact_5822_Suc__div__eq__add3__div__numeral,axiom,
% 0.30/0.60      ! [M: nat,V: num] :
% 0.30/0.60        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.60        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_div_eq_add3_div_numeral
% 0.30/0.60  thf(fact_5823_mod__Suc__eq__mod__add3,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 0.30/0.60        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_Suc_eq_mod_add3
% 0.30/0.60  thf(fact_5824_Suc__mod__eq__add3__mod__numeral,axiom,
% 0.30/0.60      ! [M: nat,V: num] :
% 0.30/0.60        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 0.30/0.60        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_mod_eq_add3_mod_numeral
% 0.30/0.60  thf(fact_5825_and__numerals_I7_J,axiom,
% 0.30/0.60      ! [X3: num,Y3: num] :
% 0.30/0.60        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.30/0.60        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(7)
% 0.30/0.60  thf(fact_5826_and__numerals_I7_J,axiom,
% 0.30/0.60      ! [X3: num,Y3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.60        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_numerals(7)
% 0.30/0.60  thf(fact_5827_DiffD2,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD2
% 0.30/0.60  thf(fact_5828_DiffD2,axiom,
% 0.30/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( member_real @ C @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD2
% 0.30/0.60  thf(fact_5829_DiffD2,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD2
% 0.30/0.60  thf(fact_5830_DiffD2,axiom,
% 0.30/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( member_int @ C @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD2
% 0.30/0.60  thf(fact_5831_DiffD2,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( member_nat @ C @ B2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD2
% 0.30/0.60  thf(fact_5832_DiffD1,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.60       => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD1
% 0.30/0.60  thf(fact_5833_DiffD1,axiom,
% 0.30/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.30/0.60       => ( member_real @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD1
% 0.30/0.60  thf(fact_5834_DiffD1,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.30/0.60       => ( member_set_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD1
% 0.30/0.60  thf(fact_5835_DiffD1,axiom,
% 0.30/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.60       => ( member_int @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD1
% 0.30/0.60  thf(fact_5836_DiffD1,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.60       => ( member_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffD1
% 0.30/0.60  thf(fact_5837_DiffE,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 0.30/0.60           => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffE
% 0.30/0.60  thf(fact_5838_DiffE,axiom,
% 0.30/0.60      ! [C: real,A2: set_real,B2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( ( member_real @ C @ A2 )
% 0.30/0.60           => ( member_real @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffE
% 0.30/0.60  thf(fact_5839_DiffE,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( ( member_set_nat @ C @ A2 )
% 0.30/0.60           => ( member_set_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffE
% 0.30/0.60  thf(fact_5840_DiffE,axiom,
% 0.30/0.60      ! [C: int,A2: set_int,B2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( ( member_int @ C @ A2 )
% 0.30/0.60           => ( member_int @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffE
% 0.30/0.60  thf(fact_5841_DiffE,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat,B2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.60       => ~ ( ( member_nat @ C @ A2 )
% 0.30/0.60           => ( member_nat @ C @ B2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % DiffE
% 0.30/0.60  thf(fact_5842_ComplD,axiom,
% 0.30/0.60      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.60        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 0.30/0.60       => ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplD
% 0.30/0.60  thf(fact_5843_ComplD,axiom,
% 0.30/0.60      ! [C: real,A2: set_real] :
% 0.30/0.60        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 0.30/0.60       => ~ ( member_real @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplD
% 0.30/0.60  thf(fact_5844_ComplD,axiom,
% 0.30/0.60      ! [C: set_nat,A2: set_set_nat] :
% 0.30/0.60        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 0.30/0.60       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplD
% 0.30/0.60  thf(fact_5845_ComplD,axiom,
% 0.30/0.60      ! [C: nat,A2: set_nat] :
% 0.30/0.60        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 0.30/0.60       => ~ ( member_nat @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplD
% 0.30/0.60  thf(fact_5846_ComplD,axiom,
% 0.30/0.60      ! [C: int,A2: set_int] :
% 0.30/0.60        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 0.30/0.60       => ~ ( member_int @ C @ A2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ComplD
% 0.30/0.60  thf(fact_5847_concat__bit__assoc,axiom,
% 0.30/0.60      ! [N2: nat,K: int,M: nat,L: int,R2: int] :
% 0.30/0.60        ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 0.30/0.60        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % concat_bit_assoc
% 0.30/0.60  thf(fact_5848_xor__num_Ocases,axiom,
% 0.30/0.60      ! [X3: product_prod_num_num] :
% 0.30/0.60        ( ( X3
% 0.30/0.60         != ( product_Pair_num_num @ one @ one ) )
% 0.30/0.60       => ( ! [N3: num] :
% 0.30/0.60              ( X3
% 0.30/0.60             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 0.30/0.60         => ( ! [N3: num] :
% 0.30/0.60                ( X3
% 0.30/0.60               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 0.30/0.60           => ( ! [M3: num] :
% 0.30/0.60                  ( X3
% 0.30/0.60                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
% 0.30/0.60             => ( ! [M3: num,N3: num] :
% 0.30/0.60                    ( X3
% 0.30/0.60                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
% 0.30/0.60               => ( ! [M3: num,N3: num] :
% 0.30/0.60                      ( X3
% 0.30/0.60                     != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
% 0.30/0.60                 => ( ! [M3: num] :
% 0.30/0.60                        ( X3
% 0.30/0.60                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
% 0.30/0.60                   => ( ! [M3: num,N3: num] :
% 0.30/0.60                          ( X3
% 0.30/0.60                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
% 0.30/0.60                     => ~ ! [M3: num,N3: num] :
% 0.30/0.60                            ( X3
% 0.30/0.60                           != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % xor_num.cases
% 0.30/0.60  thf(fact_5849_and__less__eq,axiom,
% 0.30/0.60      ! [L: int,K: int] :
% 0.30/0.60        ( ( ord_less_int @ L @ zero_zero_int )
% 0.30/0.60       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_less_eq
% 0.30/0.60  thf(fact_5850_AND__upper1_H_H,axiom,
% 0.30/0.60      ! [Y3: int,Z2: int,Ya: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.60       => ( ( ord_less_int @ Y3 @ Z2 )
% 0.30/0.60         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % AND_upper1''
% 0.30/0.60  thf(fact_5851_AND__upper2_H_H,axiom,
% 0.30/0.60      ! [Y3: int,Z2: int,X3: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.60       => ( ( ord_less_int @ Y3 @ Z2 )
% 0.30/0.60         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % AND_upper2''
% 0.30/0.60  thf(fact_5852_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5853_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5854_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5855_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5856_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5857_numeral__Bit1,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_Bit1
% 0.30/0.60  thf(fact_5858_simp__from__to,axiom,
% 0.30/0.60      ( set_or1266510415728281911st_int
% 0.30/0.60      = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % simp_from_to
% 0.30/0.60  thf(fact_5859_eval__nat__numeral_I3_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eval_nat_numeral(3)
% 0.30/0.60  thf(fact_5860_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.60      ! [N2: num,Q4: num] :
% 0.30/0.60        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.30/0.60       != zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(3)
% 0.30/0.60  thf(fact_5861_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.60      ! [N2: num,Q4: num] :
% 0.30/0.60        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.30/0.60       != zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(3)
% 0.30/0.60  thf(fact_5862_cong__exp__iff__simps_I3_J,axiom,
% 0.30/0.60      ! [N2: num,Q4: num] :
% 0.30/0.60        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.30/0.60       != zero_z3403309356797280102nteger ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(3)
% 0.30/0.60  thf(fact_5863_numeral__3__eq__3,axiom,
% 0.30/0.60      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 0.30/0.60      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_3_eq_3
% 0.30/0.60  thf(fact_5864_Suc3__eq__add__3,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( suc @ ( suc @ ( suc @ N2 ) ) )
% 0.30/0.60        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc3_eq_add_3
% 0.30/0.60  thf(fact_5865_num_Osize_I6_J,axiom,
% 0.30/0.60      ! [X32: num] :
% 0.30/0.60        ( ( size_size_num @ ( bit1 @ X32 ) )
% 0.30/0.60        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % num.size(6)
% 0.30/0.60  thf(fact_5866_num_Osize__gen_I3_J,axiom,
% 0.30/0.60      ! [X32: num] :
% 0.30/0.60        ( ( size_num @ ( bit1 @ X32 ) )
% 0.30/0.60        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % num.size_gen(3)
% 0.30/0.60  thf(fact_5867_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.60      ! [M: num,Q4: num] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.30/0.60          = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(11)
% 0.30/0.60  thf(fact_5868_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.60      ! [M: num,Q4: num] :
% 0.30/0.60        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
% 0.30/0.60          = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(11)
% 0.30/0.60  thf(fact_5869_cong__exp__iff__simps_I11_J,axiom,
% 0.30/0.60      ! [M: num,Q4: num] :
% 0.30/0.60        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.30/0.60          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(11)
% 0.30/0.60  thf(fact_5870_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.60      ! [Q4: num,N2: num] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
% 0.30/0.60          = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(7)
% 0.30/0.60  thf(fact_5871_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.60      ! [Q4: num,N2: num] :
% 0.30/0.60        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
% 0.30/0.60          = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(7)
% 0.30/0.60  thf(fact_5872_cong__exp__iff__simps_I7_J,axiom,
% 0.30/0.60      ! [Q4: num,N2: num] :
% 0.30/0.60        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 0.30/0.60          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 0.30/0.60        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 0.30/0.60          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cong_exp_iff_simps(7)
% 0.30/0.60  thf(fact_5873_periodic__finite__ex,axiom,
% 0.30/0.60      ! [D: int,P: int > $o] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.60       => ( ! [X4: int,K2: int] :
% 0.30/0.60              ( ( P @ X4 )
% 0.30/0.60              = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 0.30/0.60         => ( ( ? [X5: int] : ( P @ X5 ) )
% 0.30/0.60            = ( ? [X: int] :
% 0.30/0.60                  ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 0.30/0.60                  & ( P @ X ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % periodic_finite_ex
% 0.30/0.60  thf(fact_5874_bset_I3_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( X2 = T )
% 0.30/0.60               => ( ( minus_minus_int @ X2 @ D6 )
% 0.30/0.60                  = T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(3)
% 0.30/0.60  thf(fact_5875_bset_I4_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ T @ B2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( X2 != T )
% 0.30/0.60               => ( ( minus_minus_int @ X2 @ D6 )
% 0.30/0.60                 != T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(4)
% 0.30/0.60  thf(fact_5876_bset_I5_J,axiom,
% 0.30/0.60      ! [D6: int,B2: set_int,T: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ! [X2: int] :
% 0.30/0.60            ( ! [Xa3: int] :
% 0.30/0.60                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60               => ! [Xb2: int] :
% 0.30/0.60                    ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                   => ( X2
% 0.30/0.60                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60           => ( ( ord_less_int @ X2 @ T )
% 0.30/0.60             => ( ord_less_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(5)
% 0.30/0.60  thf(fact_5877_bset_I7_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ T @ B2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( ord_less_int @ T @ X2 )
% 0.30/0.60               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(7)
% 0.30/0.60  thf(fact_5878_aset_I3_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,A2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( X2 = T )
% 0.30/0.60               => ( ( plus_plus_int @ X2 @ D6 )
% 0.30/0.60                  = T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(3)
% 0.30/0.60  thf(fact_5879_aset_I4_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,A2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ T @ A2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( X2 != T )
% 0.30/0.60               => ( ( plus_plus_int @ X2 @ D6 )
% 0.30/0.60                 != T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(4)
% 0.30/0.60  thf(fact_5880_aset_I5_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,A2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ T @ A2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( ord_less_int @ X2 @ T )
% 0.30/0.60               => ( ord_less_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(5)
% 0.30/0.60  thf(fact_5881_aset_I7_J,axiom,
% 0.30/0.60      ! [D6: int,A2: set_int,T: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ! [X2: int] :
% 0.30/0.60            ( ! [Xa3: int] :
% 0.30/0.60                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60               => ! [Xb2: int] :
% 0.30/0.60                    ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                   => ( X2
% 0.30/0.60                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60           => ( ( ord_less_int @ T @ X2 )
% 0.30/0.60             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(7)
% 0.30/0.60  thf(fact_5882_Suc__div__eq__add3__div,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 0.30/0.60        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_div_eq_add3_div
% 0.30/0.60  thf(fact_5883_Suc__mod__eq__add3__mod,axiom,
% 0.30/0.60      ! [M: nat,N2: nat] :
% 0.30/0.60        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 0.30/0.60        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_mod_eq_add3_mod
% 0.30/0.60  thf(fact_5884_aset_I8_J,axiom,
% 0.30/0.60      ! [D6: int,A2: set_int,T: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ! [X2: int] :
% 0.30/0.60            ( ! [Xa3: int] :
% 0.30/0.60                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60               => ! [Xb2: int] :
% 0.30/0.60                    ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                   => ( X2
% 0.30/0.60                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60           => ( ( ord_less_eq_int @ T @ X2 )
% 0.30/0.60             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(8)
% 0.30/0.60  thf(fact_5885_aset_I6_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,A2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ A2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( ord_less_eq_int @ X2 @ T )
% 0.30/0.60               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % aset(6)
% 0.30/0.60  thf(fact_5886_bset_I8_J,axiom,
% 0.30/0.60      ! [D6: int,T: int,B2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 0.30/0.60         => ! [X2: int] :
% 0.30/0.60              ( ! [Xa3: int] :
% 0.30/0.60                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                 => ! [Xb2: int] :
% 0.30/0.60                      ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                     => ( X2
% 0.30/0.60                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60             => ( ( ord_less_eq_int @ T @ X2 )
% 0.30/0.60               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(8)
% 0.30/0.60  thf(fact_5887_bset_I6_J,axiom,
% 0.30/0.60      ! [D6: int,B2: set_int,T: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ! [X2: int] :
% 0.30/0.60            ( ! [Xa3: int] :
% 0.30/0.60                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60               => ! [Xb2: int] :
% 0.30/0.60                    ( ( member_int @ Xb2 @ B2 )
% 0.30/0.60                   => ( X2
% 0.30/0.60                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 0.30/0.60           => ( ( ord_less_eq_int @ X2 @ T )
% 0.30/0.60             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bset(6)
% 0.30/0.60  thf(fact_5888_cppi,axiom,
% 0.30/0.60      ! [D6: int,P: int > $o,P5: int > $o,A2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ? [Z4: int] :
% 0.30/0.60            ! [X4: int] :
% 0.30/0.60              ( ( ord_less_int @ Z4 @ X4 )
% 0.30/0.60             => ( ( P @ X4 )
% 0.30/0.60                = ( P5 @ X4 ) ) )
% 0.30/0.60         => ( ! [X4: int] :
% 0.30/0.60                ( ! [Xa: int] :
% 0.30/0.60                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                   => ! [Xb3: int] :
% 0.30/0.60                        ( ( member_int @ Xb3 @ A2 )
% 0.30/0.60                       => ( X4
% 0.30/0.60                         != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
% 0.30/0.60               => ( ( P @ X4 )
% 0.30/0.60                 => ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
% 0.30/0.60           => ( ! [X4: int,K2: int] :
% 0.30/0.60                  ( ( P5 @ X4 )
% 0.30/0.60                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 0.30/0.60             => ( ( ? [X5: int] : ( P @ X5 ) )
% 0.30/0.60                = ( ? [X: int] :
% 0.30/0.60                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                      & ( P5 @ X ) )
% 0.30/0.60                  | ? [X: int] :
% 0.30/0.60                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                      & ? [Y: int] :
% 0.30/0.60                          ( ( member_int @ Y @ A2 )
% 0.30/0.60                          & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cppi
% 0.30/0.60  thf(fact_5889_cpmi,axiom,
% 0.30/0.60      ! [D6: int,P: int > $o,P5: int > $o,B2: set_int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ D6 )
% 0.30/0.60       => ( ? [Z4: int] :
% 0.30/0.60            ! [X4: int] :
% 0.30/0.60              ( ( ord_less_int @ X4 @ Z4 )
% 0.30/0.60             => ( ( P @ X4 )
% 0.30/0.60                = ( P5 @ X4 ) ) )
% 0.30/0.60         => ( ! [X4: int] :
% 0.30/0.60                ( ! [Xa: int] :
% 0.30/0.60                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                   => ! [Xb3: int] :
% 0.30/0.60                        ( ( member_int @ Xb3 @ B2 )
% 0.30/0.60                       => ( X4
% 0.30/0.60                         != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
% 0.30/0.60               => ( ( P @ X4 )
% 0.30/0.60                 => ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
% 0.30/0.60           => ( ! [X4: int,K2: int] :
% 0.30/0.60                  ( ( P5 @ X4 )
% 0.30/0.60                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 0.30/0.60             => ( ( ? [X5: int] : ( P @ X5 ) )
% 0.30/0.60                = ( ? [X: int] :
% 0.30/0.60                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                      & ( P5 @ X ) )
% 0.30/0.60                  | ? [X: int] :
% 0.30/0.60                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 0.30/0.60                      & ? [Y: int] :
% 0.30/0.60                          ( ( member_int @ Y @ B2 )
% 0.30/0.60                          & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % cpmi
% 0.30/0.60  thf(fact_5890_take__bit__Suc__bit1,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_bit1
% 0.30/0.60  thf(fact_5891_take__bit__Suc__bit1,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_bit1
% 0.30/0.60  thf(fact_5892_odd__mod__4__div__2,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 0.30/0.60       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_mod_4_div_2
% 0.30/0.60  thf(fact_5893_mod__exhaust__less__4,axiom,
% 0.30/0.60      ! [M: nat] :
% 0.30/0.60        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = one_one_nat )
% 0.30/0.60        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.60        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.60          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mod_exhaust_less_4
% 0.30/0.60  thf(fact_5894_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(7)
% 0.30/0.60  thf(fact_5895_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(7)
% 0.30/0.60  thf(fact_5896_divmod__algorithm__code_I7_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 0.30/0.60         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(7)
% 0.30/0.60  thf(fact_5897_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(8)
% 0.30/0.60  thf(fact_5898_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(8)
% 0.30/0.60  thf(fact_5899_divmod__algorithm__code_I8_J,axiom,
% 0.30/0.60      ! [M: num,N2: num] :
% 0.30/0.60        ( ( ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 0.30/0.60        & ( ~ ( ord_less_num @ M @ N2 )
% 0.30/0.60         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 0.30/0.60            = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(8)
% 0.30/0.60  thf(fact_5900_take__bit__Suc__minus__bit1,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_minus_bit1
% 0.30/0.60  thf(fact_5901_and__int_Opsimps,axiom,
% 0.30/0.60      ! [K: int,L: int] :
% 0.30/0.60        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 0.30/0.60       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.60              = ( uminus_uminus_int
% 0.30/0.60                @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 0.30/0.60          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 0.30/0.60              = ( plus_plus_int
% 0.30/0.60                @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 0.30/0.60                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 0.30/0.60                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_int.psimps
% 0.30/0.60  thf(fact_5902_and__int_Opelims,axiom,
% 0.30/0.60      ! [X3: int,Xa2: int,Y3: int] :
% 0.30/0.60        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 0.30/0.60          = Y3 )
% 0.30/0.60       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 0.30/0.60         => ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60                 => ( Y3
% 0.30/0.60                    = ( uminus_uminus_int
% 0.30/0.60                      @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.60                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 0.30/0.60                & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60                 => ( Y3
% 0.30/0.60                    = ( plus_plus_int
% 0.30/0.60                      @ ( zero_n2684676970156552555ol_int
% 0.30/0.60                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 0.30/0.60                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 0.30/0.60                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 0.30/0.60             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_int.pelims
% 0.30/0.60  thf(fact_5903_finite__atLeastAtMost__int,axiom,
% 0.30/0.60      ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_atLeastAtMost_int
% 0.30/0.60  thf(fact_5904_and__nat__numerals_I1_J,axiom,
% 0.30/0.60      ! [Y3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_numerals(1)
% 0.30/0.60  thf(fact_5905_and__nat__numerals_I3_J,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.60        = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_numerals(3)
% 0.30/0.60  thf(fact_5906_and__nat__numerals_I2_J,axiom,
% 0.30/0.60      ! [Y3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.60        = one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_numerals(2)
% 0.30/0.60  thf(fact_5907_and__nat__numerals_I4_J,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.60        = one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_numerals(4)
% 0.30/0.60  thf(fact_5908_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( unique5052692396658037445od_int @ M @ one )
% 0.30/0.60        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(2)
% 0.30/0.60  thf(fact_5909_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( unique5055182867167087721od_nat @ M @ one )
% 0.30/0.60        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(2)
% 0.30/0.60  thf(fact_5910_divmod__algorithm__code_I2_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( unique3479559517661332726nteger @ M @ one )
% 0.30/0.60        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(2)
% 0.30/0.60  thf(fact_5911_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(6)
% 0.30/0.60  thf(fact_5912_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(6)
% 0.30/0.60  thf(fact_5913_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(6)
% 0.30/0.60  thf(fact_5914_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(6)
% 0.30/0.60  thf(fact_5915_add__neg__numeral__special_I6_J,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(6)
% 0.30/0.60  thf(fact_5916_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 0.30/0.60        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(5)
% 0.30/0.60  thf(fact_5917_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(5)
% 0.30/0.60  thf(fact_5918_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(5)
% 0.30/0.60  thf(fact_5919_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(5)
% 0.30/0.60  thf(fact_5920_add__neg__numeral__special_I5_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % add_neg_numeral_special(5)
% 0.30/0.60  thf(fact_5921_and__Suc__0__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.60        = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_Suc_0_eq
% 0.30/0.60  thf(fact_5922_Suc__0__and__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.60        = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_0_and_eq
% 0.30/0.60  thf(fact_5923_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(3)
% 0.30/0.60  thf(fact_5924_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(3)
% 0.30/0.60  thf(fact_5925_divmod__algorithm__code_I3_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(3)
% 0.30/0.60  thf(fact_5926_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(4)
% 0.30/0.60  thf(fact_5927_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(4)
% 0.30/0.60  thf(fact_5928_divmod__algorithm__code_I4_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
% 0.30/0.60        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_algorithm_code(4)
% 0.30/0.60  thf(fact_5929_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numera6690914467698888265omplex @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5930_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numera1916890842035813515d_enat @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5931_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numeral_numeral_real @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5932_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numeral_numeral_rat @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5933_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numeral_numeral_nat @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5934_numeral__inc,axiom,
% 0.30/0.60      ! [X3: num] :
% 0.30/0.60        ( ( numeral_numeral_int @ ( inc @ X3 ) )
% 0.30/0.60        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_inc
% 0.30/0.60  thf(fact_5935_divmod__int__def,axiom,
% 0.30/0.60      ( unique5052692396658037445od_int
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_int_def
% 0.30/0.60  thf(fact_5936_divmod__def,axiom,
% 0.30/0.60      ( unique5052692396658037445od_int
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_def
% 0.30/0.60  thf(fact_5937_divmod__def,axiom,
% 0.30/0.60      ( unique5055182867167087721od_nat
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_def
% 0.30/0.60  thf(fact_5938_divmod__def,axiom,
% 0.30/0.60      ( unique3479559517661332726nteger
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_def
% 0.30/0.60  thf(fact_5939_divmod_H__nat__def,axiom,
% 0.30/0.60      ( unique5055182867167087721od_nat
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod'_nat_def
% 0.30/0.60  thf(fact_5940_and__nat__unfold,axiom,
% 0.30/0.60      ( bit_se727722235901077358nd_nat
% 0.30/0.60      = ( ^ [M2: nat,N: nat] :
% 0.30/0.60            ( if_nat
% 0.30/0.60            @ ( ( M2 = zero_zero_nat )
% 0.30/0.60              | ( N = zero_zero_nat ) )
% 0.30/0.60            @ zero_zero_nat
% 0.30/0.60            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_unfold
% 0.30/0.60  thf(fact_5941_and__nat__rec,axiom,
% 0.30/0.60      ( bit_se727722235901077358nd_nat
% 0.30/0.60      = ( ^ [M2: nat,N: nat] :
% 0.30/0.60            ( plus_plus_nat
% 0.30/0.60            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.60              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
% 0.30/0.60                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.60            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_nat_rec
% 0.30/0.60  thf(fact_5942_and__int_Opinduct,axiom,
% 0.30/0.60      ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.60        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.60       => ( ! [K2: int,L4: int] :
% 0.30/0.60              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 0.30/0.60             => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 0.30/0.60                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 0.30/0.60                 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.60               => ( P @ K2 @ L4 ) ) )
% 0.30/0.60         => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_int.pinduct
% 0.30/0.60  thf(fact_5943_divmod__divmod__step,axiom,
% 0.30/0.60      ( unique5055182867167087721od_nat
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_divmod_step
% 0.30/0.60  thf(fact_5944_divmod__divmod__step,axiom,
% 0.30/0.60      ( unique5052692396658037445od_int
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_divmod_step
% 0.30/0.60  thf(fact_5945_divmod__divmod__step,axiom,
% 0.30/0.60      ( unique3479559517661332726nteger
% 0.30/0.60      = ( ^ [M2: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_divmod_step
% 0.30/0.60  thf(fact_5946_upto_Opinduct,axiom,
% 0.30/0.60      ! [A0: int,A12: int,P: int > int > $o] :
% 0.30/0.60        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 0.30/0.60       => ( ! [I3: int,J2: int] :
% 0.30/0.60              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 0.30/0.60             => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 0.30/0.60                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 0.30/0.60               => ( P @ I3 @ J2 ) ) )
% 0.30/0.60         => ( P @ A0 @ A12 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % upto.pinduct
% 0.30/0.60  thf(fact_5947_signed__take__bit__eq__take__bit__minus,axiom,
% 0.30/0.60      ( bit_ri631733984087533419it_int
% 0.30/0.60      = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_eq_take_bit_minus
% 0.30/0.60  thf(fact_5948_take__bit__numeral__bit1,axiom,
% 0.30/0.60      ! [L: num,K: num] :
% 0.30/0.60        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 0.30/0.60        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_numeral_bit1
% 0.30/0.60  thf(fact_5949_take__bit__numeral__bit1,axiom,
% 0.30/0.60      ! [L: num,K: num] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_numeral_bit1
% 0.30/0.60  thf(fact_5950_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5951_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5952_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5953_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5954_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_5955_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_5956_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_5957_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_5958_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5959_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5960_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5961_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_5962_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_5963_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_5964_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_5965_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_5966_of__int__eq__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ W2 )
% 0.30/0.60          = ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( W2 = Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_iff
% 0.30/0.60  thf(fact_5967_of__int__eq__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ W2 )
% 0.30/0.60          = ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( W2 = Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_iff
% 0.30/0.60  thf(fact_5968_bit__0__eq,axiom,
% 0.30/0.60      ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 0.30/0.60      = bot_bot_nat_o ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_0_eq
% 0.30/0.60  thf(fact_5969_bit__0__eq,axiom,
% 0.30/0.60      ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 0.30/0.60      = bot_bot_nat_o ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_0_eq
% 0.30/0.60  thf(fact_5970_of__int__0,axiom,
% 0.30/0.60      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 0.30/0.60      = zero_zero_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0
% 0.30/0.60  thf(fact_5971_of__int__0,axiom,
% 0.30/0.60      ( ( ring_1_of_int_int @ zero_zero_int )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0
% 0.30/0.60  thf(fact_5972_of__int__0,axiom,
% 0.30/0.60      ( ( ring_1_of_int_real @ zero_zero_int )
% 0.30/0.60      = zero_zero_real ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0
% 0.30/0.60  thf(fact_5973_of__int__0,axiom,
% 0.30/0.60      ( ( ring_1_of_int_rat @ zero_zero_int )
% 0.30/0.60      = zero_zero_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0
% 0.30/0.60  thf(fact_5974_of__int__0__eq__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( zero_zero_complex
% 0.30/0.60          = ( ring_17405671764205052669omplex @ Z2 ) )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_eq_iff
% 0.30/0.60  thf(fact_5975_of__int__0__eq__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( zero_zero_int
% 0.30/0.60          = ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_eq_iff
% 0.30/0.60  thf(fact_5976_of__int__0__eq__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( zero_zero_real
% 0.30/0.60          = ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_eq_iff
% 0.30/0.60  thf(fact_5977_of__int__0__eq__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( zero_zero_rat
% 0.30/0.60          = ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_eq_iff
% 0.30/0.60  thf(fact_5978_of__int__eq__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 0.30/0.60          = zero_zero_complex )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_0_iff
% 0.30/0.60  thf(fact_5979_of__int__eq__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ Z2 )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_0_iff
% 0.30/0.60  thf(fact_5980_of__int__eq__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ Z2 )
% 0.30/0.60          = zero_zero_real )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_0_iff
% 0.30/0.60  thf(fact_5981_of__int__eq__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ Z2 )
% 0.30/0.60          = zero_zero_rat )
% 0.30/0.60        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_0_iff
% 0.30/0.60  thf(fact_5982_of__int__le__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_iff
% 0.30/0.60  thf(fact_5983_of__int__le__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_iff
% 0.30/0.60  thf(fact_5984_of__int__le__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_iff
% 0.30/0.60  thf(fact_5985_of__int__eq__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ Z2 )
% 0.30/0.60          = ( numeral_numeral_real @ N2 ) )
% 0.30/0.60        = ( Z2
% 0.30/0.60          = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_iff
% 0.30/0.60  thf(fact_5986_of__int__eq__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ Z2 )
% 0.30/0.60          = ( numeral_numeral_rat @ N2 ) )
% 0.30/0.60        = ( Z2
% 0.30/0.60          = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_iff
% 0.30/0.60  thf(fact_5987_of__int__eq__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ Z2 )
% 0.30/0.60          = ( numeral_numeral_int @ N2 ) )
% 0.30/0.60        = ( Z2
% 0.30/0.60          = ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_iff
% 0.30/0.60  thf(fact_5988_of__int__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 0.30/0.60        = ( numeral_numeral_real @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral
% 0.30/0.60  thf(fact_5989_of__int__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 0.30/0.60        = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral
% 0.30/0.60  thf(fact_5990_of__int__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 0.30/0.60        = ( numeral_numeral_int @ K ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral
% 0.30/0.60  thf(fact_5991_of__int__less__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_iff
% 0.30/0.60  thf(fact_5992_of__int__less__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_iff
% 0.30/0.60  thf(fact_5993_of__int__less__iff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_iff
% 0.30/0.60  thf(fact_5994_of__int__eq__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 0.30/0.60          = one_one_complex )
% 0.30/0.60        = ( Z2 = one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_1_iff
% 0.30/0.60  thf(fact_5995_of__int__eq__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ Z2 )
% 0.30/0.60          = one_one_int )
% 0.30/0.60        = ( Z2 = one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_1_iff
% 0.30/0.60  thf(fact_5996_of__int__eq__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ Z2 )
% 0.30/0.60          = one_one_real )
% 0.30/0.60        = ( Z2 = one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_1_iff
% 0.30/0.60  thf(fact_5997_of__int__eq__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ Z2 )
% 0.30/0.60          = one_one_rat )
% 0.30/0.60        = ( Z2 = one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_1_iff
% 0.30/0.60  thf(fact_5998_of__int__1,axiom,
% 0.30/0.60      ( ( ring_17405671764205052669omplex @ one_one_int )
% 0.30/0.60      = one_one_complex ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1
% 0.30/0.60  thf(fact_5999_of__int__1,axiom,
% 0.30/0.60      ( ( ring_1_of_int_int @ one_one_int )
% 0.30/0.60      = one_one_int ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1
% 0.30/0.60  thf(fact_6000_of__int__1,axiom,
% 0.30/0.60      ( ( ring_1_of_int_real @ one_one_int )
% 0.30/0.60      = one_one_real ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1
% 0.30/0.60  thf(fact_6001_of__int__1,axiom,
% 0.30/0.60      ( ( ring_1_of_int_rat @ one_one_int )
% 0.30/0.60      = one_one_rat ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1
% 0.30/0.60  thf(fact_6002_of__int__mult,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_mult
% 0.30/0.60  thf(fact_6003_of__int__mult,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_mult
% 0.30/0.60  thf(fact_6004_of__int__mult,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_mult
% 0.30/0.60  thf(fact_6005_of__int__mult,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_mult
% 0.30/0.60  thf(fact_6006_of__int__add,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_add
% 0.30/0.60  thf(fact_6007_of__int__add,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_add
% 0.30/0.60  thf(fact_6008_of__int__add,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_add
% 0.30/0.60  thf(fact_6009_of__int__minus,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_minus
% 0.30/0.60  thf(fact_6010_of__int__minus,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_minus
% 0.30/0.60  thf(fact_6011_of__int__minus,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_minus
% 0.30/0.60  thf(fact_6012_of__int__minus,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_minus
% 0.30/0.60  thf(fact_6013_of__int__diff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_diff
% 0.30/0.60  thf(fact_6014_of__int__diff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_diff
% 0.30/0.60  thf(fact_6015_of__int__diff,axiom,
% 0.30/0.60      ! [W2: int,Z2: int] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
% 0.30/0.60        = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_diff
% 0.30/0.60  thf(fact_6016_pred__numeral__simps_I1_J,axiom,
% 0.30/0.60      ( ( pred_numeral @ one )
% 0.30/0.60      = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % pred_numeral_simps(1)
% 0.30/0.60  thf(fact_6017_eq__numeral__Suc,axiom,
% 0.30/0.60      ! [K: num,N2: nat] :
% 0.30/0.60        ( ( ( numeral_numeral_nat @ K )
% 0.30/0.60          = ( suc @ N2 ) )
% 0.30/0.60        = ( ( pred_numeral @ K )
% 0.30/0.60          = N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eq_numeral_Suc
% 0.30/0.60  thf(fact_6018_Suc__eq__numeral,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( ( suc @ N2 )
% 0.30/0.60          = ( numeral_numeral_nat @ K ) )
% 0.30/0.60        = ( N2
% 0.30/0.60          = ( pred_numeral @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_eq_numeral
% 0.30/0.60  thf(fact_6019_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ X3 )
% 0.30/0.60          = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 0.30/0.60        = ( X3
% 0.30/0.60          = ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6020_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ X3 )
% 0.30/0.60          = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 0.30/0.60        = ( X3
% 0.30/0.60          = ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6021_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ X3 )
% 0.30/0.60          = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 0.30/0.60        = ( X3
% 0.30/0.60          = ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6022_of__int__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ( ring_17405671764205052669omplex @ X3 )
% 0.30/0.60          = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
% 0.30/0.60        = ( X3
% 0.30/0.60          = ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6023_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
% 0.30/0.60          = ( ring_1_of_int_rat @ X3 ) )
% 0.30/0.60        = ( ( power_power_int @ B @ W2 )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6024_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
% 0.30/0.60          = ( ring_1_of_int_real @ X3 ) )
% 0.30/0.60        = ( ( power_power_int @ B @ W2 )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6025_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
% 0.30/0.60          = ( ring_1_of_int_int @ X3 ) )
% 0.30/0.60        = ( ( power_power_int @ B @ W2 )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6026_of__int__eq__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
% 0.30/0.60          = ( ring_17405671764205052669omplex @ X3 ) )
% 0.30/0.60        = ( ( power_power_int @ B @ W2 )
% 0.30/0.60          = X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6027_of__int__power,axiom,
% 0.30/0.60      ! [Z2: int,N2: nat] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N2 ) )
% 0.30/0.60        = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power
% 0.30/0.60  thf(fact_6028_of__int__power,axiom,
% 0.30/0.60      ! [Z2: int,N2: nat] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N2 ) )
% 0.30/0.60        = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power
% 0.30/0.60  thf(fact_6029_of__int__power,axiom,
% 0.30/0.60      ! [Z2: int,N2: nat] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N2 ) )
% 0.30/0.60        = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power
% 0.30/0.60  thf(fact_6030_of__int__power,axiom,
% 0.30/0.60      ! [Z2: int,N2: nat] :
% 0.30/0.60        ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N2 ) )
% 0.30/0.60        = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power
% 0.30/0.60  thf(fact_6031_of__int__of__bool,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.60        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_of_bool
% 0.30/0.60  thf(fact_6032_of__int__of__bool,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.60        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_of_bool
% 0.30/0.60  thf(fact_6033_of__int__of__bool,axiom,
% 0.30/0.60      ! [P: $o] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.60        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_of_bool
% 0.30/0.60  thf(fact_6034_bit__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.60      ! [M: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_numeral_Bit0_Suc_iff
% 0.30/0.60  thf(fact_6035_bit__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.60      ! [M: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_numeral_Bit0_Suc_iff
% 0.30/0.60  thf(fact_6036_bit__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.60      ! [M: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_numeral_Bit1_Suc_iff
% 0.30/0.60  thf(fact_6037_bit__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.60      ! [M: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_numeral_Bit1_Suc_iff
% 0.30/0.60  thf(fact_6038_less__numeral__Suc,axiom,
% 0.30/0.60      ! [K: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_numeral_Suc
% 0.30/0.60  thf(fact_6039_less__Suc__numeral,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60        = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_Suc_numeral
% 0.30/0.60  thf(fact_6040_le__numeral__Suc,axiom,
% 0.30/0.60      ! [K: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_numeral_Suc
% 0.30/0.60  thf(fact_6041_le__Suc__numeral,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60        = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_Suc_numeral
% 0.30/0.60  thf(fact_6042_diff__Suc__numeral,axiom,
% 0.30/0.60      ! [N2: nat,K: num] :
% 0.30/0.60        ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.60        = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_Suc_numeral
% 0.30/0.60  thf(fact_6043_diff__numeral__Suc,axiom,
% 0.30/0.60      ! [K: num,N2: nat] :
% 0.30/0.60        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % diff_numeral_Suc
% 0.30/0.60  thf(fact_6044_signed__take__bit__negative__iff,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.60        = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % signed_take_bit_negative_iff
% 0.30/0.60  thf(fact_6045_of__int__0__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_le_iff
% 0.30/0.60  thf(fact_6046_of__int__0__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_le_iff
% 0.30/0.60  thf(fact_6047_of__int__0__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_le_iff
% 0.30/0.60  thf(fact_6048_of__int__le__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_0_iff
% 0.30/0.60  thf(fact_6049_of__int__le__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_0_iff
% 0.30/0.60  thf(fact_6050_of__int__le__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_0_iff
% 0.30/0.60  thf(fact_6051_of__int__less__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 0.30/0.60        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_0_iff
% 0.30/0.60  thf(fact_6052_of__int__less__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 0.30/0.60        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_0_iff
% 0.30/0.60  thf(fact_6053_of__int__less__0__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_0_iff
% 0.30/0.60  thf(fact_6054_of__int__0__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_less_iff
% 0.30/0.60  thf(fact_6055_of__int__0__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_less_iff
% 0.30/0.60  thf(fact_6056_of__int__0__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_0_less_iff
% 0.30/0.60  thf(fact_6057_of__int__le__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_iff
% 0.30/0.60  thf(fact_6058_of__int__le__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_iff
% 0.30/0.60  thf(fact_6059_of__int__le__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_iff
% 0.30/0.60  thf(fact_6060_of__int__numeral__le__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_le_iff
% 0.30/0.60  thf(fact_6061_of__int__numeral__le__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_le_iff
% 0.30/0.60  thf(fact_6062_of__int__numeral__le__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_le_iff
% 0.30/0.60  thf(fact_6063_of__int__less__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_iff
% 0.30/0.60  thf(fact_6064_of__int__less__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_iff
% 0.30/0.60  thf(fact_6065_of__int__less__numeral__iff,axiom,
% 0.30/0.60      ! [Z2: int,N2: num] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_iff
% 0.30/0.60  thf(fact_6066_of__int__numeral__less__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_less_iff
% 0.30/0.60  thf(fact_6067_of__int__numeral__less__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_less_iff
% 0.30/0.60  thf(fact_6068_of__int__numeral__less__iff,axiom,
% 0.30/0.60      ! [N2: num,Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_numeral_less_iff
% 0.30/0.60  thf(fact_6069_of__int__le__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_1_iff
% 0.30/0.60  thf(fact_6070_of__int__le__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_1_iff
% 0.30/0.60  thf(fact_6071_of__int__le__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_1_iff
% 0.30/0.60  thf(fact_6072_of__int__1__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_le_iff
% 0.30/0.60  thf(fact_6073_of__int__1__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_le_iff
% 0.30/0.60  thf(fact_6074_of__int__1__le__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_le_iff
% 0.30/0.60  thf(fact_6075_of__int__less__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 0.30/0.60        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_1_iff
% 0.30/0.60  thf(fact_6076_of__int__less__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 0.30/0.60        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_1_iff
% 0.30/0.60  thf(fact_6077_of__int__less__1__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_1_iff
% 0.30/0.60  thf(fact_6078_of__int__1__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_less_iff
% 0.30/0.60  thf(fact_6079_of__int__1__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_less_iff
% 0.30/0.60  thf(fact_6080_of__int__1__less__iff,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 0.30/0.60        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_1_less_iff
% 0.30/0.60  thf(fact_6081_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6082_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6083_of__int__le__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6084_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6085_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6086_of__int__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6087_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.30/0.60          = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6088_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ Y3 )
% 0.30/0.60          = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6089_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ Y3 )
% 0.30/0.60          = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6090_of__int__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ Y3 )
% 0.30/0.60          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6091_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 )
% 0.30/0.60          = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6092_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6093_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6094_numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_int @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6095_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6096_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6097_of__int__less__of__int__power__cancel__iff,axiom,
% 0.30/0.60      ! [B: int,W2: nat,X3: int] :
% 0.30/0.60        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_of_int_power_cancel_iff
% 0.30/0.60  thf(fact_6098_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6099_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6100_of__int__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: int,B: int,W2: nat] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 0.30/0.60        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6101_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 0.30/0.60      ! [W2: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_minus_numeral_Bit0_Suc_iff
% 0.30/0.60  thf(fact_6102_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 0.30/0.60      ! [W2: num,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
% 0.30/0.60        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_minus_numeral_Bit1_Suc_iff
% 0.30/0.60  thf(fact_6103_bit__0,axiom,
% 0.30/0.60      ! [A: code_integer] :
% 0.30/0.60        ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 0.30/0.60        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_0
% 0.30/0.60  thf(fact_6104_bit__0,axiom,
% 0.30/0.60      ! [A: int] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 0.30/0.60        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_0
% 0.30/0.60  thf(fact_6105_bit__0,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 0.30/0.60        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_0
% 0.30/0.60  thf(fact_6106_bit__mod__2__iff,axiom,
% 0.30/0.60      ! [A: code_integer,N2: nat] :
% 0.30/0.60        ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 )
% 0.30/0.60        = ( ( N2 = zero_zero_nat )
% 0.30/0.60          & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_mod_2_iff
% 0.30/0.60  thf(fact_6107_bit__mod__2__iff,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 )
% 0.30/0.60        = ( ( N2 = zero_zero_nat )
% 0.30/0.60          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_mod_2_iff
% 0.30/0.60  thf(fact_6108_bit__mod__2__iff,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 0.30/0.60        = ( ( N2 = zero_zero_nat )
% 0.30/0.60          & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_mod_2_iff
% 0.30/0.60  thf(fact_6109_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6110_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6111_numeral__power__le__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.60        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_le_of_int_cancel_iff
% 0.30/0.60  thf(fact_6112_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6113_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6114_of__int__le__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_le_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6115_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6116_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6117_numeral__power__less__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,A: int] :
% 0.30/0.60        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 0.30/0.60        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_power_less_of_int_cancel_iff
% 0.30/0.60  thf(fact_6118_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6119_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6120_of__int__less__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [A: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 0.30/0.60        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_less_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6121_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_17405671764205052669omplex @ Y3 )
% 0.30/0.60          = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6122_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_int @ Y3 )
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6123_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_real @ Y3 )
% 0.30/0.60          = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6124_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_1_of_int_rat @ Y3 )
% 0.30/0.60          = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6125_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 0.30/0.60      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.60        ( ( ( ring_18347121197199848620nteger @ Y3 )
% 0.30/0.60          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 0.30/0.60        = ( Y3
% 0.30/0.60          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_eq_neg_numeral_power_cancel_iff
% 0.30/0.60  thf(fact_6126_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 )
% 0.30/0.60          = ( ring_17405671764205052669omplex @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6127_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_int @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6128_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6129_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 )
% 0.30/0.60          = ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6130_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 0.30/0.60      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.60        ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 )
% 0.30/0.60          = ( ring_18347121197199848620nteger @ Y3 ) )
% 0.30/0.60        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 0.30/0.60          = Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_power_eq_of_int_cancel_iff
% 0.30/0.60  thf(fact_6131_bit__disjunctive__add__iff,axiom,
% 0.30/0.60      ! [A: int,B: int,N2: nat] :
% 0.30/0.60        ( ! [N3: nat] :
% 0.30/0.60            ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
% 0.30/0.60            | ~ ( bit_se1146084159140164899it_int @ B @ N3 ) )
% 0.30/0.60       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
% 0.30/0.60          = ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.30/0.60            | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_disjunctive_add_iff
% 0.30/0.60  thf(fact_6132_bit__disjunctive__add__iff,axiom,
% 0.30/0.60      ! [A: nat,B: nat,N2: nat] :
% 0.30/0.60        ( ! [N3: nat] :
% 0.30/0.60            ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
% 0.30/0.60            | ~ ( bit_se1148574629649215175it_nat @ B @ N3 ) )
% 0.30/0.60       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 0.30/0.60          = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.30/0.60            | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_disjunctive_add_iff
% 0.30/0.60  thf(fact_6133_ex__le__of__int,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_le_of_int
% 0.30/0.60  thf(fact_6134_ex__le__of__int,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_le_of_int
% 0.30/0.60  thf(fact_6135_ex__less__of__int,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_less_of_int
% 0.30/0.60  thf(fact_6136_ex__less__of__int,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_less_of_int
% 0.30/0.60  thf(fact_6137_ex__of__int__less,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_of_int_less
% 0.30/0.60  thf(fact_6138_ex__of__int__less,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60      ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % ex_of_int_less
% 0.30/0.60  thf(fact_6139_mult__of__int__commute,axiom,
% 0.30/0.60      ! [X3: int,Y3: complex] :
% 0.30/0.60        ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X3 ) @ Y3 )
% 0.30/0.60        = ( times_times_complex @ Y3 @ ( ring_17405671764205052669omplex @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_of_int_commute
% 0.30/0.60  thf(fact_6140_mult__of__int__commute,axiom,
% 0.30/0.60      ! [X3: int,Y3: real] :
% 0.30/0.60        ( ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ Y3 )
% 0.30/0.60        = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_of_int_commute
% 0.30/0.60  thf(fact_6141_mult__of__int__commute,axiom,
% 0.30/0.60      ! [X3: int,Y3: rat] :
% 0.30/0.60        ( ( times_times_rat @ ( ring_1_of_int_rat @ X3 ) @ Y3 )
% 0.30/0.60        = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_of_int_commute
% 0.30/0.60  thf(fact_6142_mult__of__int__commute,axiom,
% 0.30/0.60      ! [X3: int,Y3: int] :
% 0.30/0.60        ( ( times_times_int @ ( ring_1_of_int_int @ X3 ) @ Y3 )
% 0.30/0.60        = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_of_int_commute
% 0.30/0.60  thf(fact_6143_not__bit__1__Suc,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % not_bit_1_Suc
% 0.30/0.60  thf(fact_6144_not__bit__1__Suc,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % not_bit_1_Suc
% 0.30/0.60  thf(fact_6145_bit__1__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_1_iff
% 0.30/0.60  thf(fact_6146_bit__1__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_1_iff
% 0.30/0.60  thf(fact_6147_bit__take__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: nat,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
% 0.30/0.60        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.60          & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_take_bit_iff
% 0.30/0.60  thf(fact_6148_bit__take__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,A: int,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
% 0.30/0.60        = ( ( ord_less_nat @ N2 @ M )
% 0.30/0.60          & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_take_bit_iff
% 0.30/0.60  thf(fact_6149_bit__of__bool__iff,axiom,
% 0.30/0.60      ! [B: $o,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N2 )
% 0.30/0.60        = ( B
% 0.30/0.60          & ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_of_bool_iff
% 0.30/0.60  thf(fact_6150_bit__of__bool__iff,axiom,
% 0.30/0.60      ! [B: $o,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N2 )
% 0.30/0.60        = ( B
% 0.30/0.60          & ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_of_bool_iff
% 0.30/0.60  thf(fact_6151_numeral__eq__Suc,axiom,
% 0.30/0.60      ( numeral_numeral_nat
% 0.30/0.60      = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_eq_Suc
% 0.30/0.60  thf(fact_6152_of__int__nonneg,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_nonneg
% 0.30/0.60  thf(fact_6153_of__int__nonneg,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_nonneg
% 0.30/0.60  thf(fact_6154_of__int__nonneg,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_nonneg
% 0.30/0.60  thf(fact_6155_of__int__pos,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_pos
% 0.30/0.60  thf(fact_6156_of__int__pos,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_pos
% 0.30/0.60  thf(fact_6157_of__int__pos,axiom,
% 0.30/0.60      ! [Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.60       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_pos
% 0.30/0.60  thf(fact_6158_floor__exists,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60      ? [Z3: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
% 0.30/0.60        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % floor_exists
% 0.30/0.60  thf(fact_6159_floor__exists,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60      ? [Z3: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
% 0.30/0.60        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % floor_exists
% 0.30/0.60  thf(fact_6160_floor__exists1,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60      ? [X4: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X3 )
% 0.30/0.60        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.30/0.60        & ! [Y6: int] :
% 0.30/0.60            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X3 )
% 0.30/0.60              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
% 0.30/0.60           => ( Y6 = X4 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % floor_exists1
% 0.30/0.60  thf(fact_6161_floor__exists1,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60      ? [X4: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X3 )
% 0.30/0.60        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 0.30/0.60        & ! [Y6: int] :
% 0.30/0.60            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X3 )
% 0.30/0.60              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
% 0.30/0.60           => ( Y6 = X4 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % floor_exists1
% 0.30/0.60  thf(fact_6162_of__int__neg__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_neg_numeral
% 0.30/0.60  thf(fact_6163_of__int__neg__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_neg_numeral
% 0.30/0.60  thf(fact_6164_of__int__neg__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_neg_numeral
% 0.30/0.60  thf(fact_6165_of__int__neg__numeral,axiom,
% 0.30/0.60      ! [K: num] :
% 0.30/0.60        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.60        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_neg_numeral
% 0.30/0.60  thf(fact_6166_bit__imp__take__bit__positive,axiom,
% 0.30/0.60      ! [N2: nat,M: nat,K: int] :
% 0.30/0.60        ( ( ord_less_nat @ N2 @ M )
% 0.30/0.60       => ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 0.30/0.60         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_imp_take_bit_positive
% 0.30/0.60  thf(fact_6167_int__le__real__less,axiom,
% 0.30/0.60      ( ord_less_eq_int
% 0.30/0.60      = ( ^ [N: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % int_le_real_less
% 0.30/0.60  thf(fact_6168_int__less__real__le,axiom,
% 0.30/0.60      ( ord_less_int
% 0.30/0.60      = ( ^ [N: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % int_less_real_le
% 0.30/0.60  thf(fact_6169_bit__concat__bit__iff,axiom,
% 0.30/0.60      ! [M: nat,K: int,L: int,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
% 0.30/0.60        = ( ( ( ord_less_nat @ N2 @ M )
% 0.30/0.60            & ( bit_se1146084159140164899it_int @ K @ N2 ) )
% 0.30/0.60          | ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.60            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_concat_bit_iff
% 0.30/0.60  thf(fact_6170_exp__eq__0__imp__not__bit,axiom,
% 0.30/0.60      ! [N2: nat,A: int] :
% 0.30/0.60        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60       => ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % exp_eq_0_imp_not_bit
% 0.30/0.60  thf(fact_6171_exp__eq__0__imp__not__bit,axiom,
% 0.30/0.60      ! [N2: nat,A: nat] :
% 0.30/0.60        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60       => ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % exp_eq_0_imp_not_bit
% 0.30/0.60  thf(fact_6172_bit__Suc,axiom,
% 0.30/0.60      ! [A: code_integer,N2: nat] :
% 0.30/0.60        ( ( bit_se9216721137139052372nteger @ A @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_Suc
% 0.30/0.60  thf(fact_6173_bit__Suc,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_Suc
% 0.30/0.60  thf(fact_6174_bit__Suc,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N2 ) )
% 0.30/0.60        = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_Suc
% 0.30/0.60  thf(fact_6175_int__bit__bound,axiom,
% 0.30/0.60      ! [K: int] :
% 0.30/0.60        ~ ! [N3: nat] :
% 0.30/0.60            ( ! [M6: nat] :
% 0.30/0.60                ( ( ord_less_eq_nat @ N3 @ M6 )
% 0.30/0.60               => ( ( bit_se1146084159140164899it_int @ K @ M6 )
% 0.30/0.60                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 0.30/0.60           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.60               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 0.30/0.60                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % int_bit_bound
% 0.30/0.60  thf(fact_6176_and__exp__eq__0__iff__not__bit,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_exp_eq_0_iff_not_bit
% 0.30/0.60  thf(fact_6177_and__exp__eq__0__iff__not__bit,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        = ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % and_exp_eq_0_iff_not_bit
% 0.30/0.60  thf(fact_6178_even__bit__succ__iff,axiom,
% 0.30/0.60      ! [A: code_integer,N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
% 0.30/0.60          = ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_bit_succ_iff
% 0.30/0.60  thf(fact_6179_even__bit__succ__iff,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
% 0.30/0.60          = ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_bit_succ_iff
% 0.30/0.60  thf(fact_6180_even__bit__succ__iff,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
% 0.30/0.60          = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % even_bit_succ_iff
% 0.30/0.60  thf(fact_6181_odd__bit__iff__bit__pred,axiom,
% 0.30/0.60      ! [A: code_integer,N2: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 0.30/0.60          = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_bit_iff_bit_pred
% 0.30/0.60  thf(fact_6182_odd__bit__iff__bit__pred,axiom,
% 0.30/0.60      ! [A: int,N2: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 0.30/0.60          = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_bit_iff_bit_pred
% 0.30/0.60  thf(fact_6183_odd__bit__iff__bit__pred,axiom,
% 0.30/0.60      ! [A: nat,N2: nat] :
% 0.30/0.60        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 0.30/0.60       => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 0.30/0.60          = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
% 0.30/0.60            | ( N2 = zero_zero_nat ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % odd_bit_iff_bit_pred
% 0.30/0.60  thf(fact_6184_bit__sum__mult__2__cases,axiom,
% 0.30/0.60      ! [A: code_integer,B: code_integer,N2: nat] :
% 0.30/0.60        ( ! [J2: nat] :
% 0.30/0.60            ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 0.30/0.60       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 0.30/0.60          = ( ( ( N2 = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60            & ( ( N2 != zero_zero_nat )
% 0.30/0.60             => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_sum_mult_2_cases
% 0.30/0.60  thf(fact_6185_bit__sum__mult__2__cases,axiom,
% 0.30/0.60      ! [A: int,B: int,N2: nat] :
% 0.30/0.60        ( ! [J2: nat] :
% 0.30/0.60            ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 0.30/0.60       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 0.30/0.60          = ( ( ( N2 = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60            & ( ( N2 != zero_zero_nat )
% 0.30/0.60             => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_sum_mult_2_cases
% 0.30/0.60  thf(fact_6186_bit__sum__mult__2__cases,axiom,
% 0.30/0.60      ! [A: nat,B: nat,N2: nat] :
% 0.30/0.60        ( ! [J2: nat] :
% 0.30/0.60            ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 0.30/0.60       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 0.30/0.60          = ( ( ( N2 = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 0.30/0.60            & ( ( N2 != zero_zero_nat )
% 0.30/0.60             => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_sum_mult_2_cases
% 0.30/0.60  thf(fact_6187_bit__rec,axiom,
% 0.30/0.60      ( bit_se9216721137139052372nteger
% 0.30/0.60      = ( ^ [A3: code_integer,N: nat] :
% 0.30/0.60            ( ( ( N = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 0.30/0.60            & ( ( N != zero_zero_nat )
% 0.30/0.60             => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_rec
% 0.30/0.60  thf(fact_6188_bit__rec,axiom,
% 0.30/0.60      ( bit_se1146084159140164899it_int
% 0.30/0.60      = ( ^ [A3: int,N: nat] :
% 0.30/0.60            ( ( ( N = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 0.30/0.60            & ( ( N != zero_zero_nat )
% 0.30/0.60             => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_rec
% 0.30/0.60  thf(fact_6189_bit__rec,axiom,
% 0.30/0.60      ( bit_se1148574629649215175it_nat
% 0.30/0.60      = ( ^ [A3: nat,N: nat] :
% 0.30/0.60            ( ( ( N = zero_zero_nat )
% 0.30/0.60             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 0.30/0.60            & ( ( N != zero_zero_nat )
% 0.30/0.60             => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_rec
% 0.30/0.60  thf(fact_6190_take__bit__Suc__from__most,axiom,
% 0.30/0.60      ! [N2: nat,K: int] :
% 0.30/0.60        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
% 0.30/0.60        = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % take_bit_Suc_from_most
% 0.30/0.60  thf(fact_6191_round__unique,axiom,
% 0.30/0.60      ! [X3: real,Y3: int] :
% 0.30/0.60        ( ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 0.30/0.60       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.60         => ( ( archim8280529875227126926d_real @ X3 )
% 0.30/0.60            = Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % round_unique
% 0.30/0.60  thf(fact_6192_round__unique,axiom,
% 0.30/0.60      ! [X3: rat,Y3: int] :
% 0.30/0.60        ( ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 0.30/0.60       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.60         => ( ( archim7778729529865785530nd_rat @ X3 )
% 0.30/0.60            = Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % round_unique
% 0.30/0.60  thf(fact_6193_of__int__round__gt,axiom,
% 0.30/0.60      ! [X3: real] : ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_gt
% 0.30/0.60  thf(fact_6194_of__int__round__gt,axiom,
% 0.30/0.60      ! [X3: rat] : ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_gt
% 0.30/0.60  thf(fact_6195_of__int__round__ge,axiom,
% 0.30/0.60      ! [X3: real] : ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_ge
% 0.30/0.60  thf(fact_6196_of__int__round__ge,axiom,
% 0.30/0.60      ! [X3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_ge
% 0.30/0.60  thf(fact_6197_of__int__round__le,axiom,
% 0.30/0.60      ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_le
% 0.30/0.60  thf(fact_6198_of__int__round__le,axiom,
% 0.30/0.60      ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_round_le
% 0.30/0.60  thf(fact_6199_divmod__BitM__2__eq,axiom,
% 0.30/0.60      ! [M: num] :
% 0.30/0.60        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 0.30/0.60        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % divmod_BitM_2_eq
% 0.30/0.60  thf(fact_6200_mask__numeral,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.60        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_numeral
% 0.30/0.60  thf(fact_6201_mask__numeral,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.60        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_numeral
% 0.30/0.60  thf(fact_6202_neg__numeral__le__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_le_ceiling
% 0.30/0.60  thf(fact_6203_neg__numeral__le__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_le_ceiling
% 0.30/0.60  thf(fact_6204_mask__nat__positive__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 0.30/0.60        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_nat_positive_iff
% 0.30/0.60  thf(fact_6205_ceiling__zero,axiom,
% 0.30/0.60      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_zero
% 0.30/0.60  thf(fact_6206_ceiling__zero,axiom,
% 0.30/0.60      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_zero
% 0.30/0.60  thf(fact_6207_mask__eq__0__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( ( bit_se2002935070580805687sk_nat @ N2 )
% 0.30/0.60          = zero_zero_nat )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_eq_0_iff
% 0.30/0.60  thf(fact_6208_mask__eq__0__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( ( bit_se2000444600071755411sk_int @ N2 )
% 0.30/0.60          = zero_zero_int )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_eq_0_iff
% 0.30/0.60  thf(fact_6209_mask__0,axiom,
% 0.30/0.60      ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 0.30/0.60      = zero_zero_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_0
% 0.30/0.60  thf(fact_6210_mask__0,axiom,
% 0.30/0.60      ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_0
% 0.30/0.60  thf(fact_6211_round__0,axiom,
% 0.30/0.60      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % round_0
% 0.30/0.60  thf(fact_6212_round__0,axiom,
% 0.30/0.60      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 0.30/0.60      = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % round_0
% 0.30/0.60  thf(fact_6213_mask__Suc__0,axiom,
% 0.30/0.60      ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 0.30/0.60      = one_one_nat ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_Suc_0
% 0.30/0.60  thf(fact_6214_mask__Suc__0,axiom,
% 0.30/0.60      ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 0.30/0.60      = one_one_int ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_Suc_0
% 0.30/0.60  thf(fact_6215_ceiling__add__of__int,axiom,
% 0.30/0.60      ! [X3: rat,Z2: int] :
% 0.30/0.60        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_of_int
% 0.30/0.60  thf(fact_6216_ceiling__add__of__int,axiom,
% 0.30/0.60      ! [X3: real,Z2: int] :
% 0.30/0.60        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_of_int
% 0.30/0.60  thf(fact_6217_ceiling__le__zero,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_zero
% 0.30/0.60  thf(fact_6218_ceiling__le__zero,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_zero
% 0.30/0.60  thf(fact_6219_zero__less__ceiling,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_ceiling
% 0.30/0.60  thf(fact_6220_zero__less__ceiling,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_less_ceiling
% 0.30/0.60  thf(fact_6221_ceiling__le__numeral,axiom,
% 0.30/0.60      ! [X3: real,V: num] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_numeral
% 0.30/0.60  thf(fact_6222_ceiling__le__numeral,axiom,
% 0.30/0.60      ! [X3: rat,V: num] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_numeral
% 0.30/0.60  thf(fact_6223_ceiling__less__one,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_one
% 0.30/0.60  thf(fact_6224_ceiling__less__one,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_one
% 0.30/0.60  thf(fact_6225_one__le__ceiling,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % one_le_ceiling
% 0.30/0.60  thf(fact_6226_one__le__ceiling,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % one_le_ceiling
% 0.30/0.60  thf(fact_6227_numeral__less__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: real] :
% 0.30/0.60        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_less_ceiling
% 0.30/0.60  thf(fact_6228_numeral__less__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_less_ceiling
% 0.30/0.60  thf(fact_6229_ceiling__le__one,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_one
% 0.30/0.60  thf(fact_6230_ceiling__le__one,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_one
% 0.30/0.60  thf(fact_6231_one__less__ceiling,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ one_one_rat @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % one_less_ceiling
% 0.30/0.60  thf(fact_6232_one__less__ceiling,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ one_one_real @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % one_less_ceiling
% 0.30/0.60  thf(fact_6233_ceiling__add__numeral,axiom,
% 0.30/0.60      ! [X3: real,V: num] :
% 0.30/0.60        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ V ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_numeral
% 0.30/0.60  thf(fact_6234_ceiling__add__numeral,axiom,
% 0.30/0.60      ! [X3: rat,V: num] :
% 0.30/0.60        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( numeral_numeral_rat @ V ) ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_numeral
% 0.30/0.60  thf(fact_6235_ceiling__add__one,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_one
% 0.30/0.60  thf(fact_6236_ceiling__add__one,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ one_one_real ) )
% 0.30/0.60        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_one
% 0.30/0.60  thf(fact_6237_ceiling__less__zero,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_zero
% 0.30/0.60  thf(fact_6238_ceiling__less__zero,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_zero
% 0.30/0.60  thf(fact_6239_zero__le__ceiling,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_ceiling
% 0.30/0.60  thf(fact_6240_zero__le__ceiling,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % zero_le_ceiling
% 0.30/0.60  thf(fact_6241_ceiling__less__numeral,axiom,
% 0.30/0.60      ! [X3: real,V: num] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_numeral
% 0.30/0.60  thf(fact_6242_ceiling__less__numeral,axiom,
% 0.30/0.60      ! [X3: rat,V: num] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_numeral
% 0.30/0.60  thf(fact_6243_numeral__le__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_le_ceiling
% 0.30/0.60  thf(fact_6244_numeral__le__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % numeral_le_ceiling
% 0.30/0.60  thf(fact_6245_ceiling__le__neg__numeral,axiom,
% 0.30/0.60      ! [X3: real,V: num] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_neg_numeral
% 0.30/0.60  thf(fact_6246_ceiling__le__neg__numeral,axiom,
% 0.30/0.60      ! [X3: rat,V: num] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_neg_numeral
% 0.30/0.60  thf(fact_6247_neg__numeral__less__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: real] :
% 0.30/0.60        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_less_ceiling
% 0.30/0.60  thf(fact_6248_neg__numeral__less__ceiling,axiom,
% 0.30/0.60      ! [V: num,X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % neg_numeral_less_ceiling
% 0.30/0.60  thf(fact_6249_ceiling__less__neg__numeral,axiom,
% 0.30/0.60      ! [X3: real,V: num] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_neg_numeral
% 0.30/0.60  thf(fact_6250_ceiling__less__neg__numeral,axiom,
% 0.30/0.60      ! [X3: rat,V: num] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_neg_numeral
% 0.30/0.60  thf(fact_6251_less__eq__mask,axiom,
% 0.30/0.60      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_eq_mask
% 0.30/0.60  thf(fact_6252_bit__Suc__0__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % bit_Suc_0_iff
% 0.30/0.60  thf(fact_6253_not__bit__Suc__0__Suc,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % not_bit_Suc_0_Suc
% 0.30/0.60  thf(fact_6254_ceiling__mono,axiom,
% 0.30/0.60      ! [Y3: real,X3: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ Y3 @ X3 )
% 0.30/0.60       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_mono
% 0.30/0.60  thf(fact_6255_ceiling__mono,axiom,
% 0.30/0.60      ! [Y3: rat,X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 0.30/0.60       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_mono
% 0.30/0.60  thf(fact_6256_le__of__int__ceiling,axiom,
% 0.30/0.60      ! [X3: real] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_of_int_ceiling
% 0.30/0.60  thf(fact_6257_le__of__int__ceiling,axiom,
% 0.30/0.60      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_of_int_ceiling
% 0.30/0.60  thf(fact_6258_ceiling__less__cancel,axiom,
% 0.30/0.60      ! [X3: rat,Y3: rat] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 0.30/0.60       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_cancel
% 0.30/0.60  thf(fact_6259_ceiling__less__cancel,axiom,
% 0.30/0.60      ! [X3: real,Y3: real] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 0.30/0.60       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_cancel
% 0.30/0.60  thf(fact_6260_not__mask__negative__int,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % not_mask_negative_int
% 0.30/0.60  thf(fact_6261_not__bit__Suc__0__numeral,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % not_bit_Suc_0_numeral
% 0.30/0.60  thf(fact_6262_ceiling__le__iff,axiom,
% 0.30/0.60      ! [X3: real,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_iff
% 0.30/0.60  thf(fact_6263_ceiling__le__iff,axiom,
% 0.30/0.60      ! [X3: rat,Z2: int] :
% 0.30/0.60        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le_iff
% 0.30/0.60  thf(fact_6264_ceiling__le,axiom,
% 0.30/0.60      ! [X3: real,A: int] :
% 0.30/0.60        ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) )
% 0.30/0.60       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le
% 0.30/0.60  thf(fact_6265_ceiling__le,axiom,
% 0.30/0.60      ! [X3: rat,A: int] :
% 0.30/0.60        ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) )
% 0.30/0.60       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ A ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_le
% 0.30/0.60  thf(fact_6266_less__ceiling__iff,axiom,
% 0.30/0.60      ! [Z2: int,X3: rat] :
% 0.30/0.60        ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_ceiling_iff
% 0.30/0.60  thf(fact_6267_less__ceiling__iff,axiom,
% 0.30/0.60      ! [Z2: int,X3: real] :
% 0.30/0.60        ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_ceiling_iff
% 0.30/0.60  thf(fact_6268_ceiling__add__le,axiom,
% 0.30/0.60      ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_le
% 0.30/0.60  thf(fact_6269_ceiling__add__le,axiom,
% 0.30/0.60      ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_add_le
% 0.30/0.60  thf(fact_6270_round__mono,axiom,
% 0.30/0.60      ! [X3: rat,Y3: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.60       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X3 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % round_mono
% 0.30/0.60  thf(fact_6271_less__mask,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.60       => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % less_mask
% 0.30/0.60  thf(fact_6272_eval__nat__numeral_I2_J,axiom,
% 0.30/0.60      ! [N2: num] :
% 0.30/0.60        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.30/0.60        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % eval_nat_numeral(2)
% 0.30/0.60  thf(fact_6273_of__int__ceiling__le__add__one,axiom,
% 0.30/0.60      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_ceiling_le_add_one
% 0.30/0.60  thf(fact_6274_of__int__ceiling__le__add__one,axiom,
% 0.30/0.60      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_ceiling_le_add_one
% 0.30/0.60  thf(fact_6275_of__int__ceiling__diff__one__le,axiom,
% 0.30/0.60      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_ceiling_diff_one_le
% 0.30/0.60  thf(fact_6276_of__int__ceiling__diff__one__le,axiom,
% 0.30/0.60      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 0.30/0.60  
% 0.30/0.60  % of_int_ceiling_diff_one_le
% 0.30/0.60  thf(fact_6277_ceiling__split,axiom,
% 0.30/0.60      ! [P: int > $o,T: real] :
% 0.30/0.60        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 0.30/0.60        = ( ! [I2: int] :
% 0.30/0.60              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 0.30/0.60                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 0.30/0.60             => ( P @ I2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_split
% 0.30/0.60  thf(fact_6278_ceiling__split,axiom,
% 0.30/0.60      ! [P: int > $o,T: rat] :
% 0.30/0.60        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 0.30/0.60        = ( ! [I2: int] :
% 0.30/0.60              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 0.30/0.60                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 0.30/0.60             => ( P @ I2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_split
% 0.30/0.60  thf(fact_6279_ceiling__eq__iff,axiom,
% 0.30/0.60      ! [X3: real,A: int] :
% 0.30/0.60        ( ( ( archim7802044766580827645g_real @ X3 )
% 0.30/0.60          = A )
% 0.30/0.60        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X3 )
% 0.30/0.60          & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_eq_iff
% 0.30/0.60  thf(fact_6280_ceiling__eq__iff,axiom,
% 0.30/0.60      ! [X3: rat,A: int] :
% 0.30/0.60        ( ( ( archim2889992004027027881ng_rat @ X3 )
% 0.30/0.60          = A )
% 0.30/0.60        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X3 )
% 0.30/0.60          & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_eq_iff
% 0.30/0.60  thf(fact_6281_ceiling__unique,axiom,
% 0.30/0.60      ! [Z2: int,X3: real] :
% 0.30/0.60        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.60         => ( ( archim7802044766580827645g_real @ X3 )
% 0.30/0.60            = Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_unique
% 0.30/0.60  thf(fact_6282_ceiling__unique,axiom,
% 0.30/0.60      ! [Z2: int,X3: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.60         => ( ( archim2889992004027027881ng_rat @ X3 )
% 0.30/0.60            = Z2 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_unique
% 0.30/0.60  thf(fact_6283_ceiling__correct,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) @ one_one_real ) @ X3 )
% 0.30/0.60        & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_correct
% 0.30/0.60  thf(fact_6284_ceiling__correct,axiom,
% 0.30/0.60      ! [X3: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) @ one_one_rat ) @ X3 )
% 0.30/0.60        & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_correct
% 0.30/0.60  thf(fact_6285_mult__ceiling__le,axiom,
% 0.30/0.60      ! [A: real,B: real] :
% 0.30/0.60        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.60       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.60         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_ceiling_le
% 0.30/0.60  thf(fact_6286_mult__ceiling__le,axiom,
% 0.30/0.60      ! [A: rat,B: rat] :
% 0.30/0.60        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.60       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.60         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mult_ceiling_le
% 0.30/0.60  thf(fact_6287_ceiling__less__iff,axiom,
% 0.30/0.60      ! [X3: real,Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
% 0.30/0.60        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_iff
% 0.30/0.60  thf(fact_6288_ceiling__less__iff,axiom,
% 0.30/0.60      ! [X3: rat,Z2: int] :
% 0.30/0.60        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
% 0.30/0.60        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_less_iff
% 0.30/0.60  thf(fact_6289_le__ceiling__iff,axiom,
% 0.30/0.60      ! [Z2: int,X3: rat] :
% 0.30/0.60        ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
% 0.30/0.60        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_ceiling_iff
% 0.30/0.60  thf(fact_6290_le__ceiling__iff,axiom,
% 0.30/0.60      ! [Z2: int,X3: real] :
% 0.30/0.60        ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
% 0.30/0.60        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % le_ceiling_iff
% 0.30/0.60  thf(fact_6291_ceiling__divide__upper,axiom,
% 0.30/0.60      ! [Q4: real,P4: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.30/0.60       => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_divide_upper
% 0.30/0.60  thf(fact_6292_ceiling__divide__upper,axiom,
% 0.30/0.60      ! [Q4: rat,P4: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.30/0.60       => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_divide_upper
% 0.30/0.60  thf(fact_6293_Suc__mask__eq__exp,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 0.30/0.60        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_mask_eq_exp
% 0.30/0.60  thf(fact_6294_mask__nat__less__exp,axiom,
% 0.30/0.60      ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % mask_nat_less_exp
% 0.30/0.60  thf(fact_6295_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_bit_operations_class.even_mask_iff
% 0.30/0.60  thf(fact_6296_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_bit_operations_class.even_mask_iff
% 0.30/0.60  thf(fact_6297_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
% 0.30/0.60        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % semiring_bit_operations_class.even_mask_iff
% 0.30/0.60  thf(fact_6298_ceiling__divide__lower,axiom,
% 0.30/0.60      ! [Q4: real,P4: real] :
% 0.30/0.60        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.30/0.60       => ( 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 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_divide_lower
% 0.30/0.60  thf(fact_6299_ceiling__divide__lower,axiom,
% 0.30/0.60      ! [Q4: rat,P4: rat] :
% 0.30/0.60        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.30/0.60       => ( 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 ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_divide_lower
% 0.30/0.60  thf(fact_6300_ceiling__eq,axiom,
% 0.30/0.60      ! [N2: int,X3: real] :
% 0.30/0.60        ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 0.30/0.60         => ( ( archim7802044766580827645g_real @ X3 )
% 0.30/0.60            = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_eq
% 0.30/0.60  thf(fact_6301_ceiling__eq,axiom,
% 0.30/0.60      ! [N2: int,X3: rat] :
% 0.30/0.60        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X3 )
% 0.30/0.60       => ( ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
% 0.30/0.60         => ( ( archim2889992004027027881ng_rat @ X3 )
% 0.30/0.60            = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % ceiling_eq
% 0.30/0.60  thf(fact_6302_Suc__0__xor__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % Suc_0_xor_eq
% 0.30/0.60  thf(fact_6303_xor__Suc__0__eq,axiom,
% 0.30/0.60      ! [N2: nat] :
% 0.30/0.60        ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.60        = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.60          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.60            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % xor_Suc_0_eq
% 0.30/0.60  thf(fact_6304_horner__sum__of__bool__2__less,axiom,
% 0.30/0.60      ! [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 ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % horner_sum_of_bool_2_less
% 0.30/0.60  thf(fact_6305_dbl__dec__simps_I2_J,axiom,
% 0.30/0.60      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 0.30/0.60      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_dec_simps(2)
% 0.30/0.60  thf(fact_6306_dbl__dec__simps_I2_J,axiom,
% 0.30/0.60      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 0.30/0.60      = ( uminus_uminus_int @ one_one_int ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_dec_simps(2)
% 0.30/0.60  thf(fact_6307_dbl__dec__simps_I2_J,axiom,
% 0.30/0.60      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 0.30/0.60      = ( uminus_uminus_real @ one_one_real ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_dec_simps(2)
% 0.30/0.60  thf(fact_6308_dbl__dec__simps_I2_J,axiom,
% 0.30/0.60      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 0.30/0.60      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_dec_simps(2)
% 0.30/0.60  thf(fact_6309_dbl__dec__simps_I2_J,axiom,
% 0.30/0.60      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 0.30/0.60      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 0.30/0.60  
% 0.30/0.60  % dbl_dec_simps(2)
% 0.30/0.60  thf(fact_6310_the__elem__eq,axiom,
% 0.30/0.60      ! [X3: produc3843707927480180839at_nat] :
% 0.30/0.60        ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 0.30/0.60        = X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % the_elem_eq
% 0.30/0.60  thf(fact_6311_the__elem__eq,axiom,
% 0.30/0.60      ! [X3: product_prod_nat_nat] :
% 0.30/0.60        ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 0.30/0.60        = X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % the_elem_eq
% 0.30/0.60  thf(fact_6312_the__elem__eq,axiom,
% 0.30/0.60      ! [X3: real] :
% 0.30/0.60        ( ( the_elem_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 0.30/0.60        = X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % the_elem_eq
% 0.30/0.60  thf(fact_6313_the__elem__eq,axiom,
% 0.30/0.60      ! [X3: nat] :
% 0.30/0.60        ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 0.30/0.60        = X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % the_elem_eq
% 0.30/0.60  thf(fact_6314_the__elem__eq,axiom,
% 0.30/0.60      ! [X3: int] :
% 0.30/0.60        ( ( the_elem_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 0.30/0.60        = X3 ) ).
% 0.30/0.60  
% 0.30/0.60  % the_elem_eq
% 0.30/0.60  thf(fact_6315_vebt__member_Oelims_I1_J,axiom,
% 0.30/0.60      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.60        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.60          = Y3 )
% 0.30/0.60       => ( ! [A4: $o,B4: $o] :
% 0.30/0.60              ( ( X3
% 0.30/0.60                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.60             => ( Y3
% 0.30/0.60                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.60                       => A4 )
% 0.30/0.60                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.60                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.60                           => B4 )
% 0.30/0.60                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.60         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.60                  ( X3
% 0.30/0.60                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.60             => Y3 )
% 0.30/0.60           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.60                    ( X3
% 0.30/0.60                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.60               => Y3 )
% 0.30/0.60             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.60                      ( X3
% 0.30/0.60                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.60                 => Y3 )
% 0.30/0.60               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.60                      ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.60                          ( X3
% 0.30/0.60                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.60                     => ( Y3
% 0.30/0.60                        = ( ~ ( ( Xa2 != Mi2 )
% 0.30/0.60                             => ( ( Xa2 != Ma2 )
% 0.30/0.60                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.60                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.60                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.60                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.60                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.60                                           => ( 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 ) ) ) ) ) )
% 0.30/0.60                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % vebt_member.elims(1)
% 0.30/0.60  thf(fact_6316_set__vebt_H__def,axiom,
% 0.30/0.60      ( vEBT_VEBT_set_vebt
% 0.30/0.60      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % set_vebt'_def
% 0.30/0.60  thf(fact_6317_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: real > $o,Q: real > $o] :
% 0.30/0.60        ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.60          | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 0.30/0.60       => ( finite_finite_real
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [X: real] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6318_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.60        ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.30/0.60          | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 0.30/0.60       => ( finite8100373058378681591st_nat
% 0.30/0.60          @ ( collect_list_nat
% 0.30/0.60            @ ^ [X: list_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6319_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.30/0.60        ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.30/0.60          | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 0.30/0.60       => ( finite1152437895449049373et_nat
% 0.30/0.60          @ ( collect_set_nat
% 0.30/0.60            @ ^ [X: set_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6320_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.60        ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.60          | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 0.30/0.60       => ( finite_finite_nat
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [X: nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6321_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: int > $o,Q: int > $o] :
% 0.30/0.60        ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.60          | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 0.30/0.60       => ( finite_finite_int
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [X: int] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6322_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.60        ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.30/0.60          | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 0.30/0.60       => ( finite3207457112153483333omplex
% 0.30/0.60          @ ( collect_complex
% 0.30/0.60            @ ^ [X: complex] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6323_finite__Collect__conjI,axiom,
% 0.30/0.60      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.30/0.60        ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.30/0.60          | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 0.30/0.60       => ( finite6177210948735845034at_nat
% 0.30/0.60          @ ( collec3392354462482085612at_nat
% 0.30/0.60            @ ^ [X: product_prod_nat_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                & ( Q @ X ) ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_conjI
% 0.30/0.60  thf(fact_6324_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: real > $o,Q: real > $o] :
% 0.30/0.60        ( ( finite_finite_real
% 0.30/0.60          @ ( collect_real
% 0.30/0.60            @ ^ [X: real] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.60          & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6325_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: list_nat > $o,Q: list_nat > $o] :
% 0.30/0.60        ( ( finite8100373058378681591st_nat
% 0.30/0.60          @ ( collect_list_nat
% 0.30/0.60            @ ^ [X: list_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.30/0.60          & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6326_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: set_nat > $o,Q: set_nat > $o] :
% 0.30/0.60        ( ( finite1152437895449049373et_nat
% 0.30/0.60          @ ( collect_set_nat
% 0.30/0.60            @ ^ [X: set_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.30/0.60          & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6327_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: nat > $o,Q: nat > $o] :
% 0.30/0.60        ( ( finite_finite_nat
% 0.30/0.60          @ ( collect_nat
% 0.30/0.60            @ ^ [X: nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.60          & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6328_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: int > $o,Q: int > $o] :
% 0.30/0.60        ( ( finite_finite_int
% 0.30/0.60          @ ( collect_int
% 0.30/0.60            @ ^ [X: int] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.60          & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6329_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: complex > $o,Q: complex > $o] :
% 0.30/0.60        ( ( finite3207457112153483333omplex
% 0.30/0.60          @ ( collect_complex
% 0.30/0.60            @ ^ [X: complex] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.30/0.60          & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6330_finite__Collect__disjI,axiom,
% 0.30/0.60      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 0.30/0.60        ( ( finite6177210948735845034at_nat
% 0.30/0.60          @ ( collec3392354462482085612at_nat
% 0.30/0.60            @ ^ [X: product_prod_nat_nat] :
% 0.30/0.60                ( ( P @ X )
% 0.30/0.60                | ( Q @ X ) ) ) )
% 0.30/0.60        = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.30/0.60          & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 0.30/0.60  
% 0.30/0.60  % finite_Collect_disjI
% 0.30/0.60  thf(fact_6331_bit_Oxor__self,axiom,
% 0.30/0.60      ! [X3: int] :
% 0.30/0.60        ( ( bit_se6526347334894502574or_int @ X3 @ X3 )
% 0.30/0.60        = zero_zero_int ) ).
% 0.30/0.60  
% 0.30/0.60  % bit.xor_self
% 0.30/0.60  thf(fact_6332_xor__self__eq,axiom,
% 0.30/0.60      ! [A: nat] :
% 0.30/0.60        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 0.30/0.61        = zero_zero_nat ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_self_eq
% 0.30/0.61  thf(fact_6333_xor__self__eq,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ A @ A )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_self_eq
% 0.30/0.61  thf(fact_6334_xor_Oleft__neutral,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % xor.left_neutral
% 0.30/0.61  thf(fact_6335_xor_Oleft__neutral,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % xor.left_neutral
% 0.30/0.61  thf(fact_6336_xor_Oright__neutral,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % xor.right_neutral
% 0.30/0.61  thf(fact_6337_xor_Oright__neutral,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 0.30/0.61        = A ) ).
% 0.30/0.61  
% 0.30/0.61  % xor.right_neutral
% 0.30/0.61  thf(fact_6338_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: real > $o,Q: real > real > $o] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                ? [Y: real] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: real] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_real
% 0.30/0.61                  @ ( collect_real
% 0.30/0.61                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6339_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: real > $o,Q: nat > real > $o] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                ? [Y: real] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: real] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_nat
% 0.30/0.61                  @ ( collect_nat
% 0.30/0.61                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6340_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: real > $o,Q: int > real > $o] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                ? [Y: real] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: real] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_int
% 0.30/0.61                  @ ( collect_int
% 0.30/0.61                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6341_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: real > $o,Q: complex > real > $o] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [X: complex] :
% 0.30/0.61                ? [Y: real] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: real] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite3207457112153483333omplex
% 0.30/0.61                  @ ( collect_complex
% 0.30/0.61                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6342_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: nat > $o,Q: real > nat > $o] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                ? [Y: nat] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: nat] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_real
% 0.30/0.61                  @ ( collect_real
% 0.30/0.61                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6343_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: nat > $o,Q: nat > nat > $o] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                ? [Y: nat] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: nat] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_nat
% 0.30/0.61                  @ ( collect_nat
% 0.30/0.61                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6344_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: nat > $o,Q: int > nat > $o] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                ? [Y: nat] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: nat] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_int
% 0.30/0.61                  @ ( collect_int
% 0.30/0.61                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6345_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: nat > $o,Q: complex > nat > $o] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [X: complex] :
% 0.30/0.61                ? [Y: nat] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: nat] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite3207457112153483333omplex
% 0.30/0.61                  @ ( collect_complex
% 0.30/0.61                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6346_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: int > $o,Q: real > int > $o] :
% 0.30/0.61        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                ? [Y: int] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: int] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_real
% 0.30/0.61                  @ ( collect_real
% 0.30/0.61                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6347_finite__Collect__bounded__ex,axiom,
% 0.30/0.61      ! [P: int > $o,Q: nat > int > $o] :
% 0.30/0.61        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                ? [Y: int] :
% 0.30/0.61                  ( ( P @ Y )
% 0.30/0.61                  & ( Q @ X @ Y ) ) ) )
% 0.30/0.61          = ( ! [Y: int] :
% 0.30/0.61                ( ( P @ Y )
% 0.30/0.61               => ( finite_finite_nat
% 0.30/0.61                  @ ( collect_nat
% 0.30/0.61                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_bounded_ex
% 0.30/0.61  thf(fact_6348_finite__Collect__subsets,axiom,
% 0.30/0.61      ! [A2: set_nat] :
% 0.30/0.61        ( ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( finite1152437895449049373et_nat
% 0.30/0.61          @ ( collect_set_nat
% 0.30/0.61            @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_subsets
% 0.30/0.61  thf(fact_6349_finite__Collect__subsets,axiom,
% 0.30/0.61      ! [A2: set_complex] :
% 0.30/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( finite6551019134538273531omplex
% 0.30/0.61          @ ( collect_set_complex
% 0.30/0.61            @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_subsets
% 0.30/0.61  thf(fact_6350_finite__Collect__subsets,axiom,
% 0.30/0.61      ! [A2: set_Pr1261947904930325089at_nat] :
% 0.30/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.61       => ( finite9047747110432174090at_nat
% 0.30/0.61          @ ( collec5514110066124741708at_nat
% 0.30/0.61            @ ^ [B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B5 @ A2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_subsets
% 0.30/0.61  thf(fact_6351_finite__Collect__subsets,axiom,
% 0.30/0.61      ! [A2: set_int] :
% 0.30/0.61        ( ( finite_finite_int @ A2 )
% 0.30/0.61       => ( finite6197958912794628473et_int
% 0.30/0.61          @ ( collect_set_int
% 0.30/0.61            @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_subsets
% 0.30/0.61  thf(fact_6352_singleton__conv2,axiom,
% 0.30/0.61      ! [A: produc3843707927480180839at_nat] :
% 0.30/0.61        ( ( collec6321179662152712658at_nat
% 0.30/0.61          @ ( ^ [Y4: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6353_singleton__conv2,axiom,
% 0.30/0.61      ! [A: list_nat] :
% 0.30/0.61        ( ( collect_list_nat
% 0.30/0.61          @ ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6354_singleton__conv2,axiom,
% 0.30/0.61      ! [A: set_nat] :
% 0.30/0.61        ( ( collect_set_nat
% 0.30/0.61          @ ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6355_singleton__conv2,axiom,
% 0.30/0.61      ! [A: product_prod_nat_nat] :
% 0.30/0.61        ( ( collec3392354462482085612at_nat
% 0.30/0.61          @ ( ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6356_singleton__conv2,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( collect_real
% 0.30/0.61          @ ( ^ [Y4: real,Z: real] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6357_singleton__conv2,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( collect_nat
% 0.30/0.61          @ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6358_singleton__conv2,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( collect_int
% 0.30/0.61          @ ( ^ [Y4: int,Z: int] : ( Y4 = Z )
% 0.30/0.61            @ A ) )
% 0.30/0.61        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv2
% 0.30/0.61  thf(fact_6359_singleton__conv,axiom,
% 0.30/0.61      ! [A: produc3843707927480180839at_nat] :
% 0.30/0.61        ( ( collec6321179662152712658at_nat
% 0.30/0.61          @ ^ [X: produc3843707927480180839at_nat] : ( X = A ) )
% 0.30/0.61        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6360_singleton__conv,axiom,
% 0.30/0.61      ! [A: list_nat] :
% 0.30/0.61        ( ( collect_list_nat
% 0.30/0.61          @ ^ [X: list_nat] : ( X = A ) )
% 0.30/0.61        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6361_singleton__conv,axiom,
% 0.30/0.61      ! [A: set_nat] :
% 0.30/0.61        ( ( collect_set_nat
% 0.30/0.61          @ ^ [X: set_nat] : ( X = A ) )
% 0.30/0.61        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6362_singleton__conv,axiom,
% 0.30/0.61      ! [A: product_prod_nat_nat] :
% 0.30/0.61        ( ( collec3392354462482085612at_nat
% 0.30/0.61          @ ^ [X: product_prod_nat_nat] : ( X = A ) )
% 0.30/0.61        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6363_singleton__conv,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( collect_real
% 0.30/0.61          @ ^ [X: real] : ( X = A ) )
% 0.30/0.61        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6364_singleton__conv,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( collect_nat
% 0.30/0.61          @ ^ [X: nat] : ( X = A ) )
% 0.30/0.61        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6365_singleton__conv,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( collect_int
% 0.30/0.61          @ ^ [X: int] : ( X = A ) )
% 0.30/0.61        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % singleton_conv
% 0.30/0.61  thf(fact_6366_finite__Collect__less__nat,axiom,
% 0.30/0.61      ! [K: nat] :
% 0.30/0.61        ( finite_finite_nat
% 0.30/0.61        @ ( collect_nat
% 0.30/0.61          @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_less_nat
% 0.30/0.61  thf(fact_6367_finite__Collect__le__nat,axiom,
% 0.30/0.61      ! [K: nat] :
% 0.30/0.61        ( finite_finite_nat
% 0.30/0.61        @ ( collect_nat
% 0.30/0.61          @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_Collect_le_nat
% 0.30/0.61  thf(fact_6368_finite__interval__int1,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( finite_finite_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [I2: int] :
% 0.30/0.61              ( ( ord_less_eq_int @ A @ I2 )
% 0.30/0.61              & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_interval_int1
% 0.30/0.61  thf(fact_6369_finite__interval__int4,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( finite_finite_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [I2: int] :
% 0.30/0.61              ( ( ord_less_int @ A @ I2 )
% 0.30/0.61              & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_interval_int4
% 0.30/0.61  thf(fact_6370_finite__interval__int2,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( finite_finite_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [I2: int] :
% 0.30/0.61              ( ( ord_less_eq_int @ A @ I2 )
% 0.30/0.61              & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_interval_int2
% 0.30/0.61  thf(fact_6371_finite__interval__int3,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( finite_finite_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [I2: int] :
% 0.30/0.61              ( ( ord_less_int @ A @ I2 )
% 0.30/0.61              & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_interval_int3
% 0.30/0.61  thf(fact_6372_xor__nat__numerals_I4_J,axiom,
% 0.30/0.61      ! [X3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.61        = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_numerals(4)
% 0.30/0.61  thf(fact_6373_xor__nat__numerals_I3_J,axiom,
% 0.30/0.61      ! [X3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.61        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_numerals(3)
% 0.30/0.61  thf(fact_6374_xor__nat__numerals_I2_J,axiom,
% 0.30/0.61      ! [Y3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.61        = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_numerals(2)
% 0.30/0.61  thf(fact_6375_xor__nat__numerals_I1_J,axiom,
% 0.30/0.61      ! [Y3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.61        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_numerals(1)
% 0.30/0.61  thf(fact_6376_xor__numerals_I6_J,axiom,
% 0.30/0.61      ! [X3: num,Y3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.61        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_numerals(6)
% 0.30/0.61  thf(fact_6377_xor__numerals_I6_J,axiom,
% 0.30/0.61      ! [X3: num,Y3: num] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 0.30/0.61        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_numerals(6)
% 0.30/0.61  thf(fact_6378_xor__numerals_I4_J,axiom,
% 0.30/0.61      ! [X3: num,Y3: num] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.61        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_numerals(4)
% 0.30/0.61  thf(fact_6379_xor__numerals_I4_J,axiom,
% 0.30/0.61      ! [X3: num,Y3: num] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 0.30/0.61        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_numerals(4)
% 0.30/0.61  thf(fact_6380_empty__def,axiom,
% 0.30/0.61      ( bot_bot_set_list_nat
% 0.30/0.61      = ( collect_list_nat
% 0.30/0.61        @ ^ [X: list_nat] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6381_empty__def,axiom,
% 0.30/0.61      ( bot_bot_set_set_nat
% 0.30/0.61      = ( collect_set_nat
% 0.30/0.61        @ ^ [X: set_nat] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6382_empty__def,axiom,
% 0.30/0.61      ( bot_bo2099793752762293965at_nat
% 0.30/0.61      = ( collec3392354462482085612at_nat
% 0.30/0.61        @ ^ [X: product_prod_nat_nat] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6383_empty__def,axiom,
% 0.30/0.61      ( bot_bot_set_real
% 0.30/0.61      = ( collect_real
% 0.30/0.61        @ ^ [X: real] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6384_empty__def,axiom,
% 0.30/0.61      ( bot_bot_set_nat
% 0.30/0.61      = ( collect_nat
% 0.30/0.61        @ ^ [X: nat] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6385_empty__def,axiom,
% 0.30/0.61      ( bot_bot_set_int
% 0.30/0.61      = ( collect_int
% 0.30/0.61        @ ^ [X: int] : $false ) ) ).
% 0.30/0.61  
% 0.30/0.61  % empty_def
% 0.30/0.61  thf(fact_6386_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collec6321179662152712658at_nat
% 0.30/0.61              @ ^ [X: produc3843707927480180839at_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collec6321179662152712658at_nat
% 0.30/0.61              @ ^ [X: produc3843707927480180839at_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6387_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: list_nat > $o,A: list_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_list_nat
% 0.30/0.61              @ ^ [X: list_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_list_nat
% 0.30/0.61              @ ^ [X: list_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_list_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6388_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: set_nat > $o,A: set_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_set_nat
% 0.30/0.61              @ ^ [X: set_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_set_nat
% 0.30/0.61              @ ^ [X: set_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_set_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6389_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collec3392354462482085612at_nat
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collec3392354462482085612at_nat
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6390_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: real > $o,A: real] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6391_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: nat > $o,A: nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6392_Collect__conv__if2,axiom,
% 0.30/0.61      ! [P: int > $o,A: int] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                  ( ( A = X )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if2
% 0.30/0.61  thf(fact_6393_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collec6321179662152712658at_nat
% 0.30/0.61              @ ^ [X: produc3843707927480180839at_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collec6321179662152712658at_nat
% 0.30/0.61              @ ^ [X: produc3843707927480180839at_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6394_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: list_nat > $o,A: list_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_list_nat
% 0.30/0.61              @ ^ [X: list_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_list_nat
% 0.30/0.61              @ ^ [X: list_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_list_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6395_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: set_nat > $o,A: set_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_set_nat
% 0.30/0.61              @ ^ [X: set_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_set_nat
% 0.30/0.61              @ ^ [X: set_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_set_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6396_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collec3392354462482085612at_nat
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collec3392354462482085612at_nat
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6397_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: real > $o,A: real] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6398_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: nat > $o,A: nat] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6399_Collect__conv__if,axiom,
% 0.30/0.61      ! [P: int > $o,A: int] :
% 0.30/0.61        ( ( ( P @ A )
% 0.30/0.61         => ( ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.61        & ( ~ ( P @ A )
% 0.30/0.61         => ( ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                  ( ( X = A )
% 0.30/0.61                  & ( P @ X ) ) )
% 0.30/0.61            = bot_bot_set_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_conv_if
% 0.30/0.61  thf(fact_6400_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.61        ( ( ord_le2646555220125990790_nat_o
% 0.30/0.61          @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6401_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 0.30/0.61        ( ( ord_le6741204236512500942_int_o
% 0.30/0.61          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6402_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
% 0.30/0.61        ( ( ord_le3636971675376928563eger_o
% 0.30/0.61          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le3216752416896350996nteger @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6403_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
% 0.30/0.61        ( ( ord_le4340812435750786203eger_o
% 0.30/0.61          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le653643898420964396nteger @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6404_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
% 0.30/0.61        ( ( ord_le5643404153117327598_int_o
% 0.30/0.61          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le8725513860283290265nt_int @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6405_pred__subset__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
% 0.30/0.61        ( ( ord_le2124322318746777828_int_o
% 0.30/0.61          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R )
% 0.30/0.61          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) )
% 0.30/0.61        = ( ord_le135402666524580259nt_int @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq2
% 0.30/0.61  thf(fact_6406_strict__subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: real,B: real] :
% 0.30/0.61        ( ( ord_less_set_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 0.30/0.61        = ( ( dvd_dvd_real @ A @ B )
% 0.30/0.61          & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % strict_subset_divisors_dvd
% 0.30/0.61  thf(fact_6407_strict__subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: nat,B: nat] :
% 0.30/0.61        ( ( ord_less_set_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 0.30/0.61        = ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.61          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % strict_subset_divisors_dvd
% 0.30/0.61  thf(fact_6408_strict__subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( ( ord_less_set_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 0.30/0.61        = ( ( dvd_dvd_int @ A @ B )
% 0.30/0.61          & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % strict_subset_divisors_dvd
% 0.30/0.61  thf(fact_6409_strict__subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: code_integer,B: code_integer] :
% 0.30/0.61        ( ( ord_le1307284697595431911nteger
% 0.30/0.61          @ ( collect_Code_integer
% 0.30/0.61            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.30/0.61          @ ( collect_Code_integer
% 0.30/0.61            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 0.30/0.61        = ( ( dvd_dvd_Code_integer @ A @ B )
% 0.30/0.61          & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % strict_subset_divisors_dvd
% 0.30/0.61  thf(fact_6410_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.61        ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6411_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 0.30/0.61        ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6412_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
% 0.30/0.61        ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6413_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
% 0.30/0.61        ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6414_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
% 0.30/0.61        ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6415_pred__equals__eq2,axiom,
% 0.30/0.61      ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
% 0.30/0.61        ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R ) )
% 0.30/0.61          = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) ) )
% 0.30/0.61        = ( R = S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_equals_eq2
% 0.30/0.61  thf(fact_6416_subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: real,B: real] :
% 0.30/0.61        ( ( ord_less_eq_set_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 0.30/0.61        = ( dvd_dvd_real @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_divisors_dvd
% 0.30/0.61  thf(fact_6417_subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: nat,B: nat] :
% 0.30/0.61        ( ( ord_less_eq_set_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 0.30/0.61        = ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_divisors_dvd
% 0.30/0.61  thf(fact_6418_subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: code_integer,B: code_integer] :
% 0.30/0.61        ( ( ord_le7084787975880047091nteger
% 0.30/0.61          @ ( collect_Code_integer
% 0.30/0.61            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 0.30/0.61          @ ( collect_Code_integer
% 0.30/0.61            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 0.30/0.61        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_divisors_dvd
% 0.30/0.61  thf(fact_6419_subset__divisors__dvd,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( ( ord_less_eq_set_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 0.30/0.61        = ( dvd_dvd_int @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % subset_divisors_dvd
% 0.30/0.61  thf(fact_6420_less__set__def,axiom,
% 0.30/0.61      ( ord_le7866589430770878221at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( ord_le549003669493604880_nat_o
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_set_def
% 0.30/0.61  thf(fact_6421_less__set__def,axiom,
% 0.30/0.61      ( ord_less_set_real
% 0.30/0.61      = ( ^ [A5: set_real,B5: set_real] :
% 0.30/0.61            ( ord_less_real_o
% 0.30/0.61            @ ^ [X: real] : ( member_real @ X @ A5 )
% 0.30/0.61            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_set_def
% 0.30/0.61  thf(fact_6422_less__set__def,axiom,
% 0.30/0.61      ( ord_less_set_set_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.30/0.61            ( ord_less_set_nat_o
% 0.30/0.61            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_set_def
% 0.30/0.61  thf(fact_6423_less__set__def,axiom,
% 0.30/0.61      ( ord_less_set_nat
% 0.30/0.61      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.30/0.61            ( ord_less_nat_o
% 0.30/0.61            @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_set_def
% 0.30/0.61  thf(fact_6424_less__set__def,axiom,
% 0.30/0.61      ( ord_less_set_int
% 0.30/0.61      = ( ^ [A5: set_int,B5: set_int] :
% 0.30/0.61            ( ord_less_int_o
% 0.30/0.61            @ ^ [X: int] : ( member_int @ X @ A5 )
% 0.30/0.61            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_set_def
% 0.30/0.61  thf(fact_6425_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.61        ( ord_le3146513528884898305at_nat
% 0.30/0.61        @ ( collec3392354462482085612at_nat
% 0.30/0.61          @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61              ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6426_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_real,P: real > $o] :
% 0.30/0.61        ( ord_less_eq_set_real
% 0.30/0.61        @ ( collect_real
% 0.30/0.61          @ ^ [X: real] :
% 0.30/0.61              ( ( member_real @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6427_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_list_nat,P: list_nat > $o] :
% 0.30/0.61        ( ord_le6045566169113846134st_nat
% 0.30/0.61        @ ( collect_list_nat
% 0.30/0.61          @ ^ [X: list_nat] :
% 0.30/0.61              ( ( member_list_nat @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6428_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_set_nat,P: set_nat > $o] :
% 0.30/0.61        ( ord_le6893508408891458716et_nat
% 0.30/0.61        @ ( collect_set_nat
% 0.30/0.61          @ ^ [X: set_nat] :
% 0.30/0.61              ( ( member_set_nat @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6429_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_nat,P: nat > $o] :
% 0.30/0.61        ( ord_less_eq_set_nat
% 0.30/0.61        @ ( collect_nat
% 0.30/0.61          @ ^ [X: nat] :
% 0.30/0.61              ( ( member_nat @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6430_Collect__subset,axiom,
% 0.30/0.61      ! [A2: set_int,P: int > $o] :
% 0.30/0.61        ( ord_less_eq_set_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [X: int] :
% 0.30/0.61              ( ( member_int @ X @ A2 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ A2 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_subset
% 0.30/0.61  thf(fact_6431_less__eq__set__def,axiom,
% 0.30/0.61      ( ord_le3146513528884898305at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( ord_le704812498762024988_nat_o
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_eq_set_def
% 0.30/0.61  thf(fact_6432_less__eq__set__def,axiom,
% 0.30/0.61      ( ord_less_eq_set_real
% 0.30/0.61      = ( ^ [A5: set_real,B5: set_real] :
% 0.30/0.61            ( ord_less_eq_real_o
% 0.30/0.61            @ ^ [X: real] : ( member_real @ X @ A5 )
% 0.30/0.61            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_eq_set_def
% 0.30/0.61  thf(fact_6433_less__eq__set__def,axiom,
% 0.30/0.61      ( ord_le6893508408891458716et_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.30/0.61            ( ord_le3964352015994296041_nat_o
% 0.30/0.61            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_eq_set_def
% 0.30/0.61  thf(fact_6434_less__eq__set__def,axiom,
% 0.30/0.61      ( ord_less_eq_set_nat
% 0.30/0.61      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.30/0.61            ( ord_less_eq_nat_o
% 0.30/0.61            @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 0.30/0.61            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_eq_set_def
% 0.30/0.61  thf(fact_6435_less__eq__set__def,axiom,
% 0.30/0.61      ( ord_less_eq_set_int
% 0.30/0.61      = ( ^ [A5: set_int,B5: set_int] :
% 0.30/0.61            ( ord_less_eq_int_o
% 0.30/0.61            @ ^ [X: int] : ( member_int @ X @ A5 )
% 0.30/0.61            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % less_eq_set_def
% 0.30/0.61  thf(fact_6436_prop__restrict,axiom,
% 0.30/0.61      ! [X3: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.61        ( ( member8440522571783428010at_nat @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_le3146513528884898305at_nat @ Z5
% 0.30/0.61            @ ( collec3392354462482085612at_nat
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                  ( ( member8440522571783428010at_nat @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6437_prop__restrict,axiom,
% 0.30/0.61      ! [X3: real,Z5: set_real,X7: set_real,P: real > $o] :
% 0.30/0.61        ( ( member_real @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_less_eq_set_real @ Z5
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [X: real] :
% 0.30/0.61                  ( ( member_real @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6438_prop__restrict,axiom,
% 0.30/0.61      ! [X3: list_nat,Z5: set_list_nat,X7: set_list_nat,P: list_nat > $o] :
% 0.30/0.61        ( ( member_list_nat @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_le6045566169113846134st_nat @ Z5
% 0.30/0.61            @ ( collect_list_nat
% 0.30/0.61              @ ^ [X: list_nat] :
% 0.30/0.61                  ( ( member_list_nat @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6439_prop__restrict,axiom,
% 0.30/0.61      ! [X3: set_nat,Z5: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
% 0.30/0.61        ( ( member_set_nat @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_le6893508408891458716et_nat @ Z5
% 0.30/0.61            @ ( collect_set_nat
% 0.30/0.61              @ ^ [X: set_nat] :
% 0.30/0.61                  ( ( member_set_nat @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6440_prop__restrict,axiom,
% 0.30/0.61      ! [X3: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
% 0.30/0.61        ( ( member_nat @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_less_eq_set_nat @ Z5
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [X: nat] :
% 0.30/0.61                  ( ( member_nat @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6441_prop__restrict,axiom,
% 0.30/0.61      ! [X3: int,Z5: set_int,X7: set_int,P: int > $o] :
% 0.30/0.61        ( ( member_int @ X3 @ Z5 )
% 0.30/0.61       => ( ( ord_less_eq_set_int @ Z5
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [X: int] :
% 0.30/0.61                  ( ( member_int @ X @ X7 )
% 0.30/0.61                  & ( P @ X ) ) ) )
% 0.30/0.61         => ( P @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prop_restrict
% 0.30/0.61  thf(fact_6442_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.61        ( ord_le3146513528884898305at_nat
% 0.30/0.61        @ ( collec3392354462482085612at_nat
% 0.30/0.61          @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61              ( ( member8440522571783428010at_nat @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6443_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_real,P: real > $o] :
% 0.30/0.61        ( ord_less_eq_set_real
% 0.30/0.61        @ ( collect_real
% 0.30/0.61          @ ^ [X: real] :
% 0.30/0.61              ( ( member_real @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6444_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_list_nat,P: list_nat > $o] :
% 0.30/0.61        ( ord_le6045566169113846134st_nat
% 0.30/0.61        @ ( collect_list_nat
% 0.30/0.61          @ ^ [X: list_nat] :
% 0.30/0.61              ( ( member_list_nat @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6445_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_set_nat,P: set_nat > $o] :
% 0.30/0.61        ( ord_le6893508408891458716et_nat
% 0.30/0.61        @ ( collect_set_nat
% 0.30/0.61          @ ^ [X: set_nat] :
% 0.30/0.61              ( ( member_set_nat @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6446_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_nat,P: nat > $o] :
% 0.30/0.61        ( ord_less_eq_set_nat
% 0.30/0.61        @ ( collect_nat
% 0.30/0.61          @ ^ [X: nat] :
% 0.30/0.61              ( ( member_nat @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6447_Collect__restrict,axiom,
% 0.30/0.61      ! [X7: set_int,P: int > $o] :
% 0.30/0.61        ( ord_less_eq_set_int
% 0.30/0.61        @ ( collect_int
% 0.30/0.61          @ ^ [X: int] :
% 0.30/0.61              ( ( member_int @ X @ X7 )
% 0.30/0.61              & ( P @ X ) ) )
% 0.30/0.61        @ X7 ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_restrict
% 0.30/0.61  thf(fact_6448_pred__subset__eq,axiom,
% 0.30/0.61      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 0.30/0.61        ( ( ord_le704812498762024988_nat_o
% 0.30/0.61          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
% 0.30/0.61          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S3 ) )
% 0.30/0.61        = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq
% 0.30/0.61  thf(fact_6449_pred__subset__eq,axiom,
% 0.30/0.61      ! [R: set_real,S3: set_real] :
% 0.30/0.61        ( ( ord_less_eq_real_o
% 0.30/0.61          @ ^ [X: real] : ( member_real @ X @ R )
% 0.30/0.61          @ ^ [X: real] : ( member_real @ X @ S3 ) )
% 0.30/0.61        = ( ord_less_eq_set_real @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq
% 0.30/0.61  thf(fact_6450_pred__subset__eq,axiom,
% 0.30/0.61      ! [R: set_set_nat,S3: set_set_nat] :
% 0.30/0.61        ( ( ord_le3964352015994296041_nat_o
% 0.30/0.61          @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
% 0.30/0.61          @ ^ [X: set_nat] : ( member_set_nat @ X @ S3 ) )
% 0.30/0.61        = ( ord_le6893508408891458716et_nat @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq
% 0.30/0.61  thf(fact_6451_pred__subset__eq,axiom,
% 0.30/0.61      ! [R: set_nat,S3: set_nat] :
% 0.30/0.61        ( ( ord_less_eq_nat_o
% 0.30/0.61          @ ^ [X: nat] : ( member_nat @ X @ R )
% 0.30/0.61          @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 0.30/0.61        = ( ord_less_eq_set_nat @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq
% 0.30/0.61  thf(fact_6452_pred__subset__eq,axiom,
% 0.30/0.61      ! [R: set_int,S3: set_int] :
% 0.30/0.61        ( ( ord_less_eq_int_o
% 0.30/0.61          @ ^ [X: int] : ( member_int @ X @ R )
% 0.30/0.61          @ ^ [X: int] : ( member_int @ X @ S3 ) )
% 0.30/0.61        = ( ord_less_eq_set_int @ R @ S3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pred_subset_eq
% 0.30/0.61  thf(fact_6453_lambda__one,axiom,
% 0.30/0.61      ( ( ^ [X: complex] : X )
% 0.30/0.61      = ( times_times_complex @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_one
% 0.30/0.61  thf(fact_6454_lambda__one,axiom,
% 0.30/0.61      ( ( ^ [X: real] : X )
% 0.30/0.61      = ( times_times_real @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_one
% 0.30/0.61  thf(fact_6455_lambda__one,axiom,
% 0.30/0.61      ( ( ^ [X: rat] : X )
% 0.30/0.61      = ( times_times_rat @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_one
% 0.30/0.61  thf(fact_6456_lambda__one,axiom,
% 0.30/0.61      ( ( ^ [X: nat] : X )
% 0.30/0.61      = ( times_times_nat @ one_one_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_one
% 0.30/0.61  thf(fact_6457_lambda__one,axiom,
% 0.30/0.61      ( ( ^ [X: int] : X )
% 0.30/0.61      = ( times_times_int @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_one
% 0.30/0.61  thf(fact_6458_lambda__zero,axiom,
% 0.30/0.61      ( ( ^ [H: complex] : zero_zero_complex )
% 0.30/0.61      = ( times_times_complex @ zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_zero
% 0.30/0.61  thf(fact_6459_lambda__zero,axiom,
% 0.30/0.61      ( ( ^ [H: real] : zero_zero_real )
% 0.30/0.61      = ( times_times_real @ zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_zero
% 0.30/0.61  thf(fact_6460_lambda__zero,axiom,
% 0.30/0.61      ( ( ^ [H: rat] : zero_zero_rat )
% 0.30/0.61      = ( times_times_rat @ zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_zero
% 0.30/0.61  thf(fact_6461_lambda__zero,axiom,
% 0.30/0.61      ( ( ^ [H: nat] : zero_zero_nat )
% 0.30/0.61      = ( times_times_nat @ zero_zero_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_zero
% 0.30/0.61  thf(fact_6462_lambda__zero,axiom,
% 0.30/0.61      ( ( ^ [H: int] : zero_zero_int )
% 0.30/0.61      = ( times_times_int @ zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % lambda_zero
% 0.30/0.61  thf(fact_6463_uminus__set__def,axiom,
% 0.30/0.61      ( uminus6524753893492686040at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( collec3392354462482085612at_nat
% 0.30/0.61            @ ( uminus8676089048583255045_nat_o
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6464_uminus__set__def,axiom,
% 0.30/0.61      ( uminus612125837232591019t_real
% 0.30/0.61      = ( ^ [A5: set_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ( uminus_uminus_real_o
% 0.30/0.61              @ ^ [X: real] : ( member_real @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6465_uminus__set__def,axiom,
% 0.30/0.61      ( uminus3195874150345416415st_nat
% 0.30/0.61      = ( ^ [A5: set_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ( uminus5770388063884162150_nat_o
% 0.30/0.61              @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6466_uminus__set__def,axiom,
% 0.30/0.61      ( uminus613421341184616069et_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ( uminus6401447641752708672_nat_o
% 0.30/0.61              @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6467_uminus__set__def,axiom,
% 0.30/0.61      ( uminus5710092332889474511et_nat
% 0.30/0.61      = ( ^ [A5: set_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ( uminus_uminus_nat_o
% 0.30/0.61              @ ^ [X: nat] : ( member_nat @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6468_uminus__set__def,axiom,
% 0.30/0.61      ( uminus1532241313380277803et_int
% 0.30/0.61      = ( ^ [A5: set_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ( uminus_uminus_int_o
% 0.30/0.61              @ ^ [X: int] : ( member_int @ X @ A5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % uminus_set_def
% 0.30/0.61  thf(fact_6469_Collect__neg__eq,axiom,
% 0.30/0.61      ! [P: real > $o] :
% 0.30/0.61        ( ( collect_real
% 0.30/0.61          @ ^ [X: real] :
% 0.30/0.61              ~ ( P @ X ) )
% 0.30/0.61        = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_neg_eq
% 0.30/0.61  thf(fact_6470_Collect__neg__eq,axiom,
% 0.30/0.61      ! [P: list_nat > $o] :
% 0.30/0.61        ( ( collect_list_nat
% 0.30/0.61          @ ^ [X: list_nat] :
% 0.30/0.61              ~ ( P @ X ) )
% 0.30/0.61        = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_neg_eq
% 0.30/0.61  thf(fact_6471_Collect__neg__eq,axiom,
% 0.30/0.61      ! [P: set_nat > $o] :
% 0.30/0.61        ( ( collect_set_nat
% 0.30/0.61          @ ^ [X: set_nat] :
% 0.30/0.61              ~ ( P @ X ) )
% 0.30/0.61        = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_neg_eq
% 0.30/0.61  thf(fact_6472_Collect__neg__eq,axiom,
% 0.30/0.61      ! [P: nat > $o] :
% 0.30/0.61        ( ( collect_nat
% 0.30/0.61          @ ^ [X: nat] :
% 0.30/0.61              ~ ( P @ X ) )
% 0.30/0.61        = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_neg_eq
% 0.30/0.61  thf(fact_6473_Collect__neg__eq,axiom,
% 0.30/0.61      ! [P: int > $o] :
% 0.30/0.61        ( ( collect_int
% 0.30/0.61          @ ^ [X: int] :
% 0.30/0.61              ~ ( P @ X ) )
% 0.30/0.61        = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Collect_neg_eq
% 0.30/0.61  thf(fact_6474_Compl__eq,axiom,
% 0.30/0.61      ( uminus6524753893492686040at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( collec3392354462482085612at_nat
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                ~ ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6475_Compl__eq,axiom,
% 0.30/0.61      ( uminus612125837232591019t_real
% 0.30/0.61      = ( ^ [A5: set_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ^ [X: real] :
% 0.30/0.61                ~ ( member_real @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6476_Compl__eq,axiom,
% 0.30/0.61      ( uminus3195874150345416415st_nat
% 0.30/0.61      = ( ^ [A5: set_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ^ [X: list_nat] :
% 0.30/0.61                ~ ( member_list_nat @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6477_Compl__eq,axiom,
% 0.30/0.61      ( uminus613421341184616069et_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ^ [X: set_nat] :
% 0.30/0.61                ~ ( member_set_nat @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6478_Compl__eq,axiom,
% 0.30/0.61      ( uminus5710092332889474511et_nat
% 0.30/0.61      = ( ^ [A5: set_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ^ [X: nat] :
% 0.30/0.61                ~ ( member_nat @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6479_Compl__eq,axiom,
% 0.30/0.61      ( uminus1532241313380277803et_int
% 0.30/0.61      = ( ^ [A5: set_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ^ [X: int] :
% 0.30/0.61                ~ ( member_int @ X @ A5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Compl_eq
% 0.30/0.61  thf(fact_6480_set__diff__eq,axiom,
% 0.30/0.61      ( minus_1356011639430497352at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( collec3392354462482085612at_nat
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                ( ( member8440522571783428010at_nat @ X @ A5 )
% 0.30/0.61                & ~ ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6481_set__diff__eq,axiom,
% 0.30/0.61      ( minus_minus_set_real
% 0.30/0.61      = ( ^ [A5: set_real,B5: set_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ^ [X: real] :
% 0.30/0.61                ( ( member_real @ X @ A5 )
% 0.30/0.61                & ~ ( member_real @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6482_set__diff__eq,axiom,
% 0.30/0.61      ( minus_7954133019191499631st_nat
% 0.30/0.61      = ( ^ [A5: set_list_nat,B5: set_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ^ [X: list_nat] :
% 0.30/0.61                ( ( member_list_nat @ X @ A5 )
% 0.30/0.61                & ~ ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6483_set__diff__eq,axiom,
% 0.30/0.61      ( minus_2163939370556025621et_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ^ [X: set_nat] :
% 0.30/0.61                ( ( member_set_nat @ X @ A5 )
% 0.30/0.61                & ~ ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6484_set__diff__eq,axiom,
% 0.30/0.61      ( minus_minus_set_int
% 0.30/0.61      = ( ^ [A5: set_int,B5: set_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ^ [X: int] :
% 0.30/0.61                ( ( member_int @ X @ A5 )
% 0.30/0.61                & ~ ( member_int @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6485_set__diff__eq,axiom,
% 0.30/0.61      ( minus_minus_set_nat
% 0.30/0.61      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ^ [X: nat] :
% 0.30/0.61                ( ( member_nat @ X @ A5 )
% 0.30/0.61                & ~ ( member_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_diff_eq
% 0.30/0.61  thf(fact_6486_insert__compr,axiom,
% 0.30/0.61      ( insert9069300056098147895at_nat
% 0.30/0.61      = ( ^ [A3: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
% 0.30/0.61            ( collec6321179662152712658at_nat
% 0.30/0.61            @ ^ [X: produc3843707927480180839at_nat] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member8757157785044589968at_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6487_insert__compr,axiom,
% 0.30/0.61      ( insert8211810215607154385at_nat
% 0.30/0.61      = ( ^ [A3: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( collec3392354462482085612at_nat
% 0.30/0.61            @ ^ [X: product_prod_nat_nat] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6488_insert__compr,axiom,
% 0.30/0.61      ( insert_real
% 0.30/0.61      = ( ^ [A3: real,B5: set_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ^ [X: real] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member_real @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6489_insert__compr,axiom,
% 0.30/0.61      ( insert_list_nat
% 0.30/0.61      = ( ^ [A3: list_nat,B5: set_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ^ [X: list_nat] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6490_insert__compr,axiom,
% 0.30/0.61      ( insert_set_nat
% 0.30/0.61      = ( ^ [A3: set_nat,B5: set_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ^ [X: set_nat] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6491_insert__compr,axiom,
% 0.30/0.61      ( insert_nat
% 0.30/0.61      = ( ^ [A3: nat,B5: set_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ^ [X: nat] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6492_insert__compr,axiom,
% 0.30/0.61      ( insert_int
% 0.30/0.61      = ( ^ [A3: int,B5: set_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ^ [X: int] :
% 0.30/0.61                ( ( X = A3 )
% 0.30/0.61                | ( member_int @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_compr
% 0.30/0.61  thf(fact_6493_minus__set__def,axiom,
% 0.30/0.61      ( minus_1356011639430497352at_nat
% 0.30/0.61      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 0.30/0.61            ( collec3392354462482085612at_nat
% 0.30/0.61            @ ( minus_2270307095948843157_nat_o
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 0.30/0.61              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6494_minus__set__def,axiom,
% 0.30/0.61      ( minus_minus_set_real
% 0.30/0.61      = ( ^ [A5: set_real,B5: set_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ( minus_minus_real_o
% 0.30/0.61              @ ^ [X: real] : ( member_real @ X @ A5 )
% 0.30/0.61              @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6495_minus__set__def,axiom,
% 0.30/0.61      ( minus_7954133019191499631st_nat
% 0.30/0.61      = ( ^ [A5: set_list_nat,B5: set_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ( minus_1139252259498527702_nat_o
% 0.30/0.61              @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
% 0.30/0.61              @ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6496_minus__set__def,axiom,
% 0.30/0.61      ( minus_2163939370556025621et_nat
% 0.30/0.61      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ( minus_6910147592129066416_nat_o
% 0.30/0.61              @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 0.30/0.61              @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6497_minus__set__def,axiom,
% 0.30/0.61      ( minus_minus_set_int
% 0.30/0.61      = ( ^ [A5: set_int,B5: set_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ( minus_minus_int_o
% 0.30/0.61              @ ^ [X: int] : ( member_int @ X @ A5 )
% 0.30/0.61              @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6498_minus__set__def,axiom,
% 0.30/0.61      ( minus_minus_set_nat
% 0.30/0.61      = ( ^ [A5: set_nat,B5: set_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ( minus_minus_nat_o
% 0.30/0.61              @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 0.30/0.61              @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % minus_set_def
% 0.30/0.61  thf(fact_6499_insert__Collect,axiom,
% 0.30/0.61      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 0.30/0.61        ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 0.30/0.61        = ( collec3392354462482085612at_nat
% 0.30/0.61          @ ^ [U2: product_prod_nat_nat] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6500_insert__Collect,axiom,
% 0.30/0.61      ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
% 0.30/0.61        ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
% 0.30/0.61        = ( collec6321179662152712658at_nat
% 0.30/0.61          @ ^ [U2: produc3843707927480180839at_nat] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6501_insert__Collect,axiom,
% 0.30/0.61      ! [A: real,P: real > $o] :
% 0.30/0.61        ( ( insert_real @ A @ ( collect_real @ P ) )
% 0.30/0.61        = ( collect_real
% 0.30/0.61          @ ^ [U2: real] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6502_insert__Collect,axiom,
% 0.30/0.61      ! [A: list_nat,P: list_nat > $o] :
% 0.30/0.61        ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
% 0.30/0.61        = ( collect_list_nat
% 0.30/0.61          @ ^ [U2: list_nat] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6503_insert__Collect,axiom,
% 0.30/0.61      ! [A: set_nat,P: set_nat > $o] :
% 0.30/0.61        ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
% 0.30/0.61        = ( collect_set_nat
% 0.30/0.61          @ ^ [U2: set_nat] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6504_insert__Collect,axiom,
% 0.30/0.61      ! [A: nat,P: nat > $o] :
% 0.30/0.61        ( ( insert_nat @ A @ ( collect_nat @ P ) )
% 0.30/0.61        = ( collect_nat
% 0.30/0.61          @ ^ [U2: nat] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6505_insert__Collect,axiom,
% 0.30/0.61      ! [A: int,P: int > $o] :
% 0.30/0.61        ( ( insert_int @ A @ ( collect_int @ P ) )
% 0.30/0.61        = ( collect_int
% 0.30/0.61          @ ^ [U2: int] :
% 0.30/0.61              ( ( U2 != A )
% 0.30/0.61             => ( P @ U2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % insert_Collect
% 0.30/0.61  thf(fact_6506_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: real > $o] :
% 0.30/0.61        ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ? [X_1: real] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6507_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: list_nat > $o] :
% 0.30/0.61        ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 0.30/0.61       => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6508_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: set_nat > $o] :
% 0.30/0.61        ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 0.30/0.61       => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6509_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: nat > $o] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6510_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: int > $o] :
% 0.30/0.61        ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.61       => ? [X_1: int] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6511_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: complex > $o] :
% 0.30/0.61        ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 0.30/0.61       => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6512_not__finite__existsD,axiom,
% 0.30/0.61      ! [P: product_prod_nat_nat > $o] :
% 0.30/0.61        ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 0.30/0.61       => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % not_finite_existsD
% 0.30/0.61  thf(fact_6513_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_real,B2: set_nat,R: real > nat > $o] :
% 0.30/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.30/0.61       => ( ( finite_finite_nat @ B2 )
% 0.30/0.61         => ( ! [X4: real] :
% 0.30/0.61                ( ( member_real @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: nat] :
% 0.30/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_real
% 0.30/0.61                    @ ( collect_real
% 0.30/0.61                      @ ^ [A3: real] :
% 0.30/0.61                          ( ( member_real @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6514_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_real,B2: set_int,R: real > int > $o] :
% 0.30/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.30/0.61       => ( ( finite_finite_int @ B2 )
% 0.30/0.61         => ( ! [X4: real] :
% 0.30/0.61                ( ( member_real @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: int] :
% 0.30/0.61                    ( ( member_int @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_real
% 0.30/0.61                    @ ( collect_real
% 0.30/0.61                      @ ^ [A3: real] :
% 0.30/0.61                          ( ( member_real @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6515_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_real,B2: set_complex,R: real > complex > $o] :
% 0.30/0.61        ( ~ ( finite_finite_real @ A2 )
% 0.30/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.61         => ( ! [X4: real] :
% 0.30/0.61                ( ( member_real @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: complex] :
% 0.30/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: complex] :
% 0.30/0.61                ( ( member_complex @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_real
% 0.30/0.61                    @ ( collect_real
% 0.30/0.61                      @ ^ [A3: real] :
% 0.30/0.61                          ( ( member_real @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6516_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( finite_finite_nat @ B2 )
% 0.30/0.61         => ( ! [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: nat] :
% 0.30/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_nat
% 0.30/0.61                    @ ( collect_nat
% 0.30/0.61                      @ ^ [A3: nat] :
% 0.30/0.61                          ( ( member_nat @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6517_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( finite_finite_int @ B2 )
% 0.30/0.61         => ( ! [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: int] :
% 0.30/0.61                    ( ( member_int @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_nat
% 0.30/0.61                    @ ( collect_nat
% 0.30/0.61                      @ ^ [A3: nat] :
% 0.30/0.61                          ( ( member_nat @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6518_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.61         => ( ! [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: complex] :
% 0.30/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: complex] :
% 0.30/0.61                ( ( member_complex @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_nat
% 0.30/0.61                    @ ( collect_nat
% 0.30/0.61                      @ ^ [A3: nat] :
% 0.30/0.61                          ( ( member_nat @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6519_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( finite_finite_nat @ B2 )
% 0.30/0.61         => ( ! [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: nat] :
% 0.30/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_int
% 0.30/0.61                    @ ( collect_int
% 0.30/0.61                      @ ^ [A3: int] :
% 0.30/0.61                          ( ( member_int @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6520_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_int,B2: set_int,R: int > int > $o] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( finite_finite_int @ B2 )
% 0.30/0.61         => ( ! [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: int] :
% 0.30/0.61                    ( ( member_int @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_int
% 0.30/0.61                    @ ( collect_int
% 0.30/0.61                      @ ^ [A3: int] :
% 0.30/0.61                          ( ( member_int @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6521_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_int,B2: set_complex,R: int > complex > $o] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.61         => ( ! [X4: int] :
% 0.30/0.61                ( ( member_int @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: complex] :
% 0.30/0.61                    ( ( member_complex @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: complex] :
% 0.30/0.61                ( ( member_complex @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite_finite_int
% 0.30/0.61                    @ ( collect_int
% 0.30/0.61                      @ ^ [A3: int] :
% 0.30/0.61                          ( ( member_int @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6522_pigeonhole__infinite__rel,axiom,
% 0.30/0.61      ! [A2: set_complex,B2: set_nat,R: complex > nat > $o] :
% 0.30/0.61        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( ( finite_finite_nat @ B2 )
% 0.30/0.61         => ( ! [X4: complex] :
% 0.30/0.61                ( ( member_complex @ X4 @ A2 )
% 0.30/0.61               => ? [Xa: nat] :
% 0.30/0.61                    ( ( member_nat @ Xa @ B2 )
% 0.30/0.61                    & ( R @ X4 @ Xa ) ) )
% 0.30/0.61           => ? [X4: nat] :
% 0.30/0.61                ( ( member_nat @ X4 @ B2 )
% 0.30/0.61                & ~ ( finite3207457112153483333omplex
% 0.30/0.61                    @ ( collect_complex
% 0.30/0.61                      @ ^ [A3: complex] :
% 0.30/0.61                          ( ( member_complex @ A3 @ A2 )
% 0.30/0.61                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % pigeonhole_infinite_rel
% 0.30/0.61  thf(fact_6523_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: real > $o,F: real > real > real] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [Uu3: real] :
% 0.30/0.61                ? [X: real,Y: real] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6524_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: real > $o,F: real > real > nat] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [Uu3: nat] :
% 0.30/0.61                ? [X: real,Y: real] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6525_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: real > $o,F: real > real > int] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [Uu3: int] :
% 0.30/0.61                ? [X: real,Y: real] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6526_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: real > $o,F: real > real > complex] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [Uu3: complex] :
% 0.30/0.61                ? [X: real,Y: real] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6527_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [Uu3: real] :
% 0.30/0.61                ? [X: real,Y: nat] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6528_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [Uu3: nat] :
% 0.30/0.61                ? [X: real,Y: nat] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6529_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [Uu3: int] :
% 0.30/0.61                ? [X: real,Y: nat] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6530_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [Uu3: complex] :
% 0.30/0.61                ? [X: real,Y: nat] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6531_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: int > $o,F: real > int > real] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [Uu3: real] :
% 0.30/0.61                ? [X: real,Y: int] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6532_finite__image__set2,axiom,
% 0.30/0.61      ! [P: real > $o,Q: int > $o,F: real > int > nat] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [Uu3: nat] :
% 0.30/0.61                ? [X: real,Y: int] :
% 0.30/0.61                  ( ( Uu3
% 0.30/0.61                    = ( F @ X @ Y ) )
% 0.30/0.61                  & ( P @ X )
% 0.30/0.61                  & ( Q @ Y ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set2
% 0.30/0.61  thf(fact_6533_finite__image__set,axiom,
% 0.30/0.61      ! [P: real > $o,F: real > real] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [Uu3: real] :
% 0.30/0.61              ? [X: real] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6534_finite__image__set,axiom,
% 0.30/0.61      ! [P: real > $o,F: real > nat] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [Uu3: nat] :
% 0.30/0.61              ? [X: real] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6535_finite__image__set,axiom,
% 0.30/0.61      ! [P: real > $o,F: real > int] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [Uu3: int] :
% 0.30/0.61              ? [X: real] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6536_finite__image__set,axiom,
% 0.30/0.61      ! [P: real > $o,F: real > complex] :
% 0.30/0.61        ( ( finite_finite_real @ ( collect_real @ P ) )
% 0.30/0.61       => ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [Uu3: complex] :
% 0.30/0.61              ? [X: real] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6537_finite__image__set,axiom,
% 0.30/0.61      ! [P: nat > $o,F: nat > real] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [Uu3: real] :
% 0.30/0.61              ? [X: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6538_finite__image__set,axiom,
% 0.30/0.61      ! [P: nat > $o,F: nat > nat] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [Uu3: nat] :
% 0.30/0.61              ? [X: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6539_finite__image__set,axiom,
% 0.30/0.61      ! [P: nat > $o,F: nat > int] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [Uu3: int] :
% 0.30/0.61              ? [X: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6540_finite__image__set,axiom,
% 0.30/0.61      ! [P: nat > $o,F: nat > complex] :
% 0.30/0.61        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.61       => ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [Uu3: complex] :
% 0.30/0.61              ? [X: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6541_finite__image__set,axiom,
% 0.30/0.61      ! [P: int > $o,F: int > real] :
% 0.30/0.61        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.61       => ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [Uu3: real] :
% 0.30/0.61              ? [X: int] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6542_finite__image__set,axiom,
% 0.30/0.61      ! [P: int > $o,F: int > nat] :
% 0.30/0.61        ( ( finite_finite_int @ ( collect_int @ P ) )
% 0.30/0.61       => ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [Uu3: nat] :
% 0.30/0.61              ? [X: int] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( F @ X ) )
% 0.30/0.61                & ( P @ X ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_image_set
% 0.30/0.61  thf(fact_6543_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bot_int_int_o
% 0.30/0.61      = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6544_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bo5358457235160185703eger_o
% 0.30/0.61      = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6545_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bo3000040243691356879eger_o
% 0.30/0.61      = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6546_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bo8662317086119403298_int_o
% 0.30/0.61      = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6547_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bo1403522918969695512_int_o
% 0.30/0.61      = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6548_bot__empty__eq2,axiom,
% 0.30/0.61      ( bot_bot_nat_nat_o
% 0.30/0.61      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bot_empty_eq2
% 0.30/0.61  thf(fact_6549_finite__M__bounded__by__nat,axiom,
% 0.30/0.61      ! [P: nat > $o,I: nat] :
% 0.30/0.61        ( finite_finite_nat
% 0.30/0.61        @ ( collect_nat
% 0.30/0.61          @ ^ [K3: nat] :
% 0.30/0.61              ( ( P @ K3 )
% 0.30/0.61              & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_M_bounded_by_nat
% 0.30/0.61  thf(fact_6550_finite__less__ub,axiom,
% 0.30/0.61      ! [F: nat > nat,U: nat] :
% 0.30/0.61        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 0.30/0.61       => ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_less_ub
% 0.30/0.61  thf(fact_6551_numeral__code_I2_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(2)
% 0.30/0.61  thf(fact_6552_numeral__code_I2_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(2)
% 0.30/0.61  thf(fact_6553_numeral__code_I2_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(2)
% 0.30/0.61  thf(fact_6554_numeral__code_I2_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(2)
% 0.30/0.61  thf(fact_6555_numeral__code_I2_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(2)
% 0.30/0.61  thf(fact_6556_set__vebt__def,axiom,
% 0.30/0.61      ( vEBT_set_vebt
% 0.30/0.61      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_vebt_def
% 0.30/0.61  thf(fact_6557_finite__divisors__int,axiom,
% 0.30/0.61      ! [I: int] :
% 0.30/0.61        ( ( I != zero_zero_int )
% 0.30/0.61       => ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_divisors_int
% 0.30/0.61  thf(fact_6558_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6559_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6560_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6561_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6562_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6563_numeral__code_I3_J,axiom,
% 0.30/0.61      ! [N2: num] :
% 0.30/0.61        ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 0.30/0.61        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % numeral_code(3)
% 0.30/0.61  thf(fact_6564_finite__divisors__nat,axiom,
% 0.30/0.61      ! [M: nat] :
% 0.30/0.61        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.61       => ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_divisors_nat
% 0.30/0.61  thf(fact_6565_set__conv__nth,axiom,
% 0.30/0.61      ( set_real2
% 0.30/0.61      = ( ^ [Xs2: list_real] :
% 0.30/0.61            ( collect_real
% 0.30/0.61            @ ^ [Uu3: real] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_real @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6566_set__conv__nth,axiom,
% 0.30/0.61      ( set_list_nat2
% 0.30/0.61      = ( ^ [Xs2: list_list_nat] :
% 0.30/0.61            ( collect_list_nat
% 0.30/0.61            @ ^ [Uu3: list_nat] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_list_nat @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6567_set__conv__nth,axiom,
% 0.30/0.61      ( set_set_nat2
% 0.30/0.61      = ( ^ [Xs2: list_set_nat] :
% 0.30/0.61            ( collect_set_nat
% 0.30/0.61            @ ^ [Uu3: set_nat] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_set_nat @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6568_set__conv__nth,axiom,
% 0.30/0.61      ( set_VEBT_VEBT2
% 0.30/0.61      = ( ^ [Xs2: list_VEBT_VEBT] :
% 0.30/0.61            ( collect_VEBT_VEBT
% 0.30/0.61            @ ^ [Uu3: vEBT_VEBT] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6569_set__conv__nth,axiom,
% 0.30/0.61      ( set_o2
% 0.30/0.61      = ( ^ [Xs2: list_o] :
% 0.30/0.61            ( collect_o
% 0.30/0.61            @ ^ [Uu3: $o] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_o @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6570_set__conv__nth,axiom,
% 0.30/0.61      ( set_nat2
% 0.30/0.61      = ( ^ [Xs2: list_nat] :
% 0.30/0.61            ( collect_nat
% 0.30/0.61            @ ^ [Uu3: nat] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_nat @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6571_set__conv__nth,axiom,
% 0.30/0.61      ( set_int2
% 0.30/0.61      = ( ^ [Xs2: list_int] :
% 0.30/0.61            ( collect_int
% 0.30/0.61            @ ^ [Uu3: int] :
% 0.30/0.61              ? [I2: nat] :
% 0.30/0.61                ( ( Uu3
% 0.30/0.61                  = ( nth_int @ Xs2 @ I2 ) )
% 0.30/0.61                & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % set_conv_nth
% 0.30/0.61  thf(fact_6572_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_complex,N2: nat] :
% 0.30/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( finite8712137658972009173omplex
% 0.30/0.61          @ ( collect_list_complex
% 0.30/0.61            @ ^ [Xs2: list_complex] :
% 0.30/0.61                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_s3451745648224563538omplex @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6573_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
% 0.30/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.61       => ( finite500796754983035824at_nat
% 0.30/0.61          @ ( collec3343600615725829874at_nat
% 0.30/0.61            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.30/0.61                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6574_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_VEBT_VEBT,N2: nat] :
% 0.30/0.61        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.30/0.61       => ( finite3004134309566078307T_VEBT
% 0.30/0.61          @ ( collec5608196760682091941T_VEBT
% 0.30/0.61            @ ^ [Xs2: list_VEBT_VEBT] :
% 0.30/0.61                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6575_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_o,N2: nat] :
% 0.30/0.61        ( ( finite_finite_o @ A2 )
% 0.30/0.61       => ( finite_finite_list_o
% 0.30/0.61          @ ( collect_list_o
% 0.30/0.61            @ ^ [Xs2: list_o] :
% 0.30/0.61                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_size_list_o @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6576_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_nat,N2: nat] :
% 0.30/0.61        ( ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( finite8100373058378681591st_nat
% 0.30/0.61          @ ( collect_list_nat
% 0.30/0.61            @ ^ [Xs2: list_nat] :
% 0.30/0.61                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_size_list_nat @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6577_finite__lists__length__eq,axiom,
% 0.30/0.61      ! [A2: set_int,N2: nat] :
% 0.30/0.61        ( ( finite_finite_int @ A2 )
% 0.30/0.61       => ( finite3922522038869484883st_int
% 0.30/0.61          @ ( collect_list_int
% 0.30/0.61            @ ^ [Xs2: list_int] :
% 0.30/0.61                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ( size_size_list_int @ Xs2 )
% 0.30/0.61                  = N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_eq
% 0.30/0.61  thf(fact_6578_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_complex,N2: nat] :
% 0.30/0.61        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( finite8712137658972009173omplex
% 0.30/0.61          @ ( collect_list_complex
% 0.30/0.61            @ ^ [Xs2: list_complex] :
% 0.30/0.61                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6579_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
% 0.30/0.61        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.61       => ( finite500796754983035824at_nat
% 0.30/0.61          @ ( collec3343600615725829874at_nat
% 0.30/0.61            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 0.30/0.61                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6580_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_VEBT_VEBT,N2: nat] :
% 0.30/0.61        ( ( finite5795047828879050333T_VEBT @ A2 )
% 0.30/0.61       => ( finite3004134309566078307T_VEBT
% 0.30/0.61          @ ( collec5608196760682091941T_VEBT
% 0.30/0.61            @ ^ [Xs2: list_VEBT_VEBT] :
% 0.30/0.61                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6581_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_o,N2: nat] :
% 0.30/0.61        ( ( finite_finite_o @ A2 )
% 0.30/0.61       => ( finite_finite_list_o
% 0.30/0.61          @ ( collect_list_o
% 0.30/0.61            @ ^ [Xs2: list_o] :
% 0.30/0.61                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6582_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_nat,N2: nat] :
% 0.30/0.61        ( ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( finite8100373058378681591st_nat
% 0.30/0.61          @ ( collect_list_nat
% 0.30/0.61            @ ^ [Xs2: list_nat] :
% 0.30/0.61                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6583_finite__lists__length__le,axiom,
% 0.30/0.61      ! [A2: set_int,N2: nat] :
% 0.30/0.61        ( ( finite_finite_int @ A2 )
% 0.30/0.61       => ( finite3922522038869484883st_int
% 0.30/0.61          @ ( collect_list_int
% 0.30/0.61            @ ^ [Xs2: list_int] :
% 0.30/0.61                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 0.30/0.61                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_lists_length_le
% 0.30/0.61  thf(fact_6584_vebt__buildup_Osimps_I3_J,axiom,
% 0.30/0.61      ! [Va: nat] :
% 0.30/0.61        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.61         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.61            = ( 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 ) ) ) ) ) ) )
% 0.30/0.61        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.61         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 0.30/0.61            = ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_buildup.simps(3)
% 0.30/0.61  thf(fact_6585_vebt__buildup_Oelims,axiom,
% 0.30/0.61      ! [X3: nat,Y3: vEBT_VEBT] :
% 0.30/0.61        ( ( ( vEBT_vebt_buildup @ X3 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( ( X3 = zero_zero_nat )
% 0.30/0.61           => ( Y3
% 0.30/0.61             != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.61         => ( ( ( X3
% 0.30/0.61                = ( suc @ zero_zero_nat ) )
% 0.30/0.61             => ( Y3
% 0.30/0.61               != ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.61           => ~ ! [Va2: nat] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                       => ( Y3
% 0.30/0.61                          = ( 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 ) ) ) ) ) ) )
% 0.30/0.61                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                       => ( Y3
% 0.30/0.61                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_buildup.elims
% 0.30/0.61  thf(fact_6586_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 0.30/0.61      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 0.30/0.61        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
% 0.30/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.61           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.simps(3)
% 0.30/0.61  thf(fact_6587_dbl__dec__def,axiom,
% 0.30/0.61      ( neg_nu6511756317524482435omplex
% 0.30/0.61      = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_dec_def
% 0.30/0.61  thf(fact_6588_dbl__dec__def,axiom,
% 0.30/0.61      ( neg_nu6075765906172075777c_real
% 0.30/0.61      = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_dec_def
% 0.30/0.61  thf(fact_6589_dbl__dec__def,axiom,
% 0.30/0.61      ( neg_nu3179335615603231917ec_rat
% 0.30/0.61      = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_dec_def
% 0.30/0.61  thf(fact_6590_dbl__dec__def,axiom,
% 0.30/0.61      ( neg_nu3811975205180677377ec_int
% 0.30/0.61      = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dbl_dec_def
% 0.30/0.61  thf(fact_6591_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 0.30/0.61      ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
% 0.30/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X3 )
% 0.30/0.61        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.61           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.61          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.simps(5)
% 0.30/0.61  thf(fact_6592_vebt__member_Osimps_I5_J,axiom,
% 0.30/0.61      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 0.30/0.61        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 0.30/0.61        = ( ( X3 != Mi )
% 0.30/0.61         => ( ( X3 != Ma )
% 0.30/0.61           => ( ~ ( ord_less_nat @ X3 @ Mi )
% 0.30/0.61              & ( ~ ( ord_less_nat @ X3 @ Mi )
% 0.30/0.61               => ( ~ ( ord_less_nat @ Ma @ X3 )
% 0.30/0.61                  & ( ~ ( ord_less_nat @ Ma @ X3 )
% 0.30/0.61                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.61                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.simps(5)
% 0.30/0.61  thf(fact_6593_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 0.30/0.61      ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 0.30/0.61        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
% 0.30/0.61        = ( ( X3 = Mi )
% 0.30/0.61          | ( X3 = Ma )
% 0.30/0.61          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 0.30/0.61             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.61            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.simps(4)
% 0.30/0.61  thf(fact_6594_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.61        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ! [A4: $o,B4: $o] :
% 0.30/0.61              ( ( X3
% 0.30/0.61                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61             => ( Y3
% 0.30/0.61                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                       => A4 )
% 0.30/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                           => B4 )
% 0.30/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 0.30/0.61         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                  ( X3
% 0.30/0.61                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.61             => Y3 )
% 0.30/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                  ( ? [S2: vEBT_VEBT] :
% 0.30/0.61                      ( X3
% 0.30/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61                 => ( Y3
% 0.30/0.61                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                           => ( 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 ) ) ) ) ) )
% 0.30/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.elims(1)
% 0.30/0.61  thf(fact_6595_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [A4: $o,B4: $o] :
% 0.30/0.61              ( ( X3
% 0.30/0.61                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                   => A4 )
% 0.30/0.61                  & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                   => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                       => B4 )
% 0.30/0.61                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                ( ? [S2: vEBT_VEBT] :
% 0.30/0.61                    ( X3
% 0.30/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                     => ( 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 ) ) ) ) ) )
% 0.30/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.elims(2)
% 0.30/0.61  thf(fact_6596_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [A4: $o,B4: $o] :
% 0.30/0.61              ( ( X3
% 0.30/0.61                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61             => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                 => A4 )
% 0.30/0.61                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                     => B4 )
% 0.30/0.61                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                ( X3
% 0.30/0.61               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                  ( ? [S2: vEBT_VEBT] :
% 0.30/0.61                      ( X3
% 0.30/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                     => ( 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 ) ) ) ) ) )
% 0.30/0.61                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.elims(3)
% 0.30/0.61  thf(fact_6597_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.61              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                  ( X3
% 0.30/0.61                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61             => ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                  | ( Xa2 = Ma2 ) ) )
% 0.30/0.61         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.61                    ( X3
% 0.30/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61               => ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                    | ( Xa2 = Ma2 )
% 0.30/0.61                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                       => ( 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 ) ) ) ) ) )
% 0.30/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.61           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                  ( ? [Vd: vEBT_VEBT] :
% 0.30/0.61                      ( X3
% 0.30/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                       => ( 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 ) ) ) ) ) )
% 0.30/0.61                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.elims(2)
% 0.30/0.61  thf(fact_6598_xor__nat__unfold,axiom,
% 0.30/0.61      ( bit_se6528837805403552850or_nat
% 0.30/0.61      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = 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 @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_unfold
% 0.30/0.61  thf(fact_6599_xor__nat__rec,axiom,
% 0.30/0.61      ( bit_se6528837805403552850or_nat
% 0.30/0.61      = ( ^ [M2: nat,N: nat] :
% 0.30/0.61            ( plus_plus_nat
% 0.30/0.61            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.61              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 0.30/0.61               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 0.30/0.61            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_nat_rec
% 0.30/0.61  thf(fact_6600_one__xor__eq,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 0.30/0.61        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n356916108424825756nteger
% 0.30/0.61            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_xor_eq
% 0.30/0.61  thf(fact_6601_one__xor__eq,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 0.30/0.61        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.61            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_xor_eq
% 0.30/0.61  thf(fact_6602_one__xor__eq,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 0.30/0.61        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n2684676970156552555ol_int
% 0.30/0.61            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % one_xor_eq
% 0.30/0.61  thf(fact_6603_xor__one__eq,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 0.30/0.61        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n356916108424825756nteger
% 0.30/0.61            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_one_eq
% 0.30/0.61  thf(fact_6604_xor__one__eq,axiom,
% 0.30/0.61      ! [A: nat] :
% 0.30/0.61        ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 0.30/0.61        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n2687167440665602831ol_nat
% 0.30/0.61            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_one_eq
% 0.30/0.61  thf(fact_6605_xor__one__eq,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 0.30/0.61        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 0.30/0.61          @ ( zero_n2684676970156552555ol_int
% 0.30/0.61            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_one_eq
% 0.30/0.61  thf(fact_6606_vebt__member_Oelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [A4: $o,B4: $o] :
% 0.30/0.61              ( ( X3
% 0.30/0.61                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61             => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                   => A4 )
% 0.30/0.61                  & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                   => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                       => B4 )
% 0.30/0.61                      & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.61                    ( X3
% 0.30/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.61               => ~ ( ( Xa2 != Mi2 )
% 0.30/0.61                   => ( ( Xa2 != Ma2 )
% 0.30/0.61                     => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                        & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                         => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                                 => ( 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 ) ) ) ) ) )
% 0.30/0.61                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.elims(2)
% 0.30/0.61  thf(fact_6607_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.61              ( X3
% 0.30/0.61             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.61         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.61                ( X3
% 0.30/0.61               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.61           => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                      ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61                 => ( ( Xa2 = Mi2 )
% 0.30/0.61                    | ( Xa2 = Ma2 ) ) )
% 0.30/0.61             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                    ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.61                        ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61                   => ( ( Xa2 = Mi2 )
% 0.30/0.61                      | ( Xa2 = Ma2 )
% 0.30/0.61                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                         => ( 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 ) ) ) ) ) )
% 0.30/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.61               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                      ( ? [Vd: vEBT_VEBT] :
% 0.30/0.61                          ( X3
% 0.30/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                         => ( 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 ) ) ) ) ) )
% 0.30/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.elims(3)
% 0.30/0.61  thf(fact_6608_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.61        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.61                ( X3
% 0.30/0.61                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.61           => Y3 )
% 0.30/0.61         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.61                  ( X3
% 0.30/0.61                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.61             => Y3 )
% 0.30/0.61           => ( ! [Mi2: nat,Ma2: nat] :
% 0.30/0.61                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                      ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61                 => ( Y3
% 0.30/0.61                    = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                          | ( Xa2 = Ma2 ) ) ) ) )
% 0.30/0.61             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                    ( ? [Vc2: vEBT_VEBT] :
% 0.30/0.61                        ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61                   => ( Y3
% 0.30/0.61                      = ( ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                            | ( Xa2 = Ma2 )
% 0.30/0.61                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                               => ( 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 ) ) ) ) ) )
% 0.30/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 0.30/0.61               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                      ( ? [Vd: vEBT_VEBT] :
% 0.30/0.61                          ( X3
% 0.30/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                     => ( Y3
% 0.30/0.61                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                               => ( 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 ) ) ) ) ) )
% 0.30/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.elims(1)
% 0.30/0.61  thf(fact_6609_bit__horner__sum__bit__iff,axiom,
% 0.30/0.61      ! [Bs: list_o,N2: nat] :
% 0.30/0.61        ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
% 0.30/0.61        = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
% 0.30/0.61          & ( nth_o @ Bs @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bit_horner_sum_bit_iff
% 0.30/0.61  thf(fact_6610_bit__horner__sum__bit__iff,axiom,
% 0.30/0.61      ! [Bs: list_o,N2: nat] :
% 0.30/0.61        ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
% 0.30/0.61        = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
% 0.30/0.61          & ( nth_o @ Bs @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % bit_horner_sum_bit_iff
% 0.30/0.61  thf(fact_6611_vebt__member_Oelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.61       => ( ! [A4: $o,B4: $o] :
% 0.30/0.61              ( ( X3
% 0.30/0.61                = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61             => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                 => A4 )
% 0.30/0.61                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                 => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                     => B4 )
% 0.30/0.61                    & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                ( X3
% 0.30/0.61               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.61           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.61                  ( X3
% 0.30/0.61                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.61             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                    ( X3
% 0.30/0.61                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.61               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 0.30/0.61                      ( ? [Summary2: vEBT_VEBT] :
% 0.30/0.61                          ( X3
% 0.30/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.61                     => ( ( Xa2 != Mi2 )
% 0.30/0.61                       => ( ( Xa2 != Ma2 )
% 0.30/0.61                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                                     => ( 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 ) ) ) ) ) )
% 0.30/0.61                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.elims(3)
% 0.30/0.61  thf(fact_6612_of__int__code__if,axiom,
% 0.30/0.61      ( ring_17405671764205052669omplex
% 0.30/0.61      = ( ^ [K3: int] :
% 0.30/0.61            ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 0.30/0.61            @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.61              @ ( if_complex
% 0.30/0.61                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = zero_zero_int )
% 0.30/0.61                @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61                @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_code_if
% 0.30/0.61  thf(fact_6613_of__int__code__if,axiom,
% 0.30/0.61      ( ring_1_of_int_int
% 0.30/0.61      = ( ^ [K3: int] :
% 0.30/0.61            ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 0.30/0.61            @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.61              @ ( if_int
% 0.30/0.61                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = zero_zero_int )
% 0.30/0.61                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61                @ ( 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 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_code_if
% 0.30/0.61  thf(fact_6614_of__int__code__if,axiom,
% 0.30/0.61      ( ring_1_of_int_real
% 0.30/0.61      = ( ^ [K3: int] :
% 0.30/0.61            ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 0.30/0.61            @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.61              @ ( if_real
% 0.30/0.61                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = zero_zero_int )
% 0.30/0.61                @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61                @ ( 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 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_code_if
% 0.30/0.61  thf(fact_6615_of__int__code__if,axiom,
% 0.30/0.61      ( ring_1_of_int_rat
% 0.30/0.61      = ( ^ [K3: int] :
% 0.30/0.61            ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 0.30/0.61            @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.61              @ ( if_rat
% 0.30/0.61                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = zero_zero_int )
% 0.30/0.61                @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61                @ ( 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 ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_code_if
% 0.30/0.61  thf(fact_6616_of__int__code__if,axiom,
% 0.30/0.61      ( ring_18347121197199848620nteger
% 0.30/0.61      = ( ^ [K3: int] :
% 0.30/0.61            ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.30/0.61            @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.61              @ ( if_Code_integer
% 0.30/0.61                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.61                  = zero_zero_int )
% 0.30/0.61                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.61                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_code_if
% 0.30/0.61  thf(fact_6617_finite__nth__roots,axiom,
% 0.30/0.61      ! [N2: nat,C: complex] :
% 0.30/0.61        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.61       => ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [Z6: complex] :
% 0.30/0.61                ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.61                = C ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_nth_roots
% 0.30/0.61  thf(fact_6618_finite__roots__unity,axiom,
% 0.30/0.61      ! [N2: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 0.30/0.61       => ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [Z6: real] :
% 0.30/0.61                ( ( power_power_real @ Z6 @ N2 )
% 0.30/0.61                = one_one_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_roots_unity
% 0.30/0.61  thf(fact_6619_finite__roots__unity,axiom,
% 0.30/0.61      ! [N2: nat] :
% 0.30/0.61        ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 0.30/0.61       => ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [Z6: complex] :
% 0.30/0.61                ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.61                = one_one_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % finite_roots_unity
% 0.30/0.61  thf(fact_6620_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6621_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6622_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 0.30/0.61        ( ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [I2: int] :
% 0.30/0.61                ( ( member_int @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6623_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 0.30/0.61        ( ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [I2: complex] :
% 0.30/0.61                ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_complex ) ) ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_complex ) ) ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6624_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_real ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6625_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_real ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6626_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 0.30/0.61        ( ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [I2: int] :
% 0.30/0.61                ( ( member_int @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_real ) ) ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6627_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 0.30/0.61        ( ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [I2: complex] :
% 0.30/0.61                ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_real ) ) ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_real ) ) ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6628_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_rat ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_rat ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6629_prod_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != one_one_rat ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != one_one_rat ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != one_one_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.finite_Collect_op
% 0.30/0.61  thf(fact_6630_vebt__member_Opelims_I1_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.61        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( Y3
% 0.30/0.61                    = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                       => A4 )
% 0.30/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                           => B4 )
% 0.30/0.61                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.61           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.61                 => ( ~ Y3
% 0.30/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.61             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.61                   => ( ~ Y3
% 0.30/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 0.30/0.61               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                      ( ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.61                     => ( ~ Y3
% 0.30/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.61                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.61                        ( ( X3
% 0.30/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.61                       => ( ( Y3
% 0.30/0.61                            = ( ( Xa2 != Mi2 )
% 0.30/0.61                             => ( ( Xa2 != Ma2 )
% 0.30/0.61                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                                           => ( 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 ) ) ) ) ) )
% 0.30/0.61                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 0.30/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.pelims(1)
% 0.30/0.61  thf(fact_6631_vebt__member_Opelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.61                 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                     => A4 )
% 0.30/0.61                    & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                     => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                         => B4 )
% 0.30/0.61                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.61           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.61             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 0.30/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 0.30/0.61               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                      ( ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 0.30/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 0.30/0.61                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.61                        ( ( X3
% 0.30/0.61                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.61                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.61                         => ( ( Xa2 != Mi2 )
% 0.30/0.61                           => ( ( Xa2 != Ma2 )
% 0.30/0.61                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                                         => ( 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 ) ) ) ) ) )
% 0.30/0.61                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.pelims(3)
% 0.30/0.61  thf(fact_6632_xor__negative__int__iff,axiom,
% 0.30/0.61      ! [K: int,L: int] :
% 0.30/0.61        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.61        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.61         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % xor_negative_int_iff
% 0.30/0.61  thf(fact_6633_XOR__upper,axiom,
% 0.30/0.61      ! [X3: int,N2: nat,Y3: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.61       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.61         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.61           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % XOR_upper
% 0.30/0.61  thf(fact_6634_vebt__member_Opelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.61                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                       => A4 )
% 0.30/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                           => B4 )
% 0.30/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.61           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 0.30/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.61                   => ~ ( ( Xa2 != Mi2 )
% 0.30/0.61                       => ( ( Xa2 != Ma2 )
% 0.30/0.61                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 0.30/0.61                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 0.30/0.61                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                                     => ( 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 ) ) ) ) ) )
% 0.30/0.61                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_member.pelims(2)
% 0.30/0.61  thf(fact_6635_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6636_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6637_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 0.30/0.61        ( ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [I2: int] :
% 0.30/0.61                ( ( member_int @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_complex ) ) ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_complex ) ) ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6638_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 0.30/0.61        ( ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [I2: complex] :
% 0.30/0.61                ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_complex ) ) ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_complex ) ) ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_complex ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6639_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_real ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6640_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_real ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6641_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 0.30/0.61        ( ( finite_finite_int
% 0.30/0.61          @ ( collect_int
% 0.30/0.61            @ ^ [I2: int] :
% 0.30/0.61                ( ( member_int @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_real ) ) ) )
% 0.30/0.61       => ( ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_real ) ) ) )
% 0.30/0.61         => ( finite_finite_int
% 0.30/0.61            @ ( collect_int
% 0.30/0.61              @ ^ [I2: int] :
% 0.30/0.61                  ( ( member_int @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6642_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 0.30/0.61        ( ( finite3207457112153483333omplex
% 0.30/0.61          @ ( collect_complex
% 0.30/0.61            @ ^ [I2: complex] :
% 0.30/0.61                ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_real ) ) ) )
% 0.30/0.61       => ( ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_real ) ) ) )
% 0.30/0.61         => ( finite3207457112153483333omplex
% 0.30/0.61            @ ( collect_complex
% 0.30/0.61              @ ^ [I2: complex] :
% 0.30/0.61                  ( ( member_complex @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6643_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 0.30/0.61        ( ( finite_finite_real
% 0.30/0.61          @ ( collect_real
% 0.30/0.61            @ ^ [I2: real] :
% 0.30/0.61                ( ( member_real @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_rat ) ) ) )
% 0.30/0.61       => ( ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_rat ) ) ) )
% 0.30/0.61         => ( finite_finite_real
% 0.30/0.61            @ ( collect_real
% 0.30/0.61              @ ^ [I2: real] :
% 0.30/0.61                  ( ( member_real @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6644_sum_Ofinite__Collect__op,axiom,
% 0.30/0.61      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 0.30/0.61        ( ( finite_finite_nat
% 0.30/0.61          @ ( collect_nat
% 0.30/0.61            @ ^ [I2: nat] :
% 0.30/0.61                ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                & ( ( X3 @ I2 )
% 0.30/0.61                 != zero_zero_rat ) ) ) )
% 0.30/0.61       => ( ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( Y3 @ I2 )
% 0.30/0.61                   != zero_zero_rat ) ) ) )
% 0.30/0.61         => ( finite_finite_nat
% 0.30/0.61            @ ( collect_nat
% 0.30/0.61              @ ^ [I2: nat] :
% 0.30/0.61                  ( ( member_nat @ I2 @ I5 )
% 0.30/0.61                  & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 0.30/0.61                   != zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.finite_Collect_op
% 0.30/0.61  thf(fact_6645_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.61        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( Y3
% 0.30/0.61                    = ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                       => A4 )
% 0.30/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                           => B4 )
% 0.30/0.61                          & ( Xa2 = one_one_nat ) ) ) ) )
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 0.30/0.61           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.61                 => ( ~ Y3
% 0.30/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 0.30/0.61             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61                   => ( ( Y3
% 0.30/0.61                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                           => ( 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 ) ) ) ) ) )
% 0.30/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.30/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.pelims(1)
% 0.30/0.61  thf(fact_6646_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.61                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                       => A4 )
% 0.30/0.61                      & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                       => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                           => B4 )
% 0.30/0.61                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.61           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 0.30/0.61                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                         => ( 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 ) ) ) ) ) )
% 0.30/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.pelims(2)
% 0.30/0.61  thf(fact_6647_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [A4: $o,B4: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 0.30/0.61                 => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.61                     => A4 )
% 0.30/0.61                    & ( ( Xa2 != zero_zero_nat )
% 0.30/0.61                     => ( ( ( Xa2 = one_one_nat )
% 0.30/0.61                         => B4 )
% 0.30/0.61                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 0.30/0.61           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 0.30/0.61             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 0.30/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 0.30/0.61                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                         => ( 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 ) ) ) ) ) )
% 0.30/0.61                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.naive_member.pelims(3)
% 0.30/0.61  thf(fact_6648_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.61               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 0.30/0.61           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 0.30/0.61             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.30/0.61                     => ( ( Xa2 = Mi2 )
% 0.30/0.61                        | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.61               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                      ( ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.30/0.61                       => ( ( Xa2 = Mi2 )
% 0.30/0.61                          | ( Xa2 = Ma2 )
% 0.30/0.61                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                             => ( 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 ) ) ) ) ) )
% 0.30/0.61                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.30/0.61                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.30/0.61                        ( ( X3
% 0.30/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
% 0.30/0.61                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                             => ( 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 ) ) ) ) ) )
% 0.30/0.61                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.pelims(3)
% 0.30/0.61  thf(fact_6649_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.61        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.61               => ( ~ Y3
% 0.30/0.61                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.61           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 0.30/0.61                 => ( ~ Y3
% 0.30/0.61                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 0.30/0.61             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61                   => ( ( Y3
% 0.30/0.61                        = ( ( Xa2 = Mi2 )
% 0.30/0.61                          | ( Xa2 = Ma2 ) ) )
% 0.30/0.61                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 0.30/0.61               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                      ( ( X3
% 0.30/0.61                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61                     => ( ( Y3
% 0.30/0.61                          = ( ( Xa2 = Mi2 )
% 0.30/0.61                            | ( Xa2 = Ma2 )
% 0.30/0.61                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                               => ( 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 ) ) ) ) ) )
% 0.30/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 0.30/0.61                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 0.30/0.61                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.30/0.61                        ( ( X3
% 0.30/0.61                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                       => ( ( Y3
% 0.30/0.61                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                               => ( 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 ) ) ) ) ) )
% 0.30/0.61                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 0.30/0.61                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.pelims(1)
% 0.30/0.61  thf(fact_6650_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 0.30/0.61      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.61        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 0.30/0.61       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.61         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 0.30/0.61                ( ( X3
% 0.30/0.61                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 0.30/0.61               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 0.30/0.61                 => ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                      | ( Xa2 = Ma2 ) ) ) )
% 0.30/0.61           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.61                  ( ( X3
% 0.30/0.61                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 0.30/0.61                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 0.30/0.61                   => ~ ( ( Xa2 = Mi2 )
% 0.30/0.61                        | ( Xa2 = Ma2 )
% 0.30/0.61                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                           => ( 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 ) ) ) ) ) )
% 0.30/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 0.30/0.61             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 0.30/0.61                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
% 0.30/0.61                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 0.30/0.61                           => ( 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 ) ) ) ) ) )
% 0.30/0.61                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % VEBT_internal.membermima.pelims(2)
% 0.30/0.61  thf(fact_6651_arcosh__1,axiom,
% 0.30/0.61      ( ( arcosh_real @ one_one_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % arcosh_1
% 0.30/0.61  thf(fact_6652_arsinh__0,axiom,
% 0.30/0.61      ( ( arsinh_real @ zero_zero_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % arsinh_0
% 0.30/0.61  thf(fact_6653_artanh__0,axiom,
% 0.30/0.61      ( ( artanh_real @ zero_zero_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % artanh_0
% 0.30/0.61  thf(fact_6654_vebt__buildup_Opelims,axiom,
% 0.30/0.61      ! [X3: nat,Y3: vEBT_VEBT] :
% 0.30/0.61        ( ( ( vEBT_vebt_buildup @ X3 )
% 0.30/0.61          = Y3 )
% 0.30/0.61       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
% 0.30/0.61         => ( ( ( X3 = zero_zero_nat )
% 0.30/0.61             => ( ( Y3
% 0.30/0.61                  = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.61               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 0.30/0.61           => ( ( ( X3
% 0.30/0.61                  = ( suc @ zero_zero_nat ) )
% 0.30/0.61               => ( ( Y3
% 0.30/0.61                    = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.61                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.61             => ~ ! [Va2: nat] :
% 0.30/0.61                    ( ( X3
% 0.30/0.61                      = ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                         => ( Y3
% 0.30/0.61                            = ( 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 ) ) ) ) ) ) )
% 0.30/0.61                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 0.30/0.61                         => ( Y3
% 0.30/0.61                            = ( 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 ) ) ) ) ) ) ) ) )
% 0.30/0.61                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % vebt_buildup.pelims
% 0.30/0.61  thf(fact_6655_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc4245557441103728435nt_int
% 0.30/0.61          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 0.30/0.61          @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(6)
% 0.30/0.61  thf(fact_6656_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc2626176000494625587at_nat
% 0.30/0.61          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 0.30/0.61          @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(6)
% 0.30/0.61  thf(fact_6657_divmod__algorithm__code_I6_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc6916734918728496179nteger
% 0.30/0.61          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 0.30/0.61          @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(6)
% 0.30/0.61  thf(fact_6658_artanh__def,axiom,
% 0.30/0.61      ( artanh_real
% 0.30/0.61      = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % artanh_def
% 0.30/0.61  thf(fact_6659_ln__inj__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ( ( ln_ln_real @ X3 )
% 0.30/0.61              = ( ln_ln_real @ Y3 ) )
% 0.30/0.61            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_inj_iff
% 0.30/0.61  thf(fact_6660_ln__less__cancel__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 0.30/0.61            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_less_cancel_iff
% 0.30/0.61  thf(fact_6661_case__prod__conv,axiom,
% 0.30/0.61      ! [F: nat > nat > nat,A: nat,B: nat] :
% 0.30/0.61        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.61        = ( F @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_conv
% 0.30/0.61  thf(fact_6662_case__prod__conv,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 0.30/0.61        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.61        = ( F @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_conv
% 0.30/0.61  thf(fact_6663_case__prod__conv,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 0.30/0.61        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 0.30/0.61        = ( F @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_conv
% 0.30/0.61  thf(fact_6664_case__prod__conv,axiom,
% 0.30/0.61      ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 0.30/0.61        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.61        = ( F @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_conv
% 0.30/0.61  thf(fact_6665_case__prod__conv,axiom,
% 0.30/0.61      ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.61        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.61        = ( F @ A @ B ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_conv
% 0.30/0.61  thf(fact_6666_ln__le__cancel__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 0.30/0.61            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_le_cancel_iff
% 0.30/0.61  thf(fact_6667_ln__less__zero__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 0.30/0.61          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_less_zero_iff
% 0.30/0.61  thf(fact_6668_ln__gt__zero__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.61          = ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_gt_zero_iff
% 0.30/0.61  thf(fact_6669_ln__eq__zero__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ( ln_ln_real @ X3 )
% 0.30/0.61            = zero_zero_real )
% 0.30/0.61          = ( X3 = one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_eq_zero_iff
% 0.30/0.61  thf(fact_6670_ln__one,axiom,
% 0.30/0.61      ( ( ln_ln_real @ one_one_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_one
% 0.30/0.61  thf(fact_6671_ln__le__zero__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 0.30/0.61          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_le_zero_iff
% 0.30/0.61  thf(fact_6672_ln__ge__zero__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.61          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_ge_zero_iff
% 0.30/0.61  thf(fact_6673_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc4245557441103728435nt_int
% 0.30/0.61          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.61          @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(5)
% 0.30/0.61  thf(fact_6674_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc2626176000494625587at_nat
% 0.30/0.61          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.61          @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(5)
% 0.30/0.61  thf(fact_6675_divmod__algorithm__code_I5_J,axiom,
% 0.30/0.61      ! [M: num,N2: num] :
% 0.30/0.61        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 0.30/0.61        = ( produc6916734918728496179nteger
% 0.30/0.61          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 0.30/0.61          @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_algorithm_code(5)
% 0.30/0.61  thf(fact_6676_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 0.30/0.61        = ( produc6842872674320459806at_nat
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6677_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
% 0.30/0.61        ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 0.30/0.61        = ( produc4947309494688390418_int_o
% 0.30/0.61          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6678_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 0.30/0.61        ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 0.30/0.61        = ( produc4947309494688390418_int_o
% 0.30/0.61          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6679_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
% 0.30/0.61        ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 0.30/0.61        = ( produc4245557441103728435nt_int
% 0.30/0.61          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6680_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 0.30/0.61        = ( produc8739625826339149834_nat_o
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6681_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 0.30/0.61        = ( produc6842872674320459806at_nat
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6682_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 0.30/0.61        ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 0.30/0.61        = ( produc4245557441103728435nt_int
% 0.30/0.61          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6683_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 0.30/0.61        = ( produc27273713700761075at_nat
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6684_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
% 0.30/0.61        = ( produc6842872674320459806at_nat
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6685_prod_Ocase__distrib,axiom,
% 0.30/0.61      ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 0.30/0.61        ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 0.30/0.61        = ( produc8739625826339149834_nat_o
% 0.30/0.61          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 0.30/0.61          @ Prod ) ) ).
% 0.30/0.61  
% 0.30/0.61  % prod.case_distrib
% 0.30/0.61  thf(fact_6686_old_Oprod_Ocase,axiom,
% 0.30/0.61      ! [F: nat > nat > nat,X1: nat,X22: nat] :
% 0.30/0.61        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.61        = ( F @ X1 @ X22 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % old.prod.case
% 0.30/0.61  thf(fact_6687_old_Oprod_Ocase,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 0.30/0.61        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.61        = ( F @ X1 @ X22 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % old.prod.case
% 0.30/0.61  thf(fact_6688_old_Oprod_Ocase,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 0.30/0.61        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 0.30/0.61        = ( F @ X1 @ X22 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % old.prod.case
% 0.30/0.61  thf(fact_6689_old_Oprod_Ocase,axiom,
% 0.30/0.61      ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 0.30/0.61        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.30/0.61        = ( F @ X1 @ X22 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % old.prod.case
% 0.30/0.61  thf(fact_6690_old_Oprod_Ocase,axiom,
% 0.30/0.61      ! [F: int > int > $o,X1: int,X22: int] :
% 0.30/0.61        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 0.30/0.61        = ( F @ X1 @ X22 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % old.prod.case
% 0.30/0.61  thf(fact_6691_split__cong,axiom,
% 0.30/0.61      ! [Q4: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P4: product_prod_nat_nat] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 0.30/0.61              = Q4 )
% 0.30/0.61           => ( ( F @ X4 @ Y5 )
% 0.30/0.61              = ( G @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( P4 = Q4 )
% 0.30/0.61         => ( ( produc6842872674320459806at_nat @ F @ P4 )
% 0.30/0.61            = ( produc6842872674320459806at_nat @ G @ Q4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_cong
% 0.30/0.61  thf(fact_6692_split__cong,axiom,
% 0.30/0.61      ! [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] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 0.30/0.61              = Q4 )
% 0.30/0.61           => ( ( F @ X4 @ Y5 )
% 0.30/0.61              = ( G @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( P4 = Q4 )
% 0.30/0.61         => ( ( produc27273713700761075at_nat @ F @ P4 )
% 0.30/0.61            = ( produc27273713700761075at_nat @ G @ Q4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_cong
% 0.30/0.61  thf(fact_6693_split__cong,axiom,
% 0.30/0.61      ! [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] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 0.30/0.61              = Q4 )
% 0.30/0.61           => ( ( F @ X4 @ Y5 )
% 0.30/0.61              = ( G @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( P4 = Q4 )
% 0.30/0.61         => ( ( produc8739625826339149834_nat_o @ F @ P4 )
% 0.30/0.61            = ( produc8739625826339149834_nat_o @ G @ Q4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_cong
% 0.30/0.61  thf(fact_6694_split__cong,axiom,
% 0.30/0.61      ! [Q4: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
% 0.30/0.61        ( ! [X4: int,Y5: int] :
% 0.30/0.61            ( ( ( product_Pair_int_int @ X4 @ Y5 )
% 0.30/0.61              = Q4 )
% 0.30/0.61           => ( ( F @ X4 @ Y5 )
% 0.30/0.61              = ( G @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( P4 = Q4 )
% 0.30/0.61         => ( ( produc4245557441103728435nt_int @ F @ P4 )
% 0.30/0.61            = ( produc4245557441103728435nt_int @ G @ Q4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_cong
% 0.30/0.61  thf(fact_6695_split__cong,axiom,
% 0.30/0.61      ! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
% 0.30/0.61        ( ! [X4: int,Y5: int] :
% 0.30/0.61            ( ( ( product_Pair_int_int @ X4 @ Y5 )
% 0.30/0.61              = Q4 )
% 0.30/0.61           => ( ( F @ X4 @ Y5 )
% 0.30/0.61              = ( G @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( P4 = Q4 )
% 0.30/0.61         => ( ( produc4947309494688390418_int_o @ F @ P4 )
% 0.30/0.61            = ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % split_cong
% 0.30/0.61  thf(fact_6696_cond__case__prod__eta,axiom,
% 0.30/0.61      ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( F @ X4 @ Y5 )
% 0.30/0.61            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( produc6842872674320459806at_nat @ F )
% 0.30/0.61          = G ) ) ).
% 0.30/0.61  
% 0.30/0.61  % cond_case_prod_eta
% 0.30/0.61  thf(fact_6697_cond__case__prod__eta,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( F @ X4 @ Y5 )
% 0.30/0.61            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( produc27273713700761075at_nat @ F )
% 0.30/0.61          = G ) ) ).
% 0.30/0.61  
% 0.30/0.61  % cond_case_prod_eta
% 0.30/0.61  thf(fact_6698_cond__case__prod__eta,axiom,
% 0.30/0.61      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.61        ( ! [X4: nat,Y5: nat] :
% 0.30/0.61            ( ( F @ X4 @ Y5 )
% 0.30/0.61            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( produc8739625826339149834_nat_o @ F )
% 0.30/0.61          = G ) ) ).
% 0.30/0.61  
% 0.30/0.61  % cond_case_prod_eta
% 0.30/0.61  thf(fact_6699_cond__case__prod__eta,axiom,
% 0.30/0.61      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 0.30/0.61        ( ! [X4: int,Y5: int] :
% 0.30/0.61            ( ( F @ X4 @ Y5 )
% 0.30/0.61            = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( produc4245557441103728435nt_int @ F )
% 0.30/0.61          = G ) ) ).
% 0.30/0.61  
% 0.30/0.61  % cond_case_prod_eta
% 0.30/0.61  thf(fact_6700_cond__case__prod__eta,axiom,
% 0.30/0.61      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 0.30/0.61        ( ! [X4: int,Y5: int] :
% 0.30/0.61            ( ( F @ X4 @ Y5 )
% 0.30/0.61            = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
% 0.30/0.61       => ( ( produc4947309494688390418_int_o @ F )
% 0.30/0.61          = G ) ) ).
% 0.30/0.61  
% 0.30/0.61  % cond_case_prod_eta
% 0.30/0.61  thf(fact_6701_case__prod__eta,axiom,
% 0.30/0.61      ! [F: product_prod_nat_nat > nat] :
% 0.30/0.61        ( ( produc6842872674320459806at_nat
% 0.30/0.61          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.61        = F ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_eta
% 0.30/0.61  thf(fact_6702_case__prod__eta,axiom,
% 0.30/0.61      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 0.30/0.61        ( ( produc27273713700761075at_nat
% 0.30/0.61          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.61        = F ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_eta
% 0.30/0.61  thf(fact_6703_case__prod__eta,axiom,
% 0.30/0.61      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 0.30/0.61        ( ( produc8739625826339149834_nat_o
% 0.30/0.61          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 0.30/0.61        = F ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_eta
% 0.30/0.61  thf(fact_6704_case__prod__eta,axiom,
% 0.30/0.61      ! [F: product_prod_int_int > product_prod_int_int] :
% 0.30/0.61        ( ( produc4245557441103728435nt_int
% 0.30/0.61          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.61        = F ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_eta
% 0.30/0.61  thf(fact_6705_case__prod__eta,axiom,
% 0.30/0.61      ! [F: product_prod_int_int > $o] :
% 0.30/0.61        ( ( produc4947309494688390418_int_o
% 0.30/0.61          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 0.30/0.61        = F ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_eta
% 0.30/0.61  thf(fact_6706_case__prodE2,axiom,
% 0.30/0.61      ! [Q: nat > $o,P: nat > nat > nat,Z2: product_prod_nat_nat] :
% 0.30/0.61        ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z2 ) )
% 0.30/0.61       => ~ ! [X4: nat,Y5: nat] :
% 0.30/0.61              ( ( Z2
% 0.30/0.61                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 0.30/0.61             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodE2
% 0.30/0.61  thf(fact_6707_case__prodE2,axiom,
% 0.30/0.61      ! [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] :
% 0.30/0.61        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
% 0.30/0.61       => ~ ! [X4: nat,Y5: nat] :
% 0.30/0.61              ( ( Z2
% 0.30/0.61                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 0.30/0.61             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodE2
% 0.30/0.61  thf(fact_6708_case__prodE2,axiom,
% 0.30/0.61      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
% 0.30/0.61        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
% 0.30/0.61       => ~ ! [X4: nat,Y5: nat] :
% 0.30/0.61              ( ( Z2
% 0.30/0.61                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 0.30/0.61             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodE2
% 0.30/0.61  thf(fact_6709_case__prodE2,axiom,
% 0.30/0.61      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
% 0.30/0.61        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
% 0.30/0.61       => ~ ! [X4: int,Y5: int] :
% 0.30/0.61              ( ( Z2
% 0.30/0.61                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.61             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodE2
% 0.30/0.61  thf(fact_6710_case__prodE2,axiom,
% 0.30/0.61      ! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
% 0.30/0.61        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
% 0.30/0.61       => ~ ! [X4: int,Y5: int] :
% 0.30/0.61              ( ( Z2
% 0.30/0.61                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.61             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodE2
% 0.30/0.61  thf(fact_6711_ln__less__self,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_less_self
% 0.30/0.61  thf(fact_6712_ln__bound,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_bound
% 0.30/0.61  thf(fact_6713_ln__gt__zero__imp__gt__one,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61         => ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_gt_zero_imp_gt_one
% 0.30/0.61  thf(fact_6714_ln__less__zero,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.61         => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_less_zero
% 0.30/0.61  thf(fact_6715_ln__gt__zero,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.61       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_gt_zero
% 0.30/0.61  thf(fact_6716_ln__div,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.61            = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_div
% 0.30/0.61  thf(fact_6717_ln__2__less__1,axiom,
% 0.30/0.61      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 0.30/0.61  
% 0.30/0.61  % ln_2_less_1
% 0.30/0.61  thf(fact_6718_ln__ge__zero__imp__ge__one,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61         => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_ge_zero_imp_ge_one
% 0.30/0.61  thf(fact_6719_ln__diff__le,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_diff_le
% 0.30/0.61  thf(fact_6720_ln__mult,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.61         => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.61            = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_mult
% 0.30/0.61  thf(fact_6721_ln__eq__minus__one,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ( ln_ln_real @ X3 )
% 0.30/0.61            = ( minus_minus_real @ X3 @ one_one_real ) )
% 0.30/0.61         => ( X3 = one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_eq_minus_one
% 0.30/0.61  thf(fact_6722_ln__le__minus__one,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_le_minus_one
% 0.30/0.61  thf(fact_6723_ln__add__one__self__le__self2,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.61       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_add_one_self_le_self2
% 0.30/0.61  thf(fact_6724_ln__one__minus__pos__upper__bound,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.61         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % ln_one_minus_pos_upper_bound
% 0.30/0.61  thf(fact_6725_divmod__step__nat__def,axiom,
% 0.30/0.61      ( unique5026877609467782581ep_nat
% 0.30/0.61      = ( ^ [L2: num] :
% 0.30/0.61            ( produc2626176000494625587at_nat
% 0.30/0.61            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_step_nat_def
% 0.30/0.61  thf(fact_6726_divmod__step__int__def,axiom,
% 0.30/0.61      ( unique5024387138958732305ep_int
% 0.30/0.61      = ( ^ [L2: num] :
% 0.30/0.61            ( produc4245557441103728435nt_int
% 0.30/0.61            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_step_int_def
% 0.30/0.61  thf(fact_6727_divmod__step__def,axiom,
% 0.30/0.61      ( unique5026877609467782581ep_nat
% 0.30/0.61      = ( ^ [L2: num] :
% 0.30/0.61            ( produc2626176000494625587at_nat
% 0.30/0.61            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_step_def
% 0.30/0.61  thf(fact_6728_divmod__step__def,axiom,
% 0.30/0.61      ( unique5024387138958732305ep_int
% 0.30/0.61      = ( ^ [L2: num] :
% 0.30/0.61            ( produc4245557441103728435nt_int
% 0.30/0.61            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_step_def
% 0.30/0.61  thf(fact_6729_divmod__step__def,axiom,
% 0.30/0.61      ( unique4921790084139445826nteger
% 0.30/0.61      = ( ^ [L2: num] :
% 0.30/0.61            ( produc6916734918728496179nteger
% 0.30/0.61            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_step_def
% 0.30/0.61  thf(fact_6730_divmod__nat__if,axiom,
% 0.30/0.61      ( divmod_nat
% 0.30/0.61      = ( ^ [M2: nat,N: nat] :
% 0.30/0.61            ( if_Pro6206227464963214023at_nat
% 0.30/0.61            @ ( ( N = zero_zero_nat )
% 0.30/0.61              | ( ord_less_nat @ M2 @ N ) )
% 0.30/0.61            @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 0.30/0.61            @ ( produc2626176000494625587at_nat
% 0.30/0.61              @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 0.30/0.61              @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % divmod_nat_if
% 0.30/0.61  thf(fact_6731_tanh__ln__real,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.61       => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.61          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_ln_real
% 0.30/0.61  thf(fact_6732_Sum__Icc__nat,axiom,
% 0.30/0.61      ! [M: nat,N2: nat] :
% 0.30/0.61        ( ( groups3542108847815614940at_nat
% 0.30/0.61          @ ^ [X: nat] : X
% 0.30/0.61          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.61        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % Sum_Icc_nat
% 0.30/0.61  thf(fact_6733_arith__series__nat,axiom,
% 0.30/0.61      ! [A: nat,D: nat,N2: nat] :
% 0.30/0.61        ( ( groups3542108847815614940at_nat
% 0.30/0.61          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 0.30/0.61          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.61        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % arith_series_nat
% 0.30/0.61  thf(fact_6734_case__prod__Pair__iden,axiom,
% 0.30/0.61      ! [P4: produc8763457246119570046nteger] :
% 0.30/0.61        ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P4 )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_Pair_iden
% 0.30/0.61  thf(fact_6735_case__prod__Pair__iden,axiom,
% 0.30/0.61      ! [P4: produc1908205239877642774nteger] :
% 0.30/0.61        ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_Pair_iden
% 0.30/0.61  thf(fact_6736_case__prod__Pair__iden,axiom,
% 0.30/0.61      ! [P4: produc2285326912895808259nt_int] :
% 0.30/0.61        ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_Pair_iden
% 0.30/0.61  thf(fact_6737_case__prod__Pair__iden,axiom,
% 0.30/0.61      ! [P4: produc7773217078559923341nt_int] :
% 0.30/0.61        ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_Pair_iden
% 0.30/0.61  thf(fact_6738_case__prod__Pair__iden,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int] :
% 0.30/0.61        ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
% 0.30/0.61        = P4 ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prod_Pair_iden
% 0.30/0.61  thf(fact_6739_round__unique_H,axiom,
% 0.30/0.61      ! [X3: real,N2: int] :
% 0.30/0.61        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.61       => ( ( archim8280529875227126926d_real @ X3 )
% 0.30/0.61          = N2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % round_unique'
% 0.30/0.61  thf(fact_6740_round__unique_H,axiom,
% 0.30/0.61      ! [X3: rat,N2: int] :
% 0.30/0.61        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.61       => ( ( archim7778729529865785530nd_rat @ X3 )
% 0.30/0.61          = N2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % round_unique'
% 0.30/0.61  thf(fact_6741_abs__abs,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.61        = ( abs_abs_int @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_abs
% 0.30/0.61  thf(fact_6742_abs__abs,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.61        = ( abs_abs_real @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_abs
% 0.30/0.61  thf(fact_6743_abs__abs,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.61        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_abs
% 0.30/0.61  thf(fact_6744_abs__abs,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.61        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_abs
% 0.30/0.61  thf(fact_6745_abs__idempotent,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 0.30/0.61        = ( abs_abs_int @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_idempotent
% 0.30/0.61  thf(fact_6746_abs__idempotent,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 0.30/0.61        = ( abs_abs_real @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_idempotent
% 0.30/0.61  thf(fact_6747_abs__idempotent,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.61        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_idempotent
% 0.30/0.61  thf(fact_6748_abs__idempotent,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.61        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_idempotent
% 0.30/0.61  thf(fact_6749_case__prodI,axiom,
% 0.30/0.61      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.61        ( ( F @ A @ B )
% 0.30/0.61       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI
% 0.30/0.61  thf(fact_6750_case__prodI,axiom,
% 0.30/0.61      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.61        ( ( F @ A @ B )
% 0.30/0.61       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI
% 0.30/0.61  thf(fact_6751_case__prodI,axiom,
% 0.30/0.61      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.61        ( ( F @ A @ B )
% 0.30/0.61       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI
% 0.30/0.61  thf(fact_6752_case__prodI,axiom,
% 0.30/0.61      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.61        ( ( F @ A @ B )
% 0.30/0.61       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI
% 0.30/0.61  thf(fact_6753_case__prodI,axiom,
% 0.30/0.61      ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.61        ( ( F @ A @ B )
% 0.30/0.61       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI
% 0.30/0.61  thf(fact_6754_case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.30/0.61        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 0.30/0.61           => ( C @ A4 @ B4 ) )
% 0.30/0.61       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2
% 0.30/0.61  thf(fact_6755_case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 0.30/0.61        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc8603105652947943368nteger @ A4 @ B4 ) )
% 0.30/0.61           => ( C @ A4 @ B4 ) )
% 0.30/0.61       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2
% 0.30/0.61  thf(fact_6756_case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.30/0.61        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
% 0.30/0.61           => ( C @ A4 @ B4 ) )
% 0.30/0.61       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2
% 0.30/0.61  thf(fact_6757_case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 0.30/0.61        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 0.30/0.61           => ( C @ A4 @ B4 ) )
% 0.30/0.61       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2
% 0.30/0.61  thf(fact_6758_case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,C: int > int > $o] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( C @ A4 @ B4 ) )
% 0.30/0.61       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2
% 0.30/0.61  thf(fact_6759_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: real,C: int > int > set_real,A: int,B: int] :
% 0.30/0.61        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6760_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
% 0.30/0.61        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6761_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: int,C: int > int > set_int,A: int,B: int] :
% 0.30/0.61        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6762_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: set_nat,C: int > int > set_set_nat,A: int,B: int] :
% 0.30/0.61        ( ( member_set_nat @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6763_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
% 0.30/0.61        ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6764_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.61        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6765_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.61        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6766_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.61        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6767_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.61        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6768_mem__case__prodI,axiom,
% 0.30/0.61      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.61        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI
% 0.30/0.61  thf(fact_6769_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,Z2: real,C: int > int > set_real] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6770_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6771_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,Z2: int,C: int > int > set_int] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6772_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,Z2: set_nat,C: int > int > set_set_nat] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_set_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6773_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
% 0.30/0.61        ( ! [A4: int,B4: int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( product_Pair_int_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member8440522571783428010at_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6774_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 0.30/0.61        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 0.30/0.61           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6775_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 0.30/0.61        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 0.30/0.61           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6776_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 0.30/0.61        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 0.30/0.61           => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6777_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc7773217078559923341nt_int,Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 0.30/0.61        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6778_mem__case__prodI2,axiom,
% 0.30/0.61      ! [P4: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 0.30/0.61        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 0.30/0.61            ( ( P4
% 0.30/0.61              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 0.30/0.61           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 0.30/0.61       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % mem_case_prodI2
% 0.30/0.61  thf(fact_6779_case__prodI2_H,axiom,
% 0.30/0.61      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 0.30/0.61        ( ! [A4: nat,B4: nat] :
% 0.30/0.61            ( ( ( product_Pair_nat_nat @ A4 @ B4 )
% 0.30/0.61              = P4 )
% 0.30/0.61           => ( C @ A4 @ B4 @ X3 ) )
% 0.30/0.61       => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % case_prodI2'
% 0.30/0.61  thf(fact_6780_abs__0__eq,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( zero_z3403309356797280102nteger
% 0.30/0.61          = ( abs_abs_Code_integer @ A ) )
% 0.30/0.61        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0_eq
% 0.30/0.61  thf(fact_6781_abs__0__eq,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( zero_zero_real
% 0.30/0.61          = ( abs_abs_real @ A ) )
% 0.30/0.61        = ( A = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0_eq
% 0.30/0.61  thf(fact_6782_abs__0__eq,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( zero_zero_rat
% 0.30/0.61          = ( abs_abs_rat @ A ) )
% 0.30/0.61        = ( A = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0_eq
% 0.30/0.61  thf(fact_6783_abs__0__eq,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( zero_zero_int
% 0.30/0.61          = ( abs_abs_int @ A ) )
% 0.30/0.61        = ( A = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0_eq
% 0.30/0.61  thf(fact_6784_abs__eq__0,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.61          = zero_z3403309356797280102nteger )
% 0.30/0.61        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_eq_0
% 0.30/0.61  thf(fact_6785_abs__eq__0,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ( abs_abs_real @ A )
% 0.30/0.61          = zero_zero_real )
% 0.30/0.61        = ( A = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_eq_0
% 0.30/0.61  thf(fact_6786_abs__eq__0,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ( abs_abs_rat @ A )
% 0.30/0.61          = zero_zero_rat )
% 0.30/0.61        = ( A = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_eq_0
% 0.30/0.61  thf(fact_6787_abs__eq__0,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ( abs_abs_int @ A )
% 0.30/0.61          = zero_zero_int )
% 0.30/0.61        = ( A = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_eq_0
% 0.30/0.61  thf(fact_6788_abs__zero,axiom,
% 0.30/0.61      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.61      = zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_zero
% 0.30/0.61  thf(fact_6789_abs__zero,axiom,
% 0.30/0.61      ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_zero
% 0.30/0.61  thf(fact_6790_abs__zero,axiom,
% 0.30/0.61      ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.61      = zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_zero
% 0.30/0.61  thf(fact_6791_abs__zero,axiom,
% 0.30/0.61      ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.61      = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_zero
% 0.30/0.61  thf(fact_6792_abs__0,axiom,
% 0.30/0.61      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 0.30/0.61      = zero_z3403309356797280102nteger ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0
% 0.30/0.61  thf(fact_6793_abs__0,axiom,
% 0.30/0.61      ( ( abs_abs_complex @ zero_zero_complex )
% 0.30/0.61      = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0
% 0.30/0.61  thf(fact_6794_abs__0,axiom,
% 0.30/0.61      ( ( abs_abs_real @ zero_zero_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0
% 0.30/0.61  thf(fact_6795_abs__0,axiom,
% 0.30/0.61      ( ( abs_abs_rat @ zero_zero_rat )
% 0.30/0.61      = zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0
% 0.30/0.61  thf(fact_6796_abs__0,axiom,
% 0.30/0.61      ( ( abs_abs_int @ zero_zero_int )
% 0.30/0.61      = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_0
% 0.30/0.61  thf(fact_6797_abs__mult__self__eq,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.61        = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_mult_self_eq
% 0.30/0.61  thf(fact_6798_abs__mult__self__eq,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 0.30/0.61        = ( times_times_real @ A @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_mult_self_eq
% 0.30/0.61  thf(fact_6799_abs__mult__self__eq,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 0.30/0.61        = ( times_times_rat @ A @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_mult_self_eq
% 0.30/0.61  thf(fact_6800_abs__mult__self__eq,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 0.30/0.61        = ( times_times_int @ A @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_mult_self_eq
% 0.30/0.61  thf(fact_6801_abs__1,axiom,
% 0.30/0.61      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.30/0.61      = one_one_Code_integer ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_1
% 0.30/0.61  thf(fact_6802_abs__1,axiom,
% 0.30/0.61      ( ( abs_abs_complex @ one_one_complex )
% 0.30/0.61      = one_one_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_1
% 0.30/0.61  thf(fact_6803_abs__1,axiom,
% 0.30/0.61      ( ( abs_abs_real @ one_one_real )
% 0.30/0.61      = one_one_real ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_1
% 0.30/0.61  thf(fact_6804_abs__1,axiom,
% 0.30/0.61      ( ( abs_abs_rat @ one_one_rat )
% 0.30/0.61      = one_one_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_1
% 0.30/0.61  thf(fact_6805_abs__1,axiom,
% 0.30/0.61      ( ( abs_abs_int @ one_one_int )
% 0.30/0.61      = one_one_int ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_1
% 0.30/0.61  thf(fact_6806_abs__add__abs,axiom,
% 0.30/0.61      ! [A: code_integer,B: code_integer] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 0.30/0.61        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_add_abs
% 0.30/0.61  thf(fact_6807_abs__add__abs,axiom,
% 0.30/0.61      ! [A: real,B: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 0.30/0.61        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_add_abs
% 0.30/0.61  thf(fact_6808_abs__add__abs,axiom,
% 0.30/0.61      ! [A: rat,B: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 0.30/0.61        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_add_abs
% 0.30/0.61  thf(fact_6809_abs__add__abs,axiom,
% 0.30/0.61      ! [A: int,B: int] :
% 0.30/0.61        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 0.30/0.61        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_add_abs
% 0.30/0.61  thf(fact_6810_abs__divide,axiom,
% 0.30/0.61      ! [A: real,B: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.61        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_divide
% 0.30/0.61  thf(fact_6811_abs__divide,axiom,
% 0.30/0.61      ! [A: rat,B: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.61        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_divide
% 0.30/0.61  thf(fact_6812_abs__minus,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.61        = ( abs_abs_int @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus
% 0.30/0.61  thf(fact_6813_abs__minus,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.61        = ( abs_abs_real @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus
% 0.30/0.61  thf(fact_6814_abs__minus,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.61        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus
% 0.30/0.61  thf(fact_6815_abs__minus,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.61        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus
% 0.30/0.61  thf(fact_6816_abs__minus__cancel,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 0.30/0.61        = ( abs_abs_int @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus_cancel
% 0.30/0.61  thf(fact_6817_abs__minus__cancel,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 0.30/0.61        = ( abs_abs_real @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus_cancel
% 0.30/0.61  thf(fact_6818_abs__minus__cancel,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 0.30/0.61        = ( abs_abs_rat @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus_cancel
% 0.30/0.61  thf(fact_6819_abs__minus__cancel,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 0.30/0.61        = ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_minus_cancel
% 0.30/0.61  thf(fact_6820_dvd__abs__iff,axiom,
% 0.30/0.61      ! [M: int,K: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 0.30/0.61        = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_abs_iff
% 0.30/0.61  thf(fact_6821_dvd__abs__iff,axiom,
% 0.30/0.61      ! [M: real,K: real] :
% 0.30/0.61        ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 0.30/0.61        = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_abs_iff
% 0.30/0.61  thf(fact_6822_dvd__abs__iff,axiom,
% 0.30/0.61      ! [M: rat,K: rat] :
% 0.30/0.61        ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 0.30/0.61        = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_abs_iff
% 0.30/0.61  thf(fact_6823_dvd__abs__iff,axiom,
% 0.30/0.61      ! [M: code_integer,K: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 0.30/0.61        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % dvd_abs_iff
% 0.30/0.61  thf(fact_6824_abs__dvd__iff,axiom,
% 0.30/0.61      ! [M: int,K: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 0.30/0.61        = ( dvd_dvd_int @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_dvd_iff
% 0.30/0.61  thf(fact_6825_abs__dvd__iff,axiom,
% 0.30/0.61      ! [M: real,K: real] :
% 0.30/0.61        ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 0.30/0.61        = ( dvd_dvd_real @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_dvd_iff
% 0.30/0.61  thf(fact_6826_abs__dvd__iff,axiom,
% 0.30/0.61      ! [M: rat,K: rat] :
% 0.30/0.61        ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 0.30/0.61        = ( dvd_dvd_rat @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_dvd_iff
% 0.30/0.61  thf(fact_6827_abs__dvd__iff,axiom,
% 0.30/0.61      ! [M: code_integer,K: code_integer] :
% 0.30/0.61        ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 0.30/0.61        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_dvd_iff
% 0.30/0.61  thf(fact_6828_of__int__abs,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ring_1_of_int_int @ ( abs_abs_int @ X3 ) )
% 0.30/0.61        = ( abs_abs_int @ ( ring_1_of_int_int @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_abs
% 0.30/0.61  thf(fact_6829_of__int__abs,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X3 ) )
% 0.30/0.61        = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_abs
% 0.30/0.61  thf(fact_6830_of__int__abs,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ring_1_of_int_real @ ( abs_abs_int @ X3 ) )
% 0.30/0.61        = ( abs_abs_real @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_abs
% 0.30/0.61  thf(fact_6831_of__int__abs,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( ring_1_of_int_rat @ ( abs_abs_int @ X3 ) )
% 0.30/0.61        = ( abs_abs_rat @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_abs
% 0.30/0.61  thf(fact_6832_abs__bool__eq,axiom,
% 0.30/0.61      ! [P: $o] :
% 0.30/0.61        ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 0.30/0.61        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_bool_eq
% 0.30/0.61  thf(fact_6833_abs__bool__eq,axiom,
% 0.30/0.61      ! [P: $o] :
% 0.30/0.61        ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 0.30/0.61        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_bool_eq
% 0.30/0.61  thf(fact_6834_abs__bool__eq,axiom,
% 0.30/0.61      ! [P: $o] :
% 0.30/0.61        ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 0.30/0.61        = ( zero_n356916108424825756nteger @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_bool_eq
% 0.30/0.61  thf(fact_6835_abs__bool__eq,axiom,
% 0.30/0.61      ! [P: $o] :
% 0.30/0.61        ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.61        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_bool_eq
% 0.30/0.61  thf(fact_6836_zdvd1__eq,axiom,
% 0.30/0.61      ! [X3: int] :
% 0.30/0.61        ( ( dvd_dvd_int @ X3 @ one_one_int )
% 0.30/0.61        = ( ( abs_abs_int @ X3 )
% 0.30/0.61          = one_one_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zdvd1_eq
% 0.30/0.61  thf(fact_6837_tanh__0,axiom,
% 0.30/0.61      ( ( tanh_complex @ zero_zero_complex )
% 0.30/0.61      = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_0
% 0.30/0.61  thf(fact_6838_tanh__0,axiom,
% 0.30/0.61      ( ( tanh_real @ zero_zero_real )
% 0.30/0.61      = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_0
% 0.30/0.61  thf(fact_6839_tanh__real__less__iff,axiom,
% 0.30/0.61      ! [X3: real,Y3: real] :
% 0.30/0.61        ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 0.30/0.61        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_real_less_iff
% 0.30/0.61  thf(fact_6840_sum_Oneutral__const,axiom,
% 0.30/0.61      ! [A2: set_nat] :
% 0.30/0.61        ( ( groups3542108847815614940at_nat
% 0.30/0.61          @ ^ [Uu3: nat] : zero_zero_nat
% 0.30/0.61          @ A2 )
% 0.30/0.61        = zero_zero_nat ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.neutral_const
% 0.30/0.61  thf(fact_6841_sum_Oneutral__const,axiom,
% 0.30/0.61      ! [A2: set_complex] :
% 0.30/0.61        ( ( groups7754918857620584856omplex
% 0.30/0.61          @ ^ [Uu3: complex] : zero_zero_complex
% 0.30/0.61          @ A2 )
% 0.30/0.61        = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.neutral_const
% 0.30/0.61  thf(fact_6842_sum_Oneutral__const,axiom,
% 0.30/0.61      ! [A2: set_nat] :
% 0.30/0.61        ( ( groups6591440286371151544t_real
% 0.30/0.61          @ ^ [Uu3: nat] : zero_zero_real
% 0.30/0.61          @ A2 )
% 0.30/0.61        = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.neutral_const
% 0.30/0.61  thf(fact_6843_of__int__sum,axiom,
% 0.30/0.61      ! [F: complex > int,A2: set_complex] :
% 0.30/0.61        ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
% 0.30/0.61        = ( groups7754918857620584856omplex
% 0.30/0.61          @ ^ [X: complex] : ( ring_17405671764205052669omplex @ ( F @ X ) )
% 0.30/0.61          @ A2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_sum
% 0.30/0.61  thf(fact_6844_of__int__sum,axiom,
% 0.30/0.61      ! [F: nat > int,A2: set_nat] :
% 0.30/0.61        ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
% 0.30/0.61        = ( groups6591440286371151544t_real
% 0.30/0.61          @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
% 0.30/0.61          @ A2 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % of_int_sum
% 0.30/0.61  thf(fact_6845_abs__of__nonneg,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.61       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_of_nonneg
% 0.30/0.61  thf(fact_6846_abs__of__nonneg,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.61       => ( ( abs_abs_real @ A )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_of_nonneg
% 0.30/0.61  thf(fact_6847_abs__of__nonneg,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.61       => ( ( abs_abs_rat @ A )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_of_nonneg
% 0.30/0.61  thf(fact_6848_abs__of__nonneg,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.61       => ( ( abs_abs_int @ A )
% 0.30/0.61          = A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_of_nonneg
% 0.30/0.61  thf(fact_6849_abs__le__self__iff,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 0.30/0.61        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_self_iff
% 0.30/0.61  thf(fact_6850_abs__le__self__iff,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 0.30/0.61        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_self_iff
% 0.30/0.61  thf(fact_6851_abs__le__self__iff,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 0.30/0.61        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_self_iff
% 0.30/0.61  thf(fact_6852_abs__le__self__iff,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 0.30/0.61        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_self_iff
% 0.30/0.61  thf(fact_6853_abs__le__zero__iff,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 0.30/0.61        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_zero_iff
% 0.30/0.61  thf(fact_6854_abs__le__zero__iff,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 0.30/0.61        = ( A = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_zero_iff
% 0.30/0.61  thf(fact_6855_abs__le__zero__iff,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 0.30/0.61        = ( A = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_zero_iff
% 0.30/0.61  thf(fact_6856_abs__le__zero__iff,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 0.30/0.61        = ( A = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % abs_le_zero_iff
% 0.30/0.61  thf(fact_6857_zero__less__abs__iff,axiom,
% 0.30/0.61      ! [A: code_integer] :
% 0.30/0.61        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 0.30/0.61        = ( A != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_abs_iff
% 0.30/0.61  thf(fact_6858_zero__less__abs__iff,axiom,
% 0.30/0.61      ! [A: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 0.30/0.61        = ( A != zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_abs_iff
% 0.30/0.61  thf(fact_6859_zero__less__abs__iff,axiom,
% 0.30/0.61      ! [A: rat] :
% 0.30/0.61        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 0.30/0.61        = ( A != zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_abs_iff
% 0.30/0.61  thf(fact_6860_zero__less__abs__iff,axiom,
% 0.30/0.61      ! [A: int] :
% 0.30/0.61        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 0.30/0.61        = ( A != zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zero_less_abs_iff
% 0.30/0.61  thf(fact_6861_sum_Oempty,axiom,
% 0.30/0.61      ! [G: real > complex] :
% 0.30/0.61        ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
% 0.30/0.61        = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6862_sum_Oempty,axiom,
% 0.30/0.61      ! [G: real > real] :
% 0.30/0.61        ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
% 0.30/0.61        = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6863_sum_Oempty,axiom,
% 0.30/0.61      ! [G: real > rat] :
% 0.30/0.61        ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
% 0.30/0.61        = zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6864_sum_Oempty,axiom,
% 0.30/0.61      ! [G: real > nat] :
% 0.30/0.61        ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
% 0.30/0.61        = zero_zero_nat ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6865_sum_Oempty,axiom,
% 0.30/0.61      ! [G: real > int] :
% 0.30/0.61        ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6866_sum_Oempty,axiom,
% 0.30/0.61      ! [G: nat > complex] :
% 0.30/0.61        ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
% 0.30/0.61        = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6867_sum_Oempty,axiom,
% 0.30/0.61      ! [G: nat > rat] :
% 0.30/0.61        ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
% 0.30/0.61        = zero_zero_rat ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6868_sum_Oempty,axiom,
% 0.30/0.61      ! [G: nat > int] :
% 0.30/0.61        ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 0.30/0.61        = zero_zero_int ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6869_sum_Oempty,axiom,
% 0.30/0.61      ! [G: int > complex] :
% 0.30/0.61        ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
% 0.30/0.61        = zero_zero_complex ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6870_sum_Oempty,axiom,
% 0.30/0.61      ! [G: int > real] :
% 0.30/0.61        ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
% 0.30/0.61        = zero_zero_real ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.empty
% 0.30/0.61  thf(fact_6871_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_nat,G: nat > complex] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.61          = zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6872_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_int,G: int > complex] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.30/0.61          = zero_zero_complex ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6873_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_int,G: int > real] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.30/0.61          = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6874_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_complex,G: complex > real] :
% 0.30/0.61        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.61          = zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6875_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_nat,G: nat > rat] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.30/0.61          = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6876_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_int,G: int > rat] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.30/0.61          = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6877_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_complex,G: complex > rat] :
% 0.30/0.61        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.61          = zero_zero_rat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6878_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_int,G: int > nat] :
% 0.30/0.61        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.61       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.30/0.61          = zero_zero_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6879_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_complex,G: complex > nat] :
% 0.30/0.61        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.61       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.61          = zero_zero_nat ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6880_sum_Oinfinite,axiom,
% 0.30/0.61      ! [A2: set_nat,G: nat > int] :
% 0.30/0.61        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.61       => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.30/0.61          = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.infinite
% 0.30/0.61  thf(fact_6881_sum__eq__0__iff,axiom,
% 0.30/0.61      ! [F3: set_int,F: int > nat] :
% 0.30/0.61        ( ( finite_finite_int @ F3 )
% 0.30/0.61       => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
% 0.30/0.61            = zero_zero_nat )
% 0.30/0.61          = ( ! [X: int] :
% 0.30/0.61                ( ( member_int @ X @ F3 )
% 0.30/0.61               => ( ( F @ X )
% 0.30/0.61                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum_eq_0_iff
% 0.30/0.61  thf(fact_6882_sum__eq__0__iff,axiom,
% 0.30/0.61      ! [F3: set_complex,F: complex > nat] :
% 0.30/0.61        ( ( finite3207457112153483333omplex @ F3 )
% 0.30/0.61       => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
% 0.30/0.61            = zero_zero_nat )
% 0.30/0.61          = ( ! [X: complex] :
% 0.30/0.61                ( ( member_complex @ X @ F3 )
% 0.30/0.61               => ( ( F @ X )
% 0.30/0.61                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum_eq_0_iff
% 0.30/0.61  thf(fact_6883_sum__eq__0__iff,axiom,
% 0.30/0.61      ! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.30/0.61        ( ( finite6177210948735845034at_nat @ F3 )
% 0.30/0.61       => ( ( ( groups977919841031483927at_nat @ F @ F3 )
% 0.30/0.61            = zero_zero_nat )
% 0.30/0.61          = ( ! [X: product_prod_nat_nat] :
% 0.30/0.61                ( ( member8440522571783428010at_nat @ X @ F3 )
% 0.30/0.61               => ( ( F @ X )
% 0.30/0.61                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum_eq_0_iff
% 0.30/0.61  thf(fact_6884_sum__eq__0__iff,axiom,
% 0.30/0.61      ! [F3: set_nat,F: nat > nat] :
% 0.30/0.61        ( ( finite_finite_nat @ F3 )
% 0.30/0.61       => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
% 0.30/0.61            = zero_zero_nat )
% 0.30/0.61          = ( ! [X: nat] :
% 0.30/0.61                ( ( member_nat @ X @ F3 )
% 0.30/0.61               => ( ( F @ X )
% 0.30/0.61                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum_eq_0_iff
% 0.30/0.61  thf(fact_6885_artanh__minus__real,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.61       => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
% 0.30/0.61          = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % artanh_minus_real
% 0.30/0.61  thf(fact_6886_zabs__less__one__iff,axiom,
% 0.30/0.61      ! [Z2: int] :
% 0.30/0.61        ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 0.30/0.61        = ( Z2 = zero_zero_int ) ) ).
% 0.30/0.61  
% 0.30/0.61  % zabs_less_one_iff
% 0.30/0.61  thf(fact_6887_tanh__real__pos__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 0.30/0.61        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_real_pos_iff
% 0.30/0.61  thf(fact_6888_tanh__real__neg__iff,axiom,
% 0.30/0.61      ! [X3: real] :
% 0.30/0.61        ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 0.30/0.61        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.61  
% 0.30/0.61  % tanh_real_neg_iff
% 0.30/0.61  thf(fact_6889_sum_Odelta,axiom,
% 0.30/0.61      ! [S3: set_real,A: real,B: real > complex] :
% 0.30/0.61        ( ( finite_finite_real @ S3 )
% 0.30/0.61       => ( ( ( member_real @ A @ S3 )
% 0.30/0.61           => ( ( groups5754745047067104278omplex
% 0.30/0.61                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = ( B @ A ) ) )
% 0.30/0.61          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.61           => ( ( groups5754745047067104278omplex
% 0.30/0.61                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.delta
% 0.30/0.61  thf(fact_6890_sum_Odelta,axiom,
% 0.30/0.61      ! [S3: set_nat,A: nat,B: nat > complex] :
% 0.30/0.61        ( ( finite_finite_nat @ S3 )
% 0.30/0.61       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.61           => ( ( groups2073611262835488442omplex
% 0.30/0.61                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = ( B @ A ) ) )
% 0.30/0.61          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.61           => ( ( groups2073611262835488442omplex
% 0.30/0.61                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.delta
% 0.30/0.61  thf(fact_6891_sum_Odelta,axiom,
% 0.30/0.61      ! [S3: set_int,A: int,B: int > complex] :
% 0.30/0.61        ( ( finite_finite_int @ S3 )
% 0.30/0.61       => ( ( ( member_int @ A @ S3 )
% 0.30/0.61           => ( ( groups3049146728041665814omplex
% 0.30/0.61                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = ( B @ A ) ) )
% 0.30/0.61          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.61           => ( ( groups3049146728041665814omplex
% 0.30/0.61                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.61                @ S3 )
% 0.30/0.61              = zero_zero_complex ) ) ) ) ).
% 0.30/0.61  
% 0.30/0.61  % sum.delta
% 0.30/0.61  thf(fact_6892_sum_Odelta,axiom,
% 0.30/0.61      ! [S3: set_real,A: real,B: real > real] :
% 0.30/0.61        ( ( finite_finite_real @ S3 )
% 0.30/0.62       => ( ( ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real
% 0.30/0.62                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real
% 0.30/0.62                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6893_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_int,A: int,B: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ S3 )
% 0.30/0.62       => ( ( ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real
% 0.30/0.62                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real
% 0.30/0.62                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6894_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_complex,A: complex,B: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real
% 0.30/0.62                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real
% 0.30/0.62                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6895_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_real,A: real,B: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ S3 )
% 0.30/0.62       => ( ( ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat
% 0.30/0.62                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat
% 0.30/0.62                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6896_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_nat,A: nat,B: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ S3 )
% 0.30/0.62       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat
% 0.30/0.62                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat
% 0.30/0.62                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6897_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_int,A: int,B: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ S3 )
% 0.30/0.62       => ( ( ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat
% 0.30/0.62                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat
% 0.30/0.62                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6898_sum_Odelta,axiom,
% 0.30/0.62      ! [S3: set_complex,A: complex,B: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat
% 0.30/0.62                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat
% 0.30/0.62                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta
% 0.30/0.62  thf(fact_6899_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_real,A: real,B: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ S3 )
% 0.30/0.62       => ( ( ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups5754745047067104278omplex
% 0.30/0.62                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups5754745047067104278omplex
% 0.30/0.62                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_complex ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6900_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_nat,A: nat,B: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ S3 )
% 0.30/0.62       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2073611262835488442omplex
% 0.30/0.62                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2073611262835488442omplex
% 0.30/0.62                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_complex ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6901_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_int,A: int,B: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ S3 )
% 0.30/0.62       => ( ( ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3049146728041665814omplex
% 0.30/0.62                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3049146728041665814omplex
% 0.30/0.62                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_complex ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6902_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_real,A: real,B: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ S3 )
% 0.30/0.62       => ( ( ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real
% 0.30/0.62                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real
% 0.30/0.62                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6903_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_int,A: int,B: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ S3 )
% 0.30/0.62       => ( ( ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real
% 0.30/0.62                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real
% 0.30/0.62                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6904_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_complex,A: complex,B: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real
% 0.30/0.62                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real
% 0.30/0.62                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_real ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6905_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_real,A: real,B: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ S3 )
% 0.30/0.62       => ( ( ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat
% 0.30/0.62                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat
% 0.30/0.62                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6906_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_nat,A: nat,B: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ S3 )
% 0.30/0.62       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat
% 0.30/0.62                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat
% 0.30/0.62                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6907_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_int,A: int,B: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ S3 )
% 0.30/0.62       => ( ( ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat
% 0.30/0.62                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat
% 0.30/0.62                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6908_sum_Odelta_H,axiom,
% 0.30/0.62      ! [S3: set_complex,A: complex,B: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat
% 0.30/0.62                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = ( B @ A ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat
% 0.30/0.62                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 0.30/0.62                @ S3 )
% 0.30/0.62              = zero_zero_rat ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.delta'
% 0.30/0.62  thf(fact_6909_sum__abs,axiom,
% 0.30/0.62      ! [F: nat > real,A2: set_nat] :
% 0.30/0.62        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 0.30/0.62        @ ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
% 0.30/0.62          @ A2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs
% 0.30/0.62  thf(fact_6910_divide__le__0__abs__iff,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 0.30/0.62        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.62          | ( B = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divide_le_0_abs_iff
% 0.30/0.62  thf(fact_6911_divide__le__0__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 0.30/0.62        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.62          | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % divide_le_0_abs_iff
% 0.30/0.62  thf(fact_6912_zero__le__divide__abs__iff,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 0.30/0.62        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62          | ( B = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_divide_abs_iff
% 0.30/0.62  thf(fact_6913_zero__le__divide__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 0.30/0.62        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62          | ( B = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_divide_abs_iff
% 0.30/0.62  thf(fact_6914_abs__of__nonpos,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.62       => ( ( abs_abs_real @ A )
% 0.30/0.62          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_nonpos
% 0.30/0.62  thf(fact_6915_abs__of__nonpos,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.62       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_nonpos
% 0.30/0.62  thf(fact_6916_abs__of__nonpos,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 0.30/0.62       => ( ( abs_abs_rat @ A )
% 0.30/0.62          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_nonpos
% 0.30/0.62  thf(fact_6917_abs__of__nonpos,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 0.30/0.62       => ( ( abs_abs_int @ A )
% 0.30/0.62          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_nonpos
% 0.30/0.62  thf(fact_6918_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6919_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6920_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6921_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6922_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6923_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6924_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6925_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6926_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6927_sum_Oinsert,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert
% 0.30/0.62  thf(fact_6928_sum__abs__ge__zero,axiom,
% 0.30/0.62      ! [F: nat > real,A2: set_nat] :
% 0.30/0.62        ( ord_less_eq_real @ zero_zero_real
% 0.30/0.62        @ ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
% 0.30/0.62          @ A2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_abs_ge_zero
% 0.30/0.62  thf(fact_6929_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: code_integer,N2: nat] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
% 0.30/0.62        = ( ( A != zero_z3403309356797280102nteger )
% 0.30/0.62          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6930_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: real,N2: nat] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 0.30/0.62        = ( ( A != zero_zero_real )
% 0.30/0.62          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6931_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: rat,N2: nat] :
% 0.30/0.62        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
% 0.30/0.62        = ( ( A != zero_zero_rat )
% 0.30/0.62          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6932_zero__less__power__abs__iff,axiom,
% 0.30/0.62      ! [A: int,N2: nat] :
% 0.30/0.62        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
% 0.30/0.62        = ( ( A != zero_zero_int )
% 0.30/0.62          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_less_power_abs_iff
% 0.30/0.62  thf(fact_6933_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N2: nat,M: nat,G: nat > complex] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = zero_zero_complex ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6934_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N2: nat,M: nat,G: nat > rat] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = zero_zero_rat ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6935_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N2: nat,M: nat,G: nat > int] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = zero_zero_int ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6936_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N2: nat,M: nat,G: nat > nat] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = zero_zero_nat ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6937_sum_Ocl__ivl__Suc,axiom,
% 0.30/0.62      ! [N2: nat,M: nat,G: nat > real] :
% 0.30/0.62        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = zero_zero_real ) )
% 0.30/0.62        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.62         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.62            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.cl_ivl_Suc
% 0.30/0.62  thf(fact_6938_sum__zero__power,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > complex] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2073611262835488442omplex
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( C @ zero_zero_nat ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2073611262835488442omplex
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_complex ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power
% 0.30/0.62  thf(fact_6939_sum__zero__power,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > rat] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( C @ zero_zero_nat ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power
% 0.30/0.62  thf(fact_6940_sum__zero__power,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > real] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( C @ zero_zero_nat ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power
% 0.30/0.62  thf(fact_6941_sum__zero__power_H,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2073611262835488442omplex
% 0.30/0.62              @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2073611262835488442omplex
% 0.30/0.62              @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_complex ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power'
% 0.30/0.62  thf(fact_6942_sum__zero__power_H,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat
% 0.30/0.62              @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups2906978787729119204at_rat
% 0.30/0.62              @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power'
% 0.30/0.62  thf(fact_6943_sum__zero__power_H,axiom,
% 0.30/0.62      ! [A2: set_nat,C: nat > real,D: nat > real] :
% 0.30/0.62        ( ( ( ( finite_finite_nat @ A2 )
% 0.30/0.62            & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 0.30/0.62        & ( ~ ( ( finite_finite_nat @ A2 )
% 0.30/0.62              & ( member_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.62         => ( ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_zero_power'
% 0.30/0.62  thf(fact_6944_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: real > complex,A2: set_real] :
% 0.30/0.62        ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.30/0.62         != zero_zero_complex )
% 0.30/0.62       => ~ ! [A4: real] :
% 0.30/0.62              ( ( member_real @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_complex ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6945_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: nat > complex,A2: set_nat] :
% 0.30/0.62        ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.62         != zero_zero_complex )
% 0.30/0.62       => ~ ! [A4: nat] :
% 0.30/0.62              ( ( member_nat @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_complex ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6946_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: int > complex,A2: set_int] :
% 0.30/0.62        ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 0.30/0.62         != zero_zero_complex )
% 0.30/0.62       => ~ ! [A4: int] :
% 0.30/0.62              ( ( member_int @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_complex ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6947_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: real > real,A2: set_real] :
% 0.30/0.62        ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.62         != zero_zero_real )
% 0.30/0.62       => ~ ! [A4: real] :
% 0.30/0.62              ( ( member_real @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6948_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: int > real,A2: set_int] :
% 0.30/0.62        ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.30/0.62         != zero_zero_real )
% 0.30/0.62       => ~ ! [A4: int] :
% 0.30/0.62              ( ( member_int @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_real ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6949_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: real > rat,A2: set_real] :
% 0.30/0.62        ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.62         != zero_zero_rat )
% 0.30/0.62       => ~ ! [A4: real] :
% 0.30/0.62              ( ( member_real @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6950_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: nat > rat,A2: set_nat] :
% 0.30/0.62        ( ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.30/0.62         != zero_zero_rat )
% 0.30/0.62       => ~ ! [A4: nat] :
% 0.30/0.62              ( ( member_nat @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6951_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: int > rat,A2: set_int] :
% 0.30/0.62        ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.30/0.62         != zero_zero_rat )
% 0.30/0.62       => ~ ! [A4: int] :
% 0.30/0.62              ( ( member_int @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_rat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6952_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: real > nat,A2: set_real] :
% 0.30/0.62        ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.62         != zero_zero_nat )
% 0.30/0.62       => ~ ! [A4: real] :
% 0.30/0.62              ( ( member_real @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6953_sum_Onot__neutral__contains__not__neutral,axiom,
% 0.30/0.62      ! [G: int > nat,A2: set_int] :
% 0.30/0.62        ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.30/0.62         != zero_zero_nat )
% 0.30/0.62       => ~ ! [A4: int] :
% 0.30/0.62              ( ( member_int @ A4 @ A2 )
% 0.30/0.62             => ( ( G @ A4 )
% 0.30/0.62                = zero_zero_nat ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.not_neutral_contains_not_neutral
% 0.30/0.62  thf(fact_6954_sum_Oneutral,axiom,
% 0.30/0.62      ! [A2: set_nat,G: nat > nat] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ( G @ X4 )
% 0.30/0.62              = zero_zero_nat ) )
% 0.30/0.62       => ( ( groups3542108847815614940at_nat @ G @ A2 )
% 0.30/0.62          = zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.neutral
% 0.30/0.62  thf(fact_6955_sum_Oneutral,axiom,
% 0.30/0.62      ! [A2: set_complex,G: complex > complex] :
% 0.30/0.62        ( ! [X4: complex] :
% 0.30/0.62            ( ( member_complex @ X4 @ A2 )
% 0.30/0.62           => ( ( G @ X4 )
% 0.30/0.62              = zero_zero_complex ) )
% 0.30/0.62       => ( ( groups7754918857620584856omplex @ G @ A2 )
% 0.30/0.62          = zero_zero_complex ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.neutral
% 0.30/0.62  thf(fact_6956_sum_Oneutral,axiom,
% 0.30/0.62      ! [A2: set_nat,G: nat > real] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ( G @ X4 )
% 0.30/0.62              = zero_zero_real ) )
% 0.30/0.62       => ( ( groups6591440286371151544t_real @ G @ A2 )
% 0.30/0.62          = zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.neutral
% 0.30/0.62  thf(fact_6957_abs__le__D1,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_real @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D1
% 0.30/0.62  thf(fact_6958_abs__le__D1,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D1
% 0.30/0.62  thf(fact_6959_abs__le__D1,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_rat @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D1
% 0.30/0.62  thf(fact_6960_abs__le__D1,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_int @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D1
% 0.30/0.62  thf(fact_6961_abs__ge__self,axiom,
% 0.30/0.62      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_self
% 0.30/0.62  thf(fact_6962_abs__ge__self,axiom,
% 0.30/0.62      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_self
% 0.30/0.62  thf(fact_6963_abs__ge__self,axiom,
% 0.30/0.62      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_self
% 0.30/0.62  thf(fact_6964_abs__ge__self,axiom,
% 0.30/0.62      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_self
% 0.30/0.62  thf(fact_6965_abs__eq__0__iff,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.62          = zero_z3403309356797280102nteger )
% 0.30/0.62        = ( A = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_0_iff
% 0.30/0.62  thf(fact_6966_abs__eq__0__iff,axiom,
% 0.30/0.62      ! [A: complex] :
% 0.30/0.62        ( ( ( abs_abs_complex @ A )
% 0.30/0.62          = zero_zero_complex )
% 0.30/0.62        = ( A = zero_zero_complex ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_0_iff
% 0.30/0.62  thf(fact_6967_abs__eq__0__iff,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ( abs_abs_real @ A )
% 0.30/0.62          = zero_zero_real )
% 0.30/0.62        = ( A = zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_0_iff
% 0.30/0.62  thf(fact_6968_abs__eq__0__iff,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ( abs_abs_rat @ A )
% 0.30/0.62          = zero_zero_rat )
% 0.30/0.62        = ( A = zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_0_iff
% 0.30/0.62  thf(fact_6969_abs__eq__0__iff,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ( abs_abs_int @ A )
% 0.30/0.62          = zero_zero_int )
% 0.30/0.62        = ( A = zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_0_iff
% 0.30/0.62  thf(fact_6970_abs__mult,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.62        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult
% 0.30/0.62  thf(fact_6971_abs__mult,axiom,
% 0.30/0.62      ! [A: complex,B: complex] :
% 0.30/0.62        ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
% 0.30/0.62        = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult
% 0.30/0.62  thf(fact_6972_abs__mult,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.30/0.62        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult
% 0.30/0.62  thf(fact_6973_abs__mult,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.62        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult
% 0.30/0.62  thf(fact_6974_abs__mult,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.30/0.62        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult
% 0.30/0.62  thf(fact_6975_abs__one,axiom,
% 0.30/0.62      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 0.30/0.62      = one_one_Code_integer ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_one
% 0.30/0.62  thf(fact_6976_abs__one,axiom,
% 0.30/0.62      ( ( abs_abs_real @ one_one_real )
% 0.30/0.62      = one_one_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_one
% 0.30/0.62  thf(fact_6977_abs__one,axiom,
% 0.30/0.62      ( ( abs_abs_rat @ one_one_rat )
% 0.30/0.62      = one_one_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_one
% 0.30/0.62  thf(fact_6978_abs__one,axiom,
% 0.30/0.62      ( ( abs_abs_int @ one_one_int )
% 0.30/0.62      = one_one_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_one
% 0.30/0.62  thf(fact_6979_abs__minus__commute,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 0.30/0.62        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_commute
% 0.30/0.62  thf(fact_6980_abs__minus__commute,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 0.30/0.62        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_commute
% 0.30/0.62  thf(fact_6981_abs__minus__commute,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 0.30/0.62        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_commute
% 0.30/0.62  thf(fact_6982_abs__minus__commute,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 0.30/0.62        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_commute
% 0.30/0.62  thf(fact_6983_abs__eq__iff,axiom,
% 0.30/0.62      ! [X3: int,Y3: int] :
% 0.30/0.62        ( ( ( abs_abs_int @ X3 )
% 0.30/0.62          = ( abs_abs_int @ Y3 ) )
% 0.30/0.62        = ( ( X3 = Y3 )
% 0.30/0.62          | ( X3
% 0.30/0.62            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff
% 0.30/0.62  thf(fact_6984_abs__eq__iff,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ( abs_abs_real @ X3 )
% 0.30/0.62          = ( abs_abs_real @ Y3 ) )
% 0.30/0.62        = ( ( X3 = Y3 )
% 0.30/0.62          | ( X3
% 0.30/0.62            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff
% 0.30/0.62  thf(fact_6985_abs__eq__iff,axiom,
% 0.30/0.62      ! [X3: rat,Y3: rat] :
% 0.30/0.62        ( ( ( abs_abs_rat @ X3 )
% 0.30/0.62          = ( abs_abs_rat @ Y3 ) )
% 0.30/0.62        = ( ( X3 = Y3 )
% 0.30/0.62          | ( X3
% 0.30/0.62            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff
% 0.30/0.62  thf(fact_6986_abs__eq__iff,axiom,
% 0.30/0.62      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.62        ( ( ( abs_abs_Code_integer @ X3 )
% 0.30/0.62          = ( abs_abs_Code_integer @ Y3 ) )
% 0.30/0.62        = ( ( X3 = Y3 )
% 0.30/0.62          | ( X3
% 0.30/0.62            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff
% 0.30/0.62  thf(fact_6987_dvd__if__abs__eq,axiom,
% 0.30/0.62      ! [L: int,K: int] :
% 0.30/0.62        ( ( ( abs_abs_int @ L )
% 0.30/0.62          = ( abs_abs_int @ K ) )
% 0.30/0.62       => ( dvd_dvd_int @ L @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_if_abs_eq
% 0.30/0.62  thf(fact_6988_dvd__if__abs__eq,axiom,
% 0.30/0.62      ! [L: real,K: real] :
% 0.30/0.62        ( ( ( abs_abs_real @ L )
% 0.30/0.62          = ( abs_abs_real @ K ) )
% 0.30/0.62       => ( dvd_dvd_real @ L @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_if_abs_eq
% 0.30/0.62  thf(fact_6989_dvd__if__abs__eq,axiom,
% 0.30/0.62      ! [L: rat,K: rat] :
% 0.30/0.62        ( ( ( abs_abs_rat @ L )
% 0.30/0.62          = ( abs_abs_rat @ K ) )
% 0.30/0.62       => ( dvd_dvd_rat @ L @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_if_abs_eq
% 0.30/0.62  thf(fact_6990_dvd__if__abs__eq,axiom,
% 0.30/0.62      ! [L: code_integer,K: code_integer] :
% 0.30/0.62        ( ( ( abs_abs_Code_integer @ L )
% 0.30/0.62          = ( abs_abs_Code_integer @ K ) )
% 0.30/0.62       => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_if_abs_eq
% 0.30/0.62  thf(fact_6991_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: real,C: int > int > set_real,P4: product_prod_int_int] :
% 0.30/0.62        ( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6992_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 0.30/0.62        ( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6993_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: int,C: int > int > set_int,P4: product_prod_int_int] :
% 0.30/0.62        ( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6994_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: set_nat,C: int > int > set_set_nat,P4: product_prod_int_int] :
% 0.30/0.62        ( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_set_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6995_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
% 0.30/0.62        ( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6996_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 0.30/0.62        ( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6997_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 0.30/0.62        ( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6998_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 0.30/0.62        ( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_6999_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 0.30/0.62        ( ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_7000_mem__case__prodE,axiom,
% 0.30/0.62      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 0.30/0.62        ( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 0.30/0.62       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % mem_case_prodE
% 0.30/0.62  thf(fact_7001_zdvd__antisym__abs,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( dvd_dvd_int @ A @ B )
% 0.30/0.62       => ( ( dvd_dvd_int @ B @ A )
% 0.30/0.62         => ( ( abs_abs_int @ A )
% 0.30/0.62            = ( abs_abs_int @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zdvd_antisym_abs
% 0.30/0.62  thf(fact_7002_case__prodD,axiom,
% 0.30/0.62      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
% 0.30/0.62       => ( F @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD
% 0.30/0.62  thf(fact_7003_case__prodD,axiom,
% 0.30/0.62      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 0.30/0.62        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
% 0.30/0.62       => ( F @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD
% 0.30/0.62  thf(fact_7004_case__prodD,axiom,
% 0.30/0.62      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.62        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
% 0.30/0.62       => ( F @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD
% 0.30/0.62  thf(fact_7005_case__prodD,axiom,
% 0.30/0.62      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 0.30/0.62        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
% 0.30/0.62       => ( F @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD
% 0.30/0.62  thf(fact_7006_case__prodD,axiom,
% 0.30/0.62      ! [F: int > int > $o,A: int,B: int] :
% 0.30/0.62        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.62       => ( F @ A @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD
% 0.30/0.62  thf(fact_7007_case__prodE,axiom,
% 0.30/0.62      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 0.30/0.62        ( ( produc127349428274296955eger_o @ C @ P4 )
% 0.30/0.62       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE
% 0.30/0.62  thf(fact_7008_case__prodE,axiom,
% 0.30/0.62      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 0.30/0.62        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 0.30/0.62       => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE
% 0.30/0.62  thf(fact_7009_case__prodE,axiom,
% 0.30/0.62      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 0.30/0.62        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 0.30/0.62       => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE
% 0.30/0.62  thf(fact_7010_case__prodE,axiom,
% 0.30/0.62      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 0.30/0.62        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 0.30/0.62       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE
% 0.30/0.62  thf(fact_7011_case__prodE,axiom,
% 0.30/0.62      ! [C: int > int > $o,P4: product_prod_int_int] :
% 0.30/0.62        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 0.30/0.62       => ~ ! [X4: int,Y5: int] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE
% 0.30/0.62  thf(fact_7012_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_real,F: real > rat,G: real > rat] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7013_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7014_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_int,F: int > rat,G: int > rat] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7015_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_real,F: real > nat,G: real > nat] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7016_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_int,F: int > nat,G: int > nat] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7017_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_real,F: real > int,G: real > int] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7018_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_nat,F: nat > int,G: nat > int] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7019_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_int,F: int > int,G: int > int] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7020_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_nat,F: nat > nat,G: nat > nat] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7021_sum__mono,axiom,
% 0.30/0.62      ! [K5: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ K5 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K5 ) @ ( groups6591440286371151544t_real @ G @ K5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono
% 0.30/0.62  thf(fact_7022_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 0.30/0.62        ( ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.62          @ A2 )
% 0.30/0.62        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_7023_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 0.30/0.62        ( ( groups7754918857620584856omplex
% 0.30/0.62          @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.62          @ A2 )
% 0.30/0.62        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_7024_sum_Odistrib,axiom,
% 0.30/0.62      ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
% 0.30/0.62          @ A2 )
% 0.30/0.62        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.distrib
% 0.30/0.62  thf(fact_7025_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups1935376822645274424al_nat
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( groups3542108847815614940at_nat @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups3542108847815614940at_nat
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups1935376822645274424al_nat
% 0.30/0.62                  @ ^ [X: real] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_real
% 0.30/0.62                    @ ^ [X: real] :
% 0.30/0.62                        ( ( member_real @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7026_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups4541462559716669496nt_nat
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( groups3542108847815614940at_nat @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups3542108847815614940at_nat
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups4541462559716669496nt_nat
% 0.30/0.62                  @ ^ [X: int] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_int
% 0.30/0.62                    @ ^ [X: int] :
% 0.30/0.62                        ( ( member_int @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7027_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups5693394587270226106ex_nat
% 0.30/0.62              @ ^ [X: complex] :
% 0.30/0.62                  ( groups3542108847815614940at_nat @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups3542108847815614940at_nat
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups5693394587270226106ex_nat
% 0.30/0.62                  @ ^ [X: complex] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_complex
% 0.30/0.62                    @ ^ [X: complex] :
% 0.30/0.62                        ( ( member_complex @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7028_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_real,B2: set_complex,G: real > complex > complex,R: real > complex > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.62         => ( ( groups5754745047067104278omplex
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( groups7754918857620584856omplex @ ( G @ X )
% 0.30/0.62                  @ ( collect_complex
% 0.30/0.62                    @ ^ [Y: complex] :
% 0.30/0.62                        ( ( member_complex @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups7754918857620584856omplex
% 0.30/0.62              @ ^ [Y: complex] :
% 0.30/0.62                  ( groups5754745047067104278omplex
% 0.30/0.62                  @ ^ [X: real] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_real
% 0.30/0.62                    @ ^ [X: real] :
% 0.30/0.62                        ( ( member_real @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7029_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_nat,B2: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.62         => ( ( groups2073611262835488442omplex
% 0.30/0.62              @ ^ [X: nat] :
% 0.30/0.62                  ( groups7754918857620584856omplex @ ( G @ X )
% 0.30/0.62                  @ ( collect_complex
% 0.30/0.62                    @ ^ [Y: complex] :
% 0.30/0.62                        ( ( member_complex @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups7754918857620584856omplex
% 0.30/0.62              @ ^ [Y: complex] :
% 0.30/0.62                  ( groups2073611262835488442omplex
% 0.30/0.62                  @ ^ [X: nat] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [X: nat] :
% 0.30/0.62                        ( ( member_nat @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7030_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_int,B2: set_complex,G: int > complex > complex,R: int > complex > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.62         => ( ( groups3049146728041665814omplex
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( groups7754918857620584856omplex @ ( G @ X )
% 0.30/0.62                  @ ( collect_complex
% 0.30/0.62                    @ ^ [Y: complex] :
% 0.30/0.62                        ( ( member_complex @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups7754918857620584856omplex
% 0.30/0.62              @ ^ [Y: complex] :
% 0.30/0.62                  ( groups3049146728041665814omplex
% 0.30/0.62                  @ ^ [X: int] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_int
% 0.30/0.62                    @ ^ [X: int] :
% 0.30/0.62                        ( ( member_int @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7031_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_real,B2: set_nat,G: real > nat > real,R: real > nat > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups8097168146408367636l_real
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( groups6591440286371151544t_real @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups8097168146408367636l_real
% 0.30/0.62                  @ ^ [X: real] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_real
% 0.30/0.62                    @ ^ [X: real] :
% 0.30/0.62                        ( ( member_real @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7032_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups8778361861064173332t_real
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( groups6591440286371151544t_real @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups8778361861064173332t_real
% 0.30/0.62                  @ ^ [X: int] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_int
% 0.30/0.62                    @ ^ [X: int] :
% 0.30/0.62                        ( ( member_int @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7033_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( finite_finite_nat @ B2 )
% 0.30/0.62         => ( ( groups5808333547571424918x_real
% 0.30/0.62              @ ^ [X: complex] :
% 0.30/0.62                  ( groups6591440286371151544t_real @ ( G @ X )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [Y: nat] :
% 0.30/0.62                        ( ( member_nat @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups6591440286371151544t_real
% 0.30/0.62              @ ^ [Y: nat] :
% 0.30/0.62                  ( groups5808333547571424918x_real
% 0.30/0.62                  @ ^ [X: complex] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_complex
% 0.30/0.62                    @ ^ [X: complex] :
% 0.30/0.62                        ( ( member_complex @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7034_sum_Oswap__restrict,axiom,
% 0.30/0.62      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( finite_finite_real @ B2 )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat
% 0.30/0.62              @ ^ [X: nat] :
% 0.30/0.62                  ( groups1935376822645274424al_nat @ ( G @ X )
% 0.30/0.62                  @ ( collect_real
% 0.30/0.62                    @ ^ [Y: real] :
% 0.30/0.62                        ( ( member_real @ Y @ B2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ A2 )
% 0.30/0.62            = ( groups1935376822645274424al_nat
% 0.30/0.62              @ ^ [Y: real] :
% 0.30/0.62                  ( groups3542108847815614940at_nat
% 0.30/0.62                  @ ^ [X: nat] : ( G @ X @ Y )
% 0.30/0.62                  @ ( collect_nat
% 0.30/0.62                    @ ^ [X: nat] :
% 0.30/0.62                        ( ( member_nat @ X @ A2 )
% 0.30/0.62                        & ( R @ X @ Y ) ) ) )
% 0.30/0.62              @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.swap_restrict
% 0.30/0.62  thf(fact_7035_case__prodD_H,axiom,
% 0.30/0.62      ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 0.30/0.62        ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 0.30/0.62       => ( R @ A @ B @ C ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodD'
% 0.30/0.62  thf(fact_7036_case__prodE_H,axiom,
% 0.30/0.62      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 0.30/0.62        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z2 )
% 0.30/0.62       => ~ ! [X4: nat,Y5: nat] :
% 0.30/0.62              ( ( P4
% 0.30/0.62                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 0.30/0.62             => ~ ( C @ X4 @ Y5 @ Z2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % case_prodE'
% 0.30/0.62  thf(fact_7037_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > real] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7038_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > real] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7039_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > rat] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7040_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > rat] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7041_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > rat] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7042_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > nat] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7043_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7044_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > int] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7045_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > int] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7046_sum__nonneg,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > int] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg
% 0.30/0.62  thf(fact_7047_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > real] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7048_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > real] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 0.30/0.62       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7049_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > rat] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7050_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > rat] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7051_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > rat] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 0.30/0.62       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7052_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > nat] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7053_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 0.30/0.62       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7054_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > int] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7055_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > int] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7056_sum__nonpos,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > int] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 0.30/0.62       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonpos
% 0.30/0.62  thf(fact_7057_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
% 0.30/0.62        ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 0.30/0.62          = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_real @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_real @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7058_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
% 0.30/0.62        ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 0.30/0.62          = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: nat] :
% 0.30/0.62              ( ( member_nat @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_nat @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_nat @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7059_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
% 0.30/0.62        ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 0.30/0.62          = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_int @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_int @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7060_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
% 0.30/0.62        ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 0.30/0.62          = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_complex @ I @ I5 )
% 0.30/0.62           => ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7061_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
% 0.30/0.62        ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 0.30/0.62          = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_real @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_real @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7062_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
% 0.30/0.62        ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 0.30/0.62          = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_int @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_int @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7063_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
% 0.30/0.62        ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 0.30/0.62          = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_complex @ I @ I5 )
% 0.30/0.62           => ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7064_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: real > int,I5: set_real,G: real > int,I: real] :
% 0.30/0.62        ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 0.30/0.62          = ( groups1932886352136224148al_int @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_real @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_real @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7065_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
% 0.30/0.62        ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 0.30/0.62          = ( groups3539618377306564664at_int @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: nat] :
% 0.30/0.62              ( ( member_nat @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_nat @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_nat @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7066_sum__mono__inv,axiom,
% 0.30/0.62      ! [F: int > int,I5: set_int,G: int > int,I: int] :
% 0.30/0.62        ( ( ( groups4538972089207619220nt_int @ F @ I5 )
% 0.30/0.62          = ( groups4538972089207619220nt_int @ G @ I5 ) )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ I5 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 0.30/0.62         => ( ( member_int @ I @ I5 )
% 0.30/0.62           => ( ( finite_finite_int @ I5 )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = ( G @ I ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_mono_inv
% 0.30/0.62  thf(fact_7067_abs__ge__zero,axiom,
% 0.30/0.62      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_zero
% 0.30/0.62  thf(fact_7068_abs__ge__zero,axiom,
% 0.30/0.62      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_zero
% 0.30/0.62  thf(fact_7069_abs__ge__zero,axiom,
% 0.30/0.62      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_zero
% 0.30/0.62  thf(fact_7070_abs__ge__zero,axiom,
% 0.30/0.62      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_zero
% 0.30/0.62  thf(fact_7071_abs__not__less__zero,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_not_less_zero
% 0.30/0.62  thf(fact_7072_abs__not__less__zero,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_not_less_zero
% 0.30/0.62  thf(fact_7073_abs__not__less__zero,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_not_less_zero
% 0.30/0.62  thf(fact_7074_abs__not__less__zero,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_not_less_zero
% 0.30/0.62  thf(fact_7075_abs__of__pos,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62          = A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_pos
% 0.30/0.62  thf(fact_7076_abs__of__pos,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.62       => ( ( abs_abs_real @ A )
% 0.30/0.62          = A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_pos
% 0.30/0.62  thf(fact_7077_abs__of__pos,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_rat @ zero_zero_rat @ A )
% 0.30/0.62       => ( ( abs_abs_rat @ A )
% 0.30/0.62          = A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_pos
% 0.30/0.62  thf(fact_7078_abs__of__pos,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.62       => ( ( abs_abs_int @ A )
% 0.30/0.62          = A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_pos
% 0.30/0.62  thf(fact_7079_abs__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq
% 0.30/0.62  thf(fact_7080_abs__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq
% 0.30/0.62  thf(fact_7081_abs__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq
% 0.30/0.62  thf(fact_7082_abs__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq
% 0.30/0.62  thf(fact_7083_abs__mult__less,axiom,
% 0.30/0.62      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 0.30/0.62       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 0.30/0.62         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_less
% 0.30/0.62  thf(fact_7084_abs__mult__less,axiom,
% 0.30/0.62      ! [A: real,C: real,B: real,D: real] :
% 0.30/0.62        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 0.30/0.62       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 0.30/0.62         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_less
% 0.30/0.62  thf(fact_7085_abs__mult__less,axiom,
% 0.30/0.62      ! [A: rat,C: rat,B: rat,D: rat] :
% 0.30/0.62        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 0.30/0.62       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 0.30/0.62         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_less
% 0.30/0.62  thf(fact_7086_abs__mult__less,axiom,
% 0.30/0.62      ! [A: int,C: int,B: int,D: int] :
% 0.30/0.62        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 0.30/0.62       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 0.30/0.62         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_less
% 0.30/0.62  thf(fact_7087_abs__triangle__ineq2,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2
% 0.30/0.62  thf(fact_7088_abs__triangle__ineq2,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2
% 0.30/0.62  thf(fact_7089_abs__triangle__ineq2,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2
% 0.30/0.62  thf(fact_7090_abs__triangle__ineq2,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2
% 0.30/0.62  thf(fact_7091_abs__triangle__ineq3,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq3
% 0.30/0.62  thf(fact_7092_abs__triangle__ineq3,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq3
% 0.30/0.62  thf(fact_7093_abs__triangle__ineq3,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq3
% 0.30/0.62  thf(fact_7094_abs__triangle__ineq3,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq3
% 0.30/0.62  thf(fact_7095_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2_sym
% 0.30/0.62  thf(fact_7096_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2_sym
% 0.30/0.62  thf(fact_7097_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2_sym
% 0.30/0.62  thf(fact_7098_abs__triangle__ineq2__sym,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq2_sym
% 0.30/0.62  thf(fact_7099_nonzero__abs__divide,axiom,
% 0.30/0.62      ! [B: real,A: real] :
% 0.30/0.62        ( ( B != zero_zero_real )
% 0.30/0.62       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.62          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % nonzero_abs_divide
% 0.30/0.62  thf(fact_7100_nonzero__abs__divide,axiom,
% 0.30/0.62      ! [B: rat,A: rat] :
% 0.30/0.62        ( ( B != zero_zero_rat )
% 0.30/0.62       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 0.30/0.62          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % nonzero_abs_divide
% 0.30/0.62  thf(fact_7101_abs__ge__minus__self,axiom,
% 0.30/0.62      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_minus_self
% 0.30/0.62  thf(fact_7102_abs__ge__minus__self,axiom,
% 0.30/0.62      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_minus_self
% 0.30/0.62  thf(fact_7103_abs__ge__minus__self,axiom,
% 0.30/0.62      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_minus_self
% 0.30/0.62  thf(fact_7104_abs__ge__minus__self,axiom,
% 0.30/0.62      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_ge_minus_self
% 0.30/0.62  thf(fact_7105_abs__le__iff,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_eq_real @ A @ B )
% 0.30/0.62          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_iff
% 0.30/0.62  thf(fact_7106_abs__le__iff,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.62          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_iff
% 0.30/0.62  thf(fact_7107_abs__le__iff,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.62          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_iff
% 0.30/0.62  thf(fact_7108_abs__le__iff,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_eq_int @ A @ B )
% 0.30/0.62          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_iff
% 0.30/0.62  thf(fact_7109_abs__le__D2,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D2
% 0.30/0.62  thf(fact_7110_abs__le__D2,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D2
% 0.30/0.62  thf(fact_7111_abs__le__D2,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D2
% 0.30/0.62  thf(fact_7112_abs__le__D2,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_le_D2
% 0.30/0.62  thf(fact_7113_abs__leI,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ A @ B )
% 0.30/0.62       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 0.30/0.62         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_leI
% 0.30/0.62  thf(fact_7114_abs__leI,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ A @ B )
% 0.30/0.62       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 0.30/0.62         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_leI
% 0.30/0.62  thf(fact_7115_abs__leI,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ A @ B )
% 0.30/0.62       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 0.30/0.62         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_leI
% 0.30/0.62  thf(fact_7116_abs__leI,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ A @ B )
% 0.30/0.62       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.30/0.62         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_leI
% 0.30/0.62  thf(fact_7117_abs__less__iff,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_int @ A @ B )
% 0.30/0.62          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_less_iff
% 0.30/0.62  thf(fact_7118_abs__less__iff,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_real @ A @ B )
% 0.30/0.62          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_less_iff
% 0.30/0.62  thf(fact_7119_abs__less__iff,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 0.30/0.62        = ( ( ord_less_rat @ A @ B )
% 0.30/0.62          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_less_iff
% 0.30/0.62  thf(fact_7120_abs__less__iff,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 0.30/0.62        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 0.30/0.62          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_less_iff
% 0.30/0.62  thf(fact_7121_sum__cong__Suc,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 0.30/0.62        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.30/0.62       => ( ! [X4: nat] :
% 0.30/0.62              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.30/0.62             => ( ( F @ ( suc @ X4 ) )
% 0.30/0.62                = ( G @ ( suc @ X4 ) ) ) )
% 0.30/0.62         => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62            = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_cong_Suc
% 0.30/0.62  thf(fact_7122_sum__cong__Suc,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 0.30/0.62        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 0.30/0.62       => ( ! [X4: nat] :
% 0.30/0.62              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 0.30/0.62             => ( ( F @ ( suc @ X4 ) )
% 0.30/0.62                = ( G @ ( suc @ X4 ) ) ) )
% 0.30/0.62         => ( ( groups6591440286371151544t_real @ F @ A2 )
% 0.30/0.62            = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_cong_Suc
% 0.30/0.62  thf(fact_7123_Collect__case__prod__mono,axiom,
% 0.30/0.62      ! [A2: int > int > $o,B2: int > int > $o] :
% 0.30/0.62        ( ( ord_le6741204236512500942_int_o @ A2 @ B2 )
% 0.30/0.62       => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % Collect_case_prod_mono
% 0.30/0.62  thf(fact_7124_abs__zmult__eq__1,axiom,
% 0.30/0.62      ! [M: int,N2: int] :
% 0.30/0.62        ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
% 0.30/0.62          = one_one_int )
% 0.30/0.62       => ( ( abs_abs_int @ M )
% 0.30/0.62          = one_one_int ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_zmult_eq_1
% 0.30/0.62  thf(fact_7125_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > complex,P: real > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups5754745047067104278omplex @ G
% 0.30/0.62            @ ( collect_real
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups5754745047067104278omplex
% 0.30/0.62            @ ^ [X: real] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7126_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( groups2073611262835488442omplex @ G
% 0.30/0.62            @ ( collect_nat
% 0.30/0.62              @ ^ [X: nat] :
% 0.30/0.62                  ( ( member_nat @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups2073611262835488442omplex
% 0.30/0.62            @ ^ [X: nat] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7127_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > complex,P: int > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups3049146728041665814omplex @ G
% 0.30/0.62            @ ( collect_int
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups3049146728041665814omplex
% 0.30/0.62            @ ^ [X: int] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7128_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > real,P: real > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups8097168146408367636l_real @ G
% 0.30/0.62            @ ( collect_real
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups8097168146408367636l_real
% 0.30/0.62            @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7129_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > real,P: int > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups8778361861064173332t_real @ G
% 0.30/0.62            @ ( collect_int
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups8778361861064173332t_real
% 0.30/0.62            @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7130_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( groups5808333547571424918x_real @ G
% 0.30/0.62            @ ( collect_complex
% 0.30/0.62              @ ^ [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups5808333547571424918x_real
% 0.30/0.62            @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7131_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > rat,P: real > $o] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups1300246762558778688al_rat @ G
% 0.30/0.62            @ ( collect_real
% 0.30/0.62              @ ^ [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups1300246762558778688al_rat
% 0.30/0.62            @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7132_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( groups2906978787729119204at_rat @ G
% 0.30/0.62            @ ( collect_nat
% 0.30/0.62              @ ^ [X: nat] :
% 0.30/0.62                  ( ( member_nat @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups2906978787729119204at_rat
% 0.30/0.62            @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7133_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > rat,P: int > $o] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups3906332499630173760nt_rat @ G
% 0.30/0.62            @ ( collect_int
% 0.30/0.62              @ ^ [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups3906332499630173760nt_rat
% 0.30/0.62            @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7134_sum_Ointer__filter,axiom,
% 0.30/0.62      ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( groups5058264527183730370ex_rat @ G
% 0.30/0.62            @ ( collect_complex
% 0.30/0.62              @ ^ [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A2 )
% 0.30/0.62                  & ( P @ X ) ) ) )
% 0.30/0.62          = ( groups5058264527183730370ex_rat
% 0.30/0.62            @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 0.30/0.62            @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.inter_filter
% 0.30/0.62  thf(fact_7135_sum__subtractf__nat,axiom,
% 0.30/0.62      ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 0.30/0.62        ( ! [X4: product_prod_nat_nat] :
% 0.30/0.62            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ( groups977919841031483927at_nat
% 0.30/0.62            @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62            @ A2 )
% 0.30/0.62          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf_nat
% 0.30/0.62  thf(fact_7136_sum__subtractf__nat,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > nat,F: real > nat] :
% 0.30/0.62        ( ! [X4: real] :
% 0.30/0.62            ( ( member_real @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ( groups1935376822645274424al_nat
% 0.30/0.62            @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62            @ A2 )
% 0.30/0.62          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf_nat
% 0.30/0.62  thf(fact_7137_sum__subtractf__nat,axiom,
% 0.30/0.62      ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 0.30/0.62        ( ! [X4: set_nat] :
% 0.30/0.62            ( ( member_set_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ( groups8294997508430121362at_nat
% 0.30/0.62            @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62            @ A2 )
% 0.30/0.62          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf_nat
% 0.30/0.62  thf(fact_7138_sum__subtractf__nat,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > nat,F: int > nat] :
% 0.30/0.62        ( ! [X4: int] :
% 0.30/0.62            ( ( member_int @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ( groups4541462559716669496nt_nat
% 0.30/0.62            @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62            @ A2 )
% 0.30/0.62          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf_nat
% 0.30/0.62  thf(fact_7139_sum__subtractf__nat,axiom,
% 0.30/0.62      ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 0.30/0.62        ( ! [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 0.30/0.62       => ( ( groups3542108847815614940at_nat
% 0.30/0.62            @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 0.30/0.62            @ A2 )
% 0.30/0.62          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_subtractf_nat
% 0.30/0.62  thf(fact_7140_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ I5 )
% 0.30/0.62           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
% 0.30/0.62            = one_one_Code_integer )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_le3102999989581377725nteger
% 0.30/0.62              @ ( abs_abs_Code_integer
% 0.30/0.62                @ ( minus_8373710615458151222nteger
% 0.30/0.62                  @ ( groups7713935264441627589nteger
% 0.30/0.62                    @ ^ [I2: real] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7141_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ I5 )
% 0.30/0.62           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
% 0.30/0.62            = one_one_Code_integer )
% 0.30/0.62         => ( ! [I3: nat] :
% 0.30/0.62                ( ( member_nat @ I3 @ I5 )
% 0.30/0.62               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_le3102999989581377725nteger
% 0.30/0.62              @ ( abs_abs_Code_integer
% 0.30/0.62                @ ( minus_8373710615458151222nteger
% 0.30/0.62                  @ ( groups7501900531339628137nteger
% 0.30/0.62                    @ ^ [I2: nat] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7142_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ I5 )
% 0.30/0.62           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
% 0.30/0.62            = one_one_Code_integer )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_le3102999989581377725nteger
% 0.30/0.62              @ ( abs_abs_Code_integer
% 0.30/0.62                @ ( minus_8373710615458151222nteger
% 0.30/0.62                  @ ( groups7873554091576472773nteger
% 0.30/0.62                    @ ^ [I2: int] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7143_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_real,X3: real > real,A: real > real,B: real,Delta: real] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
% 0.30/0.62            = one_one_real )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_real
% 0.30/0.62              @ ( abs_abs_real
% 0.30/0.62                @ ( minus_minus_real
% 0.30/0.62                  @ ( groups8097168146408367636l_real
% 0.30/0.62                    @ ^ [I2: real] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7144_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_int,X3: int > real,A: int > real,B: real,Delta: real] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
% 0.30/0.62            = one_one_real )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_real
% 0.30/0.62              @ ( abs_abs_real
% 0.30/0.62                @ ( minus_minus_real
% 0.30/0.62                  @ ( groups8778361861064173332t_real
% 0.30/0.62                    @ ^ [I2: int] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7145_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_real,X3: real > rat,A: real > rat,B: rat,Delta: rat] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
% 0.30/0.62            = one_one_rat )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_rat
% 0.30/0.62              @ ( abs_abs_rat
% 0.30/0.62                @ ( minus_minus_rat
% 0.30/0.62                  @ ( groups1300246762558778688al_rat
% 0.30/0.62                    @ ^ [I2: real] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7146_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_nat,X3: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
% 0.30/0.62            = one_one_rat )
% 0.30/0.62         => ( ! [I3: nat] :
% 0.30/0.62                ( ( member_nat @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_rat
% 0.30/0.62              @ ( abs_abs_rat
% 0.30/0.62                @ ( minus_minus_rat
% 0.30/0.62                  @ ( groups2906978787729119204at_rat
% 0.30/0.62                    @ ^ [I2: nat] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7147_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_int,X3: int > rat,A: int > rat,B: rat,Delta: rat] :
% 0.30/0.62        ( ! [I3: int] :
% 0.30/0.62            ( ( member_int @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
% 0.30/0.62            = one_one_rat )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_rat
% 0.30/0.62              @ ( abs_abs_rat
% 0.30/0.62                @ ( minus_minus_rat
% 0.30/0.62                  @ ( groups3906332499630173760nt_rat
% 0.30/0.62                    @ ^ [I2: int] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7148_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_real,X3: real > int,A: real > int,B: int,Delta: int] :
% 0.30/0.62        ( ! [I3: real] :
% 0.30/0.62            ( ( member_real @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups1932886352136224148al_int @ X3 @ I5 )
% 0.30/0.62            = one_one_int )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_int
% 0.30/0.62              @ ( abs_abs_int
% 0.30/0.62                @ ( minus_minus_int
% 0.30/0.62                  @ ( groups1932886352136224148al_int
% 0.30/0.62                    @ ^ [I2: real] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7149_convex__sum__bound__le,axiom,
% 0.30/0.62      ! [I5: set_nat,X3: nat > int,A: nat > int,B: int,Delta: int] :
% 0.30/0.62        ( ! [I3: nat] :
% 0.30/0.62            ( ( member_nat @ I3 @ I5 )
% 0.30/0.62           => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
% 0.30/0.62       => ( ( ( groups3539618377306564664at_int @ X3 @ I5 )
% 0.30/0.62            = one_one_int )
% 0.30/0.62         => ( ! [I3: nat] :
% 0.30/0.62                ( ( member_nat @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 0.30/0.62           => ( ord_less_eq_int
% 0.30/0.62              @ ( abs_abs_int
% 0.30/0.62                @ ( minus_minus_int
% 0.30/0.62                  @ ( groups3539618377306564664at_int
% 0.30/0.62                    @ ^ [I2: nat] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 0.30/0.62                    @ I5 )
% 0.30/0.62                  @ B ) )
% 0.30/0.62              @ Delta ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % convex_sum_bound_le
% 0.30/0.62  thf(fact_7150_infinite__int__iff__unbounded__le,axiom,
% 0.30/0.62      ! [S3: set_int] :
% 0.30/0.62        ( ( ~ ( finite_finite_int @ S3 ) )
% 0.30/0.62        = ( ! [M2: int] :
% 0.30/0.62            ? [N: int] :
% 0.30/0.62              ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N ) )
% 0.30/0.62              & ( member_int @ N @ S3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % infinite_int_iff_unbounded_le
% 0.30/0.62  thf(fact_7151_infinite__int__iff__unbounded,axiom,
% 0.30/0.62      ! [S3: set_int] :
% 0.30/0.62        ( ( ~ ( finite_finite_int @ S3 ) )
% 0.30/0.62        = ( ! [M2: int] :
% 0.30/0.62            ? [N: int] :
% 0.30/0.62              ( ( ord_less_int @ M2 @ ( abs_abs_int @ N ) )
% 0.30/0.62              & ( member_int @ N @ S3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % infinite_int_iff_unbounded
% 0.30/0.62  thf(fact_7152_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.62      ! [G: nat > nat,M: nat,N2: nat] :
% 0.30/0.62        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.shift_bounds_cl_Suc_ivl
% 0.30/0.62  thf(fact_7153_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 0.30/0.62      ! [G: nat > real,M: nat,N2: nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.shift_bounds_cl_Suc_ivl
% 0.30/0.62  thf(fact_7154_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.62      ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 0.30/0.62        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.shift_bounds_cl_nat_ivl
% 0.30/0.62  thf(fact_7155_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 0.30/0.62      ! [G: nat > real,M: nat,K: nat,N2: nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.shift_bounds_cl_nat_ivl
% 0.30/0.62  thf(fact_7156_tanh__real__lt__1,axiom,
% 0.30/0.62      ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).
% 0.30/0.62  
% 0.30/0.62  % tanh_real_lt_1
% 0.30/0.62  thf(fact_7157_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ( finite_finite_int @ T )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S )
% 0.30/0.62                 => ? [Xa: int] :
% 0.30/0.62                      ( ( member_int @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7158_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S )
% 0.30/0.62                 => ? [Xa: complex] :
% 0.30/0.62                      ( ( member_complex @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7159_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ( finite_finite_int @ T )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S )
% 0.30/0.62                 => ? [Xa: int] :
% 0.30/0.62                      ( ( member_int @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7160_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S )
% 0.30/0.62                 => ? [Xa: complex] :
% 0.30/0.62                      ( ( member_complex @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7161_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ S )
% 0.30/0.62       => ( ( finite_finite_nat @ T )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S )
% 0.30/0.62                 => ? [Xa: nat] :
% 0.30/0.62                      ( ( member_nat @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7162_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ S )
% 0.30/0.62       => ( ( finite_finite_int @ T )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S )
% 0.30/0.62                 => ? [Xa: int] :
% 0.30/0.62                      ( ( member_int @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7163_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ S )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S )
% 0.30/0.62                 => ? [Xa: complex] :
% 0.30/0.62                      ( ( member_complex @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7164_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ( finite_finite_nat @ T )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S )
% 0.30/0.62                 => ? [Xa: nat] :
% 0.30/0.62                      ( ( member_nat @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7165_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ( finite_finite_int @ T )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S )
% 0.30/0.62                 => ? [Xa: int] :
% 0.30/0.62                      ( ( member_int @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7166_sum__le__included,axiom,
% 0.30/0.62      ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ T )
% 0.30/0.62               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S )
% 0.30/0.62                 => ? [Xa: complex] :
% 0.30/0.62                      ( ( member_complex @ Xa @ T )
% 0.30/0.62                      & ( ( I @ Xa )
% 0.30/0.62                        = X4 )
% 0.30/0.62                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 0.30/0.62             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_le_included
% 0.30/0.62  thf(fact_7167_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ! [X4: real] :
% 0.30/0.62              ( ( member_real @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7168_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7169_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7170_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ! [X4: real] :
% 0.30/0.62              ( ( member_real @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7171_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ! [X4: nat] :
% 0.30/0.62              ( ( member_nat @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: nat] :
% 0.30/0.62                  ( ( member_nat @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7172_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7173_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7174_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ! [X4: real] :
% 0.30/0.62              ( ( member_real @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: real] :
% 0.30/0.62                  ( ( member_real @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7175_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: int] :
% 0.30/0.62                  ( ( member_int @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7176_sum__nonneg__eq__0__iff,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.62         => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62            = ( ! [X: complex] :
% 0.30/0.62                  ( ( member_complex @ X @ A2 )
% 0.30/0.62                 => ( ( F @ X )
% 0.30/0.62                    = zero_zero_nat ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_eq_0_iff
% 0.30/0.62  thf(fact_7177_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > real,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: int] :
% 0.30/0.62                ( ( member_int @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7178_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: complex] :
% 0.30/0.62                ( ( member_complex @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7179_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ! [X4: nat] :
% 0.30/0.62              ( ( member_nat @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: nat] :
% 0.30/0.62                ( ( member_nat @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7180_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: int] :
% 0.30/0.62                ( ( member_int @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7181_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: complex] :
% 0.30/0.62                ( ( member_complex @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7182_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: int] :
% 0.30/0.62                ( ( member_int @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7183_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: complex] :
% 0.30/0.62                ( ( member_complex @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7184_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ! [X4: nat] :
% 0.30/0.62              ( ( member_nat @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: nat] :
% 0.30/0.62                ( ( member_nat @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7185_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > int,G: int > int] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ! [X4: int] :
% 0.30/0.62              ( ( member_int @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: int] :
% 0.30/0.62                ( ( member_int @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7186_sum__strict__mono__ex1,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > int,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ! [X4: complex] :
% 0.30/0.62              ( ( member_complex @ X4 @ A2 )
% 0.30/0.62             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62         => ( ? [X2: complex] :
% 0.30/0.62                ( ( member_complex @ X2 @ A2 )
% 0.30/0.62                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 0.30/0.62           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono_ex1
% 0.30/0.62  thf(fact_7187_sum_Orelated,axiom,
% 0.30/0.62      ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.62        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.62       => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S3 )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7188_sum_Orelated,axiom,
% 0.30/0.62      ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
% 0.30/0.62        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 0.30/0.62       => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S3 )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7189_sum_Orelated,axiom,
% 0.30/0.62      ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
% 0.30/0.62        ( ( R @ zero_zero_real @ zero_zero_real )
% 0.30/0.62       => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S3 )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7190_sum_Orelated,axiom,
% 0.30/0.62      ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.62        ( ( R @ zero_zero_real @ zero_zero_real )
% 0.30/0.62       => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7191_sum_Orelated,axiom,
% 0.30/0.62      ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S3 )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7192_sum_Orelated,axiom,
% 0.30/0.62      ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
% 0.30/0.62        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S3 )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7193_sum_Orelated,axiom,
% 0.30/0.62      ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 0.30/0.62       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7194_sum_Orelated,axiom,
% 0.30/0.62      ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
% 0.30/0.62        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.62       => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_int @ S3 )
% 0.30/0.62           => ( ! [X4: int] :
% 0.30/0.62                  ( ( member_int @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7195_sum_Orelated,axiom,
% 0.30/0.62      ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 0.30/0.62       => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7196_sum_Orelated,axiom,
% 0.30/0.62      ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
% 0.30/0.62        ( ( R @ zero_zero_int @ zero_zero_int )
% 0.30/0.62       => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
% 0.30/0.62              ( ( ( R @ X16 @ X23 )
% 0.30/0.62                & ( R @ Y15 @ Y23 ) )
% 0.30/0.62             => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 0.30/0.62         => ( ( finite_finite_nat @ S3 )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S3 )
% 0.30/0.62                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62             => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.related
% 0.30/0.62  thf(fact_7197_dense__eq0__I,axiom,
% 0.30/0.62      ! [X3: real] :
% 0.30/0.62        ( ! [E: real] :
% 0.30/0.62            ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.62           => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E ) )
% 0.30/0.62       => ( X3 = zero_zero_real ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dense_eq0_I
% 0.30/0.62  thf(fact_7198_dense__eq0__I,axiom,
% 0.30/0.62      ! [X3: rat] :
% 0.30/0.62        ( ! [E: rat] :
% 0.30/0.62            ( ( ord_less_rat @ zero_zero_rat @ E )
% 0.30/0.62           => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E ) )
% 0.30/0.62       => ( X3 = zero_zero_rat ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dense_eq0_I
% 0.30/0.62  thf(fact_7199_abs__mult__pos,axiom,
% 0.30/0.62      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 0.30/0.62       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
% 0.30/0.62          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_pos
% 0.30/0.62  thf(fact_7200_abs__mult__pos,axiom,
% 0.30/0.62      ! [X3: real,Y3: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.62       => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
% 0.30/0.62          = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_pos
% 0.30/0.62  thf(fact_7201_abs__mult__pos,axiom,
% 0.30/0.62      ! [X3: rat,Y3: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.62       => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
% 0.30/0.62          = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_pos
% 0.30/0.62  thf(fact_7202_abs__mult__pos,axiom,
% 0.30/0.62      ! [X3: int,Y3: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.62       => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
% 0.30/0.62          = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mult_pos
% 0.30/0.62  thf(fact_7203_abs__eq__mult,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 0.30/0.62          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.62            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 0.30/0.62       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 0.30/0.62          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_mult
% 0.30/0.62  thf(fact_7204_abs__eq__mult,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 0.30/0.62          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.62            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 0.30/0.62       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 0.30/0.62          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_mult
% 0.30/0.62  thf(fact_7205_abs__eq__mult,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 0.30/0.62          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.62            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 0.30/0.62       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 0.30/0.62          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_mult
% 0.30/0.62  thf(fact_7206_abs__eq__mult,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.62            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 0.30/0.62          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.62            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 0.30/0.62       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 0.30/0.62          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_mult
% 0.30/0.62  thf(fact_7207_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7208_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7209_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > real,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7210_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7211_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > rat,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7212_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_nat )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7213_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > rat,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7214_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7215_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_real,F: real > nat,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_real )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7216_sum__strict__mono,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( A2 != bot_bot_set_int )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ A2 )
% 0.30/0.62               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 0.30/0.62           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_strict_mono
% 0.30/0.62  thf(fact_7217_abs__minus__le__zero,axiom,
% 0.30/0.62      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_le_zero
% 0.30/0.62  thf(fact_7218_abs__minus__le__zero,axiom,
% 0.30/0.62      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_le_zero
% 0.30/0.62  thf(fact_7219_abs__minus__le__zero,axiom,
% 0.30/0.62      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_le_zero
% 0.30/0.62  thf(fact_7220_abs__minus__le__zero,axiom,
% 0.30/0.62      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_minus_le_zero
% 0.30/0.62  thf(fact_7221_abs__eq__iff_H,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( ( abs_abs_real @ A )
% 0.30/0.62          = B )
% 0.30/0.62        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.62          & ( ( A = B )
% 0.30/0.62            | ( A
% 0.30/0.62              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff'
% 0.30/0.62  thf(fact_7222_abs__eq__iff_H,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( ( abs_abs_Code_integer @ A )
% 0.30/0.62          = B )
% 0.30/0.62        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 0.30/0.62          & ( ( A = B )
% 0.30/0.62            | ( A
% 0.30/0.62              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff'
% 0.30/0.62  thf(fact_7223_abs__eq__iff_H,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( ( abs_abs_rat @ A )
% 0.30/0.62          = B )
% 0.30/0.62        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.62          & ( ( A = B )
% 0.30/0.62            | ( A
% 0.30/0.62              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff'
% 0.30/0.62  thf(fact_7224_abs__eq__iff_H,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( ( abs_abs_int @ A )
% 0.30/0.62          = B )
% 0.30/0.62        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 0.30/0.62          & ( ( A = B )
% 0.30/0.62            | ( A
% 0.30/0.62              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_eq_iff'
% 0.30/0.62  thf(fact_7225_eq__abs__iff_H,axiom,
% 0.30/0.62      ! [A: real,B: real] :
% 0.30/0.62        ( ( A
% 0.30/0.62          = ( abs_abs_real @ B ) )
% 0.30/0.62        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.62          & ( ( B = A )
% 0.30/0.62            | ( B
% 0.30/0.62              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eq_abs_iff'
% 0.30/0.62  thf(fact_7226_eq__abs__iff_H,axiom,
% 0.30/0.62      ! [A: code_integer,B: code_integer] :
% 0.30/0.62        ( ( A
% 0.30/0.62          = ( abs_abs_Code_integer @ B ) )
% 0.30/0.62        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 0.30/0.62          & ( ( B = A )
% 0.30/0.62            | ( B
% 0.30/0.62              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eq_abs_iff'
% 0.30/0.62  thf(fact_7227_eq__abs__iff_H,axiom,
% 0.30/0.62      ! [A: rat,B: rat] :
% 0.30/0.62        ( ( A
% 0.30/0.62          = ( abs_abs_rat @ B ) )
% 0.30/0.62        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.62          & ( ( B = A )
% 0.30/0.62            | ( B
% 0.30/0.62              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eq_abs_iff'
% 0.30/0.62  thf(fact_7228_eq__abs__iff_H,axiom,
% 0.30/0.62      ! [A: int,B: int] :
% 0.30/0.62        ( ( A
% 0.30/0.62          = ( abs_abs_int @ B ) )
% 0.30/0.62        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 0.30/0.62          & ( ( B = A )
% 0.30/0.62            | ( B
% 0.30/0.62              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % eq_abs_iff'
% 0.30/0.62  thf(fact_7229_zero__le__power__abs,axiom,
% 0.30/0.62      ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_power_abs
% 0.30/0.62  thf(fact_7230_zero__le__power__abs,axiom,
% 0.30/0.62      ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_power_abs
% 0.30/0.62  thf(fact_7231_zero__le__power__abs,axiom,
% 0.30/0.62      ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_power_abs
% 0.30/0.62  thf(fact_7232_zero__le__power__abs,axiom,
% 0.30/0.62      ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zero_le_power_abs
% 0.30/0.62  thf(fact_7233_abs__div__pos,axiom,
% 0.30/0.62      ! [Y3: real,X3: real] :
% 0.30/0.62        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.62       => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
% 0.30/0.62          = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_div_pos
% 0.30/0.62  thf(fact_7234_abs__div__pos,axiom,
% 0.30/0.62      ! [Y3: rat,X3: rat] :
% 0.30/0.62        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 0.30/0.62       => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
% 0.30/0.62          = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_div_pos
% 0.30/0.62  thf(fact_7235_abs__if__raw,axiom,
% 0.30/0.62      ( abs_abs_int
% 0.30/0.62      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if_raw
% 0.30/0.62  thf(fact_7236_abs__if__raw,axiom,
% 0.30/0.62      ( abs_abs_real
% 0.30/0.62      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if_raw
% 0.30/0.62  thf(fact_7237_abs__if__raw,axiom,
% 0.30/0.62      ( abs_abs_rat
% 0.30/0.62      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if_raw
% 0.30/0.62  thf(fact_7238_abs__if__raw,axiom,
% 0.30/0.62      ( abs_abs_Code_integer
% 0.30/0.62      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if_raw
% 0.30/0.62  thf(fact_7239_abs__of__neg,axiom,
% 0.30/0.62      ! [A: int] :
% 0.30/0.62        ( ( ord_less_int @ A @ zero_zero_int )
% 0.30/0.62       => ( ( abs_abs_int @ A )
% 0.30/0.62          = ( uminus_uminus_int @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_neg
% 0.30/0.62  thf(fact_7240_abs__of__neg,axiom,
% 0.30/0.62      ! [A: real] :
% 0.30/0.62        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.62       => ( ( abs_abs_real @ A )
% 0.30/0.62          = ( uminus_uminus_real @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_neg
% 0.30/0.62  thf(fact_7241_abs__of__neg,axiom,
% 0.30/0.62      ! [A: rat] :
% 0.30/0.62        ( ( ord_less_rat @ A @ zero_zero_rat )
% 0.30/0.62       => ( ( abs_abs_rat @ A )
% 0.30/0.62          = ( uminus_uminus_rat @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_neg
% 0.30/0.62  thf(fact_7242_abs__of__neg,axiom,
% 0.30/0.62      ! [A: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 0.30/0.62       => ( ( abs_abs_Code_integer @ A )
% 0.30/0.62          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_of_neg
% 0.30/0.62  thf(fact_7243_abs__if,axiom,
% 0.30/0.62      ( abs_abs_int
% 0.30/0.62      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if
% 0.30/0.62  thf(fact_7244_abs__if,axiom,
% 0.30/0.62      ( abs_abs_real
% 0.30/0.62      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if
% 0.30/0.62  thf(fact_7245_abs__if,axiom,
% 0.30/0.62      ( abs_abs_rat
% 0.30/0.62      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if
% 0.30/0.62  thf(fact_7246_abs__if,axiom,
% 0.30/0.62      ( abs_abs_Code_integer
% 0.30/0.62      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_if
% 0.30/0.62  thf(fact_7247_abs__diff__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_triangle_ineq
% 0.30/0.62  thf(fact_7248_abs__diff__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_triangle_ineq
% 0.30/0.62  thf(fact_7249_abs__diff__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_triangle_ineq
% 0.30/0.62  thf(fact_7250_abs__diff__triangle__ineq,axiom,
% 0.30/0.62      ! [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 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_triangle_ineq
% 0.30/0.62  thf(fact_7251_abs__triangle__ineq4,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq4
% 0.30/0.62  thf(fact_7252_abs__triangle__ineq4,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq4
% 0.30/0.62  thf(fact_7253_abs__triangle__ineq4,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq4
% 0.30/0.62  thf(fact_7254_abs__triangle__ineq4,axiom,
% 0.30/0.62      ! [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 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_triangle_ineq4
% 0.30/0.62  thf(fact_7255_abs__diff__le__iff,axiom,
% 0.30/0.62      ! [X3: code_integer,A: code_integer,R2: code_integer] :
% 0.30/0.62        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_le_iff
% 0.30/0.62  thf(fact_7256_abs__diff__le__iff,axiom,
% 0.30/0.62      ! [X3: real,A: real,R2: real] :
% 0.30/0.62        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_le_iff
% 0.30/0.62  thf(fact_7257_abs__diff__le__iff,axiom,
% 0.30/0.62      ! [X3: rat,A: rat,R2: rat] :
% 0.30/0.62        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_le_iff
% 0.30/0.62  thf(fact_7258_abs__diff__le__iff,axiom,
% 0.30/0.62      ! [X3: int,A: int,R2: int] :
% 0.30/0.62        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_le_iff
% 0.30/0.62  thf(fact_7259_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( groups8097168146408367636l_real @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7260_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( groups8778361861064173332t_real @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7261_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( groups5808333547571424918x_real @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7262_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7263_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( ( member_nat @ X3 @ A2 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.62              = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7264_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7265_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7266_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_real,X3: real,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.62           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7267_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_int,X3: int,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.62           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7268_sum_Oinsert__if,axiom,
% 0.30/0.62      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
% 0.30/0.62          & ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.62              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.insert_if
% 0.30/0.62  thf(fact_7269_abs__diff__less__iff,axiom,
% 0.30/0.62      ! [X3: code_integer,A: code_integer,R2: code_integer] :
% 0.30/0.62        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_less_iff
% 0.30/0.62  thf(fact_7270_abs__diff__less__iff,axiom,
% 0.30/0.62      ! [X3: real,A: real,R2: real] :
% 0.30/0.62        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_less_iff
% 0.30/0.62  thf(fact_7271_abs__diff__less__iff,axiom,
% 0.30/0.62      ! [X3: rat,A: rat,R2: rat] :
% 0.30/0.62        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_less_iff
% 0.30/0.62  thf(fact_7272_abs__diff__less__iff,axiom,
% 0.30/0.62      ! [X3: int,A: int,R2: int] :
% 0.30/0.62        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
% 0.30/0.62        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
% 0.30/0.62          & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_diff_less_iff
% 0.30/0.62  thf(fact_7273_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ S6 )
% 0.30/0.62       => ( ( finite_finite_real @ T5 )
% 0.30/0.62         => ( ! [A4: real] :
% 0.30/0.62                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: real] :
% 0.30/0.62                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: real] :
% 0.30/0.62                        ( ( member_real @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_complex ) )
% 0.30/0.62                   => ( ! [B4: real] :
% 0.30/0.62                          ( ( member_real @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_complex ) )
% 0.30/0.62                     => ( ! [A4: real] :
% 0.30/0.62                            ( ( member_real @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.62                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7274_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 0.30/0.62        ( ( finite_finite_real @ S6 )
% 0.30/0.62       => ( ( finite_finite_int @ T5 )
% 0.30/0.62         => ( ! [A4: real] :
% 0.30/0.62                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: int] :
% 0.30/0.62                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: int] :
% 0.30/0.62                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: real] :
% 0.30/0.62                        ( ( member_real @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_complex ) )
% 0.30/0.62                   => ( ! [B4: int] :
% 0.30/0.62                          ( ( member_int @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_complex ) )
% 0.30/0.62                     => ( ! [A4: real] :
% 0.30/0.62                            ( ( member_real @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.62                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7275_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 0.30/0.62        ( ( finite_finite_int @ S6 )
% 0.30/0.62       => ( ( finite_finite_real @ T5 )
% 0.30/0.62         => ( ! [A4: int] :
% 0.30/0.62                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: int] :
% 0.30/0.62                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: real] :
% 0.30/0.62                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: int] :
% 0.30/0.62                        ( ( member_int @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_complex ) )
% 0.30/0.62                   => ( ! [B4: real] :
% 0.30/0.62                          ( ( member_real @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_complex ) )
% 0.30/0.62                     => ( ! [A4: int] :
% 0.30/0.62                            ( ( member_int @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.62                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7276_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ S6 )
% 0.30/0.62       => ( ( finite_finite_int @ T5 )
% 0.30/0.62         => ( ! [A4: int] :
% 0.30/0.62                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: int] :
% 0.30/0.62                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: int] :
% 0.30/0.62                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: int] :
% 0.30/0.62                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: int] :
% 0.30/0.62                        ( ( member_int @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_complex ) )
% 0.30/0.62                   => ( ! [B4: int] :
% 0.30/0.62                          ( ( member_int @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_complex ) )
% 0.30/0.62                     => ( ! [A4: int] :
% 0.30/0.62                            ( ( member_int @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.62                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7277_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ S6 )
% 0.30/0.62       => ( ( finite_finite_real @ T5 )
% 0.30/0.62         => ( ! [A4: real] :
% 0.30/0.62                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: real] :
% 0.30/0.62                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: real] :
% 0.30/0.62                        ( ( member_real @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: real] :
% 0.30/0.62                          ( ( member_real @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: real] :
% 0.30/0.62                            ( ( member_real @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.62                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7278_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 0.30/0.62        ( ( finite_finite_real @ S6 )
% 0.30/0.62       => ( ( finite_finite_int @ T5 )
% 0.30/0.62         => ( ! [A4: real] :
% 0.30/0.62                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: int] :
% 0.30/0.62                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: int] :
% 0.30/0.62                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: real] :
% 0.30/0.62                        ( ( member_real @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: int] :
% 0.30/0.62                          ( ( member_int @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: real] :
% 0.30/0.62                            ( ( member_real @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.62                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7279_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_real,T5: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 0.30/0.62        ( ( finite_finite_real @ S6 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T5 )
% 0.30/0.62         => ( ! [A4: real] :
% 0.30/0.62                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: complex] :
% 0.30/0.62                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: complex] :
% 0.30/0.62                      ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: real] :
% 0.30/0.62                        ( ( member_real @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: complex] :
% 0.30/0.62                          ( ( member_complex @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: real] :
% 0.30/0.62                            ( ( member_real @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.62                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7280_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 0.30/0.62        ( ( finite_finite_int @ S6 )
% 0.30/0.62       => ( ( finite_finite_real @ T5 )
% 0.30/0.62         => ( ! [A4: int] :
% 0.30/0.62                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: int] :
% 0.30/0.62                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: real] :
% 0.30/0.62                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: int] :
% 0.30/0.62                        ( ( member_int @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: real] :
% 0.30/0.62                          ( ( member_real @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: int] :
% 0.30/0.62                            ( ( member_int @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.62                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7281_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ S6 )
% 0.30/0.62       => ( ( finite_finite_int @ T5 )
% 0.30/0.62         => ( ! [A4: int] :
% 0.30/0.62                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: int] :
% 0.30/0.62                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: int] :
% 0.30/0.62                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: int] :
% 0.30/0.62                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: int] :
% 0.30/0.62                        ( ( member_int @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: int] :
% 0.30/0.62                          ( ( member_int @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: int] :
% 0.30/0.62                            ( ( member_int @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.62                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7282_sum_Oreindex__bij__witness__not__neutral,axiom,
% 0.30/0.62      ! [S6: set_int,T5: set_complex,S3: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 0.30/0.62        ( ( finite_finite_int @ S6 )
% 0.30/0.62       => ( ( finite3207457112153483333omplex @ T5 )
% 0.30/0.62         => ( ! [A4: int] :
% 0.30/0.62                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62               => ( ( I @ ( J @ A4 ) )
% 0.30/0.62                  = A4 ) )
% 0.30/0.62           => ( ! [A4: int] :
% 0.30/0.62                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 0.30/0.62                 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
% 0.30/0.62             => ( ! [B4: complex] :
% 0.30/0.62                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 0.30/0.62                   => ( ( J @ ( I @ B4 ) )
% 0.30/0.62                      = B4 ) )
% 0.30/0.62               => ( ! [B4: complex] :
% 0.30/0.62                      ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 0.30/0.62                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 0.30/0.62                 => ( ! [A4: int] :
% 0.30/0.62                        ( ( member_int @ A4 @ S6 )
% 0.30/0.62                       => ( ( G @ A4 )
% 0.30/0.62                          = zero_zero_real ) )
% 0.30/0.62                   => ( ! [B4: complex] :
% 0.30/0.62                          ( ( member_complex @ B4 @ T5 )
% 0.30/0.62                         => ( ( H2 @ B4 )
% 0.30/0.62                            = zero_zero_real ) )
% 0.30/0.62                     => ( ! [A4: int] :
% 0.30/0.62                            ( ( member_int @ A4 @ S3 )
% 0.30/0.62                           => ( ( H2 @ ( J @ A4 ) )
% 0.30/0.62                              = ( G @ A4 ) ) )
% 0.30/0.62                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.62                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.reindex_bij_witness_not_neutral
% 0.30/0.62  thf(fact_7283_sum__eq__Suc0__iff,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.62            = ( suc @ zero_zero_nat ) )
% 0.30/0.62          = ( ? [X: int] :
% 0.30/0.62                ( ( member_int @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = ( suc @ zero_zero_nat ) )
% 0.30/0.62                & ! [Y: int] :
% 0.30/0.62                    ( ( member_int @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_Suc0_iff
% 0.30/0.62  thf(fact_7284_sum__eq__Suc0__iff,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.62            = ( suc @ zero_zero_nat ) )
% 0.30/0.62          = ( ? [X: complex] :
% 0.30/0.62                ( ( member_complex @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = ( suc @ zero_zero_nat ) )
% 0.30/0.62                & ! [Y: complex] :
% 0.30/0.62                    ( ( member_complex @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_Suc0_iff
% 0.30/0.62  thf(fact_7285_sum__eq__Suc0__iff,axiom,
% 0.30/0.62      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.30/0.62        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.62       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 0.30/0.62            = ( suc @ zero_zero_nat ) )
% 0.30/0.62          = ( ? [X: product_prod_nat_nat] :
% 0.30/0.62                ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = ( suc @ zero_zero_nat ) )
% 0.30/0.62                & ! [Y: product_prod_nat_nat] :
% 0.30/0.62                    ( ( member8440522571783428010at_nat @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_Suc0_iff
% 0.30/0.62  thf(fact_7286_sum__eq__Suc0__iff,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > nat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62            = ( suc @ zero_zero_nat ) )
% 0.30/0.62          = ( ? [X: nat] :
% 0.30/0.62                ( ( member_nat @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = ( suc @ zero_zero_nat ) )
% 0.30/0.62                & ! [Y: nat] :
% 0.30/0.62                    ( ( member_nat @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_Suc0_iff
% 0.30/0.62  thf(fact_7287_sum__SucD,axiom,
% 0.30/0.62      ! [F: nat > nat,A2: set_nat,N2: nat] :
% 0.30/0.62        ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62          = ( suc @ N2 ) )
% 0.30/0.62       => ? [X4: nat] :
% 0.30/0.62            ( ( member_nat @ X4 @ A2 )
% 0.30/0.62            & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_SucD
% 0.30/0.62  thf(fact_7288_sum__eq__1__iff,axiom,
% 0.30/0.62      ! [A2: set_int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 0.30/0.62            = one_one_nat )
% 0.30/0.62          = ( ? [X: int] :
% 0.30/0.62                ( ( member_int @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = one_one_nat )
% 0.30/0.62                & ! [Y: int] :
% 0.30/0.62                    ( ( member_int @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_1_iff
% 0.30/0.62  thf(fact_7289_sum__eq__1__iff,axiom,
% 0.30/0.62      ! [A2: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 0.30/0.62            = one_one_nat )
% 0.30/0.62          = ( ? [X: complex] :
% 0.30/0.62                ( ( member_complex @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = one_one_nat )
% 0.30/0.62                & ! [Y: complex] :
% 0.30/0.62                    ( ( member_complex @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_1_iff
% 0.30/0.62  thf(fact_7290_sum__eq__1__iff,axiom,
% 0.30/0.62      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.30/0.62        ( ( finite6177210948735845034at_nat @ A2 )
% 0.30/0.62       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 0.30/0.62            = one_one_nat )
% 0.30/0.62          = ( ? [X: product_prod_nat_nat] :
% 0.30/0.62                ( ( member8440522571783428010at_nat @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = one_one_nat )
% 0.30/0.62                & ! [Y: product_prod_nat_nat] :
% 0.30/0.62                    ( ( member8440522571783428010at_nat @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_1_iff
% 0.30/0.62  thf(fact_7291_sum__eq__1__iff,axiom,
% 0.30/0.62      ! [A2: set_nat,F: nat > nat] :
% 0.30/0.62        ( ( finite_finite_nat @ A2 )
% 0.30/0.62       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 0.30/0.62            = one_one_nat )
% 0.30/0.62          = ( ? [X: nat] :
% 0.30/0.62                ( ( member_nat @ X @ A2 )
% 0.30/0.62                & ( ( F @ X )
% 0.30/0.62                  = one_one_nat )
% 0.30/0.62                & ! [Y: nat] :
% 0.30/0.62                    ( ( member_nat @ Y @ A2 )
% 0.30/0.62                   => ( ( X != Y )
% 0.30/0.62                     => ( ( F @ Y )
% 0.30/0.62                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_eq_1_iff
% 0.30/0.62  thf(fact_7292_abs__real__def,axiom,
% 0.30/0.62      ( abs_abs_real
% 0.30/0.62      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_real_def
% 0.30/0.62  thf(fact_7293_zabs__def,axiom,
% 0.30/0.62      ( abs_abs_int
% 0.30/0.62      = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % zabs_def
% 0.30/0.62  thf(fact_7294_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > real,B2: real,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7295_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > real,B2: real,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7296_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > real,B2: real,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7297_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > rat,B2: rat,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7298_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
% 0.30/0.62        ( ( finite_finite_nat @ S )
% 0.30/0.62       => ( ! [I3: nat] :
% 0.30/0.62              ( ( member_nat @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_nat @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7299_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > rat,B2: rat,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7300_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7301_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_real,F: real > nat,B2: nat,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7302_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_int,F: int > nat,B2: nat,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7303_sum__nonneg__leq__bound,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > nat,B2: nat,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.30/0.62              = B2 )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_leq_bound
% 0.30/0.62  thf(fact_7304_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_real,F: real > real,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_real ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7305_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_int,F: int > real,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_real ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7306_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > real,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 0.30/0.62              = zero_zero_real )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_real ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7307_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_real,F: real > rat,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7308_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_nat,F: nat > rat,I: nat] :
% 0.30/0.62        ( ( finite_finite_nat @ S )
% 0.30/0.62       => ( ! [I3: nat] :
% 0.30/0.62              ( ( member_nat @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62           => ( ( member_nat @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7309_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_int,F: int > rat,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7310_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > rat,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 0.30/0.62              = zero_zero_rat )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7311_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_real,F: real > nat,I: real] :
% 0.30/0.62        ( ( finite_finite_real @ S )
% 0.30/0.62       => ( ! [I3: real] :
% 0.30/0.62              ( ( member_real @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62           => ( ( member_real @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7312_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_int,F: int > nat,I: int] :
% 0.30/0.62        ( ( finite_finite_int @ S )
% 0.30/0.62       => ( ! [I3: int] :
% 0.30/0.62              ( ( member_int @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62           => ( ( member_int @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7313_sum__nonneg__0,axiom,
% 0.30/0.62      ! [S: set_complex,F: complex > nat,I: complex] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ S )
% 0.30/0.62       => ( ! [I3: complex] :
% 0.30/0.62              ( ( member_complex @ I3 @ S )
% 0.30/0.62             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 0.30/0.62              = zero_zero_nat )
% 0.30/0.62           => ( ( member_complex @ I @ S )
% 0.30/0.62             => ( ( F @ I )
% 0.30/0.62                = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_nonneg_0
% 0.30/0.62  thf(fact_7314_dvd__imp__le__int,axiom,
% 0.30/0.62      ! [I: int,D: int] :
% 0.30/0.62        ( ( I != zero_zero_int )
% 0.30/0.62       => ( ( dvd_dvd_int @ D @ I )
% 0.30/0.62         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % dvd_imp_le_int
% 0.30/0.62  thf(fact_7315_abs__mod__less,axiom,
% 0.30/0.62      ! [L: int,K: int] :
% 0.30/0.62        ( ( L != zero_zero_int )
% 0.30/0.62       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_mod_less
% 0.30/0.62  thf(fact_7316_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups5754745047067104278omplex @ G
% 0.30/0.62            @ ( minus_minus_set_real @ A2
% 0.30/0.62              @ ( collect_real
% 0.30/0.62                @ ^ [X: real] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_complex ) ) ) )
% 0.30/0.62          = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7317_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups3049146728041665814omplex @ G
% 0.30/0.62            @ ( minus_minus_set_int @ A2
% 0.30/0.62              @ ( collect_int
% 0.30/0.62                @ ^ [X: int] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_complex ) ) ) )
% 0.30/0.62          = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7318_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups8097168146408367636l_real @ G
% 0.30/0.62            @ ( minus_minus_set_real @ A2
% 0.30/0.62              @ ( collect_real
% 0.30/0.62                @ ^ [X: real] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) )
% 0.30/0.62          = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7319_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups8778361861064173332t_real @ G
% 0.30/0.62            @ ( minus_minus_set_int @ A2
% 0.30/0.62              @ ( collect_int
% 0.30/0.62                @ ^ [X: int] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) )
% 0.30/0.62          = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7320_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( groups5808333547571424918x_real @ G
% 0.30/0.62            @ ( minus_811609699411566653omplex @ A2
% 0.30/0.62              @ ( collect_complex
% 0.30/0.62                @ ^ [X: complex] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_real ) ) ) )
% 0.30/0.62          = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7321_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups1300246762558778688al_rat @ G
% 0.30/0.62            @ ( minus_minus_set_real @ A2
% 0.30/0.62              @ ( collect_real
% 0.30/0.62                @ ^ [X: real] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) )
% 0.30/0.62          = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7322_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups3906332499630173760nt_rat @ G
% 0.30/0.62            @ ( minus_minus_set_int @ A2
% 0.30/0.62              @ ( collect_int
% 0.30/0.62                @ ^ [X: int] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) )
% 0.30/0.62          = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7323_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.62       => ( ( groups5058264527183730370ex_rat @ G
% 0.30/0.62            @ ( minus_811609699411566653omplex @ A2
% 0.30/0.62              @ ( collect_complex
% 0.30/0.62                @ ^ [X: complex] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_rat ) ) ) )
% 0.30/0.62          = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7324_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_real,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ A2 )
% 0.30/0.62       => ( ( groups1935376822645274424al_nat @ G
% 0.30/0.62            @ ( minus_minus_set_real @ A2
% 0.30/0.62              @ ( collect_real
% 0.30/0.62                @ ^ [X: real] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_nat ) ) ) )
% 0.30/0.62          = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7325_sum_Osetdiff__irrelevant,axiom,
% 0.30/0.62      ! [A2: set_int,G: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ A2 )
% 0.30/0.62       => ( ( groups4541462559716669496nt_nat @ G
% 0.30/0.62            @ ( minus_minus_set_int @ A2
% 0.30/0.62              @ ( collect_int
% 0.30/0.62                @ ^ [X: int] :
% 0.30/0.62                    ( ( G @ X )
% 0.30/0.62                    = zero_zero_nat ) ) ) )
% 0.30/0.62          = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.setdiff_irrelevant
% 0.30/0.62  thf(fact_7326_sum__power__add,axiom,
% 0.30/0.62      ! [X3: complex,M: nat,I5: set_nat] :
% 0.30/0.62        ( ( groups2073611262835488442omplex
% 0.30/0.62          @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 0.30/0.62          @ I5 )
% 0.30/0.62        = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_power_add
% 0.30/0.62  thf(fact_7327_sum__power__add,axiom,
% 0.30/0.62      ! [X3: rat,M: nat,I5: set_nat] :
% 0.30/0.62        ( ( groups2906978787729119204at_rat
% 0.30/0.62          @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 0.30/0.62          @ I5 )
% 0.30/0.62        = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_power_add
% 0.30/0.62  thf(fact_7328_sum__power__add,axiom,
% 0.30/0.62      ! [X3: int,M: nat,I5: set_nat] :
% 0.30/0.62        ( ( groups3539618377306564664at_int
% 0.30/0.62          @ ^ [I2: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 0.30/0.62          @ I5 )
% 0.30/0.62        = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_power_add
% 0.30/0.62  thf(fact_7329_sum__power__add,axiom,
% 0.30/0.62      ! [X3: real,M: nat,I5: set_nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 0.30/0.62          @ I5 )
% 0.30/0.62        = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_power_add
% 0.30/0.62  thf(fact_7330_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.62      ! [G: nat > nat,N2: nat,M: nat] :
% 0.30/0.62        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.30/0.62        = ( groups3542108847815614940at_nat
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.atLeastAtMost_rev
% 0.30/0.62  thf(fact_7331_sum_OatLeastAtMost__rev,axiom,
% 0.30/0.62      ! [G: nat > real,N2: nat,M: nat] :
% 0.30/0.62        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 0.30/0.62        = ( groups6591440286371151544t_real
% 0.30/0.62          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
% 0.30/0.62          @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.atLeastAtMost_rev
% 0.30/0.62  thf(fact_7332_tanh__real__gt__neg1,axiom,
% 0.30/0.62      ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).
% 0.30/0.62  
% 0.30/0.62  % tanh_real_gt_neg1
% 0.30/0.62  thf(fact_7333_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_real,I: real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( member_real @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: real] :
% 0.30/0.62                  ( ( member_real @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7334_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_int,I: int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( member_int @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: int] :
% 0.30/0.62                  ( ( member_int @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7335_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_complex,I: complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( member_complex @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: complex] :
% 0.30/0.62                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7336_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_real,I: real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( member_real @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: real] :
% 0.30/0.62                  ( ( member_real @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7337_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_nat,I: nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ I5 )
% 0.30/0.62       => ( ( member_nat @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: nat] :
% 0.30/0.62                  ( ( member_nat @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7338_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_int,I: int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( member_int @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: int] :
% 0.30/0.62                  ( ( member_int @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7339_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_complex,I: complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( member_complex @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: complex] :
% 0.30/0.62                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7340_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_real,I: real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( member_real @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: real] :
% 0.30/0.62                  ( ( member_real @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7341_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_int,I: int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( member_int @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: int] :
% 0.30/0.62                  ( ( member_int @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7342_sum__pos2,axiom,
% 0.30/0.62      ! [I5: set_complex,I: complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( member_complex @ I @ I5 )
% 0.30/0.62         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.30/0.62           => ( ! [I3: complex] :
% 0.30/0.62                  ( ( member_complex @ I3 @ I5 )
% 0.30/0.62                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62             => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos2
% 0.30/0.62  thf(fact_7343_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_complex,F: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [I3: complex] :
% 0.30/0.62                ( ( member_complex @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7344_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_real,F: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_real )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7345_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_int,F: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_int )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7346_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_complex,F: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [I3: complex] :
% 0.30/0.62                ( ( member_complex @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7347_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_real,F: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_real )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7348_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_nat,F: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_nat )
% 0.30/0.62         => ( ! [I3: nat] :
% 0.30/0.62                ( ( member_nat @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7349_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_int,F: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_int )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7350_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_complex,F: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_complex )
% 0.30/0.62         => ( ! [I3: complex] :
% 0.30/0.62                ( ( member_complex @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7351_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_real,F: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_real )
% 0.30/0.62         => ( ! [I3: real] :
% 0.30/0.62                ( ( member_real @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7352_sum__pos,axiom,
% 0.30/0.62      ! [I5: set_int,F: int > nat] :
% 0.30/0.62        ( ( finite_finite_int @ I5 )
% 0.30/0.62       => ( ( I5 != bot_bot_set_int )
% 0.30/0.62         => ( ! [I3: int] :
% 0.30/0.62                ( ( member_int @ I3 @ I5 )
% 0.30/0.62               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 0.30/0.62           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum_pos
% 0.30/0.62  thf(fact_7353_abs__add__one__gt__zero,axiom,
% 0.30/0.62      ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_one_gt_zero
% 0.30/0.62  thf(fact_7354_abs__add__one__gt__zero,axiom,
% 0.30/0.62      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_one_gt_zero
% 0.30/0.62  thf(fact_7355_abs__add__one__gt__zero,axiom,
% 0.30/0.62      ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_one_gt_zero
% 0.30/0.62  thf(fact_7356_abs__add__one__gt__zero,axiom,
% 0.30/0.62      ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % abs_add_one_gt_zero
% 0.30/0.62  thf(fact_7357_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 0.30/0.62                = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7358_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 0.30/0.62                = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7359_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.62                = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7360_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 0.30/0.62                = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7361_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.62                = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7362_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 0.30/0.62                = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7363_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.62                = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7364_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 0.30/0.62                = ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7365_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.30/0.62                = ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7366_sum_Omono__neutral__cong__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.62                = ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_right
% 0.30/0.62  thf(fact_7367_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5754745047067104278omplex @ G @ S3 )
% 0.30/0.62                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7368_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups8097168146408367636l_real @ G @ S3 )
% 0.30/0.62                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7369_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5808333547571424918x_real @ G @ S3 )
% 0.30/0.62                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7370_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1300246762558778688al_rat @ G @ S3 )
% 0.30/0.62                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7371_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 0.30/0.62                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7372_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1935376822645274424al_nat @ G @ S3 )
% 0.30/0.62                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7373_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 0.30/0.62                = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7374_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
% 0.30/0.62        ( ( finite_finite_real @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: real] :
% 0.30/0.62                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X4: real] :
% 0.30/0.62                  ( ( member_real @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups1932886352136224148al_int @ G @ S3 )
% 0.30/0.62                = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7375_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ! [X4: complex] :
% 0.30/0.62                  ( ( member_complex @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 0.30/0.62                = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7376_sum_Omono__neutral__cong__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( H2 @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ! [X4: nat] :
% 0.30/0.62                  ( ( member_nat @ X4 @ S3 )
% 0.30/0.62                 => ( ( G @ X4 )
% 0.30/0.62                    = ( H2 @ X4 ) ) )
% 0.30/0.62             => ( ( groups2073611262835488442omplex @ G @ S3 )
% 0.30/0.62                = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_cong_left
% 0.30/0.62  thf(fact_7377_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 0.30/0.62              = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7378_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 0.30/0.62              = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7379_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 0.30/0.62              = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7380_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 0.30/0.62              = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7381_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 0.30/0.62              = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7382_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 0.30/0.62              = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7383_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 0.30/0.62              = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7384_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 0.30/0.62              = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7385_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 0.30/0.62              = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7386_sum_Omono__neutral__right,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 0.30/0.62              = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_right
% 0.30/0.62  thf(fact_7387_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups5808333547571424918x_real @ G @ S3 )
% 0.30/0.62              = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7388_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 0.30/0.62              = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7389_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_nat ) )
% 0.30/0.62           => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 0.30/0.62              = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7390_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ T3 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: complex] :
% 0.30/0.62                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 0.30/0.62              = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7391_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups2073611262835488442omplex @ G @ S3 )
% 0.30/0.62              = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7392_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups2906978787729119204at_rat @ G @ S3 )
% 0.30/0.62              = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7393_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 0.30/0.62        ( ( finite_finite_nat @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: nat] :
% 0.30/0.62                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_int ) )
% 0.30/0.62           => ( ( groups3539618377306564664at_int @ G @ S3 )
% 0.30/0.62              = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7394_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > complex] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_complex ) )
% 0.30/0.62           => ( ( groups3049146728041665814omplex @ G @ S3 )
% 0.30/0.62              = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7395_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > real] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_real ) )
% 0.30/0.62           => ( ( groups8778361861064173332t_real @ G @ S3 )
% 0.30/0.62              = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7396_sum_Omono__neutral__left,axiom,
% 0.30/0.62      ! [T3: set_int,S3: set_int,G: int > rat] :
% 0.30/0.62        ( ( finite_finite_int @ T3 )
% 0.30/0.62       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 0.30/0.62         => ( ! [X4: int] :
% 0.30/0.62                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 0.30/0.62               => ( ( G @ X4 )
% 0.30/0.62                  = zero_zero_rat ) )
% 0.30/0.62           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
% 0.30/0.62              = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.mono_neutral_left
% 0.30/0.62  thf(fact_7397_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62                 => ( ( G @ A4 )
% 0.30/0.62                    = zero_zero_complex ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.62                   => ( ( H2 @ B4 )
% 0.30/0.62                      = zero_zero_complex ) )
% 0.30/0.62               => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.30/0.62                    = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.30/0.62                    = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_7398_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62                 => ( ( G @ A4 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.62                   => ( ( H2 @ B4 )
% 0.30/0.62                      = zero_zero_real ) )
% 0.30/0.62               => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.30/0.62                    = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.62                    = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_7399_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.62        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.62       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.62         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.62           => ( ! [A4: complex] :
% 0.30/0.62                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.62                 => ( ( G @ A4 )
% 0.30/0.62                    = zero_zero_real ) )
% 0.30/0.62             => ( ! [B4: complex] :
% 0.30/0.62                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.62                   => ( ( H2 @ B4 )
% 0.30/0.62                      = zero_zero_real ) )
% 0.30/0.62               => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.30/0.62                    = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 0.30/0.62                 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.62                    = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.62  
% 0.30/0.62  % sum.same_carrierI
% 0.30/0.62  thf(fact_7400_sum_Osame__carrierI,axiom,
% 0.30/0.62      ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.62        ( ( finite_finite_real @ C4 )
% 0.30/0.62       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.62         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.62           => ( ! [A4: real] :
% 0.30/0.62                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.62                 => ( ( G @ A4 )
% 0.30/0.62                    = zero_zero_rat ) )
% 0.30/0.62             => ( ! [B4: real] :
% 0.30/0.62                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_rat ) )
% 0.30/0.63               => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.63                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.63                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7401_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_rat ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_rat ) )
% 0.30/0.63               => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.63                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.63                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7402_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_nat ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_nat ) )
% 0.30/0.63               => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.63                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.63                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7403_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_nat ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_nat ) )
% 0.30/0.63               => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.63                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.63                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7404_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_int ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_int ) )
% 0.30/0.63               => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.63                    = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.30/0.63                    = ( groups1932886352136224148al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7405_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_int ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_int ) )
% 0.30/0.63               => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.30/0.63                    = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.63                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7406_sum_Osame__carrierI,axiom,
% 0.30/0.63      ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.63        ( ( finite_finite_nat @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: nat] :
% 0.30/0.63                  ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_complex ) )
% 0.30/0.63             => ( ! [B4: nat] :
% 0.30/0.63                    ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_complex ) )
% 0.30/0.63               => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.63                    = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 0.30/0.63                 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.63                    = ( groups2073611262835488442omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrierI
% 0.30/0.63  thf(fact_7407_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_complex ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_complex ) )
% 0.30/0.63               => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 0.30/0.63                    = ( groups5754745047067104278omplex @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups5754745047067104278omplex @ G @ C4 )
% 0.30/0.63                    = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7408_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_real ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_real ) )
% 0.30/0.63               => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.63                    = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups8097168146408367636l_real @ G @ C4 )
% 0.30/0.63                    = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7409_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_real ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_real ) )
% 0.30/0.63               => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.63                    = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups5808333547571424918x_real @ G @ C4 )
% 0.30/0.63                    = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7410_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_rat ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_rat ) )
% 0.30/0.63               => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.63                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 0.30/0.63                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7411_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_rat ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_rat ) )
% 0.30/0.63               => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.63                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 0.30/0.63                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7412_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_nat ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_nat ) )
% 0.30/0.63               => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.63                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 0.30/0.63                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7413_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_nat ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_nat ) )
% 0.30/0.63               => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.63                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 0.30/0.63                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7414_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: real] :
% 0.30/0.63                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_int ) )
% 0.30/0.63             => ( ! [B4: real] :
% 0.30/0.63                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_int ) )
% 0.30/0.63               => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.30/0.63                    = ( groups1932886352136224148al_int @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups1932886352136224148al_int @ G @ C4 )
% 0.30/0.63                    = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7415_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ C4 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: complex] :
% 0.30/0.63                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_int ) )
% 0.30/0.63             => ( ! [B4: complex] :
% 0.30/0.63                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_int ) )
% 0.30/0.63               => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.63                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 0.30/0.63                    = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7416_sum_Osame__carrier,axiom,
% 0.30/0.63      ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 0.30/0.63        ( ( finite_finite_nat @ C4 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 0.30/0.63         => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 0.30/0.63           => ( ! [A4: nat] :
% 0.30/0.63                  ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 0.30/0.63                 => ( ( G @ A4 )
% 0.30/0.63                    = zero_zero_complex ) )
% 0.30/0.63             => ( ! [B4: nat] :
% 0.30/0.63                    ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
% 0.30/0.63                   => ( ( H2 @ B4 )
% 0.30/0.63                      = zero_zero_complex ) )
% 0.30/0.63               => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 0.30/0.63                    = ( groups2073611262835488442omplex @ H2 @ B2 ) )
% 0.30/0.63                  = ( ( groups2073611262835488442omplex @ G @ C4 )
% 0.30/0.63                    = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.same_carrier
% 0.30/0.63  thf(fact_7417_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,G: complex > real] :
% 0.30/0.63        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63       => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.63            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7418_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,G: complex > rat] :
% 0.30/0.63        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63       => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7419_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,G: complex > nat] :
% 0.30/0.63        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63       => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7420_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,G: complex > int] :
% 0.30/0.63        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63       => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.63            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7421_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_nat @ A2 )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7422_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_nat @ A2 )
% 0.30/0.63         => ( ( groups3539618377306564664at_int @ G @ A2 )
% 0.30/0.63            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7423_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_int,A2: set_int,G: int > real] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_int @ A2 )
% 0.30/0.63         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.30/0.63            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7424_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_int,A2: set_int,G: int > rat] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_int @ A2 )
% 0.30/0.63         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7425_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_int,A2: set_int,G: int > nat] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_int @ A2 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7426_sum_Osubset__diff,axiom,
% 0.30/0.63      ! [B2: set_int,A2: set_int,G: int > int] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63       => ( ( finite_finite_int @ A2 )
% 0.30/0.63         => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 0.30/0.63            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.subset_diff
% 0.30/0.63  thf(fact_7427_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_complex,B2: set_complex,F: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7428_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_complex,B2: set_complex,F: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7429_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_complex,B2: set_complex,F: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7430_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_nat,B2: set_nat,F: nat > rat] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7431_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7432_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_int,B2: set_int,F: int > real] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7433_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_int,B2: set_int,F: int > rat] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7434_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_int,B2: set_int,F: int > int] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7435_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_complex,B2: set_complex,F: complex > complex] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7436_sum__diff,axiom,
% 0.30/0.63      ! [A2: set_nat,B2: set_nat,F: nat > real] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63         => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff
% 0.30/0.63  thf(fact_7437_of__int__leD,axiom,
% 0.30/0.63      ! [N2: int,X3: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_leD
% 0.30/0.63  thf(fact_7438_of__int__leD,axiom,
% 0.30/0.63      ! [N2: int,X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_leD
% 0.30/0.63  thf(fact_7439_of__int__leD,axiom,
% 0.30/0.63      ! [N2: int,X3: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_leD
% 0.30/0.63  thf(fact_7440_of__int__leD,axiom,
% 0.30/0.63      ! [N2: int,X3: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_eq_int @ one_one_int @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_leD
% 0.30/0.63  thf(fact_7441_sum__diff__nat,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 0.30/0.63         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_7442_sum__diff__nat,axiom,
% 0.30/0.63      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.30/0.63        ( ( finite6177210948735845034at_nat @ B2 )
% 0.30/0.63       => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
% 0.30/0.63         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_7443_sum__diff__nat,axiom,
% 0.30/0.63      ! [B2: set_int,A2: set_int,F: int > nat] :
% 0.30/0.63        ( ( finite_finite_int @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_7444_sum__diff__nat,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 0.30/0.63        ( ( finite_finite_nat @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 0.30/0.63         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_nat
% 0.30/0.63  thf(fact_7445_of__int__lessD,axiom,
% 0.30/0.63      ! [N2: int,X3: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_lessD
% 0.30/0.63  thf(fact_7446_of__int__lessD,axiom,
% 0.30/0.63      ! [N2: int,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_lessD
% 0.30/0.63  thf(fact_7447_of__int__lessD,axiom,
% 0.30/0.63      ! [N2: int,X3: rat] :
% 0.30/0.63        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_lessD
% 0.30/0.63  thf(fact_7448_of__int__lessD,axiom,
% 0.30/0.63      ! [N2: int,X3: int] :
% 0.30/0.63        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
% 0.30/0.63       => ( ( N2 = zero_zero_int )
% 0.30/0.63          | ( ord_less_int @ one_one_int @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_lessD
% 0.30/0.63  thf(fact_7449_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 0.30/0.63        ( ( ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 0.30/0.63            = ( groups3860910324918113789at_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7450_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
% 0.30/0.63        ( ( ( member_set_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member_set_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 0.30/0.63            = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7451_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 0.30/0.63        ( ( ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 0.30/0.63            = ( groups977919841031483927at_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7452_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: real,A2: set_real,F: real > nat] :
% 0.30/0.63        ( ( ( member_real @ A @ A2 )
% 0.30/0.63         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member_real @ A @ A2 )
% 0.30/0.63         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63            = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7453_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: int,A2: set_int,F: int > nat] :
% 0.30/0.63        ( ( ( member_int @ A @ A2 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member_int @ A @ A2 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63            = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7454_sum__diff1__nat,axiom,
% 0.30/0.63      ! [A: nat,A2: set_nat,F: nat > nat] :
% 0.30/0.63        ( ( ( member_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.63            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63        & ( ~ ( member_nat @ A @ A2 )
% 0.30/0.63         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.63            = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1_nat
% 0.30/0.63  thf(fact_7455_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.63      ! [F: nat > complex,K: nat] :
% 0.30/0.63        ( ( ( F @ zero_zero_nat )
% 0.30/0.63          = zero_zero_complex )
% 0.30/0.63       => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.63          = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_shift_lb_Suc0_0
% 0.30/0.63  thf(fact_7456_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.63      ! [F: nat > rat,K: nat] :
% 0.30/0.63        ( ( ( F @ zero_zero_nat )
% 0.30/0.63          = zero_zero_rat )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.63          = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_shift_lb_Suc0_0
% 0.30/0.63  thf(fact_7457_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.63      ! [F: nat > int,K: nat] :
% 0.30/0.63        ( ( ( F @ zero_zero_nat )
% 0.30/0.63          = zero_zero_int )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.63          = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_shift_lb_Suc0_0
% 0.30/0.63  thf(fact_7458_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.63      ! [F: nat > nat,K: nat] :
% 0.30/0.63        ( ( ( F @ zero_zero_nat )
% 0.30/0.63          = zero_zero_nat )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.63          = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_shift_lb_Suc0_0
% 0.30/0.63  thf(fact_7459_sum__shift__lb__Suc0__0,axiom,
% 0.30/0.63      ! [F: nat > real,K: nat] :
% 0.30/0.63        ( ( ( F @ zero_zero_nat )
% 0.30/0.63          = zero_zero_real )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 0.30/0.63          = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_shift_lb_Suc0_0
% 0.30/0.63  thf(fact_7460_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > rat,N2: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_7461_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > int,N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_7462_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_7463_sum_OatLeast0__atMost__Suc,axiom,
% 0.30/0.63      ! [G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast0_atMost_Suc
% 0.30/0.63  thf(fact_7464_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_7465_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_7466_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_7467_sum_OatLeast__Suc__atMost,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast_Suc_atMost
% 0.30/0.63  thf(fact_7468_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_7469_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_7470_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_7471_sum_Onat__ivl__Suc_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_ivl_Suc'
% 0.30/0.63  thf(fact_7472_round__diff__minimal,axiom,
% 0.30/0.63      ! [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 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % round_diff_minimal
% 0.30/0.63  thf(fact_7473_round__diff__minimal,axiom,
% 0.30/0.63      ! [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 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % round_diff_minimal
% 0.30/0.63  thf(fact_7474_zdvd__mult__cancel1,axiom,
% 0.30/0.63      ! [M: int,N2: int] :
% 0.30/0.63        ( ( M != zero_zero_int )
% 0.30/0.63       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
% 0.30/0.63          = ( ( abs_abs_int @ N2 )
% 0.30/0.63            = one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zdvd_mult_cancel1
% 0.30/0.63  thf(fact_7475_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ M )
% 0.30/0.63            @ ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_7476_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ M )
% 0.30/0.63            @ ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_7477_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ M )
% 0.30/0.63            @ ( groups3542108847815614940at_nat
% 0.30/0.63              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_7478_sum_OSuc__reindex__ivl,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ M )
% 0.30/0.63            @ ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.Suc_reindex_ivl
% 0.30/0.63  thf(fact_7479_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_7480_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_7481_sum__Suc__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_Suc_diff
% 0.30/0.63  thf(fact_7482_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,F: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: real] :
% 0.30/0.63                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7483_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,F: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: complex] :
% 0.30/0.63                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7484_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,F: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: real] :
% 0.30/0.63                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7485_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,F: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: complex] :
% 0.30/0.63                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7486_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,F: nat > rat] :
% 0.30/0.63        ( ( finite_finite_nat @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: nat] :
% 0.30/0.63                ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7487_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,F: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: real] :
% 0.30/0.63                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7488_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: complex] :
% 0.30/0.63                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7489_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,F: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: real] :
% 0.30/0.63                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7490_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: complex] :
% 0.30/0.63                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7491_sum__mono2,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.63         => ( ! [B4: nat] :
% 0.30/0.63                ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.30/0.63               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 0.30/0.63           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_mono2
% 0.30/0.63  thf(fact_7492_abs__le__square__iff,axiom,
% 0.30/0.63      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 0.30/0.63        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_square_iff
% 0.30/0.63  thf(fact_7493_abs__le__square__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_square_iff
% 0.30/0.63  thf(fact_7494_abs__le__square__iff,axiom,
% 0.30/0.63      ! [X3: rat,Y3: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_square_iff
% 0.30/0.63  thf(fact_7495_abs__le__square__iff,axiom,
% 0.30/0.63      ! [X3: int,Y3: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_le_square_iff
% 0.30/0.63  thf(fact_7496_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_complex,X3: complex,G: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( member_complex @ X3 @ A2 )
% 0.30/0.63         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 0.30/0.63            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7497_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( member_complex @ X3 @ A2 )
% 0.30/0.63         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7498_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( member_complex @ X3 @ A2 )
% 0.30/0.63         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7499_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_complex,X3: complex,G: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( member_complex @ X3 @ A2 )
% 0.30/0.63         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 0.30/0.63            = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7500_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_real,X3: real,G: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( member_real @ X3 @ A2 )
% 0.30/0.63         => ( ( groups8097168146408367636l_real @ G @ A2 )
% 0.30/0.63            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7501_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_real,X3: real,G: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( member_real @ X3 @ A2 )
% 0.30/0.63         => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7502_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_real,X3: real,G: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( member_real @ X3 @ A2 )
% 0.30/0.63         => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7503_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_real,X3: real,G: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( member_real @ X3 @ A2 )
% 0.30/0.63         => ( ( groups1932886352136224148al_int @ G @ A2 )
% 0.30/0.63            = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7504_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_int,X3: int,G: int > real] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( member_int @ X3 @ A2 )
% 0.30/0.63         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 0.30/0.63            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7505_sum_Oremove,axiom,
% 0.30/0.63      ! [A2: set_int,X3: int,G: int > rat] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( member_int @ X3 @ A2 )
% 0.30/0.63         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 0.30/0.63            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.remove
% 0.30/0.63  thf(fact_7506_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_complex,G: complex > real,X3: complex] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7507_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_complex,G: complex > rat,X3: complex] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7508_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_complex,G: complex > nat,X3: complex] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7509_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_complex,G: complex > int,X3: complex] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7510_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_real,G: real > real,X3: real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7511_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_real,G: real > rat,X3: real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7512_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_real,G: real > nat,X3: real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7513_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_real,G: real > int,X3: real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7514_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_int,G: int > real,X3: int] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7515_sum_Oinsert__remove,axiom,
% 0.30/0.63      ! [A2: set_int,G: int > rat,X3: int] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.63          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.insert_remove
% 0.30/0.63  thf(fact_7516_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_complex,A: complex,F: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7517_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_real,A: real,F: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7518_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_int,A: int,F: int > real] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7519_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_complex,A: complex,F: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7520_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_real,A: real,F: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7521_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_int,A: int,F: int > rat] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7522_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_complex,A: complex,F: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63       => ( ( ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ A2 )
% 0.30/0.63           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 0.30/0.63              = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7523_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_real,A: real,F: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ A2 )
% 0.30/0.63       => ( ( ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ A2 )
% 0.30/0.63           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 0.30/0.63              = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7524_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_int,A: int,F: int > int] :
% 0.30/0.63        ( ( finite_finite_int @ A2 )
% 0.30/0.63       => ( ( ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_int @ A @ A2 )
% 0.30/0.63           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 0.30/0.63              = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7525_sum__diff1,axiom,
% 0.30/0.63      ! [A2: set_nat,A: nat,F: nat > rat] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( ( ( member_nat @ A @ A2 )
% 0.30/0.63           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.63              = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 0.30/0.63          & ( ~ ( member_nat @ A @ A2 )
% 0.30/0.63           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 0.30/0.63              = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff1
% 0.30/0.63  thf(fact_7526_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_7527_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > int,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.30/0.63       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_7528_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.30/0.63       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_7529_sum_Oub__add__nat,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,G: nat > real,P4: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 0.30/0.63       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 0.30/0.63          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.ub_add_nat
% 0.30/0.63  thf(fact_7530_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.63       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5808333547571424918x_real
% 0.30/0.63                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5808333547571424918x_real
% 0.30/0.63                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7531_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.63       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5058264527183730370ex_rat
% 0.30/0.63                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5058264527183730370ex_rat
% 0.30/0.63                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7532_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.63       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5693394587270226106ex_nat
% 0.30/0.63                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5693394587270226106ex_nat
% 0.30/0.63                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7533_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.63       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5690904116761175830ex_int
% 0.30/0.63                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.63           => ( ( groups5690904116761175830ex_int
% 0.30/0.63                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7534_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_real,A: real,B: real > real,C: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ S3 )
% 0.30/0.63       => ( ( ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups8097168146408367636l_real
% 0.30/0.63                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups8097168146408367636l_real
% 0.30/0.63                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7535_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ S3 )
% 0.30/0.63       => ( ( ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1300246762558778688al_rat
% 0.30/0.63                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1300246762558778688al_rat
% 0.30/0.63                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7536_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ S3 )
% 0.30/0.63       => ( ( ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1935376822645274424al_nat
% 0.30/0.63                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1935376822645274424al_nat
% 0.30/0.63                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7537_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_real,A: real,B: real > int,C: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ S3 )
% 0.30/0.63       => ( ( ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1932886352136224148al_int
% 0.30/0.63                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.63           => ( ( groups1932886352136224148al_int
% 0.30/0.63                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7538_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_int,A: int,B: int > real,C: int > real] :
% 0.30/0.63        ( ( finite_finite_int @ S3 )
% 0.30/0.63       => ( ( ( member_int @ A @ S3 )
% 0.30/0.63           => ( ( groups8778361861064173332t_real
% 0.30/0.63                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.63           => ( ( groups8778361861064173332t_real
% 0.30/0.63                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7539_sum_Odelta__remove,axiom,
% 0.30/0.63      ! [S3: set_int,A: int,B: int > rat,C: int > rat] :
% 0.30/0.63        ( ( finite_finite_int @ S3 )
% 0.30/0.63       => ( ( ( member_int @ A @ S3 )
% 0.30/0.63           => ( ( groups3906332499630173760nt_rat
% 0.30/0.63                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 0.30/0.63          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.63           => ( ( groups3906332499630173760nt_rat
% 0.30/0.63                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 0.30/0.63                @ S3 )
% 0.30/0.63              = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.delta_remove
% 0.30/0.63  thf(fact_7540_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_complex,X7: set_complex] :
% 0.30/0.63        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite3207457112153483333omplex @ X7 )
% 0.30/0.63         => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
% 0.30/0.63            = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7541_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_P6011104703257516679at_nat,X7: set_Pr1261947904930325089at_nat] :
% 0.30/0.63        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite6177210948735845034at_nat @ X7 )
% 0.30/0.63         => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X7 )
% 0.30/0.63            = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7542_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
% 0.30/0.63        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite5795047828879050333T_VEBT @ X7 )
% 0.30/0.63         => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
% 0.30/0.63            = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7543_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_o,X7: set_o] :
% 0.30/0.63        ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite_finite_o @ X7 )
% 0.30/0.63         => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X7 )
% 0.30/0.63            = ( size_size_list_o @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7544_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_int,X7: set_int] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite_finite_int @ X7 )
% 0.30/0.63         => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
% 0.30/0.63            = ( size_size_list_int @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7545_sum__count__set,axiom,
% 0.30/0.63      ! [Xs: list_nat,X7: set_nat] :
% 0.30/0.63        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
% 0.30/0.63       => ( ( finite_finite_nat @ X7 )
% 0.30/0.63         => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
% 0.30/0.63            = ( size_size_list_nat @ Xs ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_count_set
% 0.30/0.63  thf(fact_7546_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,B: real,F: real > real] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: real] :
% 0.30/0.63                    ( ( member_real @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7547_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: complex] :
% 0.30/0.63                    ( ( member_complex @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7548_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: real] :
% 0.30/0.63                    ( ( member_real @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7549_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: complex] :
% 0.30/0.63                    ( ( member_complex @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7550_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 0.30/0.63        ( ( finite_finite_nat @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.63         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: nat] :
% 0.30/0.63                    ( ( member_nat @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7551_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: real] :
% 0.30/0.63                    ( ( member_real @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7552_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: complex] :
% 0.30/0.63                    ( ( member_complex @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7553_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_real,A2: set_real,B: real,F: real > int] :
% 0.30/0.63        ( ( finite_finite_real @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 0.30/0.63         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: real] :
% 0.30/0.63                    ( ( member_real @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7554_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 0.30/0.63        ( ( finite3207457112153483333omplex @ B2 )
% 0.30/0.63       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 0.30/0.63         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: complex] :
% 0.30/0.63                    ( ( member_complex @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7555_sum__strict__mono2,axiom,
% 0.30/0.63      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ B2 )
% 0.30/0.63       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 0.30/0.63         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 0.30/0.63           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 0.30/0.63             => ( ! [X4: nat] :
% 0.30/0.63                    ( ( member_nat @ X4 @ B2 )
% 0.30/0.63                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 0.30/0.63               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_strict_mono2
% 0.30/0.63  thf(fact_7556_member__le__sum,axiom,
% 0.30/0.63      ! [I: complex,A2: set_complex,F: complex > real] :
% 0.30/0.63        ( ( member_complex @ I @ A2 )
% 0.30/0.63       => ( ! [X4: complex] :
% 0.30/0.63              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.30/0.63             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7557_member__le__sum,axiom,
% 0.30/0.63      ! [I: real,A2: set_real,F: real > real] :
% 0.30/0.63        ( ( member_real @ I @ A2 )
% 0.30/0.63       => ( ! [X4: real] :
% 0.30/0.63              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.30/0.63             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_real @ A2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7558_member__le__sum,axiom,
% 0.30/0.63      ! [I: int,A2: set_int,F: int > real] :
% 0.30/0.63        ( ( member_int @ I @ A2 )
% 0.30/0.63       => ( ! [X4: int] :
% 0.30/0.63              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.30/0.63             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_int @ A2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7559_member__le__sum,axiom,
% 0.30/0.63      ! [I: complex,A2: set_complex,F: complex > rat] :
% 0.30/0.63        ( ( member_complex @ I @ A2 )
% 0.30/0.63       => ( ! [X4: complex] :
% 0.30/0.63              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.30/0.63             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7560_member__le__sum,axiom,
% 0.30/0.63      ! [I: real,A2: set_real,F: real > rat] :
% 0.30/0.63        ( ( member_real @ I @ A2 )
% 0.30/0.63       => ( ! [X4: real] :
% 0.30/0.63              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.30/0.63             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_real @ A2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7561_member__le__sum,axiom,
% 0.30/0.63      ! [I: int,A2: set_int,F: int > rat] :
% 0.30/0.63        ( ( member_int @ I @ A2 )
% 0.30/0.63       => ( ! [X4: int] :
% 0.30/0.63              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.30/0.63             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_int @ A2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7562_member__le__sum,axiom,
% 0.30/0.63      ! [I: nat,A2: set_nat,F: nat > rat] :
% 0.30/0.63        ( ( member_nat @ I @ A2 )
% 0.30/0.63       => ( ! [X4: nat] :
% 0.30/0.63              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 0.30/0.63             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_nat @ A2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7563_member__le__sum,axiom,
% 0.30/0.63      ! [I: complex,A2: set_complex,F: complex > nat] :
% 0.30/0.63        ( ( member_complex @ I @ A2 )
% 0.30/0.63       => ( ! [X4: complex] :
% 0.30/0.63              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 0.30/0.63             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7564_member__le__sum,axiom,
% 0.30/0.63      ! [I: real,A2: set_real,F: real > nat] :
% 0.30/0.63        ( ( member_real @ I @ A2 )
% 0.30/0.63       => ( ! [X4: real] :
% 0.30/0.63              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 0.30/0.63             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_real @ A2 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7565_member__le__sum,axiom,
% 0.30/0.63      ! [I: int,A2: set_int,F: int > nat] :
% 0.30/0.63        ( ( member_int @ I @ A2 )
% 0.30/0.63       => ( ! [X4: int] :
% 0.30/0.63              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 0.30/0.63             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 0.30/0.63         => ( ( finite_finite_int @ A2 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % member_le_sum
% 0.30/0.63  thf(fact_7566_power2__le__iff__abs__le,axiom,
% 0.30/0.63      ! [Y3: code_integer,X3: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 0.30/0.63       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.63          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power2_le_iff_abs_le
% 0.30/0.63  thf(fact_7567_power2__le__iff__abs__le,axiom,
% 0.30/0.63      ! [Y3: real,X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.63          = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power2_le_iff_abs_le
% 0.30/0.63  thf(fact_7568_power2__le__iff__abs__le,axiom,
% 0.30/0.63      ! [Y3: rat,X3: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 0.30/0.63       => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.63          = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power2_le_iff_abs_le
% 0.30/0.63  thf(fact_7569_power2__le__iff__abs__le,axiom,
% 0.30/0.63      ! [Y3: int,X3: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.63          = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power2_le_iff_abs_le
% 0.30/0.63  thf(fact_7570_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: code_integer > code_integer > $o,X3: code_integer] :
% 0.30/0.63        ( ! [X4: code_integer] :
% 0.30/0.63            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 0.30/0.63           => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.63       => ( P @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_sqrt_wlog
% 0.30/0.63  thf(fact_7571_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: real > real > $o,X3: real] :
% 0.30/0.63        ( ! [X4: real] :
% 0.30/0.63            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.30/0.63           => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.63       => ( P @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_sqrt_wlog
% 0.30/0.63  thf(fact_7572_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: rat > rat > $o,X3: rat] :
% 0.30/0.63        ( ! [X4: rat] :
% 0.30/0.63            ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 0.30/0.63           => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.63       => ( P @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_sqrt_wlog
% 0.30/0.63  thf(fact_7573_abs__sqrt__wlog,axiom,
% 0.30/0.63      ! [P: int > int > $o,X3: int] :
% 0.30/0.63        ( ! [X4: int] :
% 0.30/0.63            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 0.30/0.63           => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.63       => ( P @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_sqrt_wlog
% 0.30/0.63  thf(fact_7574_abs__square__le__1,axiom,
% 0.30/0.63      ! [X3: code_integer] :
% 0.30/0.63        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.30/0.63        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_le_1
% 0.30/0.63  thf(fact_7575_abs__square__le__1,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.30/0.63        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_le_1
% 0.30/0.63  thf(fact_7576_abs__square__le__1,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.30/0.63        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_le_1
% 0.30/0.63  thf(fact_7577_abs__square__le__1,axiom,
% 0.30/0.63      ! [X3: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.30/0.63        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_le_1
% 0.30/0.63  thf(fact_7578_abs__square__less__1,axiom,
% 0.30/0.63      ! [X3: code_integer] :
% 0.30/0.63        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 0.30/0.63        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_less_1
% 0.30/0.63  thf(fact_7579_abs__square__less__1,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 0.30/0.63        = ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_less_1
% 0.30/0.63  thf(fact_7580_abs__square__less__1,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 0.30/0.63        = ( ord_less_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_less_1
% 0.30/0.63  thf(fact_7581_abs__square__less__1,axiom,
% 0.30/0.63      ! [X3: int] :
% 0.30/0.63        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 0.30/0.63        = ( ord_less_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_square_less_1
% 0.30/0.63  thf(fact_7582_power__mono__even,axiom,
% 0.30/0.63      ! [N2: nat,A: code_integer,B: code_integer] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.63       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 0.30/0.63         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_7583_power__mono__even,axiom,
% 0.30/0.63      ! [N2: nat,A: real,B: real] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 0.30/0.63         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_7584_power__mono__even,axiom,
% 0.30/0.63      ! [N2: nat,A: rat,B: rat] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.63       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 0.30/0.63         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_7585_power__mono__even,axiom,
% 0.30/0.63      ! [N2: nat,A: int,B: int] :
% 0.30/0.63        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 0.30/0.63         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_mono_even
% 0.30/0.63  thf(fact_7586_nat__intermed__int__val,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > int,K: int] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( ( ord_less_eq_nat @ M @ I3 )
% 0.30/0.63              & ( ord_less_nat @ I3 @ N2 ) )
% 0.30/0.63           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.63       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 0.30/0.63           => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.30/0.63             => ? [I3: nat] :
% 0.30/0.63                  ( ( ord_less_eq_nat @ M @ I3 )
% 0.30/0.63                  & ( ord_less_eq_nat @ I3 @ N2 )
% 0.30/0.63                  & ( ( F @ I3 )
% 0.30/0.63                    = K ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_intermed_int_val
% 0.30/0.63  thf(fact_7587_incr__lemma,axiom,
% 0.30/0.63      ! [D: int,Z2: int,X3: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.63       => ( ord_less_int @ Z2 @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % incr_lemma
% 0.30/0.63  thf(fact_7588_decr__lemma,axiom,
% 0.30/0.63      ! [D: int,X3: int,Z2: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ D )
% 0.30/0.63       => ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % decr_lemma
% 0.30/0.63  thf(fact_7589_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > complex] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_7590_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > rat] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_7591_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > int] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups3539618377306564664at_int
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_7592_sum__natinterval__diff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > real] :
% 0.30/0.63        ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 0.30/0.63        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63         => ( ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_natinterval_diff
% 0.30/0.63  thf(fact_7593_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.30/0.63          = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_7594_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.30/0.63          = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_7595_sum__telescope_H_H,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,F: nat > real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 0.30/0.63          = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_telescope''
% 0.30/0.63  thf(fact_7596_mask__eq__sum__exp,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
% 0.30/0.63        = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.63          @ ( collect_nat
% 0.30/0.63            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp
% 0.30/0.63  thf(fact_7597_mask__eq__sum__exp,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
% 0.30/0.63        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          @ ( collect_nat
% 0.30/0.63            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp
% 0.30/0.63  thf(fact_7598_nat__ivt__aux,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > int,K: int] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.63           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.30/0.63         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.30/0.63           => ? [I3: nat] :
% 0.30/0.63                ( ( ord_less_eq_nat @ I3 @ N2 )
% 0.30/0.63                & ( ( F @ I3 )
% 0.30/0.63                  = K ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_ivt_aux
% 0.30/0.63  thf(fact_7599_divmod__nat__def,axiom,
% 0.30/0.63      ( divmod_nat
% 0.30/0.63      = ( ^ [M2: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % divmod_nat_def
% 0.30/0.63  thf(fact_7600_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,X3: complex] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.30/0.63          = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_7601_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,X3: rat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.30/0.63          = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_7602_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,X3: int] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.30/0.63          = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_7603_sum__gp__multiplied,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,X3: real] :
% 0.30/0.63        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.63       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 0.30/0.63          = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_multiplied
% 0.30/0.63  thf(fact_7604_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > rat,M: nat,N2: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.63        = ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_7605_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > int,M: nat,N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.63        = ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_7606_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > nat,M: nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_7607_sum_Oin__pairs,axiom,
% 0.30/0.63      ! [G: nat > real,M: nat,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.in_pairs
% 0.30/0.63  thf(fact_7608_nat0__intermed__int__val,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > int,K: int] :
% 0.30/0.63        ( ! [I3: nat] :
% 0.30/0.63            ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.63           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 0.30/0.63       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 0.30/0.63         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 0.30/0.63           => ? [I3: nat] :
% 0.30/0.63                ( ( ord_less_eq_nat @ I3 @ N2 )
% 0.30/0.63                & ( ( F @ I3 )
% 0.30/0.63                  = K ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat0_intermed_int_val
% 0.30/0.63  thf(fact_7609_mask__eq__sum__exp__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.63          @ ( collect_nat
% 0.30/0.63            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mask_eq_sum_exp_nat
% 0.30/0.63  thf(fact_7610_gauss__sum__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [X: nat] : X
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum_nat
% 0.30/0.63  thf(fact_7611_of__int__round__abs__le,axiom,
% 0.30/0.63      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ X3 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_round_abs_le
% 0.30/0.63  thf(fact_7612_of__int__round__abs__le,axiom,
% 0.30/0.63      ! [X3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ X3 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_round_abs_le
% 0.30/0.63  thf(fact_7613_listrel1p__def,axiom,
% 0.30/0.63      ( listrel1p_int
% 0.30/0.63      = ( ^ [R5: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R5 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % listrel1p_def
% 0.30/0.63  thf(fact_7614_lemma__interval,axiom,
% 0.30/0.63      ! [A: real,X3: real,B: real] :
% 0.30/0.63        ( ( ord_less_real @ A @ X3 )
% 0.30/0.63       => ( ( ord_less_real @ X3 @ B )
% 0.30/0.63         => ? [D4: real] :
% 0.30/0.63              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.63              & ! [Y6: real] :
% 0.30/0.63                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
% 0.30/0.63                 => ( ( ord_less_eq_real @ A @ Y6 )
% 0.30/0.63                    & ( ord_less_eq_real @ Y6 @ B ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_interval
% 0.30/0.63  thf(fact_7615_sum__gp,axiom,
% 0.30/0.63      ! [N2: nat,M: nat,X3: complex] :
% 0.30/0.63        ( ( ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_complex ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_complex )
% 0.30/0.63             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.30/0.63            & ( ( X3 != one_one_complex )
% 0.30/0.63             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_7616_sum__gp,axiom,
% 0.30/0.63      ! [N2: nat,M: nat,X3: rat] :
% 0.30/0.63        ( ( ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_rat ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_rat )
% 0.30/0.63             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.30/0.63            & ( ( X3 != one_one_rat )
% 0.30/0.63             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_7617_sum__gp,axiom,
% 0.30/0.63      ! [N2: nat,M: nat,X3: real] :
% 0.30/0.63        ( ( ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63            = zero_zero_real ) )
% 0.30/0.63        & ( ~ ( ord_less_nat @ N2 @ M )
% 0.30/0.63         => ( ( ( X3 = one_one_real )
% 0.30/0.63             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 0.30/0.63            & ( ( X3 != one_one_real )
% 0.30/0.63             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 0.30/0.63                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp
% 0.30/0.63  thf(fact_7618_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.63        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7619_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7620_gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7621_lemma__interval__lt,axiom,
% 0.30/0.63      ! [A: real,X3: real,B: real] :
% 0.30/0.63        ( ( ord_less_real @ A @ X3 )
% 0.30/0.63       => ( ( ord_less_real @ X3 @ B )
% 0.30/0.63         => ? [D4: real] :
% 0.30/0.63              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.63              & ! [Y6: real] :
% 0.30/0.63                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
% 0.30/0.63                 => ( ( ord_less_real @ A @ Y6 )
% 0.30/0.63                    & ( ord_less_real @ Y6 @ B ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_interval_lt
% 0.30/0.63  thf(fact_7622_arctan__double,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.63       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X3 ) )
% 0.30/0.63          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_double
% 0.30/0.63  thf(fact_7623_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( M = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_7624_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( M = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_7625_of__nat__eq__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( M = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_iff
% 0.30/0.63  thf(fact_7626_split__part,axiom,
% 0.30/0.63      ! [P: $o,Q: int > int > $o] :
% 0.30/0.63        ( ( produc4947309494688390418_int_o
% 0.30/0.63          @ ^ [A3: int,B3: int] :
% 0.30/0.63              ( P
% 0.30/0.63              & ( Q @ A3 @ B3 ) ) )
% 0.30/0.63        = ( ^ [Ab: product_prod_int_int] :
% 0.30/0.63              ( P
% 0.30/0.63              & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % split_part
% 0.30/0.63  thf(fact_7627_int__eq__iff__numeral,axiom,
% 0.30/0.63      ! [M: nat,V: num] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.63          = ( numeral_numeral_int @ V ) )
% 0.30/0.63        = ( M
% 0.30/0.63          = ( numeral_numeral_nat @ V ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_eq_iff_numeral
% 0.30/0.63  thf(fact_7628_abs__of__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_7629_abs__of__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
% 0.30/0.63        = ( semiri4939895301339042750nteger @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_7630_abs__of__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_7631_abs__of__nat,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % abs_of_nat
% 0.30/0.63  thf(fact_7632_negative__eq__positive,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ M ) )
% 0.30/0.63        = ( ( N2 = zero_zero_nat )
% 0.30/0.63          & ( M = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_eq_positive
% 0.30/0.63  thf(fact_7633_of__int__of__nat__eq,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_of_nat_eq
% 0.30/0.63  thf(fact_7634_of__int__of__nat__eq,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_of_nat_eq
% 0.30/0.63  thf(fact_7635_of__int__of__nat__eq,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_int_of_nat_eq
% 0.30/0.63  thf(fact_7636_negative__zle,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zle
% 0.30/0.63  thf(fact_7637_int__dvd__int__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_dvd_int_iff
% 0.30/0.63  thf(fact_7638_of__nat__eq__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ( semiri8010041392384452111omplex @ M )
% 0.30/0.63          = zero_zero_complex )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_0_iff
% 0.30/0.63  thf(fact_7639_of__nat__eq__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ( semiri681578069525770553at_rat @ M )
% 0.30/0.63          = zero_zero_rat )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_0_iff
% 0.30/0.63  thf(fact_7640_of__nat__eq__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.63          = zero_zero_int )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_0_iff
% 0.30/0.63  thf(fact_7641_of__nat__eq__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.63          = zero_zero_real )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_0_iff
% 0.30/0.63  thf(fact_7642_of__nat__eq__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ M )
% 0.30/0.63          = zero_zero_nat )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_0_iff
% 0.30/0.63  thf(fact_7643_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( zero_zero_complex
% 0.30/0.63          = ( semiri8010041392384452111omplex @ N2 ) )
% 0.30/0.63        = ( zero_zero_nat = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_7644_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( zero_zero_rat
% 0.30/0.63          = ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( zero_zero_nat = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_7645_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( zero_zero_int
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( zero_zero_nat = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_7646_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( zero_zero_real
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( zero_zero_nat = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_7647_of__nat__0__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( zero_zero_nat
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( zero_zero_nat = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_eq_iff
% 0.30/0.63  thf(fact_7648_of__nat__0,axiom,
% 0.30/0.63      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 0.30/0.63      = zero_zero_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0
% 0.30/0.63  thf(fact_7649_of__nat__0,axiom,
% 0.30/0.63      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 0.30/0.63      = zero_zero_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0
% 0.30/0.63  thf(fact_7650_of__nat__0,axiom,
% 0.30/0.63      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.30/0.63      = zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0
% 0.30/0.63  thf(fact_7651_of__nat__0,axiom,
% 0.30/0.63      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 0.30/0.63      = zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0
% 0.30/0.63  thf(fact_7652_of__nat__0,axiom,
% 0.30/0.63      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 0.30/0.63      = zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0
% 0.30/0.63  thf(fact_7653_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_7654_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_7655_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_7656_of__nat__less__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_iff
% 0.30/0.63  thf(fact_7657_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_7658_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_7659_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_7660_of__nat__le__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_iff
% 0.30/0.63  thf(fact_7661_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.63        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_7662_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_7663_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.63        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_7664_of__nat__add,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.63        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_add
% 0.30/0.63  thf(fact_7665_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.63        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_7666_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_7667_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_7668_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_7669_of__nat__mult,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mult
% 0.30/0.63  thf(fact_7670_of__nat__1,axiom,
% 0.30/0.63      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 0.30/0.63      = one_one_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1
% 0.30/0.63  thf(fact_7671_of__nat__1,axiom,
% 0.30/0.63      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 0.30/0.63      = one_one_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1
% 0.30/0.63  thf(fact_7672_of__nat__1,axiom,
% 0.30/0.63      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 0.30/0.63      = one_one_int ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1
% 0.30/0.63  thf(fact_7673_of__nat__1,axiom,
% 0.30/0.63      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 0.30/0.63      = one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1
% 0.30/0.63  thf(fact_7674_of__nat__1,axiom,
% 0.30/0.63      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 0.30/0.63      = one_one_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1
% 0.30/0.63  thf(fact_7675_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( one_one_complex
% 0.30/0.63          = ( semiri8010041392384452111omplex @ N2 ) )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_7676_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( one_one_rat
% 0.30/0.63          = ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_7677_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( one_one_int
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_7678_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( one_one_real
% 0.30/0.63          = ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_7679_of__nat__1__eq__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( one_one_nat
% 0.30/0.63          = ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_1_eq_iff
% 0.30/0.63  thf(fact_7680_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( semiri8010041392384452111omplex @ N2 )
% 0.30/0.63          = one_one_complex )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_7681_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( semiri681578069525770553at_rat @ N2 )
% 0.30/0.63          = one_one_rat )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_7682_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ N2 )
% 0.30/0.63          = one_one_int )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_7683_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( semiri5074537144036343181t_real @ N2 )
% 0.30/0.63          = one_one_real )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_7684_of__nat__eq__1__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( semiri1316708129612266289at_nat @ N2 )
% 0.30/0.63          = one_one_nat )
% 0.30/0.63        = ( N2 = one_one_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_eq_1_iff
% 0.30/0.63  thf(fact_7685_negative__zless,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zless
% 0.30/0.63  thf(fact_7686_arctan__less__zero__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_less_zero_iff
% 0.30/0.63  thf(fact_7687_zero__less__arctan__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
% 0.30/0.63        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_less_arctan_iff
% 0.30/0.63  thf(fact_7688_of__nat__of__bool,axiom,
% 0.30/0.63      ! [P: $o] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_of_bool
% 0.30/0.63  thf(fact_7689_of__nat__of__bool,axiom,
% 0.30/0.63      ! [P: $o] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_of_bool
% 0.30/0.63  thf(fact_7690_of__nat__of__bool,axiom,
% 0.30/0.63      ! [P: $o] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 0.30/0.63        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_of_bool
% 0.30/0.63  thf(fact_7691_of__nat__sum,axiom,
% 0.30/0.63      ! [F: complex > nat,A2: set_complex] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
% 0.30/0.63        = ( groups7754918857620584856omplex
% 0.30/0.63          @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
% 0.30/0.63          @ A2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_sum
% 0.30/0.63  thf(fact_7692_of__nat__sum,axiom,
% 0.30/0.63      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.30/0.63        = ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 0.30/0.63          @ A2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_sum
% 0.30/0.63  thf(fact_7693_of__nat__sum,axiom,
% 0.30/0.63      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
% 0.30/0.63          @ A2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_sum
% 0.30/0.63  thf(fact_7694_of__nat__sum,axiom,
% 0.30/0.63      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
% 0.30/0.63          @ A2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_sum
% 0.30/0.63  thf(fact_7695_of__nat__le__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_0_iff
% 0.30/0.63  thf(fact_7696_of__nat__le__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_0_iff
% 0.30/0.63  thf(fact_7697_of__nat__le__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_0_iff
% 0.30/0.63  thf(fact_7698_of__nat__le__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 0.30/0.63        = ( M = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_0_iff
% 0.30/0.63  thf(fact_7699_of__nat__Suc,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 0.30/0.63        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_Suc
% 0.30/0.63  thf(fact_7700_of__nat__Suc,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 0.30/0.63        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_Suc
% 0.30/0.63  thf(fact_7701_of__nat__Suc,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 0.30/0.63        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_Suc
% 0.30/0.63  thf(fact_7702_of__nat__Suc,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 0.30/0.63        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_Suc
% 0.30/0.63  thf(fact_7703_of__nat__Suc,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 0.30/0.63        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_Suc
% 0.30/0.63  thf(fact_7704_real__of__nat__less__numeral__iff,axiom,
% 0.30/0.63      ! [N2: nat,W2: num] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
% 0.30/0.63        = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_of_nat_less_numeral_iff
% 0.30/0.63  thf(fact_7705_numeral__less__real__of__nat__iff,axiom,
% 0.30/0.63      ! [W2: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_less_real_of_nat_iff
% 0.30/0.63  thf(fact_7706_numeral__le__real__of__nat__iff,axiom,
% 0.30/0.63      ! [N2: num,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_le_real_of_nat_iff
% 0.30/0.63  thf(fact_7707_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7708_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7709_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7710_of__nat__0__less__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_less_iff
% 0.30/0.63  thf(fact_7711_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7712_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7713_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7714_of__nat__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7715_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7716_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7717_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7718_of__nat__less__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7719_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7720_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7721_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7722_of__nat__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,B: nat,W2: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7723_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7724_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7725_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7726_of__nat__le__of__nat__power__cancel__iff,axiom,
% 0.30/0.63      ! [B: nat,W2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_of_nat_power_cancel_iff
% 0.30/0.63  thf(fact_7727_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N2 ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7728_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N2 ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7729_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N2 ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7730_of__nat__zero__less__power__iff,axiom,
% 0.30/0.63      ! [X3: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N2 ) )
% 0.30/0.63        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_zero_less_power_iff
% 0.30/0.63  thf(fact_7731_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7732_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7733_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7734_numeral__power__less__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_less_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7735_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7736_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7737_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7738_of__nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7739_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7740_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7741_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7742_of__nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.63      ! [X3: nat,I: num,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_le_numeral_power_cancel_iff
% 0.30/0.63  thf(fact_7743_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7744_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7745_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7746_numeral__power__le__of__nat__cancel__iff,axiom,
% 0.30/0.63      ! [I: num,N2: nat,X3: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % numeral_power_le_of_nat_cancel_iff
% 0.30/0.63  thf(fact_7747_prod_Odisc__eq__case,axiom,
% 0.30/0.63      ! [Prod: product_prod_int_int] :
% 0.30/0.63        ( produc4947309494688390418_int_o
% 0.30/0.63        @ ^ [Uu3: int,Uv3: int] : $true
% 0.30/0.63        @ Prod ) ).
% 0.30/0.63  
% 0.30/0.63  % prod.disc_eq_case
% 0.30/0.63  thf(fact_7748_int__sum,axiom,
% 0.30/0.63      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 0.30/0.63        = ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 0.30/0.63          @ A2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_sum
% 0.30/0.63  thf(fact_7749_real__arch__simple,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63      ? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_arch_simple
% 0.30/0.63  thf(fact_7750_real__arch__simple,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63      ? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_arch_simple
% 0.30/0.63  thf(fact_7751_reals__Archimedean2,axiom,
% 0.30/0.63      ! [X3: rat] :
% 0.30/0.63      ? [N3: nat] : ( ord_less_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % reals_Archimedean2
% 0.30/0.63  thf(fact_7752_reals__Archimedean2,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63      ? [N3: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % reals_Archimedean2
% 0.30/0.63  thf(fact_7753_mult__of__nat__commute,axiom,
% 0.30/0.63      ! [X3: nat,Y3: complex] :
% 0.30/0.63        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X3 ) @ Y3 )
% 0.30/0.63        = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mult_of_nat_commute
% 0.30/0.63  thf(fact_7754_mult__of__nat__commute,axiom,
% 0.30/0.63      ! [X3: nat,Y3: rat] :
% 0.30/0.63        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y3 )
% 0.30/0.63        = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mult_of_nat_commute
% 0.30/0.63  thf(fact_7755_mult__of__nat__commute,axiom,
% 0.30/0.63      ! [X3: nat,Y3: int] :
% 0.30/0.63        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
% 0.30/0.63        = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mult_of_nat_commute
% 0.30/0.63  thf(fact_7756_mult__of__nat__commute,axiom,
% 0.30/0.63      ! [X3: nat,Y3: real] :
% 0.30/0.63        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y3 )
% 0.30/0.63        = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mult_of_nat_commute
% 0.30/0.63  thf(fact_7757_mult__of__nat__commute,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
% 0.30/0.63        = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mult_of_nat_commute
% 0.30/0.63  thf(fact_7758_int__cases2,axiom,
% 0.30/0.63      ! [Z2: int] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( Z2
% 0.30/0.63           != ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( Z2
% 0.30/0.63             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_cases2
% 0.30/0.63  thf(fact_7759_arctan__less__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 0.30/0.63        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_less_iff
% 0.30/0.63  thf(fact_7760_arctan__monotone,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.63       => ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_monotone
% 0.30/0.63  thf(fact_7761_int__diff__cases,axiom,
% 0.30/0.63      ! [Z2: int] :
% 0.30/0.63        ~ ! [M3: nat,N3: nat] :
% 0.30/0.63            ( Z2
% 0.30/0.63           != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_diff_cases
% 0.30/0.63  thf(fact_7762_of__nat__less__of__int__iff,axiom,
% 0.30/0.63      ! [N2: nat,X3: int] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 0.30/0.63        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_int_iff
% 0.30/0.63  thf(fact_7763_of__nat__less__of__int__iff,axiom,
% 0.30/0.63      ! [N2: nat,X3: int] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X3 ) )
% 0.30/0.63        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_int_iff
% 0.30/0.63  thf(fact_7764_of__nat__less__of__int__iff,axiom,
% 0.30/0.63      ! [N2: nat,X3: int] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X3 ) )
% 0.30/0.63        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_of_int_iff
% 0.30/0.63  thf(fact_7765_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7766_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7767_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7768_of__nat__0__le__iff,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_0_le_iff
% 0.30/0.63  thf(fact_7769_of__nat__less__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_0_iff
% 0.30/0.63  thf(fact_7770_of__nat__less__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_0_iff
% 0.30/0.63  thf(fact_7771_of__nat__less__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_0_iff
% 0.30/0.63  thf(fact_7772_of__nat__less__0__iff,axiom,
% 0.30/0.63      ! [M: nat] :
% 0.30/0.63        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_0_iff
% 0.30/0.63  thf(fact_7773_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
% 0.30/0.63       != zero_zero_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_neq_0
% 0.30/0.63  thf(fact_7774_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
% 0.30/0.63       != zero_zero_rat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_neq_0
% 0.30/0.63  thf(fact_7775_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 0.30/0.63       != zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_neq_0
% 0.30/0.63  thf(fact_7776_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
% 0.30/0.63       != zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_neq_0
% 0.30/0.63  thf(fact_7777_of__nat__neq__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
% 0.30/0.63       != zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_neq_0
% 0.30/0.63  thf(fact_7778_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7779_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7780_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7781_of__nat__less__imp__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 0.30/0.63       => ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_imp_less
% 0.30/0.63  thf(fact_7782_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.63       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7783_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.63       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7784_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.63       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7785_less__imp__of__nat__less,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ M @ N2 )
% 0.30/0.63       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % less_imp_of_nat_less
% 0.30/0.63  thf(fact_7786_of__nat__mono,axiom,
% 0.30/0.63      ! [I: nat,J: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.63       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mono
% 0.30/0.63  thf(fact_7787_of__nat__mono,axiom,
% 0.30/0.63      ! [I: nat,J: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.63       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mono
% 0.30/0.63  thf(fact_7788_of__nat__mono,axiom,
% 0.30/0.63      ! [I: nat,J: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.63       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mono
% 0.30/0.63  thf(fact_7789_of__nat__mono,axiom,
% 0.30/0.63      ! [I: nat,J: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.63       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_mono
% 0.30/0.63  thf(fact_7790_int__ops_I1_J,axiom,
% 0.30/0.63      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 0.30/0.63      = zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(1)
% 0.30/0.63  thf(fact_7791_nat__int__comparison_I2_J,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_int_comparison(2)
% 0.30/0.63  thf(fact_7792_int__of__nat__induct,axiom,
% 0.30/0.63      ! [P: int > $o,Z2: int] :
% 0.30/0.63        ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63       => ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
% 0.30/0.63         => ( P @ Z2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_of_nat_induct
% 0.30/0.63  thf(fact_7793_int__cases,axiom,
% 0.30/0.63      ! [Z2: int] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( Z2
% 0.30/0.63           != ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( Z2
% 0.30/0.63             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_cases
% 0.30/0.63  thf(fact_7794_zle__int,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zle_int
% 0.30/0.63  thf(fact_7795_nat__int__comparison_I3_J,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_int_comparison(3)
% 0.30/0.63  thf(fact_7796_zero__le__imp__eq__int,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.63       => ? [N3: nat] :
% 0.30/0.63            ( K
% 0.30/0.63            = ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_imp_eq_int
% 0.30/0.63  thf(fact_7797_nonneg__int__cases,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( K
% 0.30/0.63             != ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonneg_int_cases
% 0.30/0.63  thf(fact_7798_zadd__int__left,axiom,
% 0.30/0.63      ! [M: nat,N2: nat,Z2: int] :
% 0.30/0.63        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z2 ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zadd_int_left
% 0.30/0.63  thf(fact_7799_int__ops_I5_J,axiom,
% 0.30/0.63      ! [A: nat,B: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(5)
% 0.30/0.63  thf(fact_7800_int__plus,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_plus
% 0.30/0.63  thf(fact_7801_zle__iff__zadd,axiom,
% 0.30/0.63      ( ord_less_eq_int
% 0.30/0.63      = ( ^ [W3: int,Z6: int] :
% 0.30/0.63          ? [N: nat] :
% 0.30/0.63            ( Z6
% 0.30/0.63            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zle_iff_zadd
% 0.30/0.63  thf(fact_7802_not__int__zless__negative,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % not_int_zless_negative
% 0.30/0.63  thf(fact_7803_nat__less__as__int,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_less_as_int
% 0.30/0.63  thf(fact_7804_nat__leq__as__int,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_leq_as_int
% 0.30/0.63  thf(fact_7805_ex__less__of__nat__mult,axiom,
% 0.30/0.63      ! [X3: rat,Y3: rat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.63       => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ex_less_of_nat_mult
% 0.30/0.63  thf(fact_7806_ex__less__of__nat__mult,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ex_less_of_nat_mult
% 0.30/0.63  thf(fact_7807_of__nat__diff,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7808_of__nat__diff,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7809_of__nat__diff,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7810_of__nat__diff,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
% 0.30/0.63          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_diff
% 0.30/0.63  thf(fact_7811_reals__Archimedean3,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ! [Y6: real] :
% 0.30/0.63          ? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % reals_Archimedean3
% 0.30/0.63  thf(fact_7812_int__cases4,axiom,
% 0.30/0.63      ! [M: int] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( M
% 0.30/0.63           != ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.63             => ( M
% 0.30/0.63               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_cases4
% 0.30/0.63  thf(fact_7813_int__zle__neg,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.63        = ( ( N2 = zero_zero_nat )
% 0.30/0.63          & ( M = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_zle_neg
% 0.30/0.63  thf(fact_7814_int__ops_I4_J,axiom,
% 0.30/0.63      ! [A: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(4)
% 0.30/0.63  thf(fact_7815_int__Suc,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 0.30/0.63        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_Suc
% 0.30/0.63  thf(fact_7816_zless__iff__Suc__zadd,axiom,
% 0.30/0.63      ( ord_less_int
% 0.30/0.63      = ( ^ [W3: int,Z6: int] :
% 0.30/0.63          ? [N: nat] :
% 0.30/0.63            ( Z6
% 0.30/0.63            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zless_iff_Suc_zadd
% 0.30/0.63  thf(fact_7817_negative__zle__0,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zle_0
% 0.30/0.63  thf(fact_7818_nonpos__int__cases,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( K
% 0.30/0.63             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonpos_int_cases
% 0.30/0.63  thf(fact_7819_sum__roots__unity,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.30/0.63       => ( ( groups7754918857620584856omplex
% 0.30/0.63            @ ^ [X: complex] : X
% 0.30/0.63            @ ( collect_complex
% 0.30/0.63              @ ^ [Z6: complex] :
% 0.30/0.63                  ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.63                  = one_one_complex ) ) )
% 0.30/0.63          = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_roots_unity
% 0.30/0.63  thf(fact_7820_sum__nth__roots,axiom,
% 0.30/0.63      ! [N2: nat,C: complex] :
% 0.30/0.63        ( ( ord_less_nat @ one_one_nat @ N2 )
% 0.30/0.63       => ( ( groups7754918857620584856omplex
% 0.30/0.63            @ ^ [X: complex] : X
% 0.30/0.63            @ ( collect_complex
% 0.30/0.63              @ ^ [Z6: complex] :
% 0.30/0.63                  ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.63                  = C ) ) )
% 0.30/0.63          = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_nth_roots
% 0.30/0.63  thf(fact_7821_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: code_integer,M: nat,N2: nat] :
% 0.30/0.63        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
% 0.30/0.63        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7822_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: int,M: nat,N2: nat] :
% 0.30/0.63        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7823_mod__mult2__eq_H,axiom,
% 0.30/0.63      ! [A: nat,M: nat,N2: nat] :
% 0.30/0.63        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % mod_mult2_eq'
% 0.30/0.63  thf(fact_7824_pos__int__cases,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( ( K
% 0.30/0.63                = ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % pos_int_cases
% 0.30/0.63  thf(fact_7825_zero__less__imp__eq__int,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_int @ zero_zero_int @ K )
% 0.30/0.63       => ? [N3: nat] :
% 0.30/0.63            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 0.30/0.63            & ( K
% 0.30/0.63              = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_less_imp_eq_int
% 0.30/0.63  thf(fact_7826_int__cases3,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( K != zero_zero_int )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ( K
% 0.30/0.63                = ( semiri1314217659103216013at_int @ N3 ) )
% 0.30/0.63             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 0.30/0.63         => ~ ! [N3: nat] :
% 0.30/0.63                ( ( K
% 0.30/0.63                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.30/0.63               => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_cases3
% 0.30/0.63  thf(fact_7827_nat__less__real__le,axiom,
% 0.30/0.63      ( ord_less_nat
% 0.30/0.63      = ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_less_real_le
% 0.30/0.63  thf(fact_7828_nat__le__real__less,axiom,
% 0.30/0.63      ( ord_less_eq_nat
% 0.30/0.63      = ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_le_real_less
% 0.30/0.63  thf(fact_7829_zmult__zless__mono2__lemma,axiom,
% 0.30/0.63      ! [I: int,J: int,K: nat] :
% 0.30/0.63        ( ( ord_less_int @ I @ J )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.63         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zmult_zless_mono2_lemma
% 0.30/0.63  thf(fact_7830_not__zle__0__negative,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % not_zle_0_negative
% 0.30/0.63  thf(fact_7831_negative__zless__0,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % negative_zless_0
% 0.30/0.63  thf(fact_7832_negD,axiom,
% 0.30/0.63      ! [X3: int] :
% 0.30/0.63        ( ( ord_less_int @ X3 @ zero_zero_int )
% 0.30/0.63       => ? [N3: nat] :
% 0.30/0.63            ( X3
% 0.30/0.63            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % negD
% 0.30/0.63  thf(fact_7833_int__ops_I6_J,axiom,
% 0.30/0.63      ! [A: nat,B: nat] :
% 0.30/0.63        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.63            = zero_zero_int ) )
% 0.30/0.63        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 0.30/0.63         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 0.30/0.63            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_ops(6)
% 0.30/0.63  thf(fact_7834_nat__approx__posE,axiom,
% 0.30/0.63      ! [E2: rat] :
% 0.30/0.63        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_approx_posE
% 0.30/0.63  thf(fact_7835_nat__approx__posE,axiom,
% 0.30/0.63      ! [E2: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nat_approx_posE
% 0.30/0.63  thf(fact_7836_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7837_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7838_of__nat__less__two__power,axiom,
% 0.30/0.63      ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_less_two_power
% 0.30/0.63  thf(fact_7839_inverse__of__nat__le,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( N2 != zero_zero_nat )
% 0.30/0.63         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % inverse_of_nat_le
% 0.30/0.63  thf(fact_7840_inverse__of__nat__le,axiom,
% 0.30/0.63      ! [N2: nat,M: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.63       => ( ( N2 != zero_zero_nat )
% 0.30/0.63         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % inverse_of_nat_le
% 0.30/0.63  thf(fact_7841_real__archimedian__rdiv__eq__0,axiom,
% 0.30/0.63      ! [X3: real,C: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 0.30/0.63         => ( ! [M3: nat] :
% 0.30/0.63                ( ( ord_less_nat @ zero_zero_nat @ M3 )
% 0.30/0.63               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X3 ) @ C ) )
% 0.30/0.63           => ( X3 = zero_zero_real ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_archimedian_rdiv_eq_0
% 0.30/0.63  thf(fact_7842_neg__int__cases,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.63       => ~ ! [N3: nat] :
% 0.30/0.63              ( ( K
% 0.30/0.63                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 0.30/0.63             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % neg_int_cases
% 0.30/0.63  thf(fact_7843_zdiff__int__split,axiom,
% 0.30/0.63      ! [P: int > $o,X3: nat,Y3: nat] :
% 0.30/0.63        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y3 ) ) )
% 0.30/0.63        = ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 0.30/0.63           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 0.30/0.63          & ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.63           => ( P @ zero_zero_int ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zdiff_int_split
% 0.30/0.63  thf(fact_7844_ln__realpow,axiom,
% 0.30/0.63      ! [X3: real,N2: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ln_ln_real @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.63          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ln_realpow
% 0.30/0.63  thf(fact_7845_arctan__add,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.63       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.63         => ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 0.30/0.63            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_add
% 0.30/0.63  thf(fact_7846_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7847_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7848_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7849_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7850_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7851_double__gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum
% 0.30/0.63  thf(fact_7852_double__arith__series,axiom,
% 0.30/0.63      ! [A: complex,D: complex,N2: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7853_double__arith__series,axiom,
% 0.30/0.63      ! [A: extended_enat,D: extended_enat,N2: nat] :
% 0.30/0.63        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups7108830773950497114d_enat
% 0.30/0.63            @ ^ [I2: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7854_double__arith__series,axiom,
% 0.30/0.63      ! [A: rat,D: rat,N2: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7855_double__arith__series,axiom,
% 0.30/0.63      ! [A: int,D: int,N2: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7856_double__arith__series,axiom,
% 0.30/0.63      ! [A: nat,D: nat,N2: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups3542108847815614940at_nat
% 0.30/0.63            @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7857_double__arith__series,axiom,
% 0.30/0.63      ! [A: real,D: real,N2: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I2 ) @ D ) )
% 0.30/0.63            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.63        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_arith_series
% 0.30/0.63  thf(fact_7858_gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum
% 0.30/0.63  thf(fact_7859_gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum
% 0.30/0.63  thf(fact_7860_gauss__sum,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % gauss_sum
% 0.30/0.63  thf(fact_7861_arith__series,axiom,
% 0.30/0.63      ! [A: code_integer,D: code_integer,N2: nat] :
% 0.30/0.63        ( ( groups7501900531339628137nteger
% 0.30/0.63          @ ^ [I2: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I2 ) @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series
% 0.30/0.63  thf(fact_7862_arith__series,axiom,
% 0.30/0.63      ! [A: int,D: int,N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series
% 0.30/0.63  thf(fact_7863_arith__series,axiom,
% 0.30/0.63      ! [A: nat,D: nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
% 0.30/0.63          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.63        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arith_series
% 0.30/0.63  thf(fact_7864_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7865_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7866_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7867_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7868_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7869_double__gauss__sum__from__Suc__0,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 0.30/0.63        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % double_gauss_sum_from_Suc_0
% 0.30/0.63  thf(fact_7870_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: complex,M: nat,N2: nat] :
% 0.30/0.63        ( ( ( X3 = one_one_complex )
% 0.30/0.63         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
% 0.30/0.63        & ( ( X3 != one_one_complex )
% 0.30/0.63         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7871_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: rat,M: nat,N2: nat] :
% 0.30/0.63        ( ( ( X3 = one_one_rat )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
% 0.30/0.63        & ( ( X3 != one_one_rat )
% 0.30/0.63         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7872_sum__gp__offset,axiom,
% 0.30/0.63      ! [X3: real,M: nat,N2: nat] :
% 0.30/0.63        ( ( ( X3 = one_one_real )
% 0.30/0.63         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
% 0.30/0.63        & ( ( X3 != one_one_real )
% 0.30/0.63         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 0.30/0.63            = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_gp_offset
% 0.30/0.63  thf(fact_7873_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri8010041392384452111omplex
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
% 0.30/0.63            @ ( produc1917071388513777916omplex
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7874_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri4216267220026989637d_enat
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_Extended_enat @ ( N = zero_zero_nat ) @ zero_z5237406670263579293d_enat
% 0.30/0.63            @ ( produc2676513652042109336d_enat
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7875_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri681578069525770553at_rat
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
% 0.30/0.63            @ ( produc6207742614233964070at_rat
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7876_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri1314217659103216013at_int
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
% 0.30/0.63            @ ( produc6840382203811409530at_int
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7877_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri5074537144036343181t_real
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 0.30/0.63            @ ( produc1703576794950452218t_real
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7878_of__nat__code__if,axiom,
% 0.30/0.63      ( semiri1316708129612266289at_nat
% 0.30/0.63      = ( ^ [N: nat] :
% 0.30/0.63            ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.63            @ ( produc6842872674320459806at_nat
% 0.30/0.63              @ ^ [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 ) )
% 0.30/0.63              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % of_nat_code_if
% 0.30/0.63  thf(fact_7879_monoseq__arctan__series,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.63       => ( topolo6980174941875973593q_real
% 0.30/0.63          @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % monoseq_arctan_series
% 0.30/0.63  thf(fact_7880_lemma__termdiff3,axiom,
% 0.30/0.63      ! [H2: real,Z2: real,K5: real,N2: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_real )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff3
% 0.30/0.63  thf(fact_7881_lemma__termdiff3,axiom,
% 0.30/0.63      ! [H2: complex,Z2: complex,K5: real,N2: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_complex )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff3
% 0.30/0.63  thf(fact_7882_ln__series,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.63         => ( ( ln_ln_real @ X3 )
% 0.30/0.63            = ( suminf_real
% 0.30/0.63              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ln_series
% 0.30/0.63  thf(fact_7883_lemma__termdiff2,axiom,
% 0.30/0.63      ! [H2: complex,Z2: complex,N2: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_complex )
% 0.30/0.63       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63          = ( times_times_complex @ H2
% 0.30/0.63            @ ( groups2073611262835488442omplex
% 0.30/0.63              @ ^ [P6: nat] :
% 0.30/0.63                  ( groups2073611262835488442omplex
% 0.30/0.63                  @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.30/0.63                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.30/0.63              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff2
% 0.30/0.63  thf(fact_7884_lemma__termdiff2,axiom,
% 0.30/0.63      ! [H2: rat,Z2: rat,N2: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_rat )
% 0.30/0.63       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N2 ) @ ( power_power_rat @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63          = ( times_times_rat @ H2
% 0.30/0.63            @ ( groups2906978787729119204at_rat
% 0.30/0.63              @ ^ [P6: nat] :
% 0.30/0.63                  ( groups2906978787729119204at_rat
% 0.30/0.63                  @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q5 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.30/0.63                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.30/0.63              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff2
% 0.30/0.63  thf(fact_7885_lemma__termdiff2,axiom,
% 0.30/0.63      ! [H2: real,Z2: real,N2: nat] :
% 0.30/0.63        ( ( H2 != zero_zero_real )
% 0.30/0.63       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 0.30/0.63          = ( times_times_real @ H2
% 0.30/0.63            @ ( groups6591440286371151544t_real
% 0.30/0.63              @ ^ [P6: nat] :
% 0.30/0.63                  ( groups6591440286371151544t_real
% 0.30/0.63                  @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q5 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 0.30/0.63                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 0.30/0.63              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff2
% 0.30/0.63  thf(fact_7886_arctan__series,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.63       => ( ( arctan @ X3 )
% 0.30/0.63          = ( suminf_real
% 0.30/0.63            @ ^ [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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % arctan_series
% 0.30/0.63  thf(fact_7887_lessThan__iff,axiom,
% 0.30/0.63      ! [I: set_nat,K: set_nat] :
% 0.30/0.63        ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 0.30/0.63        = ( ord_less_set_nat @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7888_lessThan__iff,axiom,
% 0.30/0.63      ! [I: rat,K: rat] :
% 0.30/0.63        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 0.30/0.63        = ( ord_less_rat @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7889_lessThan__iff,axiom,
% 0.30/0.63      ! [I: num,K: num] :
% 0.30/0.63        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 0.30/0.63        = ( ord_less_num @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7890_lessThan__iff,axiom,
% 0.30/0.63      ! [I: nat,K: nat] :
% 0.30/0.63        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 0.30/0.63        = ( ord_less_nat @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7891_lessThan__iff,axiom,
% 0.30/0.63      ! [I: int,K: int] :
% 0.30/0.63        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 0.30/0.63        = ( ord_less_int @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7892_lessThan__iff,axiom,
% 0.30/0.63      ! [I: real,K: real] :
% 0.30/0.63        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 0.30/0.63        = ( ord_less_real @ I @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_iff
% 0.30/0.63  thf(fact_7893_finite__lessThan,axiom,
% 0.30/0.63      ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 0.30/0.63  
% 0.30/0.63  % finite_lessThan
% 0.30/0.63  thf(fact_7894_lessThan__subset__iff,axiom,
% 0.30/0.63      ! [X3: rat,Y3: rat] :
% 0.30/0.63        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_subset_iff
% 0.30/0.63  thf(fact_7895_lessThan__subset__iff,axiom,
% 0.30/0.63      ! [X3: num,Y3: num] :
% 0.30/0.63        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_subset_iff
% 0.30/0.63  thf(fact_7896_lessThan__subset__iff,axiom,
% 0.30/0.63      ! [X3: nat,Y3: nat] :
% 0.30/0.63        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_subset_iff
% 0.30/0.63  thf(fact_7897_lessThan__subset__iff,axiom,
% 0.30/0.63      ! [X3: int,Y3: int] :
% 0.30/0.63        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_subset_iff
% 0.30/0.63  thf(fact_7898_lessThan__subset__iff,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 0.30/0.63        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_subset_iff
% 0.30/0.63  thf(fact_7899_lessThan__0,axiom,
% 0.30/0.63      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 0.30/0.63      = bot_bot_set_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_0
% 0.30/0.63  thf(fact_7900_single__Diff__lessThan,axiom,
% 0.30/0.63      ! [K: nat] :
% 0.30/0.63        ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 0.30/0.63        = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % single_Diff_lessThan
% 0.30/0.63  thf(fact_7901_single__Diff__lessThan,axiom,
% 0.30/0.63      ! [K: int] :
% 0.30/0.63        ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 0.30/0.63        = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % single_Diff_lessThan
% 0.30/0.63  thf(fact_7902_single__Diff__lessThan,axiom,
% 0.30/0.63      ! [K: real] :
% 0.30/0.63        ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 0.30/0.63        = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % single_Diff_lessThan
% 0.30/0.63  thf(fact_7903_sum_OlessThan__Suc,axiom,
% 0.30/0.63      ! [G: nat > rat,N2: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc
% 0.30/0.63  thf(fact_7904_sum_OlessThan__Suc,axiom,
% 0.30/0.63      ! [G: nat > int,N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc
% 0.30/0.63  thf(fact_7905_sum_OlessThan__Suc,axiom,
% 0.30/0.63      ! [G: nat > nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc
% 0.30/0.63  thf(fact_7906_sum_OlessThan__Suc,axiom,
% 0.30/0.63      ! [G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc
% 0.30/0.63  thf(fact_7907_powser__zero,axiom,
% 0.30/0.63      ! [F: nat > complex] :
% 0.30/0.63        ( ( suminf_complex
% 0.30/0.63          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
% 0.30/0.63        = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_zero
% 0.30/0.63  thf(fact_7908_powser__zero,axiom,
% 0.30/0.63      ! [F: nat > real] :
% 0.30/0.63        ( ( suminf_real
% 0.30/0.63          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
% 0.30/0.63        = ( F @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_zero
% 0.30/0.63  thf(fact_7909_int__int__eq,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.63          = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.63        = ( M = N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % int_int_eq
% 0.30/0.63  thf(fact_7910_lessThan__non__empty,axiom,
% 0.30/0.63      ! [X3: int] :
% 0.30/0.63        ( ( set_ord_lessThan_int @ X3 )
% 0.30/0.63       != bot_bot_set_int ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_non_empty
% 0.30/0.63  thf(fact_7911_lessThan__non__empty,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( set_or5984915006950818249n_real @ X3 )
% 0.30/0.63       != bot_bot_set_real ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_non_empty
% 0.30/0.63  thf(fact_7912_infinite__Iio,axiom,
% 0.30/0.63      ! [A: int] :
% 0.30/0.63        ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % infinite_Iio
% 0.30/0.63  thf(fact_7913_infinite__Iio,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% 0.30/0.63  
% 0.30/0.63  % infinite_Iio
% 0.30/0.63  thf(fact_7914_lessThan__def,axiom,
% 0.30/0.63      ( set_or890127255671739683et_nat
% 0.30/0.63      = ( ^ [U2: set_nat] :
% 0.30/0.63            ( collect_set_nat
% 0.30/0.63            @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7915_lessThan__def,axiom,
% 0.30/0.63      ( set_ord_lessThan_rat
% 0.30/0.63      = ( ^ [U2: rat] :
% 0.30/0.63            ( collect_rat
% 0.30/0.63            @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7916_lessThan__def,axiom,
% 0.30/0.63      ( set_ord_lessThan_num
% 0.30/0.63      = ( ^ [U2: num] :
% 0.30/0.63            ( collect_num
% 0.30/0.63            @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7917_lessThan__def,axiom,
% 0.30/0.63      ( set_ord_lessThan_nat
% 0.30/0.63      = ( ^ [U2: nat] :
% 0.30/0.63            ( collect_nat
% 0.30/0.63            @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7918_lessThan__def,axiom,
% 0.30/0.63      ( set_ord_lessThan_int
% 0.30/0.63      = ( ^ [U2: int] :
% 0.30/0.63            ( collect_int
% 0.30/0.63            @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7919_lessThan__def,axiom,
% 0.30/0.63      ( set_or5984915006950818249n_real
% 0.30/0.63      = ( ^ [U2: real] :
% 0.30/0.63            ( collect_real
% 0.30/0.63            @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_def
% 0.30/0.63  thf(fact_7920_Iio__eq__empty__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( set_ord_lessThan_nat @ N2 )
% 0.30/0.63          = bot_bot_set_nat )
% 0.30/0.63        = ( N2 = bot_bot_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % Iio_eq_empty_iff
% 0.30/0.63  thf(fact_7921_lessThan__strict__subset__iff,axiom,
% 0.30/0.63      ! [M: rat,N2: rat] :
% 0.30/0.63        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
% 0.30/0.63        = ( ord_less_rat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_strict_subset_iff
% 0.30/0.63  thf(fact_7922_lessThan__strict__subset__iff,axiom,
% 0.30/0.63      ! [M: num,N2: num] :
% 0.30/0.63        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
% 0.30/0.63        = ( ord_less_num @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_strict_subset_iff
% 0.30/0.63  thf(fact_7923_lessThan__strict__subset__iff,axiom,
% 0.30/0.63      ! [M: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.63        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_strict_subset_iff
% 0.30/0.63  thf(fact_7924_lessThan__strict__subset__iff,axiom,
% 0.30/0.63      ! [M: int,N2: int] :
% 0.30/0.63        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
% 0.30/0.63        = ( ord_less_int @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_strict_subset_iff
% 0.30/0.63  thf(fact_7925_lessThan__strict__subset__iff,axiom,
% 0.30/0.63      ! [M: real,N2: real] :
% 0.30/0.63        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
% 0.30/0.63        = ( ord_less_real @ M @ N2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_strict_subset_iff
% 0.30/0.63  thf(fact_7926_lessThan__Suc,axiom,
% 0.30/0.63      ! [K: nat] :
% 0.30/0.63        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.30/0.63        = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_Suc
% 0.30/0.63  thf(fact_7927_lessThan__empty__iff,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( ( set_ord_lessThan_nat @ N2 )
% 0.30/0.63          = bot_bot_set_nat )
% 0.30/0.63        = ( N2 = zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lessThan_empty_iff
% 0.30/0.63  thf(fact_7928_finite__nat__iff__bounded,axiom,
% 0.30/0.63      ( finite_finite_nat
% 0.30/0.63      = ( ^ [S7: set_nat] :
% 0.30/0.63          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % finite_nat_iff_bounded
% 0.30/0.63  thf(fact_7929_finite__nat__bounded,axiom,
% 0.30/0.63      ! [S3: set_nat] :
% 0.30/0.63        ( ( finite_finite_nat @ S3 )
% 0.30/0.63       => ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % finite_nat_bounded
% 0.30/0.63  thf(fact_7930_sum_Onat__diff__reindex,axiom,
% 0.30/0.63      ! [G: nat > nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_diff_reindex
% 0.30/0.63  thf(fact_7931_sum_Onat__diff__reindex,axiom,
% 0.30/0.63      ! [G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.63        = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.nat_diff_reindex
% 0.30/0.63  thf(fact_7932_sum__diff__distrib,axiom,
% 0.30/0.63      ! [Q: int > nat,P: int > nat,N2: int] :
% 0.30/0.63        ( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.30/0.63       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
% 0.30/0.63          = ( groups4541462559716669496nt_nat
% 0.30/0.63            @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63            @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_distrib
% 0.30/0.63  thf(fact_7933_sum__diff__distrib,axiom,
% 0.30/0.63      ! [Q: real > nat,P: real > nat,N2: real] :
% 0.30/0.63        ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.30/0.63       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
% 0.30/0.63          = ( groups1935376822645274424al_nat
% 0.30/0.63            @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63            @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_distrib
% 0.30/0.63  thf(fact_7934_sum__diff__distrib,axiom,
% 0.30/0.63      ! [Q: nat > nat,P: nat > nat,N2: nat] :
% 0.30/0.63        ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 0.30/0.63       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
% 0.30/0.63          = ( groups3542108847815614940at_nat
% 0.30/0.63            @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_diff_distrib
% 0.30/0.63  thf(fact_7935_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63      ! [G: nat > rat,N2: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc_shift
% 0.30/0.63  thf(fact_7936_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63      ! [G: nat > int,N2: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc_shift
% 0.30/0.63  thf(fact_7937_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63      ! [G: nat > nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 0.30/0.63          @ ( groups3542108847815614940at_nat
% 0.30/0.63            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc_shift
% 0.30/0.63  thf(fact_7938_sum_OlessThan__Suc__shift,axiom,
% 0.30/0.63      ! [G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.lessThan_Suc_shift
% 0.30/0.63  thf(fact_7939_sum__lessThan__telescope,axiom,
% 0.30/0.63      ! [F: nat > rat,M: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope
% 0.30/0.63  thf(fact_7940_sum__lessThan__telescope,axiom,
% 0.30/0.63      ! [F: nat > int,M: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope
% 0.30/0.63  thf(fact_7941_sum__lessThan__telescope,axiom,
% 0.30/0.63      ! [F: nat > real,M: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope
% 0.30/0.63  thf(fact_7942_sum__lessThan__telescope_H,axiom,
% 0.30/0.63      ! [F: nat > rat,M: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope'
% 0.30/0.63  thf(fact_7943_sum__lessThan__telescope_H,axiom,
% 0.30/0.63      ! [F: nat > int,M: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope'
% 0.30/0.63  thf(fact_7944_sum__lessThan__telescope_H,axiom,
% 0.30/0.63      ! [F: nat > real,M: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_lessThan_telescope'
% 0.30/0.63  thf(fact_7945_sum_OatLeast1__atMost__eq,axiom,
% 0.30/0.63      ! [G: nat > nat,N2: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast1_atMost_eq
% 0.30/0.63  thf(fact_7946_sum_OatLeast1__atMost__eq,axiom,
% 0.30/0.63      ! [G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum.atLeast1_atMost_eq
% 0.30/0.63  thf(fact_7947_lemma__termdiff1,axiom,
% 0.30/0.63      ! [Z2: complex,H2: complex,M: nat] :
% 0.30/0.63        ( ( groups2073611262835488442omplex
% 0.30/0.63          @ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ P6 ) ) @ ( power_power_complex @ Z2 @ M ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( groups2073611262835488442omplex
% 0.30/0.63          @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff1
% 0.30/0.63  thf(fact_7948_lemma__termdiff1,axiom,
% 0.30/0.63      ! [Z2: rat,H2: rat,M: nat] :
% 0.30/0.63        ( ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [P6: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ P6 ) ) @ ( power_power_rat @ Z2 @ M ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( groups2906978787729119204at_rat
% 0.30/0.63          @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P6 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff1
% 0.30/0.63  thf(fact_7949_lemma__termdiff1,axiom,
% 0.30/0.63      ! [Z2: int,H2: int,M: nat] :
% 0.30/0.63        ( ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ P6 ) ) @ ( power_power_int @ Z2 @ M ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( groups3539618377306564664at_int
% 0.30/0.63          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff1
% 0.30/0.63  thf(fact_7950_lemma__termdiff1,axiom,
% 0.30/0.63      ! [Z2: real,H2: real,M: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ P6 ) ) @ ( power_power_real @ Z2 @ M ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) )
% 0.30/0.63        = ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % lemma_termdiff1
% 0.30/0.63  thf(fact_7951_diff__power__eq__sum,axiom,
% 0.30/0.63      ! [X3: complex,N2: nat,Y3: complex] :
% 0.30/0.63        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) @ ( power_power_complex @ Y3 @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.63          @ ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P6 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % diff_power_eq_sum
% 0.30/0.63  thf(fact_7952_diff__power__eq__sum,axiom,
% 0.30/0.63      ! [X3: rat,N2: nat,Y3: rat] :
% 0.30/0.63        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) @ ( power_power_rat @ Y3 @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P6 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % diff_power_eq_sum
% 0.30/0.63  thf(fact_7953_diff__power__eq__sum,axiom,
% 0.30/0.63      ! [X3: int,N2: nat,Y3: int] :
% 0.30/0.63        ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N2 ) ) @ ( power_power_int @ Y3 @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X3 @ P6 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % diff_power_eq_sum
% 0.30/0.63  thf(fact_7954_diff__power__eq__sum,axiom,
% 0.30/0.63      ! [X3: real,N2: nat,Y3: real] :
% 0.30/0.63        ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) @ ( power_power_real @ Y3 @ ( suc @ N2 ) ) )
% 0.30/0.63        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X3 @ P6 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % diff_power_eq_sum
% 0.30/0.63  thf(fact_7955_power__diff__sumr2,axiom,
% 0.30/0.63      ! [X3: complex,N2: nat,Y3: complex] :
% 0.30/0.63        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ N2 ) )
% 0.30/0.63        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 0.30/0.63          @ ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [I2: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_complex @ X3 @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_diff_sumr2
% 0.30/0.63  thf(fact_7956_power__diff__sumr2,axiom,
% 0.30/0.63      ! [X3: rat,N2: nat,Y3: rat] :
% 0.30/0.63        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N2 ) @ ( power_power_rat @ Y3 @ N2 ) )
% 0.30/0.63        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I2: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_rat @ X3 @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_diff_sumr2
% 0.30/0.63  thf(fact_7957_power__diff__sumr2,axiom,
% 0.30/0.63      ! [X3: int,N2: nat,Y3: int] :
% 0.30/0.63        ( ( minus_minus_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ N2 ) )
% 0.30/0.63        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I2: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_int @ X3 @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_diff_sumr2
% 0.30/0.63  thf(fact_7958_power__diff__sumr2,axiom,
% 0.30/0.63      ! [X3: real,N2: nat,Y3: real] :
% 0.30/0.63        ( ( minus_minus_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ N2 ) )
% 0.30/0.63        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_real @ X3 @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_diff_sumr2
% 0.30/0.63  thf(fact_7959_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
% 0.30/0.63        ( ! [P7: nat] :
% 0.30/0.63            ( ( ord_less_nat @ P7 @ N2 )
% 0.30/0.63           => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63       => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 0.30/0.63         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_sum_nat_ivl_bounded2
% 0.30/0.63  thf(fact_7960_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > int,K5: int,K: nat] :
% 0.30/0.63        ( ! [P7: nat] :
% 0.30/0.63            ( ( ord_less_nat @ P7 @ N2 )
% 0.30/0.63           => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63       => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 0.30/0.63         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_sum_nat_ivl_bounded2
% 0.30/0.63  thf(fact_7961_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
% 0.30/0.63        ( ! [P7: nat] :
% 0.30/0.63            ( ( ord_less_nat @ P7 @ N2 )
% 0.30/0.63           => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63       => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 0.30/0.63         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_sum_nat_ivl_bounded2
% 0.30/0.63  thf(fact_7962_real__sum__nat__ivl__bounded2,axiom,
% 0.30/0.63      ! [N2: nat,F: nat > real,K5: real,K: nat] :
% 0.30/0.63        ( ! [P7: nat] :
% 0.30/0.63            ( ( ord_less_nat @ P7 @ N2 )
% 0.30/0.63           => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 0.30/0.63       => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 0.30/0.63         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % real_sum_nat_ivl_bounded2
% 0.30/0.63  thf(fact_7963_one__diff__power__eq_H,axiom,
% 0.30/0.63      ! [X3: complex,N2: nat] :
% 0.30/0.63        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N2 ) )
% 0.30/0.63        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
% 0.30/0.63          @ ( groups2073611262835488442omplex
% 0.30/0.63            @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_diff_power_eq'
% 0.30/0.63  thf(fact_7964_one__diff__power__eq_H,axiom,
% 0.30/0.63      ! [X3: rat,N2: nat] :
% 0.30/0.63        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N2 ) )
% 0.30/0.63        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
% 0.30/0.63          @ ( groups2906978787729119204at_rat
% 0.30/0.63            @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_diff_power_eq'
% 0.30/0.63  thf(fact_7965_one__diff__power__eq_H,axiom,
% 0.30/0.63      ! [X3: int,N2: nat] :
% 0.30/0.63        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N2 ) )
% 0.30/0.63        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
% 0.30/0.63          @ ( groups3539618377306564664at_int
% 0.30/0.63            @ ^ [I2: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_diff_power_eq'
% 0.30/0.63  thf(fact_7966_one__diff__power__eq_H,axiom,
% 0.30/0.63      ! [X3: real,N2: nat] :
% 0.30/0.63        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.63        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_diff_power_eq'
% 0.30/0.63  thf(fact_7967_sum__split__even__odd,axiom,
% 0.30/0.63      ! [F: nat > real,G: nat > real,N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.63        = ( plus_plus_real
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.63          @ ( groups6591440286371151544t_real
% 0.30/0.63            @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 0.30/0.63            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_split_even_odd
% 0.30/0.63  thf(fact_7968_norm__le__zero__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_le_zero_iff
% 0.30/0.63  thf(fact_7969_norm__le__zero__iff,axiom,
% 0.30/0.63      ! [X3: complex] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
% 0.30/0.63        = ( X3 = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_le_zero_iff
% 0.30/0.63  thf(fact_7970_zero__less__norm__iff,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
% 0.30/0.63        = ( X3 != zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_less_norm_iff
% 0.30/0.63  thf(fact_7971_zero__less__norm__iff,axiom,
% 0.30/0.63      ! [X3: complex] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
% 0.30/0.63        = ( X3 != zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_less_norm_iff
% 0.30/0.63  thf(fact_7972_suminf__geometric,axiom,
% 0.30/0.63      ! [C: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.63       => ( ( suminf_real @ ( power_power_real @ C ) )
% 0.30/0.63          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_geometric
% 0.30/0.63  thf(fact_7973_suminf__geometric,axiom,
% 0.30/0.63      ! [C: complex] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.63       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 0.30/0.63          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_geometric
% 0.30/0.63  thf(fact_7974_norm__zero,axiom,
% 0.30/0.63      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 0.30/0.63      = zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_zero
% 0.30/0.63  thf(fact_7975_norm__zero,axiom,
% 0.30/0.63      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 0.30/0.63      = zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_zero
% 0.30/0.63  thf(fact_7976_norm__eq__zero,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ( real_V7735802525324610683m_real @ X3 )
% 0.30/0.63          = zero_zero_real )
% 0.30/0.63        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_eq_zero
% 0.30/0.63  thf(fact_7977_norm__eq__zero,axiom,
% 0.30/0.63      ! [X3: complex] :
% 0.30/0.63        ( ( ( real_V1022390504157884413omplex @ X3 )
% 0.30/0.63          = zero_zero_real )
% 0.30/0.63        = ( X3 = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_eq_zero
% 0.30/0.63  thf(fact_7978_suminf__zero,axiom,
% 0.30/0.63      ( ( suminf_complex
% 0.30/0.63        @ ^ [N: nat] : zero_zero_complex )
% 0.30/0.63      = zero_zero_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_zero
% 0.30/0.63  thf(fact_7979_suminf__zero,axiom,
% 0.30/0.63      ( ( suminf_real
% 0.30/0.63        @ ^ [N: nat] : zero_zero_real )
% 0.30/0.63      = zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_zero
% 0.30/0.63  thf(fact_7980_suminf__zero,axiom,
% 0.30/0.63      ( ( suminf_nat
% 0.30/0.63        @ ^ [N: nat] : zero_zero_nat )
% 0.30/0.63      = zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_zero
% 0.30/0.63  thf(fact_7981_suminf__zero,axiom,
% 0.30/0.63      ( ( suminf_int
% 0.30/0.63        @ ^ [N: nat] : zero_zero_int )
% 0.30/0.63      = zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_zero
% 0.30/0.63  thf(fact_7982_norm__not__less__zero,axiom,
% 0.30/0.63      ! [X3: complex] :
% 0.30/0.63        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_not_less_zero
% 0.30/0.63  thf(fact_7983_norm__uminus__minus,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] :
% 0.30/0.63        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) )
% 0.30/0.63        = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_uminus_minus
% 0.30/0.63  thf(fact_7984_norm__uminus__minus,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex] :
% 0.30/0.63        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 ) )
% 0.30/0.63        = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_uminus_minus
% 0.30/0.63  thf(fact_7985_nonzero__norm__divide,axiom,
% 0.30/0.63      ! [B: real,A: real] :
% 0.30/0.63        ( ( B != zero_zero_real )
% 0.30/0.63       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 0.30/0.63          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_norm_divide
% 0.30/0.63  thf(fact_7986_nonzero__norm__divide,axiom,
% 0.30/0.63      ! [B: complex,A: complex] :
% 0.30/0.63        ( ( B != zero_zero_complex )
% 0.30/0.63       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 0.30/0.63          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % nonzero_norm_divide
% 0.30/0.63  thf(fact_7987_power__eq__imp__eq__norm,axiom,
% 0.30/0.63      ! [W2: real,N2: nat,Z2: real] :
% 0.30/0.63        ( ( ( power_power_real @ W2 @ N2 )
% 0.30/0.63          = ( power_power_real @ Z2 @ N2 ) )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.63         => ( ( real_V7735802525324610683m_real @ W2 )
% 0.30/0.63            = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_imp_eq_norm
% 0.30/0.63  thf(fact_7988_power__eq__imp__eq__norm,axiom,
% 0.30/0.63      ! [W2: complex,N2: nat,Z2: complex] :
% 0.30/0.63        ( ( ( power_power_complex @ W2 @ N2 )
% 0.30/0.63          = ( power_power_complex @ Z2 @ N2 ) )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.63         => ( ( real_V1022390504157884413omplex @ W2 )
% 0.30/0.63            = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_imp_eq_norm
% 0.30/0.63  thf(fact_7989_norm__mult__less,axiom,
% 0.30/0.63      ! [X3: real,R2: real,Y3: real,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.30/0.63         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_less
% 0.30/0.63  thf(fact_7990_norm__mult__less,axiom,
% 0.30/0.63      ! [X3: complex,R2: real,Y3: complex,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.30/0.63         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_mult_less
% 0.30/0.63  thf(fact_7991_norm__triangle__lt,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E2: real] :
% 0.30/0.63        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 0.30/0.63       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_lt
% 0.30/0.63  thf(fact_7992_norm__triangle__lt,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E2: real] :
% 0.30/0.63        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 0.30/0.63       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_lt
% 0.30/0.63  thf(fact_7993_norm__add__less,axiom,
% 0.30/0.63      ! [X3: real,R2: real,Y3: real,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 0.30/0.63         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_less
% 0.30/0.63  thf(fact_7994_norm__add__less,axiom,
% 0.30/0.63      ! [X3: complex,R2: real,Y3: complex,S: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 0.30/0.63         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_less
% 0.30/0.63  thf(fact_7995_norm__triangle__mono,axiom,
% 0.30/0.63      ! [A: real,R2: real,B: real,S: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_mono
% 0.30/0.63  thf(fact_7996_norm__triangle__mono,axiom,
% 0.30/0.63      ! [A: complex,R2: real,B: complex,S: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 0.30/0.63       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 0.30/0.63         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_mono
% 0.30/0.63  thf(fact_7997_norm__triangle__ineq,axiom,
% 0.30/0.63      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq
% 0.30/0.63  thf(fact_7998_norm__triangle__ineq,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_ineq
% 0.30/0.63  thf(fact_7999_norm__triangle__le,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le
% 0.30/0.63  thf(fact_8000_norm__triangle__le,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E2: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_triangle_le
% 0.30/0.63  thf(fact_8001_norm__add__leD,axiom,
% 0.30/0.63      ! [A: real,B: real,C: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_leD
% 0.30/0.63  thf(fact_8002_norm__add__leD,axiom,
% 0.30/0.63      ! [A: complex,B: complex,C: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 0.30/0.63       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_add_leD
% 0.30/0.63  thf(fact_8003_norm__diff__triangle__less,axiom,
% 0.30/0.63      ! [X3: real,Y3: real,E1: real,Z2: real,E22: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z2 ) ) @ E22 )
% 0.30/0.63         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_less
% 0.30/0.63  thf(fact_8004_norm__diff__triangle__less,axiom,
% 0.30/0.63      ! [X3: complex,Y3: complex,E1: real,Z2: complex,E22: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
% 0.30/0.63       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z2 ) ) @ E22 )
% 0.30/0.63         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_less
% 0.30/0.63  thf(fact_8005_norm__diff__ineq,axiom,
% 0.30/0.63      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_ineq
% 0.30/0.63  thf(fact_8006_norm__diff__ineq,axiom,
% 0.30/0.63      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_ineq
% 0.30/0.63  thf(fact_8007_suminf__finite,axiom,
% 0.30/0.63      ! [N7: set_nat,F: nat > complex] :
% 0.30/0.63        ( ( finite_finite_nat @ N7 )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.63             => ( ( F @ N3 )
% 0.30/0.63                = zero_zero_complex ) )
% 0.30/0.63         => ( ( suminf_complex @ F )
% 0.30/0.63            = ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_finite
% 0.30/0.63  thf(fact_8008_suminf__finite,axiom,
% 0.30/0.63      ! [N7: set_nat,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ N7 )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.63             => ( ( F @ N3 )
% 0.30/0.63                = zero_zero_int ) )
% 0.30/0.63         => ( ( suminf_int @ F )
% 0.30/0.63            = ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_finite
% 0.30/0.63  thf(fact_8009_suminf__finite,axiom,
% 0.30/0.63      ! [N7: set_nat,F: nat > nat] :
% 0.30/0.63        ( ( finite_finite_nat @ N7 )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.63             => ( ( F @ N3 )
% 0.30/0.63                = zero_zero_nat ) )
% 0.30/0.63         => ( ( suminf_nat @ F )
% 0.30/0.63            = ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_finite
% 0.30/0.63  thf(fact_8010_suminf__finite,axiom,
% 0.30/0.63      ! [N7: set_nat,F: nat > real] :
% 0.30/0.63        ( ( finite_finite_nat @ N7 )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.63             => ( ( F @ N3 )
% 0.30/0.63                = zero_zero_real ) )
% 0.30/0.63         => ( ( suminf_real @ F )
% 0.30/0.63            = ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % suminf_finite
% 0.30/0.63  thf(fact_8011_power__eq__1__iff,axiom,
% 0.30/0.63      ! [W2: real,N2: nat] :
% 0.30/0.63        ( ( ( power_power_real @ W2 @ N2 )
% 0.30/0.63          = one_one_real )
% 0.30/0.63       => ( ( ( real_V7735802525324610683m_real @ W2 )
% 0.30/0.63            = one_one_real )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_1_iff
% 0.30/0.63  thf(fact_8012_power__eq__1__iff,axiom,
% 0.30/0.63      ! [W2: complex,N2: nat] :
% 0.30/0.63        ( ( ( power_power_complex @ W2 @ N2 )
% 0.30/0.63          = one_one_complex )
% 0.30/0.63       => ( ( ( real_V1022390504157884413omplex @ W2 )
% 0.30/0.63            = one_one_real )
% 0.30/0.63          | ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % power_eq_1_iff
% 0.30/0.63  thf(fact_8013_norm__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_ineq
% 0.30/0.63  thf(fact_8014_norm__diff__triangle__ineq,axiom,
% 0.30/0.63      ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % norm_diff_triangle_ineq
% 0.30/0.63  thf(fact_8015_sum__bounds__lt__plus1,axiom,
% 0.30/0.63      ! [F: nat > nat,Mm: nat] :
% 0.30/0.63        ( ( groups3542108847815614940at_nat
% 0.30/0.63          @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ Mm ) )
% 0.30/0.63        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_bounds_lt_plus1
% 0.30/0.63  thf(fact_8016_sum__bounds__lt__plus1,axiom,
% 0.30/0.63      ! [F: nat > real,Mm: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ Mm ) )
% 0.30/0.63        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sum_bounds_lt_plus1
% 0.30/0.63  thf(fact_8017_sumr__cos__zero__one,axiom,
% 0.30/0.63      ! [N2: nat] :
% 0.30/0.63        ( ( groups6591440286371151544t_real
% 0.30/0.63          @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
% 0.30/0.63          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.63        = one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % sumr_cos_zero_one
% 0.30/0.63  thf(fact_8018_pi__series,axiom,
% 0.30/0.63      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 0.30/0.63      = ( suminf_real
% 0.30/0.63        @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % pi_series
% 0.30/0.63  thf(fact_8019_summable__arctan__series,axiom,
% 0.30/0.63      ! [X3: real] :
% 0.30/0.63        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.63       => ( summable_real
% 0.30/0.63          @ ^ [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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_arctan_series
% 0.30/0.63  thf(fact_8020_ceiling__log__nat__eq__powr__iff,axiom,
% 0.30/0.63      ! [B: nat,K: nat,N2: nat] :
% 0.30/0.63        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.63       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.63         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.63              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
% 0.30/0.63            = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.63              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % ceiling_log_nat_eq_powr_iff
% 0.30/0.63  thf(fact_8021_geometric__deriv__sums,axiom,
% 0.30/0.63      ! [Z2: real] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 0.30/0.63       => ( sums_real
% 0.30/0.63          @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) )
% 0.30/0.63          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % geometric_deriv_sums
% 0.30/0.63  thf(fact_8022_geometric__deriv__sums,axiom,
% 0.30/0.63      ! [Z2: complex] :
% 0.30/0.63        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 0.30/0.63       => ( sums_complex
% 0.30/0.63          @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) )
% 0.30/0.63          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % geometric_deriv_sums
% 0.30/0.63  thf(fact_8023_summable__zero,axiom,
% 0.30/0.63      ( summable_complex
% 0.30/0.63      @ ^ [N: nat] : zero_zero_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_zero
% 0.30/0.63  thf(fact_8024_summable__zero,axiom,
% 0.30/0.63      ( summable_real
% 0.30/0.63      @ ^ [N: nat] : zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_zero
% 0.30/0.63  thf(fact_8025_summable__zero,axiom,
% 0.30/0.63      ( summable_nat
% 0.30/0.63      @ ^ [N: nat] : zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_zero
% 0.30/0.63  thf(fact_8026_summable__zero,axiom,
% 0.30/0.63      ( summable_int
% 0.30/0.63      @ ^ [N: nat] : zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_zero
% 0.30/0.63  thf(fact_8027_summable__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > complex] :
% 0.30/0.63        ( summable_complex
% 0.30/0.63        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_single
% 0.30/0.63  thf(fact_8028_summable__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > real] :
% 0.30/0.63        ( summable_real
% 0.30/0.63        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_single
% 0.30/0.63  thf(fact_8029_summable__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > nat] :
% 0.30/0.63        ( summable_nat
% 0.30/0.63        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_single
% 0.30/0.63  thf(fact_8030_summable__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > int] :
% 0.30/0.63        ( summable_int
% 0.30/0.63        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_single
% 0.30/0.63  thf(fact_8031_sums__zero,axiom,
% 0.30/0.63      ( sums_complex
% 0.30/0.63      @ ^ [N: nat] : zero_zero_complex
% 0.30/0.63      @ zero_zero_complex ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_zero
% 0.30/0.63  thf(fact_8032_sums__zero,axiom,
% 0.30/0.63      ( sums_real
% 0.30/0.63      @ ^ [N: nat] : zero_zero_real
% 0.30/0.63      @ zero_zero_real ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_zero
% 0.30/0.63  thf(fact_8033_sums__zero,axiom,
% 0.30/0.63      ( sums_nat
% 0.30/0.63      @ ^ [N: nat] : zero_zero_nat
% 0.30/0.63      @ zero_zero_nat ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_zero
% 0.30/0.63  thf(fact_8034_sums__zero,axiom,
% 0.30/0.63      ( sums_int
% 0.30/0.63      @ ^ [N: nat] : zero_zero_int
% 0.30/0.63      @ zero_zero_int ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_zero
% 0.30/0.63  thf(fact_8035_summable__iff__shift,axiom,
% 0.30/0.63      ! [F: nat > real,K: nat] :
% 0.30/0.63        ( ( summable_real
% 0.30/0.63          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.63        = ( summable_real @ F ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_iff_shift
% 0.30/0.63  thf(fact_8036_cos__coeff__0,axiom,
% 0.30/0.63      ( ( cos_coeff @ zero_zero_nat )
% 0.30/0.63      = one_one_real ) ).
% 0.30/0.63  
% 0.30/0.63  % cos_coeff_0
% 0.30/0.63  thf(fact_8037_summable__cmult__iff,axiom,
% 0.30/0.63      ! [C: complex,F: nat > complex] :
% 0.30/0.63        ( ( summable_complex
% 0.30/0.63          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 0.30/0.63        = ( ( C = zero_zero_complex )
% 0.30/0.63          | ( summable_complex @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_cmult_iff
% 0.30/0.63  thf(fact_8038_summable__cmult__iff,axiom,
% 0.30/0.63      ! [C: real,F: nat > real] :
% 0.30/0.63        ( ( summable_real
% 0.30/0.63          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 0.30/0.63        = ( ( C = zero_zero_real )
% 0.30/0.63          | ( summable_real @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_cmult_iff
% 0.30/0.63  thf(fact_8039_summable__divide__iff,axiom,
% 0.30/0.63      ! [F: nat > complex,C: complex] :
% 0.30/0.63        ( ( summable_complex
% 0.30/0.63          @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 0.30/0.63        = ( ( C = zero_zero_complex )
% 0.30/0.63          | ( summable_complex @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_divide_iff
% 0.30/0.63  thf(fact_8040_summable__divide__iff,axiom,
% 0.30/0.63      ! [F: nat > real,C: real] :
% 0.30/0.63        ( ( summable_real
% 0.30/0.63          @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 0.30/0.63        = ( ( C = zero_zero_real )
% 0.30/0.63          | ( summable_real @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_divide_iff
% 0.30/0.63  thf(fact_8041_summable__If__finite,axiom,
% 0.30/0.63      ! [P: nat > $o,F: nat > complex] :
% 0.30/0.63        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.63       => ( summable_complex
% 0.30/0.63          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite
% 0.30/0.63  thf(fact_8042_summable__If__finite,axiom,
% 0.30/0.63      ! [P: nat > $o,F: nat > real] :
% 0.30/0.63        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.63       => ( summable_real
% 0.30/0.63          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite
% 0.30/0.63  thf(fact_8043_summable__If__finite,axiom,
% 0.30/0.63      ! [P: nat > $o,F: nat > nat] :
% 0.30/0.63        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.63       => ( summable_nat
% 0.30/0.63          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite
% 0.30/0.63  thf(fact_8044_summable__If__finite,axiom,
% 0.30/0.63      ! [P: nat > $o,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.63       => ( summable_int
% 0.30/0.63          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite
% 0.30/0.63  thf(fact_8045_summable__If__finite__set,axiom,
% 0.30/0.63      ! [A2: set_nat,F: nat > complex] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( summable_complex
% 0.30/0.63          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite_set
% 0.30/0.63  thf(fact_8046_summable__If__finite__set,axiom,
% 0.30/0.63      ! [A2: set_nat,F: nat > real] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( summable_real
% 0.30/0.63          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite_set
% 0.30/0.63  thf(fact_8047_summable__If__finite__set,axiom,
% 0.30/0.63      ! [A2: set_nat,F: nat > nat] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( summable_nat
% 0.30/0.63          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite_set
% 0.30/0.63  thf(fact_8048_summable__If__finite__set,axiom,
% 0.30/0.63      ! [A2: set_nat,F: nat > int] :
% 0.30/0.63        ( ( finite_finite_nat @ A2 )
% 0.30/0.63       => ( summable_int
% 0.30/0.63          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_If_finite_set
% 0.30/0.63  thf(fact_8049_log__eq__one,axiom,
% 0.30/0.63      ! [A: real] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63       => ( ( A != one_one_real )
% 0.30/0.63         => ( ( log @ A @ A )
% 0.30/0.63            = one_one_real ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_eq_one
% 0.30/0.63  thf(fact_8050_log__less__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63           => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 0.30/0.63              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_less_cancel_iff
% 0.30/0.63  thf(fact_8051_log__less__one__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
% 0.30/0.63            = ( ord_less_real @ X3 @ A ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_less_one_cancel_iff
% 0.30/0.63  thf(fact_8052_one__less__log__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
% 0.30/0.63            = ( ord_less_real @ A @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_less_log_cancel_iff
% 0.30/0.63  thf(fact_8053_log__less__zero__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 0.30/0.63            = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_less_zero_cancel_iff
% 0.30/0.63  thf(fact_8054_zero__less__log__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 0.30/0.63            = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_less_log_cancel_iff
% 0.30/0.63  thf(fact_8055_log__le__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real,Y3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.63           => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 0.30/0.63              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_le_cancel_iff
% 0.30/0.63  thf(fact_8056_log__le__one__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
% 0.30/0.63            = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_le_one_cancel_iff
% 0.30/0.63  thf(fact_8057_one__le__log__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
% 0.30/0.63            = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % one_le_log_cancel_iff
% 0.30/0.63  thf(fact_8058_log__le__zero__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 0.30/0.63            = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_le_zero_cancel_iff
% 0.30/0.63  thf(fact_8059_zero__le__log__cancel__iff,axiom,
% 0.30/0.63      ! [A: real,X3: real] :
% 0.30/0.63        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.63       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.63         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 0.30/0.63            = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % zero_le_log_cancel_iff
% 0.30/0.63  thf(fact_8060_powser__sums__zero__iff,axiom,
% 0.30/0.63      ! [A: nat > complex,X3: complex] :
% 0.30/0.63        ( ( sums_complex
% 0.30/0.63          @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 0.30/0.63          @ X3 )
% 0.30/0.63        = ( ( A @ zero_zero_nat )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_sums_zero_iff
% 0.30/0.63  thf(fact_8061_powser__sums__zero__iff,axiom,
% 0.30/0.63      ! [A: nat > real,X3: real] :
% 0.30/0.63        ( ( sums_real
% 0.30/0.63          @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 0.30/0.63          @ X3 )
% 0.30/0.63        = ( ( A @ zero_zero_nat )
% 0.30/0.63          = X3 ) ) ).
% 0.30/0.63  
% 0.30/0.63  % powser_sums_zero_iff
% 0.30/0.63  thf(fact_8062_summable__geometric__iff,axiom,
% 0.30/0.63      ! [C: real] :
% 0.30/0.63        ( ( summable_real @ ( power_power_real @ C ) )
% 0.30/0.63        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_geometric_iff
% 0.30/0.63  thf(fact_8063_summable__geometric__iff,axiom,
% 0.30/0.63      ! [C: complex] :
% 0.30/0.63        ( ( summable_complex @ ( power_power_complex @ C ) )
% 0.30/0.63        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_geometric_iff
% 0.30/0.63  thf(fact_8064_log__pow__cancel,axiom,
% 0.30/0.63      ! [A: real,B: nat] :
% 0.30/0.63        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.63       => ( ( A != one_one_real )
% 0.30/0.63         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 0.30/0.63            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % log_pow_cancel
% 0.30/0.63  thf(fact_8065_sums__0,axiom,
% 0.30/0.63      ! [F: nat > complex] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( ( F @ N3 )
% 0.30/0.63            = zero_zero_complex )
% 0.30/0.63       => ( sums_complex @ F @ zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_0
% 0.30/0.63  thf(fact_8066_sums__0,axiom,
% 0.30/0.63      ! [F: nat > real] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( ( F @ N3 )
% 0.30/0.63            = zero_zero_real )
% 0.30/0.63       => ( sums_real @ F @ zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_0
% 0.30/0.63  thf(fact_8067_sums__0,axiom,
% 0.30/0.63      ! [F: nat > nat] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( ( F @ N3 )
% 0.30/0.63            = zero_zero_nat )
% 0.30/0.63       => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_0
% 0.30/0.63  thf(fact_8068_sums__0,axiom,
% 0.30/0.63      ! [F: nat > int] :
% 0.30/0.63        ( ! [N3: nat] :
% 0.30/0.63            ( ( F @ N3 )
% 0.30/0.63            = zero_zero_int )
% 0.30/0.63       => ( sums_int @ F @ zero_zero_int ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_0
% 0.30/0.63  thf(fact_8069_sums__le,axiom,
% 0.30/0.63      ! [F: nat > real,G: nat > real,S: real,T: real] :
% 0.30/0.63        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.63       => ( ( sums_real @ F @ S )
% 0.30/0.63         => ( ( sums_real @ G @ T )
% 0.30/0.63           => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_le
% 0.30/0.63  thf(fact_8070_sums__le,axiom,
% 0.30/0.63      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 0.30/0.63        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.63       => ( ( sums_nat @ F @ S )
% 0.30/0.63         => ( ( sums_nat @ G @ T )
% 0.30/0.63           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_le
% 0.30/0.63  thf(fact_8071_sums__le,axiom,
% 0.30/0.63      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 0.30/0.63        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.63       => ( ( sums_int @ F @ S )
% 0.30/0.63         => ( ( sums_int @ G @ T )
% 0.30/0.63           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_le
% 0.30/0.63  thf(fact_8072_sums__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > complex] :
% 0.30/0.63        ( sums_complex
% 0.30/0.63        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.30/0.63        @ ( F @ I ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_single
% 0.30/0.63  thf(fact_8073_sums__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > real] :
% 0.30/0.63        ( sums_real
% 0.30/0.63        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 0.30/0.63        @ ( F @ I ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_single
% 0.30/0.63  thf(fact_8074_sums__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > nat] :
% 0.30/0.63        ( sums_nat
% 0.30/0.63        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.30/0.63        @ ( F @ I ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_single
% 0.30/0.63  thf(fact_8075_sums__single,axiom,
% 0.30/0.63      ! [I: nat,F: nat > int] :
% 0.30/0.63        ( sums_int
% 0.30/0.63        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 0.30/0.63        @ ( F @ I ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_single
% 0.30/0.63  thf(fact_8076_sums__add,axiom,
% 0.30/0.63      ! [F: nat > real,A: real,G: nat > real,B: real] :
% 0.30/0.63        ( ( sums_real @ F @ A )
% 0.30/0.63       => ( ( sums_real @ G @ B )
% 0.30/0.63         => ( sums_real
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.63            @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_add
% 0.30/0.63  thf(fact_8077_sums__add,axiom,
% 0.30/0.63      ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 0.30/0.63        ( ( sums_nat @ F @ A )
% 0.30/0.63       => ( ( sums_nat @ G @ B )
% 0.30/0.63         => ( sums_nat
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.63            @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_add
% 0.30/0.63  thf(fact_8078_sums__add,axiom,
% 0.30/0.63      ! [F: nat > int,A: int,G: nat > int,B: int] :
% 0.30/0.63        ( ( sums_int @ F @ A )
% 0.30/0.63       => ( ( sums_int @ G @ B )
% 0.30/0.63         => ( sums_int
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.63            @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % sums_add
% 0.30/0.63  thf(fact_8079_summable__comparison__test,axiom,
% 0.30/0.63      ! [F: nat > real,G: nat > real] :
% 0.30/0.63        ( ? [N8: nat] :
% 0.30/0.63          ! [N3: nat] :
% 0.30/0.63            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.63       => ( ( summable_real @ G )
% 0.30/0.63         => ( summable_real @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_comparison_test
% 0.30/0.63  thf(fact_8080_summable__comparison__test,axiom,
% 0.30/0.63      ! [F: nat > complex,G: nat > real] :
% 0.30/0.63        ( ? [N8: nat] :
% 0.30/0.63          ! [N3: nat] :
% 0.30/0.63            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.30/0.63           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.63       => ( ( summable_real @ G )
% 0.30/0.63         => ( summable_complex @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_comparison_test
% 0.30/0.63  thf(fact_8081_summable__comparison__test_H,axiom,
% 0.30/0.63      ! [G: nat > real,N7: nat,F: nat > real] :
% 0.30/0.63        ( ( summable_real @ G )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.30/0.63             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.63         => ( summable_real @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_comparison_test'
% 0.30/0.63  thf(fact_8082_summable__comparison__test_H,axiom,
% 0.30/0.63      ! [G: nat > real,N7: nat,F: nat > complex] :
% 0.30/0.63        ( ( summable_real @ G )
% 0.30/0.63       => ( ! [N3: nat] :
% 0.30/0.63              ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.30/0.63             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.63         => ( summable_complex @ F ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_comparison_test'
% 0.30/0.63  thf(fact_8083_summable__const__iff,axiom,
% 0.30/0.63      ! [C: complex] :
% 0.30/0.63        ( ( summable_complex
% 0.30/0.63          @ ^ [Uu3: nat] : C )
% 0.30/0.63        = ( C = zero_zero_complex ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_const_iff
% 0.30/0.63  thf(fact_8084_summable__const__iff,axiom,
% 0.30/0.63      ! [C: real] :
% 0.30/0.63        ( ( summable_real
% 0.30/0.63          @ ^ [Uu3: nat] : C )
% 0.30/0.63        = ( C = zero_zero_real ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_const_iff
% 0.30/0.63  thf(fact_8085_summable__add,axiom,
% 0.30/0.63      ! [F: nat > real,G: nat > real] :
% 0.30/0.63        ( ( summable_real @ F )
% 0.30/0.63       => ( ( summable_real @ G )
% 0.30/0.63         => ( summable_real
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_add
% 0.30/0.63  thf(fact_8086_summable__add,axiom,
% 0.30/0.63      ! [F: nat > nat,G: nat > nat] :
% 0.30/0.63        ( ( summable_nat @ F )
% 0.30/0.63       => ( ( summable_nat @ G )
% 0.30/0.63         => ( summable_nat
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.30/0.63  
% 0.30/0.63  % summable_add
% 0.30/0.63  thf(fact_8087_summable__add,axiom,
% 0.30/0.63      ! [F: nat > int,G: nat > int] :
% 0.30/0.63        ( ( summable_int @ F )
% 0.30/0.63       => ( ( summable_int @ G )
% 0.30/0.63         => ( summable_int
% 0.30/0.63            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_add
% 0.30/0.64  thf(fact_8088_summable__Suc__iff,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.30/0.64        = ( summable_real @ F ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_Suc_iff
% 0.30/0.64  thf(fact_8089_summable__ignore__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,K: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_ignore_initial_segment
% 0.30/0.64  thf(fact_8090_powser__insidea,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_insidea
% 0.30/0.64  thf(fact_8091_powser__insidea,axiom,
% 0.30/0.64      ! [F: nat > complex,X3: complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_insidea
% 0.30/0.64  thf(fact_8092_suminf__le,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( summable_real @ F )
% 0.30/0.64         => ( ( summable_real @ G )
% 0.30/0.64           => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le
% 0.30/0.64  thf(fact_8093_suminf__le,axiom,
% 0.30/0.64      ! [F: nat > nat,G: nat > nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( summable_nat @ F )
% 0.30/0.64         => ( ( summable_nat @ G )
% 0.30/0.64           => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le
% 0.30/0.64  thf(fact_8094_suminf__le,axiom,
% 0.30/0.64      ! [F: nat > int,G: nat > int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.64       => ( ( summable_int @ F )
% 0.30/0.64         => ( ( summable_int @ G )
% 0.30/0.64           => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le
% 0.30/0.64  thf(fact_8095_summable__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > complex] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_complex ) )
% 0.30/0.64         => ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_finite
% 0.30/0.64  thf(fact_8096_summable__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > real] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_real ) )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_finite
% 0.30/0.64  thf(fact_8097_summable__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > nat] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_nat ) )
% 0.30/0.64         => ( summable_nat @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_finite
% 0.30/0.64  thf(fact_8098_summable__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > int] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_int ) )
% 0.30/0.64         => ( summable_int @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_finite
% 0.30/0.64  thf(fact_8099_sums__mult__iff,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex,D: complex] :
% 0.30/0.64        ( ( C != zero_zero_complex )
% 0.30/0.64       => ( ( sums_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 0.30/0.64            @ ( times_times_complex @ C @ D ) )
% 0.30/0.64          = ( sums_complex @ F @ D ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult_iff
% 0.30/0.64  thf(fact_8100_sums__mult__iff,axiom,
% 0.30/0.64      ! [C: real,F: nat > real,D: real] :
% 0.30/0.64        ( ( C != zero_zero_real )
% 0.30/0.64       => ( ( sums_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 0.30/0.64            @ ( times_times_real @ C @ D ) )
% 0.30/0.64          = ( sums_real @ F @ D ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult_iff
% 0.30/0.64  thf(fact_8101_sums__mult2__iff,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex,D: complex] :
% 0.30/0.64        ( ( C != zero_zero_complex )
% 0.30/0.64       => ( ( sums_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 0.30/0.64            @ ( times_times_complex @ D @ C ) )
% 0.30/0.64          = ( sums_complex @ F @ D ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult2_iff
% 0.30/0.64  thf(fact_8102_sums__mult2__iff,axiom,
% 0.30/0.64      ! [C: real,F: nat > real,D: real] :
% 0.30/0.64        ( ( C != zero_zero_real )
% 0.30/0.64       => ( ( sums_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 0.30/0.64            @ ( times_times_real @ D @ C ) )
% 0.30/0.64          = ( sums_real @ F @ D ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult2_iff
% 0.30/0.64  thf(fact_8103_summable__mult__D,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 0.30/0.64       => ( ( C != zero_zero_complex )
% 0.30/0.64         => ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_mult_D
% 0.30/0.64  thf(fact_8104_summable__mult__D,axiom,
% 0.30/0.64      ! [C: real,F: nat > real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 0.30/0.64       => ( ( C != zero_zero_real )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_mult_D
% 0.30/0.64  thf(fact_8105_summable__zero__power,axiom,
% 0.30/0.64      summable_real @ ( power_power_real @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power
% 0.30/0.64  thf(fact_8106_summable__zero__power,axiom,
% 0.30/0.64      summable_int @ ( power_power_int @ zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power
% 0.30/0.64  thf(fact_8107_summable__zero__power,axiom,
% 0.30/0.64      summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power
% 0.30/0.64  thf(fact_8108_pi__not__less__zero,axiom,
% 0.30/0.64      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_not_less_zero
% 0.30/0.64  thf(fact_8109_pi__gt__zero,axiom,
% 0.30/0.64      ord_less_real @ zero_zero_real @ pi ).
% 0.30/0.64  
% 0.30/0.64  % pi_gt_zero
% 0.30/0.64  thf(fact_8110_suminf__add,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 0.30/0.64            = ( suminf_real
% 0.30/0.64              @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_8111_suminf__add,axiom,
% 0.30/0.64      ! [F: nat > nat,G: nat > nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ( summable_nat @ G )
% 0.30/0.64         => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 0.30/0.64            = ( suminf_nat
% 0.30/0.64              @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_8112_suminf__add,axiom,
% 0.30/0.64      ! [F: nat > int,G: nat > int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ( summable_int @ G )
% 0.30/0.64         => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 0.30/0.64            = ( suminf_int
% 0.30/0.64              @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_add
% 0.30/0.64  thf(fact_8113_suminf__eq__zero__iff,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ( suminf_real @ F )
% 0.30/0.64              = zero_zero_real )
% 0.30/0.64            = ( ! [N: nat] :
% 0.30/0.64                  ( ( F @ N )
% 0.30/0.64                  = zero_zero_real ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_eq_zero_iff
% 0.30/0.64  thf(fact_8114_suminf__eq__zero__iff,axiom,
% 0.30/0.64      ! [F: nat > nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ( suminf_nat @ F )
% 0.30/0.64              = zero_zero_nat )
% 0.30/0.64            = ( ! [N: nat] :
% 0.30/0.64                  ( ( F @ N )
% 0.30/0.64                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_eq_zero_iff
% 0.30/0.64  thf(fact_8115_suminf__eq__zero__iff,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ( suminf_int @ F )
% 0.30/0.64              = zero_zero_int )
% 0.30/0.64            = ( ! [N: nat] :
% 0.30/0.64                  ( ( F @ N )
% 0.30/0.64                  = zero_zero_int ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_eq_zero_iff
% 0.30/0.64  thf(fact_8116_suminf__nonneg,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_nonneg
% 0.30/0.64  thf(fact_8117_suminf__nonneg,axiom,
% 0.30/0.64      ! [F: nat > nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_nonneg
% 0.30/0.64  thf(fact_8118_suminf__nonneg,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_nonneg
% 0.30/0.64  thf(fact_8119_suminf__pos,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos
% 0.30/0.64  thf(fact_8120_suminf__pos,axiom,
% 0.30/0.64      ! [F: nat > nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos
% 0.30/0.64  thf(fact_8121_suminf__pos,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos
% 0.30/0.64  thf(fact_8122_sums__mult__D,axiom,
% 0.30/0.64      ! [C: complex,F: nat > complex,A: complex] :
% 0.30/0.64        ( ( sums_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 0.30/0.64          @ A )
% 0.30/0.64       => ( ( C != zero_zero_complex )
% 0.30/0.64         => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult_D
% 0.30/0.64  thf(fact_8123_sums__mult__D,axiom,
% 0.30/0.64      ! [C: real,F: nat > real,A: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 0.30/0.64          @ A )
% 0.30/0.64       => ( ( C != zero_zero_real )
% 0.30/0.64         => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_mult_D
% 0.30/0.64  thf(fact_8124_sums__Suc__imp,axiom,
% 0.30/0.64      ! [F: nat > complex,S: complex] :
% 0.30/0.64        ( ( ( F @ zero_zero_nat )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64       => ( ( sums_complex
% 0.30/0.64            @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64            @ S )
% 0.30/0.64         => ( sums_complex @ F @ S ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc_imp
% 0.30/0.64  thf(fact_8125_sums__Suc__imp,axiom,
% 0.30/0.64      ! [F: nat > real,S: real] :
% 0.30/0.64        ( ( ( F @ zero_zero_nat )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64       => ( ( sums_real
% 0.30/0.64            @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64            @ S )
% 0.30/0.64         => ( sums_real @ F @ S ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc_imp
% 0.30/0.64  thf(fact_8126_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > real,L: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64          @ L )
% 0.30/0.64       => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc
% 0.30/0.64  thf(fact_8127_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > nat,L: nat] :
% 0.30/0.64        ( ( sums_nat
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64          @ L )
% 0.30/0.64       => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc
% 0.30/0.64  thf(fact_8128_sums__Suc,axiom,
% 0.30/0.64      ! [F: nat > int,L: int] :
% 0.30/0.64        ( ( sums_int
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64          @ L )
% 0.30/0.64       => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc
% 0.30/0.64  thf(fact_8129_sums__Suc__iff,axiom,
% 0.30/0.64      ! [F: nat > real,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 0.30/0.64          @ S )
% 0.30/0.64        = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_Suc_iff
% 0.30/0.64  thf(fact_8130_sums__zero__iff__shift,axiom,
% 0.30/0.64      ! [N2: nat,F: nat > complex,S: complex] :
% 0.30/0.64        ( ! [I3: nat] :
% 0.30/0.64            ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.64           => ( ( F @ I3 )
% 0.30/0.64              = zero_zero_complex ) )
% 0.30/0.64       => ( ( sums_complex
% 0.30/0.64            @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 0.30/0.64            @ S )
% 0.30/0.64          = ( sums_complex @ F @ S ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_zero_iff_shift
% 0.30/0.64  thf(fact_8131_sums__zero__iff__shift,axiom,
% 0.30/0.64      ! [N2: nat,F: nat > real,S: real] :
% 0.30/0.64        ( ! [I3: nat] :
% 0.30/0.64            ( ( ord_less_nat @ I3 @ N2 )
% 0.30/0.64           => ( ( F @ I3 )
% 0.30/0.64              = zero_zero_real ) )
% 0.30/0.64       => ( ( sums_real
% 0.30/0.64            @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 0.30/0.64            @ S )
% 0.30/0.64          = ( sums_real @ F @ S ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_zero_iff_shift
% 0.30/0.64  thf(fact_8132_summable__0__powser,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( summable_complex
% 0.30/0.64        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_0_powser
% 0.30/0.64  thf(fact_8133_summable__0__powser,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( summable_real
% 0.30/0.64        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_0_powser
% 0.30/0.64  thf(fact_8134_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > complex] :
% 0.30/0.64        ( summable_complex
% 0.30/0.64        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_8135_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( summable_real
% 0.30/0.64        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_8136_summable__zero__power_H,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( summable_int
% 0.30/0.64        @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_zero_power'
% 0.30/0.64  thf(fact_8137_summable__powser__split__head,axiom,
% 0.30/0.64      ! [F: nat > complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64        = ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_split_head
% 0.30/0.64  thf(fact_8138_summable__powser__split__head,axiom,
% 0.30/0.64      ! [F: nat > real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64        = ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_split_head
% 0.30/0.64  thf(fact_8139_powser__split__head_I3_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64       => ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(3)
% 0.30/0.64  thf(fact_8140_powser__split__head_I3_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64       => ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(3)
% 0.30/0.64  thf(fact_8141_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > complex,M: nat,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64        = ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_ignore_initial_segment
% 0.30/0.64  thf(fact_8142_summable__powser__ignore__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,M: nat,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64        = ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_powser_ignore_initial_segment
% 0.30/0.64  thf(fact_8143_sums__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > complex] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_complex ) )
% 0.30/0.64         => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_finite
% 0.30/0.64  thf(fact_8144_sums__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > int] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_int ) )
% 0.30/0.64         => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_finite
% 0.30/0.64  thf(fact_8145_sums__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > nat] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_nat ) )
% 0.30/0.64         => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_finite
% 0.30/0.64  thf(fact_8146_sums__finite,axiom,
% 0.30/0.64      ! [N7: set_nat,F: nat > real] :
% 0.30/0.64        ( ( finite_finite_nat @ N7 )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ~ ( member_nat @ N3 @ N7 )
% 0.30/0.64             => ( ( F @ N3 )
% 0.30/0.64                = zero_zero_real ) )
% 0.30/0.64         => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_finite
% 0.30/0.64  thf(fact_8147_sums__If__finite,axiom,
% 0.30/0.64      ! [P: nat > $o,F: nat > complex] :
% 0.30/0.64        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.30/0.64          @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite
% 0.30/0.64  thf(fact_8148_sums__If__finite,axiom,
% 0.30/0.64      ! [P: nat > $o,F: nat > int] :
% 0.30/0.64        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.64       => ( sums_int
% 0.30/0.64          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.30/0.64          @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite
% 0.30/0.64  thf(fact_8149_sums__If__finite,axiom,
% 0.30/0.64      ! [P: nat > $o,F: nat > nat] :
% 0.30/0.64        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.64       => ( sums_nat
% 0.30/0.64          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.30/0.64          @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite
% 0.30/0.64  thf(fact_8150_sums__If__finite,axiom,
% 0.30/0.64      ! [P: nat > $o,F: nat > real] :
% 0.30/0.64        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.30/0.64          @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite
% 0.30/0.64  thf(fact_8151_sums__If__finite__set,axiom,
% 0.30/0.64      ! [A2: set_nat,F: nat > complex] :
% 0.30/0.64        ( ( finite_finite_nat @ A2 )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 0.30/0.64          @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set
% 0.30/0.64  thf(fact_8152_sums__If__finite__set,axiom,
% 0.30/0.64      ! [A2: set_nat,F: nat > int] :
% 0.30/0.64        ( ( finite_finite_nat @ A2 )
% 0.30/0.64       => ( sums_int
% 0.30/0.64          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 0.30/0.64          @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set
% 0.30/0.64  thf(fact_8153_sums__If__finite__set,axiom,
% 0.30/0.64      ! [A2: set_nat,F: nat > nat] :
% 0.30/0.64        ( ( finite_finite_nat @ A2 )
% 0.30/0.64       => ( sums_nat
% 0.30/0.64          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 0.30/0.64          @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set
% 0.30/0.64  thf(fact_8154_sums__If__finite__set,axiom,
% 0.30/0.64      ! [A2: set_nat,F: nat > real] :
% 0.30/0.64        ( ( finite_finite_nat @ A2 )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 0.30/0.64          @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set
% 0.30/0.64  thf(fact_8155_summable__norm__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > complex,G: nat > real] :
% 0.30/0.64        ( ? [N8: nat] :
% 0.30/0.64          ! [N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_norm_comparison_test
% 0.30/0.64  thf(fact_8156_summable__rabs__comparison__test,axiom,
% 0.30/0.64      ! [F: nat > real,G: nat > real] :
% 0.30/0.64        ( ? [N8: nat] :
% 0.30/0.64          ! [N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 0.30/0.64       => ( ( summable_real @ G )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_rabs_comparison_test
% 0.30/0.64  thf(fact_8157_suminf__pos__iff,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 0.30/0.64            = ( ? [I2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_8158_suminf__pos__iff,axiom,
% 0.30/0.64      ! [F: nat > nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 0.30/0.64            = ( ? [I2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_8159_suminf__pos__iff,axiom,
% 0.30/0.64      ! [F: nat > int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 0.30/0.64            = ( ? [I2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos_iff
% 0.30/0.64  thf(fact_8160_suminf__pos2,axiom,
% 0.30/0.64      ! [F: nat > real,I: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.30/0.64           => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos2
% 0.30/0.64  thf(fact_8161_suminf__pos2,axiom,
% 0.30/0.64      ! [F: nat > nat,I: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.30/0.64           => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos2
% 0.30/0.64  thf(fact_8162_suminf__pos2,axiom,
% 0.30/0.64      ! [F: nat > int,I: nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.30/0.64           => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_pos2
% 0.30/0.64  thf(fact_8163_suminf__le__const,axiom,
% 0.30/0.64      ! [F: nat > int,X3: int] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le_const
% 0.30/0.64  thf(fact_8164_suminf__le__const,axiom,
% 0.30/0.64      ! [F: nat > nat,X3: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le_const
% 0.30/0.64  thf(fact_8165_suminf__le__const,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_le_const
% 0.30/0.64  thf(fact_8166_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z2: complex] :
% 0.30/0.64        ( sums_complex
% 0.30/0.64        @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N ) )
% 0.30/0.64        @ ( power_power_complex @ Z2 @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_8167_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z2: real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N ) )
% 0.30/0.64        @ ( power_power_real @ Z2 @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_8168_powser__sums__if,axiom,
% 0.30/0.64      ! [M: nat,Z2: int] :
% 0.30/0.64        ( sums_int
% 0.30/0.64        @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N ) )
% 0.30/0.64        @ ( power_power_int @ Z2 @ M ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_if
% 0.30/0.64  thf(fact_8169_powser__sums__zero,axiom,
% 0.30/0.64      ! [A: nat > complex] :
% 0.30/0.64        ( sums_complex
% 0.30/0.64        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 0.30/0.64        @ ( A @ zero_zero_nat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_zero
% 0.30/0.64  thf(fact_8170_powser__sums__zero,axiom,
% 0.30/0.64      ! [A: nat > real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 0.30/0.64        @ ( A @ zero_zero_nat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_sums_zero
% 0.30/0.64  thf(fact_8171_log__base__change,axiom,
% 0.30/0.64      ! [A: real,B: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( log @ B @ X3 )
% 0.30/0.64            = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_base_change
% 0.30/0.64  thf(fact_8172_log__of__power__eq,axiom,
% 0.30/0.64      ! [M: nat,B: real,N2: nat] :
% 0.30/0.64        ( ( ( semiri5074537144036343181t_real @ M )
% 0.30/0.64          = ( power_power_real @ B @ N2 ) )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ( semiri5074537144036343181t_real @ N2 )
% 0.30/0.64            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_of_power_eq
% 0.30/0.64  thf(fact_8173_less__log__of__power,axiom,
% 0.30/0.64      ! [B: real,N2: nat,M: real] :
% 0.30/0.64        ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_log_of_power
% 0.30/0.64  thf(fact_8174_powser__inside,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_inside
% 0.30/0.64  thf(fact_8175_powser__inside,axiom,
% 0.30/0.64      ! [F: nat > complex,X3: complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 0.30/0.64         => ( summable_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_inside
% 0.30/0.64  thf(fact_8176_sums__iff__shift,axiom,
% 0.30/0.64      ! [F: nat > real,N2: nat,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 0.30/0.64          @ S )
% 0.30/0.64        = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_iff_shift
% 0.30/0.64  thf(fact_8177_summableI__nonneg__bounded,axiom,
% 0.30/0.64      ! [F: nat > int,X3: int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( summable_int @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summableI_nonneg_bounded
% 0.30/0.64  thf(fact_8178_summableI__nonneg__bounded,axiom,
% 0.30/0.64      ! [F: nat > nat,X3: nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( summable_nat @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summableI_nonneg_bounded
% 0.30/0.64  thf(fact_8179_summableI__nonneg__bounded,axiom,
% 0.30/0.64      ! [F: nat > real,X3: real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 0.30/0.64       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summableI_nonneg_bounded
% 0.30/0.64  thf(fact_8180_sums__split__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,S: real,N2: nat] :
% 0.30/0.64        ( ( sums_real @ F @ S )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 0.30/0.64          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_split_initial_segment
% 0.30/0.64  thf(fact_8181_sums__iff__shift_H,axiom,
% 0.30/0.64      ! [F: nat > real,N2: nat,S: real] :
% 0.30/0.64        ( ( sums_real
% 0.30/0.64          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 0.30/0.64          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 0.30/0.64        = ( sums_real @ F @ S ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_iff_shift'
% 0.30/0.64  thf(fact_8182_complete__algebra__summable__geometric,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ one_one_real )
% 0.30/0.64       => ( summable_real @ ( power_power_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % complete_algebra_summable_geometric
% 0.30/0.64  thf(fact_8183_complete__algebra__summable__geometric,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ one_one_real )
% 0.30/0.64       => ( summable_complex @ ( power_power_complex @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % complete_algebra_summable_geometric
% 0.30/0.64  thf(fact_8184_summable__geometric,axiom,
% 0.30/0.64      ! [C: real] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.64       => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_geometric
% 0.30/0.64  thf(fact_8185_summable__geometric,axiom,
% 0.30/0.64      ! [C: complex] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.64       => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_geometric
% 0.30/0.64  thf(fact_8186_sums__If__finite__set_H,axiom,
% 0.30/0.64      ! [G: nat > real,S3: real,A2: set_nat,S6: real,F: nat > real] :
% 0.30/0.64        ( ( sums_real @ G @ S3 )
% 0.30/0.64       => ( ( finite_finite_nat @ A2 )
% 0.30/0.64         => ( ( S6
% 0.30/0.64              = ( plus_plus_real @ S3
% 0.30/0.64                @ ( groups6591440286371151544t_real
% 0.30/0.64                  @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.64                  @ A2 ) ) )
% 0.30/0.64           => ( sums_real
% 0.30/0.64              @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.64              @ S6 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sums_If_finite_set'
% 0.30/0.64  thf(fact_8187_suminf__split__head,axiom,
% 0.30/0.64      ! [F: nat > real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 0.30/0.64          = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_split_head
% 0.30/0.64  thf(fact_8188_pi__less__4,axiom,
% 0.30/0.64      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_less_4
% 0.30/0.64  thf(fact_8189_sum__le__suminf,axiom,
% 0.30/0.64      ! [F: nat > int,I5: set_nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ( finite_finite_nat @ I5 )
% 0.30/0.64         => ( ! [N3: nat] :
% 0.30/0.64                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.64               => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 0.30/0.64           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_le_suminf
% 0.30/0.64  thf(fact_8190_sum__le__suminf,axiom,
% 0.30/0.64      ! [F: nat > nat,I5: set_nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ( finite_finite_nat @ I5 )
% 0.30/0.64         => ( ! [N3: nat] :
% 0.30/0.64                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.64               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 0.30/0.64           => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_le_suminf
% 0.30/0.64  thf(fact_8191_sum__le__suminf,axiom,
% 0.30/0.64      ! [F: nat > real,I5: set_nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( finite_finite_nat @ I5 )
% 0.30/0.64         => ( ! [N3: nat] :
% 0.30/0.64                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 0.30/0.64               => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 0.30/0.64           => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_le_suminf
% 0.30/0.64  thf(fact_8192_log__mult,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64             => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
% 0.30/0.64                = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_mult
% 0.30/0.64  thf(fact_8193_log__divide,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64             => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
% 0.30/0.64                = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_divide
% 0.30/0.64  thf(fact_8194_le__log__of__power,axiom,
% 0.30/0.64      ! [B: real,N2: nat,M: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_log_of_power
% 0.30/0.64  thf(fact_8195_log__base__pow,axiom,
% 0.30/0.64      ! [A: real,N2: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( log @ ( power_power_real @ A @ N2 ) @ X3 )
% 0.30/0.64          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_base_pow
% 0.30/0.64  thf(fact_8196_log__nat__power,axiom,
% 0.30/0.64      ! [X3: real,B: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( log @ B @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.64          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_nat_power
% 0.30/0.64  thf(fact_8197_suminf__split__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,K: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( suminf_real @ F )
% 0.30/0.64          = ( plus_plus_real
% 0.30/0.64            @ ( suminf_real
% 0.30/0.64              @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.64            @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_split_initial_segment
% 0.30/0.64  thf(fact_8198_suminf__minus__initial__segment,axiom,
% 0.30/0.64      ! [F: nat > real,K: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 0.30/0.64          = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_minus_initial_segment
% 0.30/0.64  thf(fact_8199_pi__half__less__two,axiom,
% 0.30/0.64      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_half_less_two
% 0.30/0.64  thf(fact_8200_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > int,N2: nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_8201_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > nat,N2: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_8202_sum__less__suminf,axiom,
% 0.30/0.64      ! [F: nat > real,N2: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf
% 0.30/0.64  thf(fact_8203_powser__split__head_I1_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64       => ( ( suminf_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64          = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 0.30/0.64            @ ( times_times_complex
% 0.30/0.64              @ ( suminf_complex
% 0.30/0.64                @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64              @ Z2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(1)
% 0.30/0.64  thf(fact_8204_powser__split__head_I1_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64       => ( ( suminf_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64          = ( plus_plus_real @ ( F @ zero_zero_nat )
% 0.30/0.64            @ ( times_times_real
% 0.30/0.64              @ ( suminf_real
% 0.30/0.64                @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64              @ Z2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(1)
% 0.30/0.64  thf(fact_8205_powser__split__head_I2_J,axiom,
% 0.30/0.64      ! [F: nat > complex,Z2: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64       => ( ( times_times_complex
% 0.30/0.64            @ ( suminf_complex
% 0.30/0.64              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64            @ Z2 )
% 0.30/0.64          = ( minus_minus_complex
% 0.30/0.64            @ ( suminf_complex
% 0.30/0.64              @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 0.30/0.64            @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(2)
% 0.30/0.64  thf(fact_8206_powser__split__head_I2_J,axiom,
% 0.30/0.64      ! [F: nat > real,Z2: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64       => ( ( times_times_real
% 0.30/0.64            @ ( suminf_real
% 0.30/0.64              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64            @ Z2 )
% 0.30/0.64          = ( minus_minus_real
% 0.30/0.64            @ ( suminf_real
% 0.30/0.64              @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 0.30/0.64            @ ( F @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powser_split_head(2)
% 0.30/0.64  thf(fact_8207_log__of__power__less,axiom,
% 0.30/0.64      ! [M: nat,B: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_of_power_less
% 0.30/0.64  thf(fact_8208_log__eq__div__ln__mult__log,axiom,
% 0.30/0.64      ! [A: real,B: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64           => ( ( B != one_one_real )
% 0.30/0.64             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64               => ( ( log @ A @ X3 )
% 0.30/0.64                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_eq_div_ln_mult_log
% 0.30/0.64  thf(fact_8209_summable__partial__sum__bound,axiom,
% 0.30/0.64      ! [F: nat > complex,E2: real] :
% 0.30/0.64        ( ( summable_complex @ F )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.64         => ~ ! [N9: nat] :
% 0.30/0.64                ~ ! [M6: nat] :
% 0.30/0.64                    ( ( ord_less_eq_nat @ N9 @ M6 )
% 0.30/0.64                   => ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_partial_sum_bound
% 0.30/0.64  thf(fact_8210_summable__partial__sum__bound,axiom,
% 0.30/0.64      ! [F: nat > real,E2: real] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.64         => ~ ! [N9: nat] :
% 0.30/0.64                ~ ! [M6: nat] :
% 0.30/0.64                    ( ( ord_less_eq_nat @ N9 @ M6 )
% 0.30/0.64                   => ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_partial_sum_bound
% 0.30/0.64  thf(fact_8211_suminf__exist__split,axiom,
% 0.30/0.64      ! [R2: real,F: nat > real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.30/0.64       => ( ( summable_real @ F )
% 0.30/0.64         => ? [N9: nat] :
% 0.30/0.64            ! [N6: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N9 @ N6 )
% 0.30/0.64             => ( ord_less_real
% 0.30/0.64                @ ( real_V7735802525324610683m_real
% 0.30/0.64                  @ ( suminf_real
% 0.30/0.64                    @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
% 0.30/0.64                @ R2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_exist_split
% 0.30/0.64  thf(fact_8212_suminf__exist__split,axiom,
% 0.30/0.64      ! [R2: real,F: nat > complex] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ R2 )
% 0.30/0.64       => ( ( summable_complex @ F )
% 0.30/0.64         => ? [N9: nat] :
% 0.30/0.64            ! [N6: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N9 @ N6 )
% 0.30/0.64             => ( ord_less_real
% 0.30/0.64                @ ( real_V1022390504157884413omplex
% 0.30/0.64                  @ ( suminf_complex
% 0.30/0.64                    @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
% 0.30/0.64                @ R2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % suminf_exist_split
% 0.30/0.64  thf(fact_8213_summable__power__series,axiom,
% 0.30/0.64      ! [F: nat > real,Z2: real] :
% 0.30/0.64        ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 0.30/0.64       => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 0.30/0.64         => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 0.30/0.64           => ( ( ord_less_real @ Z2 @ one_one_real )
% 0.30/0.64             => ( summable_real
% 0.30/0.64                @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z2 @ I2 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_power_series
% 0.30/0.64  thf(fact_8214_Abel__lemma,axiom,
% 0.30/0.64      ! [R2: real,R0: real,A: nat > complex,M7: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 0.30/0.64       => ( ( ord_less_real @ R2 @ R0 )
% 0.30/0.64         => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
% 0.30/0.64           => ( summable_real
% 0.30/0.64              @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R2 @ N ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Abel_lemma
% 0.30/0.64  thf(fact_8215_pi__half__gt__zero,axiom,
% 0.30/0.64      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pi_half_gt_zero
% 0.30/0.64  thf(fact_8216_m2pi__less__pi,axiom,
% 0.30/0.64      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 0.30/0.64  
% 0.30/0.64  % m2pi_less_pi
% 0.30/0.64  thf(fact_8217_summable__ratio__test,axiom,
% 0.30/0.64      ! [C: real,N7: nat,F: nat > real] :
% 0.30/0.64        ( ( ord_less_real @ C @ one_one_real )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.30/0.64             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 0.30/0.64         => ( summable_real @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_ratio_test
% 0.30/0.64  thf(fact_8218_summable__ratio__test,axiom,
% 0.30/0.64      ! [C: real,N7: nat,F: nat > complex] :
% 0.30/0.64        ( ( ord_less_real @ C @ one_one_real )
% 0.30/0.64       => ( ! [N3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N7 @ N3 )
% 0.30/0.64             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 0.30/0.64         => ( summable_complex @ F ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % summable_ratio_test
% 0.30/0.64  thf(fact_8219_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > int,N2: nat,I: nat] :
% 0.30/0.64        ( ( summable_int @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N2 @ I )
% 0.30/0.64           => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 0.30/0.64             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_8220_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > nat,N2: nat,I: nat] :
% 0.30/0.64        ( ( summable_nat @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N2 @ I )
% 0.30/0.64           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 0.30/0.64             => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_8221_sum__less__suminf2,axiom,
% 0.30/0.64      ! [F: nat > real,N2: nat,I: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [M3: nat] :
% 0.30/0.64              ( ( ord_less_eq_nat @ N2 @ M3 )
% 0.30/0.64             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ N2 @ I )
% 0.30/0.64           => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 0.30/0.64             => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_less_suminf2
% 0.30/0.64  thf(fact_8222_arctan__ubound,axiom,
% 0.30/0.64      ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_ubound
% 0.30/0.64  thf(fact_8223_geometric__sums,axiom,
% 0.30/0.64      ! [C: real] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 0.30/0.64       => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_sums
% 0.30/0.64  thf(fact_8224_geometric__sums,axiom,
% 0.30/0.64      ! [C: complex] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 0.30/0.64       => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % geometric_sums
% 0.30/0.64  thf(fact_8225_power__half__series,axiom,
% 0.30/0.64      ( sums_real
% 0.30/0.64      @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
% 0.30/0.64      @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % power_half_series
% 0.30/0.64  thf(fact_8226_log__of__power__le,axiom,
% 0.30/0.64      ! [M: nat,B: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_of_power_le
% 0.30/0.64  thf(fact_8227_minus__pi__half__less__zero,axiom,
% 0.30/0.64      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 0.30/0.64  
% 0.30/0.64  % minus_pi_half_less_zero
% 0.30/0.64  thf(fact_8228_arctan__lbound,axiom,
% 0.30/0.64      ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_lbound
% 0.30/0.64  thf(fact_8229_arctan__bounded,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.30/0.64        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_bounded
% 0.30/0.64  thf(fact_8230_less__log2__of__power,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 0.30/0.64       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_log2_of_power
% 0.30/0.64  thf(fact_8231_le__log2__of__power,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 0.30/0.64       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_log2_of_power
% 0.30/0.64  thf(fact_8232_log2__of__power__less,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log2_of_power_less
% 0.30/0.64  thf(fact_8233_log2__of__power__le,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log2_of_power_le
% 0.30/0.64  thf(fact_8234_sum__pos__lt__pair,axiom,
% 0.30/0.64      ! [F: nat > real,K: nat] :
% 0.30/0.64        ( ( summable_real @ F )
% 0.30/0.64       => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
% 0.30/0.64         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sum_pos_lt_pair
% 0.30/0.64  thf(fact_8235_ceiling__log__nat__eq__if,axiom,
% 0.30/0.64      ! [B: nat,N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.64       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.64           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.64              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ceiling_log_nat_eq_if
% 0.30/0.64  thf(fact_8236_ceiling__log2__div2,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.30/0.64          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ceiling_log2_div2
% 0.30/0.64  thf(fact_8237_sin__cos__npi,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_npi
% 0.30/0.64  thf(fact_8238_diffs__equiv,axiom,
% 0.30/0.64      ! [C: nat > complex,X3: complex] :
% 0.30/0.64        ( ( summable_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 0.30/0.64       => ( sums_complex
% 0.30/0.64          @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64          @ ( suminf_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_equiv
% 0.30/0.64  thf(fact_8239_diffs__equiv,axiom,
% 0.30/0.64      ! [C: nat > real,X3: real] :
% 0.30/0.64        ( ( summable_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 0.30/0.64       => ( sums_real
% 0.30/0.64          @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.64          @ ( suminf_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_equiv
% 0.30/0.64  thf(fact_8240_cos__pi__eq__zero,axiom,
% 0.30/0.64      ! [M: nat] :
% 0.30/0.64        ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_pi_eq_zero
% 0.30/0.64  thf(fact_8241_ceiling__log__eq__powr__iff,axiom,
% 0.30/0.64      ! [X3: real,B: real,K: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X3 ) )
% 0.30/0.64              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 0.30/0.64            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
% 0.30/0.64              & ( ord_less_eq_real @ X3 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ceiling_log_eq_powr_iff
% 0.30/0.64  thf(fact_8242_floor__log__nat__eq__powr__iff,axiom,
% 0.30/0.64      ! [B: nat,K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.64              = ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.64            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.64              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_log_nat_eq_powr_iff
% 0.30/0.64  thf(fact_8243_sin__zero,axiom,
% 0.30/0.64      ( ( sin_complex @ zero_zero_complex )
% 0.30/0.64      = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero
% 0.30/0.64  thf(fact_8244_sin__zero,axiom,
% 0.30/0.64      ( ( sin_real @ zero_zero_real )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero
% 0.30/0.64  thf(fact_8245_powr__eq__0__iff,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( ( powr_real @ W2 @ Z2 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        = ( W2 = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_eq_0_iff
% 0.30/0.64  thf(fact_8246_powr__0,axiom,
% 0.30/0.64      ! [Z2: real] :
% 0.30/0.64        ( ( powr_real @ zero_zero_real @ Z2 )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_0
% 0.30/0.64  thf(fact_8247_cos__zero,axiom,
% 0.30/0.64      ( ( cos_complex @ zero_zero_complex )
% 0.30/0.64      = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_zero
% 0.30/0.64  thf(fact_8248_cos__zero,axiom,
% 0.30/0.64      ( ( cos_real @ zero_zero_real )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_zero
% 0.30/0.64  thf(fact_8249_powr__zero__eq__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( X3 = zero_zero_real )
% 0.30/0.64         => ( ( powr_real @ X3 @ zero_zero_real )
% 0.30/0.64            = zero_zero_real ) )
% 0.30/0.64        & ( ( X3 != zero_zero_real )
% 0.30/0.64         => ( ( powr_real @ X3 @ zero_zero_real )
% 0.30/0.64            = one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_zero_eq_one
% 0.30/0.64  thf(fact_8250_floor__zero,axiom,
% 0.30/0.64      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 0.30/0.64      = zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_zero
% 0.30/0.64  thf(fact_8251_floor__zero,axiom,
% 0.30/0.64      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 0.30/0.64      = zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_zero
% 0.30/0.64  thf(fact_8252_powr__gt__zero,axiom,
% 0.30/0.64      ! [X3: real,A: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
% 0.30/0.64        = ( X3 != zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_gt_zero
% 0.30/0.64  thf(fact_8253_powr__less__cancel__iff,axiom,
% 0.30/0.64      ! [X3: real,A: real,B: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 0.30/0.64          = ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_cancel_iff
% 0.30/0.64  thf(fact_8254_powr__eq__one__iff,axiom,
% 0.30/0.64      ! [A: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ A )
% 0.30/0.64       => ( ( ( powr_real @ A @ X3 )
% 0.30/0.64            = one_one_real )
% 0.30/0.64          = ( X3 = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_eq_one_iff
% 0.30/0.64  thf(fact_8255_powr__le__cancel__iff,axiom,
% 0.30/0.64      ! [X3: real,A: real,B: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 0.30/0.64          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_le_cancel_iff
% 0.30/0.64  thf(fact_8256_zero__le__floor,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_le_floor
% 0.30/0.64  thf(fact_8257_zero__le__floor,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ zero_zero_rat @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_le_floor
% 0.30/0.64  thf(fact_8258_sin__cos__squared__add3,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add3
% 0.30/0.64  thf(fact_8259_sin__cos__squared__add3,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add3
% 0.30/0.64  thf(fact_8260_floor__less__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
% 0.30/0.64        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_zero
% 0.30/0.64  thf(fact_8261_floor__less__zero,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
% 0.30/0.64        = ( ord_less_rat @ X3 @ zero_zero_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_zero
% 0.30/0.64  thf(fact_8262_numeral__le__floor,axiom,
% 0.30/0.64      ! [V: num,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % numeral_le_floor
% 0.30/0.64  thf(fact_8263_numeral__le__floor,axiom,
% 0.30/0.64      ! [V: num,X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % numeral_le_floor
% 0.30/0.64  thf(fact_8264_zero__less__floor,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_less_floor
% 0.30/0.64  thf(fact_8265_zero__less__floor,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_less_floor
% 0.30/0.64  thf(fact_8266_floor__le__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
% 0.30/0.64        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_zero
% 0.30/0.64  thf(fact_8267_floor__le__zero,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
% 0.30/0.64        = ( ord_less_rat @ X3 @ one_one_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_zero
% 0.30/0.64  thf(fact_8268_floor__less__numeral,axiom,
% 0.30/0.64      ! [X3: real,V: num] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_numeral
% 0.30/0.64  thf(fact_8269_floor__less__numeral,axiom,
% 0.30/0.64      ! [X3: rat,V: num] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_numeral
% 0.30/0.64  thf(fact_8270_one__le__floor,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_le_floor
% 0.30/0.64  thf(fact_8271_one__le__floor,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_le_floor
% 0.30/0.64  thf(fact_8272_floor__less__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 0.30/0.64        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_one
% 0.30/0.64  thf(fact_8273_floor__less__one,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 0.30/0.64        = ( ord_less_rat @ X3 @ one_one_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_one
% 0.30/0.64  thf(fact_8274_powr__log__cancel,axiom,
% 0.30/0.64      ! [A: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( powr_real @ A @ ( log @ A @ X3 ) )
% 0.30/0.64              = X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_log_cancel
% 0.30/0.64  thf(fact_8275_log__powr__cancel,axiom,
% 0.30/0.64      ! [A: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 0.30/0.64            = Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_powr_cancel
% 0.30/0.64  thf(fact_8276_numeral__less__floor,axiom,
% 0.30/0.64      ! [V: num,X3: real] :
% 0.30/0.64        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % numeral_less_floor
% 0.30/0.64  thf(fact_8277_numeral__less__floor,axiom,
% 0.30/0.64      ! [V: num,X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % numeral_less_floor
% 0.30/0.64  thf(fact_8278_floor__le__numeral,axiom,
% 0.30/0.64      ! [X3: real,V: num] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_numeral
% 0.30/0.64  thf(fact_8279_floor__le__numeral,axiom,
% 0.30/0.64      ! [X3: rat,V: num] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_numeral
% 0.30/0.64  thf(fact_8280_one__less__floor,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_less_floor
% 0.30/0.64  thf(fact_8281_one__less__floor,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_less_floor
% 0.30/0.64  thf(fact_8282_floor__le__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_one
% 0.30/0.64  thf(fact_8283_floor__le__one,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_one
% 0.30/0.64  thf(fact_8284_neg__numeral__le__floor,axiom,
% 0.30/0.64      ! [V: num,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % neg_numeral_le_floor
% 0.30/0.64  thf(fact_8285_neg__numeral__le__floor,axiom,
% 0.30/0.64      ! [V: num,X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % neg_numeral_le_floor
% 0.30/0.64  thf(fact_8286_floor__less__neg__numeral,axiom,
% 0.30/0.64      ! [X3: real,V: num] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_neg_numeral
% 0.30/0.64  thf(fact_8287_floor__less__neg__numeral,axiom,
% 0.30/0.64      ! [X3: rat,V: num] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_neg_numeral
% 0.30/0.64  thf(fact_8288_sin__cos__squared__add2,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add2
% 0.30/0.64  thf(fact_8289_sin__cos__squared__add2,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add2
% 0.30/0.64  thf(fact_8290_sin__cos__squared__add,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add
% 0.30/0.64  thf(fact_8291_sin__cos__squared__add,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_cos_squared_add
% 0.30/0.64  thf(fact_8292_neg__numeral__less__floor,axiom,
% 0.30/0.64      ! [V: num,X3: real] :
% 0.30/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % neg_numeral_less_floor
% 0.30/0.64  thf(fact_8293_neg__numeral__less__floor,axiom,
% 0.30/0.64      ! [V: num,X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % neg_numeral_less_floor
% 0.30/0.64  thf(fact_8294_floor__le__neg__numeral,axiom,
% 0.30/0.64      ! [X3: real,V: num] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_neg_numeral
% 0.30/0.64  thf(fact_8295_floor__le__neg__numeral,axiom,
% 0.30/0.64      ! [X3: rat,V: num] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_neg_numeral
% 0.30/0.64  thf(fact_8296_cos__one__sin__zero,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64          = one_one_complex )
% 0.30/0.64       => ( ( sin_complex @ X3 )
% 0.30/0.64          = zero_zero_complex ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_one_sin_zero
% 0.30/0.64  thf(fact_8297_cos__one__sin__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64          = one_one_real )
% 0.30/0.64       => ( ( sin_real @ X3 )
% 0.30/0.64          = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_one_sin_zero
% 0.30/0.64  thf(fact_8298_sin__add,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_add
% 0.30/0.64  thf(fact_8299_sin__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_add
% 0.30/0.64  thf(fact_8300_cos__diff,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff
% 0.30/0.64  thf(fact_8301_cos__diff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff
% 0.30/0.64  thf(fact_8302_cos__add,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 0.30/0.64        = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_add
% 0.30/0.64  thf(fact_8303_cos__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_add
% 0.30/0.64  thf(fact_8304_sin__zero__norm__cos__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( sin_real @ X3 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X3 ) )
% 0.30/0.64          = one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero_norm_cos_one
% 0.30/0.64  thf(fact_8305_sin__zero__norm__cos__one,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( ( sin_complex @ X3 )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X3 ) )
% 0.30/0.64          = one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero_norm_cos_one
% 0.30/0.64  thf(fact_8306_sincos__principal__value,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64      ? [Y5: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y5 )
% 0.30/0.64        & ( ord_less_eq_real @ Y5 @ pi )
% 0.30/0.64        & ( ( sin_real @ Y5 )
% 0.30/0.64          = ( sin_real @ X3 ) )
% 0.30/0.64        & ( ( cos_real @ Y5 )
% 0.30/0.64          = ( cos_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_principal_value
% 0.30/0.64  thf(fact_8307_floor__mono,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.64       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_mono
% 0.30/0.64  thf(fact_8308_floor__mono,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] :
% 0.30/0.64        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 0.30/0.64       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_mono
% 0.30/0.64  thf(fact_8309_of__int__floor__le,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 ) ).
% 0.30/0.64  
% 0.30/0.64  % of_int_floor_le
% 0.30/0.64  thf(fact_8310_of__int__floor__le,axiom,
% 0.30/0.64      ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 ) ).
% 0.30/0.64  
% 0.30/0.64  % of_int_floor_le
% 0.30/0.64  thf(fact_8311_floor__less__cancel,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 0.30/0.64       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_cancel
% 0.30/0.64  thf(fact_8312_floor__less__cancel,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 0.30/0.64       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_cancel
% 0.30/0.64  thf(fact_8313_powr__non__neg,axiom,
% 0.30/0.64      ! [A: real,X3: real] :
% 0.30/0.64        ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_non_neg
% 0.30/0.64  thf(fact_8314_powr__less__mono2__neg,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ A @ zero_zero_real )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.64           => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_mono2_neg
% 0.30/0.64  thf(fact_8315_powr__less__cancel,axiom,
% 0.30/0.64      ! [X3: real,A: real,B: real] :
% 0.30/0.64        ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64         => ( ord_less_real @ A @ B ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_cancel
% 0.30/0.64  thf(fact_8316_powr__less__mono,axiom,
% 0.30/0.64      ! [A: real,B: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ A @ B )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_mono
% 0.30/0.64  thf(fact_8317_le__floor__iff,axiom,
% 0.30/0.64      ! [Z2: int,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_floor_iff
% 0.30/0.64  thf(fact_8318_le__floor__iff,axiom,
% 0.30/0.64      ! [Z2: int,X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_floor_iff
% 0.30/0.64  thf(fact_8319_floor__less__iff,axiom,
% 0.30/0.64      ! [X3: real,Z2: int] :
% 0.30/0.64        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_iff
% 0.30/0.64  thf(fact_8320_floor__less__iff,axiom,
% 0.30/0.64      ! [X3: rat,Z2: int] :
% 0.30/0.64        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_less_iff
% 0.30/0.64  thf(fact_8321_powr__mono2_H,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_mono2'
% 0.30/0.64  thf(fact_8322_powr__less__mono2,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.64           => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_mono2
% 0.30/0.64  thf(fact_8323_le__floor__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_floor_add
% 0.30/0.64  thf(fact_8324_le__floor__add,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_floor_add
% 0.30/0.64  thf(fact_8325_gr__one__powr,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64         => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gr_one_powr
% 0.30/0.64  thf(fact_8326_powr__inj,axiom,
% 0.30/0.64      ! [A: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( A != one_one_real )
% 0.30/0.64         => ( ( ( powr_real @ A @ X3 )
% 0.30/0.64              = ( powr_real @ A @ Y3 ) )
% 0.30/0.64            = ( X3 = Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_inj
% 0.30/0.64  thf(fact_8327_int__add__floor,axiom,
% 0.30/0.64      ! [Z2: int,X3: real] :
% 0.30/0.64        ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % int_add_floor
% 0.30/0.64  thf(fact_8328_int__add__floor,axiom,
% 0.30/0.64      ! [Z2: int,X3: rat] :
% 0.30/0.64        ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % int_add_floor
% 0.30/0.64  thf(fact_8329_floor__add__int,axiom,
% 0.30/0.64      ! [X3: real,Z2: int] :
% 0.30/0.64        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 0.30/0.64        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_add_int
% 0.30/0.64  thf(fact_8330_floor__add__int,axiom,
% 0.30/0.64      ! [X3: rat,Z2: int] :
% 0.30/0.64        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 0.30/0.64        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_add_int
% 0.30/0.64  thf(fact_8331_sin__gt__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ pi )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_gt_zero
% 0.30/0.64  thf(fact_8332_sin__times__sin,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_sin
% 0.30/0.64  thf(fact_8333_sin__times__sin,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_sin
% 0.30/0.64  thf(fact_8334_sin__times__cos,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_cos
% 0.30/0.64  thf(fact_8335_sin__times__cos,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_times_cos
% 0.30/0.64  thf(fact_8336_cos__times__sin,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_sin
% 0.30/0.64  thf(fact_8337_cos__times__sin,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_sin
% 0.30/0.64  thf(fact_8338_sin__plus__sin,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_plus_sin
% 0.30/0.64  thf(fact_8339_sin__plus__sin,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_plus_sin
% 0.30/0.64  thf(fact_8340_sin__diff__sin,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_diff_sin
% 0.30/0.64  thf(fact_8341_sin__diff__sin,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_diff_sin
% 0.30/0.64  thf(fact_8342_cos__diff__cos,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff_cos
% 0.30/0.64  thf(fact_8343_cos__diff__cos,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_diff_cos
% 0.30/0.64  thf(fact_8344_powr__add,axiom,
% 0.30/0.64      ! [X3: real,A: real,B: real] :
% 0.30/0.64        ( ( powr_real @ X3 @ ( plus_plus_real @ A @ B ) )
% 0.30/0.64        = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_add
% 0.30/0.64  thf(fact_8345_one__add__floor,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 0.30/0.64        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_add_floor
% 0.30/0.64  thf(fact_8346_one__add__floor,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 0.30/0.64        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_add_floor
% 0.30/0.64  thf(fact_8347_floor__log__eq__powr__iff,axiom,
% 0.30/0.64      ! [X3: real,B: real,K: int] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X3 ) )
% 0.30/0.64              = K )
% 0.30/0.64            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X3 )
% 0.30/0.64              & ( ord_less_real @ X3 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_log_eq_powr_iff
% 0.30/0.64  thf(fact_8348_powr__realpow,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.64          = ( power_power_real @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_realpow
% 0.30/0.64  thf(fact_8349_less__log__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ Y3 @ ( log @ B @ X3 ) )
% 0.30/0.64            = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_log_iff
% 0.30/0.64  thf(fact_8350_log__less__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ ( log @ B @ X3 ) @ Y3 )
% 0.30/0.64            = ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_less_iff
% 0.30/0.64  thf(fact_8351_less__powr__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) )
% 0.30/0.64            = ( ord_less_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_powr_iff
% 0.30/0.64  thf(fact_8352_powr__less__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 )
% 0.30/0.64            = ( ord_less_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_less_iff
% 0.30/0.64  thf(fact_8353_cos__mono__less__eq,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ X3 @ pi )
% 0.30/0.64         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64           => ( ( ord_less_eq_real @ Y3 @ pi )
% 0.30/0.64             => ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_real @ Y3 @ X3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_mono_less_eq
% 0.30/0.64  thf(fact_8354_cos__monotone__0__pi,axiom,
% 0.30/0.64      ! [Y3: real,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ pi )
% 0.30/0.64           => ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_monotone_0_pi
% 0.30/0.64  thf(fact_8355_sin__eq__0__pi,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ pi )
% 0.30/0.64         => ( ( ( sin_real @ X3 )
% 0.30/0.64              = zero_zero_real )
% 0.30/0.64           => ( X3 = zero_zero_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_eq_0_pi
% 0.30/0.64  thf(fact_8356_real__of__int__floor__add__one__gt,axiom,
% 0.30/0.64      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_of_int_floor_add_one_gt
% 0.30/0.64  thf(fact_8357_floor__eq,axiom,
% 0.30/0.64      ! [N2: int,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 0.30/0.64         => ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.64            = N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_eq
% 0.30/0.64  thf(fact_8358_sin__zero__pi__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
% 0.30/0.64       => ( ( ( sin_real @ X3 )
% 0.30/0.64            = zero_zero_real )
% 0.30/0.64          = ( X3 = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_zero_pi_iff
% 0.30/0.64  thf(fact_8359_real__of__int__floor__gt__diff__one,axiom,
% 0.30/0.64      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_of_int_floor_gt_diff_one
% 0.30/0.64  thf(fact_8360_diffs__def,axiom,
% 0.30/0.64      ( diffs_complex
% 0.30/0.64      = ( ^ [C3: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_8361_diffs__def,axiom,
% 0.30/0.64      ( diffs_rat
% 0.30/0.64      = ( ^ [C3: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_8362_diffs__def,axiom,
% 0.30/0.64      ( diffs_int
% 0.30/0.64      = ( ^ [C3: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_8363_diffs__def,axiom,
% 0.30/0.64      ( diffs_real
% 0.30/0.64      = ( ^ [C3: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % diffs_def
% 0.30/0.64  thf(fact_8364_sin__expansion__lemma,axiom,
% 0.30/0.64      ! [X3: real,M: nat] :
% 0.30/0.64        ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64        = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_expansion_lemma
% 0.30/0.64  thf(fact_8365_floor__unique,axiom,
% 0.30/0.64      ! [Z2: int,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 0.30/0.64         => ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.64            = Z2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_unique
% 0.30/0.64  thf(fact_8366_floor__unique,axiom,
% 0.30/0.64      ! [Z2: int,X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
% 0.30/0.64       => ( ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 0.30/0.64         => ( ( archim3151403230148437115or_rat @ X3 )
% 0.30/0.64            = Z2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_unique
% 0.30/0.64  thf(fact_8367_floor__eq__iff,axiom,
% 0.30/0.64      ! [X3: real,A: int] :
% 0.30/0.64        ( ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.64          = A )
% 0.30/0.64        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X3 )
% 0.30/0.64          & ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_eq_iff
% 0.30/0.64  thf(fact_8368_floor__eq__iff,axiom,
% 0.30/0.64      ! [X3: rat,A: int] :
% 0.30/0.64        ( ( ( archim3151403230148437115or_rat @ X3 )
% 0.30/0.64          = A )
% 0.30/0.64        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X3 )
% 0.30/0.64          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_eq_iff
% 0.30/0.64  thf(fact_8369_floor__split,axiom,
% 0.30/0.64      ! [P: int > $o,T: real] :
% 0.30/0.64        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 0.30/0.64        = ( ! [I2: int] :
% 0.30/0.64              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 0.30/0.64                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 0.30/0.64             => ( P @ I2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_split
% 0.30/0.64  thf(fact_8370_floor__split,axiom,
% 0.30/0.64      ! [P: int > $o,T: rat] :
% 0.30/0.64        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 0.30/0.64        = ( ! [I2: int] :
% 0.30/0.64              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 0.30/0.64                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 0.30/0.64             => ( P @ I2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_split
% 0.30/0.64  thf(fact_8371_le__mult__floor,axiom,
% 0.30/0.64      ! [A: real,B: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 0.30/0.64       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 0.30/0.64         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_mult_floor
% 0.30/0.64  thf(fact_8372_le__mult__floor,axiom,
% 0.30/0.64      ! [A: rat,B: rat] :
% 0.30/0.64        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 0.30/0.64       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 0.30/0.64         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_mult_floor
% 0.30/0.64  thf(fact_8373_less__floor__iff,axiom,
% 0.30/0.64      ! [Z2: int,X3: real] :
% 0.30/0.64        ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_floor_iff
% 0.30/0.64  thf(fact_8374_less__floor__iff,axiom,
% 0.30/0.64      ! [Z2: int,X3: rat] :
% 0.30/0.64        ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 0.30/0.64        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % less_floor_iff
% 0.30/0.64  thf(fact_8375_floor__le__iff,axiom,
% 0.30/0.64      ! [X3: real,Z2: int] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 0.30/0.64        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_iff
% 0.30/0.64  thf(fact_8376_floor__le__iff,axiom,
% 0.30/0.64      ! [X3: rat,Z2: int] :
% 0.30/0.64        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 0.30/0.64        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_le_iff
% 0.30/0.64  thf(fact_8377_floor__correct,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 )
% 0.30/0.64        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_correct
% 0.30/0.64  thf(fact_8378_floor__correct,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 )
% 0.30/0.64        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_correct
% 0.30/0.64  thf(fact_8379_cos__expansion__lemma,axiom,
% 0.30/0.64      ! [X3: real,M: nat] :
% 0.30/0.64        ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64        = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_expansion_lemma
% 0.30/0.64  thf(fact_8380_powr__neg__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 0.30/0.64          = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_neg_one
% 0.30/0.64  thf(fact_8381_sin__gt__zero__02,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_gt_zero_02
% 0.30/0.64  thf(fact_8382_le__log__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) )
% 0.30/0.64            = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_log_iff
% 0.30/0.64  thf(fact_8383_log__le__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 )
% 0.30/0.64            = ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_le_iff
% 0.30/0.64  thf(fact_8384_le__powr__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) )
% 0.30/0.64            = ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % le_powr_iff
% 0.30/0.64  thf(fact_8385_powr__le__iff,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 )
% 0.30/0.64            = ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_le_iff
% 0.30/0.64  thf(fact_8386_cos__two__less__zero,axiom,
% 0.30/0.64      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 0.30/0.64  
% 0.30/0.64  % cos_two_less_zero
% 0.30/0.64  thf(fact_8387_floor__eq2,axiom,
% 0.30/0.64      ! [N2: int,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 0.30/0.64         => ( ( archim6058952711729229775r_real @ X3 )
% 0.30/0.64            = N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_eq2
% 0.30/0.64  thf(fact_8388_cos__monotone__minus__pi__0,axiom,
% 0.30/0.64      ! [Y3: real,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.64           => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_monotone_minus_pi_0
% 0.30/0.64  thf(fact_8389_floor__divide__lower,axiom,
% 0.30/0.64      ! [Q4: real,P4: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.30/0.64       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_divide_lower
% 0.30/0.64  thf(fact_8390_floor__divide__lower,axiom,
% 0.30/0.64      ! [Q4: rat,P4: rat] :
% 0.30/0.64        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.30/0.64       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_divide_lower
% 0.30/0.64  thf(fact_8391_sincos__total__2pi,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64          = one_one_real )
% 0.30/0.64       => ~ ! [T6: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.64             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64               => ( ( X3
% 0.30/0.64                    = ( cos_real @ T6 ) )
% 0.30/0.64                 => ( Y3
% 0.30/0.64                   != ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sincos_total_2pi
% 0.30/0.64  thf(fact_8392_ln__powr__bound,axiom,
% 0.30/0.64      ! [X3: real,A: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64         => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ln_powr_bound
% 0.30/0.64  thf(fact_8393_ln__powr__bound2,axiom,
% 0.30/0.64      ! [X3: real,A: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.64         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ln_powr_bound2
% 0.30/0.64  thf(fact_8394_log__add__eq__powr,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64       => ( ( B != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( plus_plus_real @ ( log @ B @ X3 ) @ Y3 )
% 0.30/0.64              = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_add_eq_powr
% 0.30/0.64  thf(fact_8395_add__log__eq__powr,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64       => ( ( B != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( plus_plus_real @ Y3 @ ( log @ B @ X3 ) )
% 0.30/0.64              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % add_log_eq_powr
% 0.30/0.64  thf(fact_8396_minus__log__eq__powr,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64       => ( ( B != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( minus_minus_real @ Y3 @ ( log @ B @ X3 ) )
% 0.30/0.64              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % minus_log_eq_powr
% 0.30/0.64  thf(fact_8397_sin__pi__divide__n__ge__0,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( N2 != zero_zero_nat )
% 0.30/0.64       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_pi_divide_n_ge_0
% 0.30/0.64  thf(fact_8398_cos__times__cos,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 0.30/0.64        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_cos
% 0.30/0.64  thf(fact_8399_cos__times__cos,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_times_cos
% 0.30/0.64  thf(fact_8400_cos__plus__cos,axiom,
% 0.30/0.64      ! [W2: complex,Z2: complex] :
% 0.30/0.64        ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_plus_cos
% 0.30/0.64  thf(fact_8401_cos__plus__cos,axiom,
% 0.30/0.64      ! [W2: real,Z2: real] :
% 0.30/0.64        ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_plus_cos
% 0.30/0.64  thf(fact_8402_floor__divide__upper,axiom,
% 0.30/0.64      ! [Q4: real,P4: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 0.30/0.64       => ( 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 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_divide_upper
% 0.30/0.64  thf(fact_8403_floor__divide__upper,axiom,
% 0.30/0.64      ! [Q4: rat,P4: rat] :
% 0.30/0.64        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 0.30/0.64       => ( 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 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_divide_upper
% 0.30/0.64  thf(fact_8404_round__def,axiom,
% 0.30/0.64      ( archim8280529875227126926d_real
% 0.30/0.64      = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % round_def
% 0.30/0.64  thf(fact_8405_round__def,axiom,
% 0.30/0.64      ( archim7778729529865785530nd_rat
% 0.30/0.64      = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % round_def
% 0.30/0.64  thf(fact_8406_sin__gt__zero2,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_gt_zero2
% 0.30/0.64  thf(fact_8407_sin__lt__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ pi @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_lt_zero
% 0.30/0.64  thf(fact_8408_cos__double__less__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 0.30/0.64         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_double_less_one
% 0.30/0.64  thf(fact_8409_cos__gt__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_gt_zero
% 0.30/0.64  thf(fact_8410_log__minus__eq__powr,axiom,
% 0.30/0.64      ! [B: real,X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.64       => ( ( B != one_one_real )
% 0.30/0.64         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( minus_minus_real @ ( log @ B @ X3 ) @ Y3 )
% 0.30/0.64              = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_minus_eq_powr
% 0.30/0.64  thf(fact_8411_termdiff__converges,axiom,
% 0.30/0.64      ! [X3: real,K5: real,C: nat > real] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
% 0.30/0.64       => ( ! [X4: real] :
% 0.30/0.64              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
% 0.30/0.64             => ( summable_real
% 0.30/0.64                @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) ) )
% 0.30/0.64         => ( summable_real
% 0.30/0.64            @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges
% 0.30/0.64  thf(fact_8412_termdiff__converges,axiom,
% 0.30/0.64      ! [X3: complex,K5: real,C: nat > complex] :
% 0.30/0.64        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
% 0.30/0.64       => ( ! [X4: complex] :
% 0.30/0.64              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
% 0.30/0.64             => ( summable_complex
% 0.30/0.64                @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) )
% 0.30/0.64         => ( summable_complex
% 0.30/0.64            @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % termdiff_converges
% 0.30/0.64  thf(fact_8413_powr__neg__numeral,axiom,
% 0.30/0.64      ! [X3: real,N2: num] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 0.30/0.64          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_neg_numeral
% 0.30/0.64  thf(fact_8414_sin__le__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ pi @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64         => ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_le_zero
% 0.30/0.64  thf(fact_8415_sin__less__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.64         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_less_zero
% 0.30/0.64  thf(fact_8416_sin__mono__less__eq,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64             => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_mono_less_eq
% 0.30/0.64  thf(fact_8417_sin__monotone__2pi,axiom,
% 0.30/0.64      ! [Y3: real,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64           => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_monotone_2pi
% 0.30/0.64  thf(fact_8418_cos__gt__zero__pi,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_gt_zero_pi
% 0.30/0.64  thf(fact_8419_sin__pi__divide__n__gt__0,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_pi_divide_n_gt_0
% 0.30/0.64  thf(fact_8420_floor__log2__div2,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.30/0.64          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_log2_div2
% 0.30/0.64  thf(fact_8421_floor__log__nat__eq__if,axiom,
% 0.30/0.64      ! [B: nat,N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 0.30/0.64       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 0.30/0.64           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 0.30/0.64              = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_log_nat_eq_if
% 0.30/0.64  thf(fact_8422_Maclaurin__minus__cos__expansion,axiom,
% 0.30/0.64      ! [N2: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.64         => ? [T6: real] :
% 0.30/0.64              ( ( ord_less_real @ X3 @ T6 )
% 0.30/0.64              & ( ord_less_real @ T6 @ zero_zero_real )
% 0.30/0.64              & ( ( cos_real @ X3 )
% 0.30/0.64                = ( plus_plus_real
% 0.30/0.64                  @ ( groups6591440286371151544t_real
% 0.30/0.64                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_minus_cos_expansion
% 0.30/0.64  thf(fact_8423_Maclaurin__cos__expansion2,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64         => ? [T6: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.30/0.64              & ( ord_less_real @ T6 @ X3 )
% 0.30/0.64              & ( ( cos_real @ X3 )
% 0.30/0.64                = ( plus_plus_real
% 0.30/0.64                  @ ( groups6591440286371151544t_real
% 0.30/0.64                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_cos_expansion2
% 0.30/0.64  thf(fact_8424_sin__paired,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( sums_real
% 0.30/0.64        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.30/0.64        @ ( sin_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_paired
% 0.30/0.64  thf(fact_8425_tan__double,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64           != zero_zero_complex )
% 0.30/0.64         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_double
% 0.30/0.64  thf(fact_8426_tan__double,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64           != zero_zero_real )
% 0.30/0.64         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 0.30/0.64            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X3 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_double
% 0.30/0.64  thf(fact_8427_arcosh__def,axiom,
% 0.30/0.64      ( arcosh_real
% 0.30/0.64      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arcosh_def
% 0.30/0.64  thf(fact_8428_tan__zero,axiom,
% 0.30/0.64      ( ( tan_complex @ zero_zero_complex )
% 0.30/0.64      = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_zero
% 0.30/0.64  thf(fact_8429_tan__zero,axiom,
% 0.30/0.64      ( ( tan_real @ zero_zero_real )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_zero
% 0.30/0.64  thf(fact_8430_of__real__eq__0__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( real_V1803761363581548252l_real @ X3 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_eq_0_iff
% 0.30/0.64  thf(fact_8431_of__real__eq__0__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( real_V4546457046886955230omplex @ X3 )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64        = ( X3 = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_eq_0_iff
% 0.30/0.64  thf(fact_8432_of__real__0,axiom,
% 0.30/0.64      ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_0
% 0.30/0.64  thf(fact_8433_of__real__0,axiom,
% 0.30/0.64      ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 0.30/0.64      = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_0
% 0.30/0.64  thf(fact_8434_fact__0,axiom,
% 0.30/0.64      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 0.30/0.64      = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_0
% 0.30/0.64  thf(fact_8435_fact__0,axiom,
% 0.30/0.64      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 0.30/0.64      = one_one_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_0
% 0.30/0.64  thf(fact_8436_fact__0,axiom,
% 0.30/0.64      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 0.30/0.64      = one_one_int ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_0
% 0.30/0.64  thf(fact_8437_fact__0,axiom,
% 0.30/0.64      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_0
% 0.30/0.64  thf(fact_8438_fact__0,axiom,
% 0.30/0.64      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 0.30/0.64      = one_one_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_0
% 0.30/0.64  thf(fact_8439_of__real__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_add
% 0.30/0.64  thf(fact_8440_of__real__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_real_add
% 0.30/0.64  thf(fact_8441_fact__Suc__0,axiom,
% 0.30/0.64      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 0.30/0.64      = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc_0
% 0.30/0.64  thf(fact_8442_fact__Suc__0,axiom,
% 0.30/0.64      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 0.30/0.64      = one_one_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc_0
% 0.30/0.64  thf(fact_8443_fact__Suc__0,axiom,
% 0.30/0.64      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 0.30/0.64      = one_one_int ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc_0
% 0.30/0.64  thf(fact_8444_fact__Suc__0,axiom,
% 0.30/0.64      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc_0
% 0.30/0.64  thf(fact_8445_fact__Suc__0,axiom,
% 0.30/0.64      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 0.30/0.64      = one_one_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc_0
% 0.30/0.64  thf(fact_8446_fact__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_8447_fact__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_8448_fact__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_8449_fact__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_8450_fact__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_Suc
% 0.30/0.64  thf(fact_8451_sin__of__real__pi,axiom,
% 0.30/0.64      ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_of_real_pi
% 0.30/0.64  thf(fact_8452_sin__of__real__pi,axiom,
% 0.30/0.64      ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 0.30/0.64      = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_of_real_pi
% 0.30/0.64  thf(fact_8453_norm__of__real__add1,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ one_one_real ) )
% 0.30/0.64        = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_of_real_add1
% 0.30/0.64  thf(fact_8454_norm__of__real__add1,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ one_one_complex ) )
% 0.30/0.64        = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_of_real_add1
% 0.30/0.64  thf(fact_8455_norm__of__real__addn,axiom,
% 0.30/0.64      ! [X3: real,B: num] :
% 0.30/0.64        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( numeral_numeral_real @ B ) ) )
% 0.30/0.64        = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_of_real_addn
% 0.30/0.64  thf(fact_8456_norm__of__real__addn,axiom,
% 0.30/0.64      ! [X3: real,B: num] :
% 0.30/0.64        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( numera6690914467698888265omplex @ B ) ) )
% 0.30/0.64        = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_of_real_addn
% 0.30/0.64  thf(fact_8457_cos__of__real__pi__half,axiom,
% 0.30/0.64      ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_of_real_pi_half
% 0.30/0.64  thf(fact_8458_cos__of__real__pi__half,axiom,
% 0.30/0.64      ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.30/0.64      = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_of_real_pi_half
% 0.30/0.64  thf(fact_8459_fact__nonzero,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri5044797733671781792omplex @ N2 )
% 0.30/0.64       != zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_nonzero
% 0.30/0.64  thf(fact_8460_fact__nonzero,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri773545260158071498ct_rat @ N2 )
% 0.30/0.64       != zero_zero_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_nonzero
% 0.30/0.64  thf(fact_8461_fact__nonzero,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri1406184849735516958ct_int @ N2 )
% 0.30/0.64       != zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_nonzero
% 0.30/0.64  thf(fact_8462_fact__nonzero,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri2265585572941072030t_real @ N2 )
% 0.30/0.64       != zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_nonzero
% 0.30/0.64  thf(fact_8463_fact__nonzero,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( semiri1408675320244567234ct_nat @ N2 )
% 0.30/0.64       != zero_zero_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_nonzero
% 0.30/0.64  thf(fact_8464_fact__ge__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_8465_fact__ge__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_8466_fact__ge__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_8467_fact__ge__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_zero
% 0.30/0.64  thf(fact_8468_fact__not__neg,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_8469_fact__not__neg,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_8470_fact__not__neg,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_8471_fact__not__neg,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_not_neg
% 0.30/0.64  thf(fact_8472_fact__gt__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_8473_fact__gt__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_8474_fact__gt__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_8475_fact__gt__zero,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_gt_zero
% 0.30/0.64  thf(fact_8476_fact__ge__1,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_8477_fact__ge__1,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_8478_fact__ge__1,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_8479_fact__ge__1,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_1
% 0.30/0.64  thf(fact_8480_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_8481_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_8482_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_8483_fact__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono
% 0.30/0.64  thf(fact_8484_fact__dvd,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_8485_fact__dvd,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_8486_fact__dvd,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_8487_fact__dvd,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_dvd
% 0.30/0.64  thf(fact_8488_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.64         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_8489_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.64         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_8490_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.64         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_8491_fact__less__mono,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.64         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono
% 0.30/0.64  thf(fact_8492_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_8493_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_8494_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_8495_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_8496_fact__fact__dvd__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_fact_dvd_fact
% 0.30/0.64  thf(fact_8497_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
% 0.30/0.64          = zero_zero_int ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mod
% 0.30/0.64  thf(fact_8498_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
% 0.30/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mod
% 0.30/0.64  thf(fact_8499_fact__mod,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 0.30/0.64          = zero_zero_nat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mod
% 0.30/0.64  thf(fact_8500_fact__le__power,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_8501_fact__le__power,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_8502_fact__le__power,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_8503_fact__le__power,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_le_power
% 0.30/0.64  thf(fact_8504_norm__less__p1,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X3 ) ) @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_less_p1
% 0.30/0.64  thf(fact_8505_norm__less__p1,axiom,
% 0.30/0.64      ! [X3: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X3 ) ) @ one_one_complex ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % norm_less_p1
% 0.30/0.64  thf(fact_8506_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_8507_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_8508_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_8509_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_8510_choose__dvd,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_dvd
% 0.30/0.64  thf(fact_8511_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri5044797733671781792omplex
% 0.30/0.64      = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_8512_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri773545260158071498ct_rat
% 0.30/0.64      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_8513_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri1406184849735516958ct_int
% 0.30/0.64      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_8514_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri2265585572941072030t_real
% 0.30/0.64      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_8515_fact__num__eq__if,axiom,
% 0.30/0.64      ( semiri1408675320244567234ct_nat
% 0.30/0.64      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_num_eq_if
% 0.30/0.64  thf(fact_8516_fact__reduce,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( semiri5044797733671781792omplex @ N2 )
% 0.30/0.64          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_8517_fact__reduce,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( semiri773545260158071498ct_rat @ N2 )
% 0.30/0.64          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_8518_fact__reduce,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( semiri1406184849735516958ct_int @ N2 )
% 0.30/0.64          = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_8519_fact__reduce,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( semiri2265585572941072030t_real @ N2 )
% 0.30/0.64          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_8520_fact__reduce,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( semiri1408675320244567234ct_nat @ N2 )
% 0.30/0.64          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_reduce
% 0.30/0.64  thf(fact_8521_tan__gt__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_gt_zero
% 0.30/0.64  thf(fact_8522_lemma__tan__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64       => ? [X4: real] :
% 0.30/0.64            ( ( ord_less_real @ zero_zero_real @ X4 )
% 0.30/0.64            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64            & ( ord_less_real @ Y3 @ ( tan_real @ X4 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_total
% 0.30/0.64  thf(fact_8523_lemma__tan__total1,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64      ? [X4: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.30/0.64        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        & ( ( tan_real @ X4 )
% 0.30/0.64          = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_total1
% 0.30/0.64  thf(fact_8524_tan__mono__lt__eq,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64             => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_mono_lt_eq
% 0.30/0.64  thf(fact_8525_tan__monotone_H,axiom,
% 0.30/0.64      ! [Y3: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64           => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64             => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.64                = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_monotone'
% 0.30/0.64  thf(fact_8526_tan__monotone,axiom,
% 0.30/0.64      ! [Y3: real,X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ X3 )
% 0.30/0.64         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64           => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_monotone
% 0.30/0.64  thf(fact_8527_tan__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64      ? [X4: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 0.30/0.64        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        & ( ( tan_real @ X4 )
% 0.30/0.64          = Y3 )
% 0.30/0.64        & ! [Y6: real] :
% 0.30/0.64            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
% 0.30/0.64              & ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64              & ( ( tan_real @ Y6 )
% 0.30/0.64                = Y3 ) )
% 0.30/0.64           => ( Y6 = X4 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total
% 0.30/0.64  thf(fact_8528_add__tan__eq,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ( cos_complex @ Y3 )
% 0.30/0.64           != zero_zero_complex )
% 0.30/0.64         => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) )
% 0.30/0.64            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % add_tan_eq
% 0.30/0.64  thf(fact_8529_add__tan__eq,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ( cos_real @ Y3 )
% 0.30/0.64           != zero_zero_real )
% 0.30/0.64         => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % add_tan_eq
% 0.30/0.64  thf(fact_8530_tan__total__pos,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64       => ? [X4: real] :
% 0.30/0.64            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 0.30/0.64            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64            & ( ( tan_real @ X4 )
% 0.30/0.64              = Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total_pos
% 0.30/0.64  thf(fact_8531_tan__pos__pi2__le,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_pos_pi2_le
% 0.30/0.64  thf(fact_8532_tan__less__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.64         => ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_less_zero
% 0.30/0.64  thf(fact_8533_tan__mono__le__eq,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 0.30/0.64           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64             => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 0.30/0.64                = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_mono_le_eq
% 0.30/0.64  thf(fact_8534_tan__mono__le,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.64         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64           => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_mono_le
% 0.30/0.64  thf(fact_8535_tan__bound__pi2,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X3 ) ) @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_bound_pi2
% 0.30/0.64  thf(fact_8536_arctan,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 0.30/0.64        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64        & ( ( tan_real @ ( arctan @ Y3 ) )
% 0.30/0.64          = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan
% 0.30/0.64  thf(fact_8537_arctan__tan,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( arctan @ ( tan_real @ X3 ) )
% 0.30/0.64            = X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_tan
% 0.30/0.64  thf(fact_8538_arctan__unique,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ( tan_real @ X3 )
% 0.30/0.64              = Y3 )
% 0.30/0.64           => ( ( arctan @ Y3 )
% 0.30/0.64              = X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arctan_unique
% 0.30/0.64  thf(fact_8539_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N2: nat,Diff: nat > complex > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64            = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_zero
% 0.30/0.64  thf(fact_8540_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N2: nat,Diff: nat > real > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64            = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_zero
% 0.30/0.64  thf(fact_8541_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N2: nat,Diff: nat > rat > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64            = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_zero
% 0.30/0.64  thf(fact_8542_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N2: nat,Diff: nat > nat > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64            = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_zero
% 0.30/0.64  thf(fact_8543_Maclaurin__zero,axiom,
% 0.30/0.64      ! [X3: real,N2: nat,Diff: nat > int > real] :
% 0.30/0.64        ( ( X3 = zero_zero_real )
% 0.30/0.64       => ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( groups6591440286371151544t_real
% 0.30/0.64              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64              @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64            = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_zero
% 0.30/0.64  thf(fact_8544_Maclaurin__lemma,axiom,
% 0.30/0.64      ! [H2: real,F: real > real,J: nat > real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.64       => ? [B9: real] :
% 0.30/0.64            ( ( F @ H2 )
% 0.30/0.64            = ( plus_plus_real
% 0.30/0.64              @ ( groups6591440286371151544t_real
% 0.30/0.64                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.30/0.64                @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64              @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_lemma
% 0.30/0.64  thf(fact_8545_tan__add,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ( cos_complex @ Y3 )
% 0.30/0.64           != zero_zero_complex )
% 0.30/0.64         => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 0.30/0.64             != zero_zero_complex )
% 0.30/0.64           => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 0.30/0.64              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_add
% 0.30/0.64  thf(fact_8546_tan__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ( cos_real @ Y3 )
% 0.30/0.64           != zero_zero_real )
% 0.30/0.64         => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64             != zero_zero_real )
% 0.30/0.64           => ( ( tan_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_add
% 0.30/0.64  thf(fact_8547_tan__diff,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ( cos_complex @ Y3 )
% 0.30/0.64           != zero_zero_complex )
% 0.30/0.64         => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 0.30/0.64             != zero_zero_complex )
% 0.30/0.64           => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 0.30/0.64              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_diff
% 0.30/0.64  thf(fact_8548_tan__diff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ( cos_real @ Y3 )
% 0.30/0.64           != zero_zero_real )
% 0.30/0.64         => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 0.30/0.64             != zero_zero_real )
% 0.30/0.64           => ( ( tan_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 0.30/0.64              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_diff
% 0.30/0.64  thf(fact_8549_lemma__tan__add1,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( ( cos_complex @ X3 )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ( cos_complex @ Y3 )
% 0.30/0.64           != zero_zero_complex )
% 0.30/0.64         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) )
% 0.30/0.64            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_add1
% 0.30/0.64  thf(fact_8550_lemma__tan__add1,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( cos_real @ X3 )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ( cos_real @ Y3 )
% 0.30/0.64           != zero_zero_real )
% 0.30/0.64         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) )
% 0.30/0.64            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_tan_add1
% 0.30/0.64  thf(fact_8551_minus__sin__cos__eq,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( uminus_uminus_real @ ( sin_real @ X3 ) )
% 0.30/0.64        = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % minus_sin_cos_eq
% 0.30/0.64  thf(fact_8552_minus__sin__cos__eq,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( uminus1482373934393186551omplex @ ( sin_complex @ X3 ) )
% 0.30/0.64        = ( cos_complex @ ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % minus_sin_cos_eq
% 0.30/0.64  thf(fact_8553_tan__total__pi4,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.64       => ? [Z3: real] :
% 0.30/0.64            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
% 0.30/0.64            & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64            & ( ( tan_real @ Z3 )
% 0.30/0.64              = X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_total_pi4
% 0.30/0.64  thf(fact_8554_tan__half,axiom,
% 0.30/0.64      ( tan_complex
% 0.30/0.64      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_half
% 0.30/0.64  thf(fact_8555_tan__half,axiom,
% 0.30/0.64      ( tan_real
% 0.30/0.64      = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tan_half
% 0.30/0.64  thf(fact_8556_arsinh__def,axiom,
% 0.30/0.64      ( arsinh_real
% 0.30/0.64      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arsinh_def
% 0.30/0.64  thf(fact_8557_Maclaurin__sin__expansion3,axiom,
% 0.30/0.64      ! [N2: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64         => ? [T6: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.30/0.64              & ( ord_less_real @ T6 @ X3 )
% 0.30/0.64              & ( ( sin_real @ X3 )
% 0.30/0.64                = ( plus_plus_real
% 0.30/0.64                  @ ( groups6591440286371151544t_real
% 0.30/0.64                    @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion3
% 0.30/0.64  thf(fact_8558_Maclaurin__sin__expansion4,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ? [T6: real] :
% 0.30/0.64            ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.30/0.64            & ( ord_less_eq_real @ T6 @ X3 )
% 0.30/0.64            & ( ( sin_real @ X3 )
% 0.30/0.64              = ( plus_plus_real
% 0.30/0.64                @ ( groups6591440286371151544t_real
% 0.30/0.64                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.64                  @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_sin_expansion4
% 0.30/0.64  thf(fact_8559_sin__coeff__def,axiom,
% 0.30/0.64      ( sin_coeff
% 0.30/0.64      = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_coeff_def
% 0.30/0.64  thf(fact_8560_Maclaurin__exp__lt,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( X3 != zero_zero_real )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64         => ? [T6: real] :
% 0.30/0.64              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.30/0.64              & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.64              & ( ( exp_real @ X3 )
% 0.30/0.64                = ( plus_plus_real
% 0.30/0.64                  @ ( groups6591440286371151544t_real
% 0.30/0.64                    @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 0.30/0.64                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.64                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Maclaurin_exp_lt
% 0.30/0.64  thf(fact_8561_round__altdef,axiom,
% 0.30/0.64      ( archim8280529875227126926d_real
% 0.30/0.64      = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % round_altdef
% 0.30/0.64  thf(fact_8562_round__altdef,axiom,
% 0.30/0.64      ( archim7778729529865785530nd_rat
% 0.30/0.64      = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % round_altdef
% 0.30/0.64  thf(fact_8563_exp__less__mono,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.64       => ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_less_mono
% 0.30/0.64  thf(fact_8564_exp__less__cancel__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_less_cancel_iff
% 0.30/0.64  thf(fact_8565_exp__zero,axiom,
% 0.30/0.64      ( ( exp_complex @ zero_zero_complex )
% 0.30/0.64      = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_zero
% 0.30/0.64  thf(fact_8566_exp__zero,axiom,
% 0.30/0.64      ( ( exp_real @ zero_zero_real )
% 0.30/0.64      = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_zero
% 0.30/0.64  thf(fact_8567_frac__of__int,axiom,
% 0.30/0.64      ! [Z2: int] :
% 0.30/0.64        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_of_int
% 0.30/0.64  thf(fact_8568_frac__of__int,axiom,
% 0.30/0.64      ! [Z2: int] :
% 0.30/0.64        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 0.30/0.64        = zero_zero_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_of_int
% 0.30/0.64  thf(fact_8569_sin__coeff__0,axiom,
% 0.30/0.64      ( ( sin_coeff @ zero_zero_nat )
% 0.30/0.64      = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_coeff_0
% 0.30/0.64  thf(fact_8570_one__less__exp__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
% 0.30/0.64        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % one_less_exp_iff
% 0.30/0.64  thf(fact_8571_exp__less__one__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
% 0.30/0.64        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_less_one_iff
% 0.30/0.64  thf(fact_8572_exp__ln__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.64          = X3 )
% 0.30/0.64        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ln_iff
% 0.30/0.64  thf(fact_8573_exp__ln,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.64          = X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ln
% 0.30/0.64  thf(fact_8574_fact__mono__nat,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_mono_nat
% 0.30/0.64  thf(fact_8575_fact__ge__self,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_self
% 0.30/0.64  thf(fact_8576_exp__less__cancel,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_less_cancel
% 0.30/0.64  thf(fact_8577_exp__not__eq__zero,axiom,
% 0.30/0.64      ! [X3: complex] :
% 0.30/0.64        ( ( exp_complex @ X3 )
% 0.30/0.64       != zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_not_eq_zero
% 0.30/0.64  thf(fact_8578_exp__not__eq__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( exp_real @ X3 )
% 0.30/0.64       != zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_not_eq_zero
% 0.30/0.64  thf(fact_8579_fact__less__mono__nat,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.64       => ( ( ord_less_nat @ M @ N2 )
% 0.30/0.64         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_less_mono_nat
% 0.30/0.64  thf(fact_8580_exp__total,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 0.30/0.64       => ? [X4: real] :
% 0.30/0.64            ( ( exp_real @ X4 )
% 0.30/0.64            = Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_total
% 0.30/0.64  thf(fact_8581_exp__gt__zero,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_gt_zero
% 0.30/0.64  thf(fact_8582_not__exp__less__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % not_exp_less_zero
% 0.30/0.64  thf(fact_8583_mult__exp__exp,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) )
% 0.30/0.64        = ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mult_exp_exp
% 0.30/0.64  thf(fact_8584_mult__exp__exp,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 0.30/0.64        = ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mult_exp_exp
% 0.30/0.64  thf(fact_8585_exp__add__commuting,axiom,
% 0.30/0.64      ! [X3: complex,Y3: complex] :
% 0.30/0.64        ( ( ( times_times_complex @ X3 @ Y3 )
% 0.30/0.64          = ( times_times_complex @ Y3 @ X3 ) )
% 0.30/0.64       => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 0.30/0.64          = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_add_commuting
% 0.30/0.64  thf(fact_8586_exp__add__commuting,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( times_times_real @ X3 @ Y3 )
% 0.30/0.64          = ( times_times_real @ Y3 @ X3 ) )
% 0.30/0.64       => ( ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64          = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_add_commuting
% 0.30/0.64  thf(fact_8587_frac__ge__0,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_ge_0
% 0.30/0.64  thf(fact_8588_frac__ge__0,axiom,
% 0.30/0.64      ! [X3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_ge_0
% 0.30/0.64  thf(fact_8589_frac__lt__1,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X3 ) @ one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_lt_1
% 0.30/0.64  thf(fact_8590_frac__lt__1,axiom,
% 0.30/0.64      ! [X3: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X3 ) @ one_one_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_lt_1
% 0.30/0.64  thf(fact_8591_frac__1__eq,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ one_one_real ) )
% 0.30/0.64        = ( archim2898591450579166408c_real @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_1_eq
% 0.30/0.64  thf(fact_8592_frac__1__eq,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
% 0.30/0.64        = ( archimedean_frac_rat @ X3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_1_eq
% 0.30/0.64  thf(fact_8593_fact__ge__Suc__0__nat,axiom,
% 0.30/0.64      ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_ge_Suc_0_nat
% 0.30/0.64  thf(fact_8594_exp__gt__one,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_gt_one
% 0.30/0.64  thf(fact_8595_dvd__fact,axiom,
% 0.30/0.64      ! [M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.64       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % dvd_fact
% 0.30/0.64  thf(fact_8596_fact__diff__Suc,axiom,
% 0.30/0.64      ! [N2: nat,M: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 0.30/0.64       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
% 0.30/0.64          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_diff_Suc
% 0.30/0.64  thf(fact_8597_ln__ge__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X3 ) )
% 0.30/0.64          = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ln_ge_iff
% 0.30/0.64  thf(fact_8598_fact__div__fact__le__pow,axiom,
% 0.30/0.64      ! [R2: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ R2 @ N2 )
% 0.30/0.64       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_div_fact_le_pow
% 0.30/0.64  thf(fact_8599_powr__def,axiom,
% 0.30/0.64      ( powr_real
% 0.30/0.64      = ( ^ [X: real,A3: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % powr_def
% 0.30/0.64  thf(fact_8600_exp__divide__power__eq,axiom,
% 0.30/0.64      ! [N2: nat,X3: complex] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
% 0.30/0.64          = ( exp_complex @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_divide_power_eq
% 0.30/0.64  thf(fact_8601_exp__divide__power__eq,axiom,
% 0.30/0.64      ! [N2: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 0.30/0.64          = ( exp_real @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_divide_power_eq
% 0.30/0.64  thf(fact_8602_frac__eq,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ( archim2898591450579166408c_real @ X3 )
% 0.30/0.64          = X3 )
% 0.30/0.64        = ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64          & ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_eq
% 0.30/0.64  thf(fact_8603_frac__eq,axiom,
% 0.30/0.64      ! [X3: rat] :
% 0.30/0.64        ( ( ( archimedean_frac_rat @ X3 )
% 0.30/0.64          = X3 )
% 0.30/0.64        = ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 0.30/0.64          & ( ord_less_rat @ X3 @ one_one_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_eq
% 0.30/0.64  thf(fact_8604_frac__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.30/0.64         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 0.30/0.64        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.30/0.64         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_add
% 0.30/0.64  thf(fact_8605_frac__add,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] :
% 0.30/0.64        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.30/0.64         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 0.30/0.64        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.30/0.64         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 0.30/0.64            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % frac_add
% 0.30/0.64  thf(fact_8606_tanh__altdef,axiom,
% 0.30/0.64      ( tanh_real
% 0.30/0.64      = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % tanh_altdef
% 0.30/0.64  thf(fact_8607_sin__coeff__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( sin_coeff @ ( suc @ N2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_coeff_Suc
% 0.30/0.64  thf(fact_8608_floor__add,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.30/0.64         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
% 0.30/0.64        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 0.30/0.64         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_add
% 0.30/0.64  thf(fact_8609_floor__add,axiom,
% 0.30/0.64      ! [X3: rat,Y3: rat] :
% 0.30/0.64        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.30/0.64         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
% 0.30/0.64        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 0.30/0.64         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 0.30/0.64            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % floor_add
% 0.30/0.64  thf(fact_8610_cos__coeff__Suc,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( cos_coeff @ ( suc @ N2 ) )
% 0.30/0.64        = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_coeff_Suc
% 0.30/0.64  thf(fact_8611_exp__ge__one__plus__x__over__n__power__n,axiom,
% 0.30/0.64      ! [N2: nat,X3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X3 )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ge_one_plus_x_over_n_power_n
% 0.30/0.64  thf(fact_8612_exp__ge__one__minus__x__over__n__power__n,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % exp_ge_one_minus_x_over_n_power_n
% 0.30/0.64  thf(fact_8613_log__base__10__eq2,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_base_10_eq2
% 0.30/0.64  thf(fact_8614_log__base__10__eq1,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 0.30/0.64          = ( 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 @ X3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % log_base_10_eq1
% 0.30/0.64  thf(fact_8615_sin__tan,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ( sin_real @ X3 )
% 0.30/0.64          = ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sin_tan
% 0.30/0.64  thf(fact_8616_cos__tan,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ( cos_real @ X3 )
% 0.30/0.64          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % cos_tan
% 0.30/0.64  thf(fact_8617_monoseq__def,axiom,
% 0.30/0.64      ( topolo6980174941875973593q_real
% 0.30/0.64      = ( ^ [X5: nat > real] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8618_monoseq__def,axiom,
% 0.30/0.64      ( topolo3100542954746470799et_int
% 0.30/0.64      = ( ^ [X5: nat > set_int] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_set_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8619_monoseq__def,axiom,
% 0.30/0.64      ( topolo4267028734544971653eq_rat
% 0.30/0.64      = ( ^ [X5: nat > rat] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_rat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8620_monoseq__def,axiom,
% 0.30/0.64      ( topolo1459490580787246023eq_num
% 0.30/0.64      = ( ^ [X5: nat > num] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_num @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8621_monoseq__def,axiom,
% 0.30/0.64      ( topolo4902158794631467389eq_nat
% 0.30/0.64      = ( ^ [X5: nat > nat] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_nat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8622_monoseq__def,axiom,
% 0.30/0.64      ( topolo4899668324122417113eq_int
% 0.30/0.64      = ( ^ [X5: nat > int] :
% 0.30/0.64            ( ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 0.30/0.64            | ! [M2: nat,N: nat] :
% 0.30/0.64                ( ( ord_less_eq_nat @ M2 @ N )
% 0.30/0.64               => ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_def
% 0.30/0.64  thf(fact_8623_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > real] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8624_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > set_int] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8625_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > rat] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8626_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > num] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8627_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > nat] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8628_monoI2,axiom,
% 0.30/0.64      ! [X7: nat > int] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI2
% 0.30/0.64  thf(fact_8629_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > real] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8630_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > set_int] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8631_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > rat] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8632_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > num] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8633_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > nat] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8634_monoI1,axiom,
% 0.30/0.64      ! [X7: nat > int] :
% 0.30/0.64        ( ! [M3: nat,N3: nat] :
% 0.30/0.64            ( ( ord_less_eq_nat @ M3 @ N3 )
% 0.30/0.64           => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoI1
% 0.30/0.64  thf(fact_8635_real__sqrt__less__iff,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
% 0.30/0.64        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_less_iff
% 0.30/0.64  thf(fact_8636_real__sqrt__lt__0__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
% 0.30/0.64        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_lt_0_iff
% 0.30/0.64  thf(fact_8637_real__sqrt__gt__0__iff,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 0.30/0.64        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_gt_0_iff
% 0.30/0.64  thf(fact_8638_real__sqrt__lt__1__iff,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( sqrt @ X3 ) @ one_one_real )
% 0.30/0.64        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_lt_1_iff
% 0.30/0.64  thf(fact_8639_real__sqrt__gt__1__iff,axiom,
% 0.30/0.64      ! [Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 0.30/0.64        = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_gt_1_iff
% 0.30/0.64  thf(fact_8640_real__sqrt__less__mono,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.64       => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_less_mono
% 0.30/0.64  thf(fact_8641_real__sqrt__gt__zero,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_gt_zero
% 0.30/0.64  thf(fact_8642_sqrt2__less__2,axiom,
% 0.30/0.64      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sqrt2_less_2
% 0.30/0.64  thf(fact_8643_real__less__rsqrt,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 0.30/0.64       => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_less_rsqrt
% 0.30/0.64  thf(fact_8644_lemma__real__divide__sqrt__less,axiom,
% 0.30/0.64      ! [U: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ U )
% 0.30/0.64       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 0.30/0.64  
% 0.30/0.64  % lemma_real_divide_sqrt_less
% 0.30/0.64  thf(fact_8645_real__less__lsqrt,axiom,
% 0.30/0.64      ! [X3: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64         => ( ( ord_less_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64           => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_less_lsqrt
% 0.30/0.64  thf(fact_8646_ln__sqrt,axiom,
% 0.30/0.64      ! [X3: real] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ( ln_ln_real @ ( sqrt @ X3 ) )
% 0.30/0.64          = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % ln_sqrt
% 0.30/0.64  thf(fact_8647_arsinh__real__aux,axiom,
% 0.30/0.64      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % arsinh_real_aux
% 0.30/0.64  thf(fact_8648_real__sqrt__sum__squares__less,axiom,
% 0.30/0.64      ! [X3: real,U: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % real_sqrt_sum_squares_less
% 0.30/0.64  thf(fact_8649_sqrt__sum__squares__half__less,axiom,
% 0.30/0.64      ! [X3: real,U: real,Y3: real] :
% 0.30/0.64        ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.64         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.64           => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.64             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % sqrt_sum_squares_half_less
% 0.30/0.64  thf(fact_8650_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo6980174941875973593q_real
% 0.30/0.64      = ( ^ [X5: nat > real] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_real @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8651_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo3100542954746470799et_int
% 0.30/0.64      = ( ^ [X5: nat > set_int] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8652_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4267028734544971653eq_rat
% 0.30/0.64      = ( ^ [X5: nat > rat] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8653_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo1459490580787246023eq_num
% 0.30/0.64      = ( ^ [X5: nat > num] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_num @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8654_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4902158794631467389eq_nat
% 0.30/0.64      = ( ^ [X5: nat > nat] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8655_monoseq__Suc,axiom,
% 0.30/0.64      ( topolo4899668324122417113eq_int
% 0.30/0.64      = ( ^ [X5: nat > int] :
% 0.30/0.64            ( ! [N: nat] : ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 0.30/0.64            | ! [N: nat] : ( ord_less_eq_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % monoseq_Suc
% 0.30/0.64  thf(fact_8656_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8657_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > set_int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8658_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > rat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8659_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > num] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8660_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8661_mono__SucI2,axiom,
% 0.30/0.64      ! [X7: nat > int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI2
% 0.30/0.64  thf(fact_8662_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > real] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8663_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > set_int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8664_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > rat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8665_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > num] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8666_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > nat] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8667_mono__SucI1,axiom,
% 0.30/0.64      ! [X7: nat > int] :
% 0.30/0.64        ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 0.30/0.64       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % mono_SucI1
% 0.30/0.64  thf(fact_8668_pochhammer__double,axiom,
% 0.30/0.64      ! [Z2: complex,N2: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.64        = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8669_pochhammer__double,axiom,
% 0.30/0.64      ! [Z2: rat,N2: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.64        = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8670_pochhammer__double,axiom,
% 0.30/0.64      ! [Z2: real,N2: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.64        = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_double
% 0.30/0.64  thf(fact_8671_of__nat__code,axiom,
% 0.30/0.64      ( semiri8010041392384452111omplex
% 0.30/0.64      = ( ^ [N: nat] :
% 0.30/0.64            ( semiri2816024913162550771omplex
% 0.30/0.64            @ ^ [I2: complex] : ( plus_plus_complex @ I2 @ one_one_complex )
% 0.30/0.64            @ N
% 0.30/0.64            @ zero_zero_complex ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_nat_code
% 0.30/0.64  thf(fact_8672_of__nat__code,axiom,
% 0.30/0.64      ( semiri681578069525770553at_rat
% 0.30/0.64      = ( ^ [N: nat] :
% 0.30/0.64            ( semiri7787848453975740701ux_rat
% 0.30/0.64            @ ^ [I2: rat] : ( plus_plus_rat @ I2 @ one_one_rat )
% 0.30/0.64            @ N
% 0.30/0.64            @ zero_zero_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_nat_code
% 0.30/0.64  thf(fact_8673_of__nat__code,axiom,
% 0.30/0.64      ( semiri1314217659103216013at_int
% 0.30/0.64      = ( ^ [N: nat] :
% 0.30/0.64            ( semiri8420488043553186161ux_int
% 0.30/0.64            @ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
% 0.30/0.64            @ N
% 0.30/0.64            @ zero_zero_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_nat_code
% 0.30/0.64  thf(fact_8674_of__nat__code,axiom,
% 0.30/0.64      ( semiri5074537144036343181t_real
% 0.30/0.64      = ( ^ [N: nat] :
% 0.30/0.64            ( semiri7260567687927622513x_real
% 0.30/0.64            @ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
% 0.30/0.64            @ N
% 0.30/0.64            @ zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_nat_code
% 0.30/0.64  thf(fact_8675_of__nat__code,axiom,
% 0.30/0.64      ( semiri1316708129612266289at_nat
% 0.30/0.64      = ( ^ [N: nat] :
% 0.30/0.64            ( semiri8422978514062236437ux_nat
% 0.30/0.64            @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
% 0.30/0.64            @ N
% 0.30/0.64            @ zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % of_nat_code
% 0.30/0.64  thf(fact_8676_gchoose__row__sum__weighted,axiom,
% 0.30/0.64      ! [R2: complex,M: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 0.30/0.64          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.64        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gchoose_row_sum_weighted
% 0.30/0.64  thf(fact_8677_gchoose__row__sum__weighted,axiom,
% 0.30/0.64      ! [R2: rat,M: nat] :
% 0.30/0.64        ( ( groups2906978787729119204at_rat
% 0.30/0.64          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 0.30/0.64          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.64        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gchoose_row_sum_weighted
% 0.30/0.64  thf(fact_8678_gchoose__row__sum__weighted,axiom,
% 0.30/0.64      ! [R2: real,M: nat] :
% 0.30/0.64        ( ( groups6591440286371151544t_real
% 0.30/0.64          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 0.30/0.64          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 0.30/0.64        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gchoose_row_sum_weighted
% 0.30/0.64  thf(fact_8679_central__binomial__lower__bound,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % central_binomial_lower_bound
% 0.30/0.64  thf(fact_8680_complex__unimodular__polar,axiom,
% 0.30/0.64      ! [Z2: complex] :
% 0.30/0.64        ( ( ( real_V1022390504157884413omplex @ Z2 )
% 0.30/0.64          = one_one_real )
% 0.30/0.64       => ~ ! [T6: real] :
% 0.30/0.64              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.64             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 0.30/0.64               => ( Z2
% 0.30/0.64                 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % complex_unimodular_polar
% 0.30/0.64  thf(fact_8681_binomial__Suc__n,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( binomial @ ( suc @ N2 ) @ N2 )
% 0.30/0.64        = ( suc @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_Suc_n
% 0.30/0.64  thf(fact_8682_gbinomial__0_I2_J,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(2)
% 0.30/0.64  thf(fact_8683_gbinomial__0_I2_J,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_real ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(2)
% 0.30/0.64  thf(fact_8684_gbinomial__0_I2_J,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(2)
% 0.30/0.64  thf(fact_8685_gbinomial__0_I2_J,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(2)
% 0.30/0.64  thf(fact_8686_gbinomial__0_I2_J,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_int ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(2)
% 0.30/0.64  thf(fact_8687_binomial__1,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.64        = N2 ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_1
% 0.30/0.64  thf(fact_8688_binomial__0__Suc,axiom,
% 0.30/0.64      ! [K: nat] :
% 0.30/0.64        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 0.30/0.64        = zero_zero_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_0_Suc
% 0.30/0.64  thf(fact_8689_binomial__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( binomial @ N2 @ K )
% 0.30/0.64          = zero_zero_nat )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_eq_0_iff
% 0.30/0.64  thf(fact_8690_gbinomial__0_I1_J,axiom,
% 0.30/0.64      ! [A: complex] :
% 0.30/0.64        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(1)
% 0.30/0.64  thf(fact_8691_gbinomial__0_I1_J,axiom,
% 0.30/0.64      ! [A: real] :
% 0.30/0.64        ( ( gbinomial_real @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(1)
% 0.30/0.64  thf(fact_8692_gbinomial__0_I1_J,axiom,
% 0.30/0.64      ! [A: rat] :
% 0.30/0.64        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(1)
% 0.30/0.64  thf(fact_8693_gbinomial__0_I1_J,axiom,
% 0.30/0.64      ! [A: nat] :
% 0.30/0.64        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(1)
% 0.30/0.64  thf(fact_8694_gbinomial__0_I1_J,axiom,
% 0.30/0.64      ! [A: int] :
% 0.30/0.64        ( ( gbinomial_int @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_int ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_0(1)
% 0.30/0.64  thf(fact_8695_binomial__Suc__Suc,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_Suc_Suc
% 0.30/0.64  thf(fact_8696_pochhammer__0,axiom,
% 0.30/0.64      ! [A: complex] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_complex ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0
% 0.30/0.64  thf(fact_8697_pochhammer__0,axiom,
% 0.30/0.64      ! [A: real] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_real ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0
% 0.30/0.64  thf(fact_8698_pochhammer__0,axiom,
% 0.30/0.64      ! [A: rat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_rat ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0
% 0.30/0.64  thf(fact_8699_pochhammer__0,axiom,
% 0.30/0.64      ! [A: nat] :
% 0.30/0.64        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0
% 0.30/0.64  thf(fact_8700_pochhammer__0,axiom,
% 0.30/0.64      ! [A: int] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 0.30/0.64        = one_one_int ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0
% 0.30/0.64  thf(fact_8701_binomial__n__0,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( binomial @ N2 @ zero_zero_nat )
% 0.30/0.64        = one_one_nat ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_n_0
% 0.30/0.64  thf(fact_8702_zero__less__binomial__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
% 0.30/0.64        = ( ord_less_eq_nat @ K @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_less_binomial_iff
% 0.30/0.64  thf(fact_8703_binomial__eq__0,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( binomial @ N2 @ K )
% 0.30/0.64          = zero_zero_nat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_eq_0
% 0.30/0.64  thf(fact_8704_Suc__times__binomial,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_binomial
% 0.30/0.64  thf(fact_8705_Suc__times__binomial__eq,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
% 0.30/0.64        = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_binomial_eq
% 0.30/0.64  thf(fact_8706_binomial__symmetric,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( binomial @ N2 @ K )
% 0.30/0.64          = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_symmetric
% 0.30/0.64  thf(fact_8707_choose__mult__lemma,axiom,
% 0.30/0.64      ! [M: nat,R2: nat,K: nat] :
% 0.30/0.64        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 0.30/0.64        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_mult_lemma
% 0.30/0.64  thf(fact_8708_binomial__le__pow,axiom,
% 0.30/0.64      ! [R2: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ R2 @ N2 )
% 0.30/0.64       => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_le_pow
% 0.30/0.64  thf(fact_8709_pochhammer__pos,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_pos
% 0.30/0.64  thf(fact_8710_pochhammer__pos,axiom,
% 0.30/0.64      ! [X3: rat,N2: nat] :
% 0.30/0.64        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.64       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_pos
% 0.30/0.64  thf(fact_8711_pochhammer__pos,axiom,
% 0.30/0.64      ! [X3: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.64       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_pos
% 0.30/0.64  thf(fact_8712_pochhammer__pos,axiom,
% 0.30/0.64      ! [X3: int,N2: nat] :
% 0.30/0.64        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.64       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_pos
% 0.30/0.64  thf(fact_8713_pochhammer__neq__0__mono,axiom,
% 0.30/0.64      ! [A: complex,M: nat,N2: nat] :
% 0.30/0.64        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.64         != zero_zero_complex )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.30/0.64           != zero_zero_complex ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_neq_0_mono
% 0.30/0.64  thf(fact_8714_pochhammer__neq__0__mono,axiom,
% 0.30/0.64      ! [A: real,M: nat,N2: nat] :
% 0.30/0.64        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.64         != zero_zero_real )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.30/0.64           != zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_neq_0_mono
% 0.30/0.64  thf(fact_8715_pochhammer__neq__0__mono,axiom,
% 0.30/0.64      ! [A: rat,M: nat,N2: nat] :
% 0.30/0.64        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.64         != zero_zero_rat )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.30/0.64           != zero_zero_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_neq_0_mono
% 0.30/0.64  thf(fact_8716_pochhammer__eq__0__mono,axiom,
% 0.30/0.64      ! [A: complex,N2: nat,M: nat] :
% 0.30/0.64        ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 0.30/0.64            = zero_zero_complex ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_mono
% 0.30/0.64  thf(fact_8717_pochhammer__eq__0__mono,axiom,
% 0.30/0.64      ! [A: real,N2: nat,M: nat] :
% 0.30/0.64        ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 0.30/0.64            = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_mono
% 0.30/0.64  thf(fact_8718_pochhammer__eq__0__mono,axiom,
% 0.30/0.64      ! [A: rat,N2: nat,M: nat] :
% 0.30/0.64        ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.30/0.64          = zero_zero_rat )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.64         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 0.30/0.64            = zero_zero_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_mono
% 0.30/0.64  thf(fact_8719_zero__less__binomial,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % zero_less_binomial
% 0.30/0.64  thf(fact_8720_Suc__times__binomial__add,axiom,
% 0.30/0.64      ! [A: nat,B: nat] :
% 0.30/0.64        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 0.30/0.64        = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_binomial_add
% 0.30/0.64  thf(fact_8721_choose__mult,axiom,
% 0.30/0.64      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.64       => ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64         => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
% 0.30/0.64            = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_mult
% 0.30/0.64  thf(fact_8722_binomial__Suc__Suc__eq__times,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 0.30/0.64        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_Suc_Suc_eq_times
% 0.30/0.64  thf(fact_8723_gbinomial__pochhammer_H,axiom,
% 0.30/0.64      ( gbinomial_complex
% 0.30/0.64      = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_pochhammer'
% 0.30/0.64  thf(fact_8724_gbinomial__pochhammer_H,axiom,
% 0.30/0.64      ( gbinomial_rat
% 0.30/0.64      = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_pochhammer'
% 0.30/0.64  thf(fact_8725_gbinomial__pochhammer_H,axiom,
% 0.30/0.64      ( gbinomial_real
% 0.30/0.64      = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_pochhammer'
% 0.30/0.64  thf(fact_8726_gbinomial__Suc__Suc,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_Suc_Suc
% 0.30/0.64  thf(fact_8727_gbinomial__Suc__Suc,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_Suc_Suc
% 0.30/0.64  thf(fact_8728_gbinomial__Suc__Suc,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_Suc_Suc
% 0.30/0.64  thf(fact_8729_gbinomial__of__nat__symmetric,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
% 0.30/0.64          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_of_nat_symmetric
% 0.30/0.64  thf(fact_8730_pochhammer__nonneg,axiom,
% 0.30/0.64      ! [X3: real,N2: nat] :
% 0.30/0.64        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.64       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_nonneg
% 0.30/0.64  thf(fact_8731_pochhammer__nonneg,axiom,
% 0.30/0.64      ! [X3: rat,N2: nat] :
% 0.30/0.64        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 0.30/0.64       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_nonneg
% 0.30/0.64  thf(fact_8732_pochhammer__nonneg,axiom,
% 0.30/0.64      ! [X3: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 0.30/0.64       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_nonneg
% 0.30/0.64  thf(fact_8733_pochhammer__nonneg,axiom,
% 0.30/0.64      ! [X3: int,N2: nat] :
% 0.30/0.64        ( ( ord_less_int @ zero_zero_int @ X3 )
% 0.30/0.64       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_nonneg
% 0.30/0.64  thf(fact_8734_pochhammer__0__left,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ( N2 = zero_zero_nat )
% 0.30/0.64         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 0.30/0.64            = one_one_complex ) )
% 0.30/0.64        & ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 0.30/0.64            = zero_zero_complex ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0_left
% 0.30/0.64  thf(fact_8735_pochhammer__0__left,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ( N2 = zero_zero_nat )
% 0.30/0.64         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 0.30/0.64            = one_one_real ) )
% 0.30/0.64        & ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 0.30/0.64            = zero_zero_real ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0_left
% 0.30/0.64  thf(fact_8736_pochhammer__0__left,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ( N2 = zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 0.30/0.64            = one_one_rat ) )
% 0.30/0.64        & ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 0.30/0.64            = zero_zero_rat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0_left
% 0.30/0.64  thf(fact_8737_pochhammer__0__left,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ( N2 = zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 0.30/0.64            = one_one_nat ) )
% 0.30/0.64        & ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 0.30/0.64            = zero_zero_nat ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0_left
% 0.30/0.64  thf(fact_8738_pochhammer__0__left,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ( ( N2 = zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 0.30/0.64            = one_one_int ) )
% 0.30/0.64        & ( ( N2 != zero_zero_nat )
% 0.30/0.64         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 0.30/0.64            = zero_zero_int ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_0_left
% 0.30/0.64  thf(fact_8739_binomial__absorption,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_absorption
% 0.30/0.64  thf(fact_8740_gbinomial__addition__formula,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_addition_formula
% 0.30/0.64  thf(fact_8741_gbinomial__addition__formula,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_addition_formula
% 0.30/0.64  thf(fact_8742_gbinomial__addition__formula,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 0.30/0.64        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_addition_formula
% 0.30/0.64  thf(fact_8743_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64      ! [K: nat,A: real] :
% 0.30/0.64        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 0.30/0.64       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_ge_n_over_k_pow_k
% 0.30/0.64  thf(fact_8744_gbinomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64      ! [K: nat,A: rat] :
% 0.30/0.64        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 0.30/0.64       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_ge_n_over_k_pow_k
% 0.30/0.64  thf(fact_8745_gbinomial__mult__1_H,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 0.30/0.64        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1'
% 0.30/0.64  thf(fact_8746_gbinomial__mult__1_H,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 0.30/0.64        = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1'
% 0.30/0.64  thf(fact_8747_gbinomial__mult__1_H,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 0.30/0.64        = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1'
% 0.30/0.64  thf(fact_8748_gbinomial__mult__1,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 0.30/0.64        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1
% 0.30/0.64  thf(fact_8749_gbinomial__mult__1,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 0.30/0.64        = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1
% 0.30/0.64  thf(fact_8750_gbinomial__mult__1,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 0.30/0.64        = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_mult_1
% 0.30/0.64  thf(fact_8751_binomial__fact__lemma,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
% 0.30/0.64          = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_fact_lemma
% 0.30/0.64  thf(fact_8752_pochhammer__rec,axiom,
% 0.30/0.64      ! [A: complex,N2: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec
% 0.30/0.64  thf(fact_8753_pochhammer__rec,axiom,
% 0.30/0.64      ! [A: real,N2: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec
% 0.30/0.64  thf(fact_8754_pochhammer__rec,axiom,
% 0.30/0.64      ! [A: rat,N2: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec
% 0.30/0.64  thf(fact_8755_pochhammer__rec,axiom,
% 0.30/0.64      ! [A: nat,N2: nat] :
% 0.30/0.64        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec
% 0.30/0.64  thf(fact_8756_pochhammer__rec,axiom,
% 0.30/0.64      ! [A: int,N2: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec
% 0.30/0.64  thf(fact_8757_pochhammer__rec_H,axiom,
% 0.30/0.64      ! [Z2: complex,N2: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec'
% 0.30/0.64  thf(fact_8758_pochhammer__rec_H,axiom,
% 0.30/0.64      ! [Z2: rat,N2: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec'
% 0.30/0.64  thf(fact_8759_pochhammer__rec_H,axiom,
% 0.30/0.64      ! [Z2: int,N2: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec'
% 0.30/0.64  thf(fact_8760_pochhammer__rec_H,axiom,
% 0.30/0.64      ! [Z2: real,N2: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec'
% 0.30/0.64  thf(fact_8761_pochhammer__rec_H,axiom,
% 0.30/0.64      ! [Z2: nat,N2: nat] :
% 0.30/0.64        ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_rec'
% 0.30/0.64  thf(fact_8762_pochhammer__Suc,axiom,
% 0.30/0.64      ! [A: complex,N2: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_Suc
% 0.30/0.64  thf(fact_8763_pochhammer__Suc,axiom,
% 0.30/0.64      ! [A: rat,N2: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_Suc
% 0.30/0.64  thf(fact_8764_pochhammer__Suc,axiom,
% 0.30/0.64      ! [A: int,N2: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_Suc
% 0.30/0.64  thf(fact_8765_pochhammer__Suc,axiom,
% 0.30/0.64      ! [A: real,N2: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_Suc
% 0.30/0.64  thf(fact_8766_pochhammer__Suc,axiom,
% 0.30/0.64      ! [A: nat,N2: nat] :
% 0.30/0.64        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 0.30/0.64        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_Suc
% 0.30/0.64  thf(fact_8767_pochhammer__eq__0__iff,axiom,
% 0.30/0.64      ! [A: complex,N2: nat] :
% 0.30/0.64        ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64        = ( ? [K3: nat] :
% 0.30/0.64              ( ( ord_less_nat @ K3 @ N2 )
% 0.30/0.64              & ( A
% 0.30/0.64                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_iff
% 0.30/0.64  thf(fact_8768_pochhammer__eq__0__iff,axiom,
% 0.30/0.64      ! [A: rat,N2: nat] :
% 0.30/0.64        ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 0.30/0.64          = zero_zero_rat )
% 0.30/0.64        = ( ? [K3: nat] :
% 0.30/0.64              ( ( ord_less_nat @ K3 @ N2 )
% 0.30/0.64              & ( A
% 0.30/0.64                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_iff
% 0.30/0.64  thf(fact_8769_pochhammer__eq__0__iff,axiom,
% 0.30/0.64      ! [A: real,N2: nat] :
% 0.30/0.64        ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        = ( ? [K3: nat] :
% 0.30/0.64              ( ( ord_less_nat @ K3 @ N2 )
% 0.30/0.64              & ( A
% 0.30/0.64                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_eq_0_iff
% 0.30/0.64  thf(fact_8770_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_complex )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_iff
% 0.30/0.64  thf(fact_8771_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_rat )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_iff
% 0.30/0.64  thf(fact_8772_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.30/0.64          = zero_z3403309356797280102nteger )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_iff
% 0.30/0.64  thf(fact_8773_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_int )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_iff
% 0.30/0.64  thf(fact_8774_pochhammer__of__nat__eq__0__iff,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_real )
% 0.30/0.64        = ( ord_less_nat @ N2 @ K ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_iff
% 0.30/0.64  thf(fact_8775_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_complex ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.64  thf(fact_8776_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.64  thf(fact_8777_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.30/0.64          = zero_z3403309356797280102nteger ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.64  thf(fact_8778_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_int ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.64  thf(fact_8779_pochhammer__of__nat__eq__0__lemma,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ N2 @ K )
% 0.30/0.64       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.30/0.64          = zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma
% 0.30/0.64  thf(fact_8780_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 0.30/0.64         != zero_zero_complex ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.64  thf(fact_8781_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 0.30/0.64         != zero_zero_rat ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.64  thf(fact_8782_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 0.30/0.64         != zero_z3403309356797280102nteger ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.64  thf(fact_8783_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 0.30/0.64         != zero_zero_int ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.64  thf(fact_8784_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 0.30/0.64         != zero_zero_real ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_of_nat_eq_0_lemma'
% 0.30/0.64  thf(fact_8785_pochhammer__product_H,axiom,
% 0.30/0.64      ! [Z2: complex,N2: nat,M: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.64        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product'
% 0.30/0.64  thf(fact_8786_pochhammer__product_H,axiom,
% 0.30/0.64      ! [Z2: rat,N2: nat,M: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.64        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product'
% 0.30/0.64  thf(fact_8787_pochhammer__product_H,axiom,
% 0.30/0.64      ! [Z2: int,N2: nat,M: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.64        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product'
% 0.30/0.64  thf(fact_8788_pochhammer__product_H,axiom,
% 0.30/0.64      ! [Z2: real,N2: nat,M: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.64        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product'
% 0.30/0.64  thf(fact_8789_pochhammer__product_H,axiom,
% 0.30/0.64      ! [Z2: nat,N2: nat,M: nat] :
% 0.30/0.64        ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 0.30/0.64        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product'
% 0.30/0.64  thf(fact_8790_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_ge_n_over_k_pow_k
% 0.30/0.64  thf(fact_8791_binomial__ge__n__over__k__pow__k,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_ge_n_over_k_pow_k
% 0.30/0.64  thf(fact_8792_binomial__maximum_H,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_maximum'
% 0.30/0.64  thf(fact_8793_binomial__mono,axiom,
% 0.30/0.64      ! [K: nat,K6: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.64       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 0.30/0.64         => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_mono
% 0.30/0.64  thf(fact_8794_binomial__antimono,axiom,
% 0.30/0.64      ! [K: nat,K6: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ K6 )
% 0.30/0.64       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 0.30/0.64         => ( ( ord_less_eq_nat @ K6 @ N2 )
% 0.30/0.64           => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_antimono
% 0.30/0.64  thf(fact_8795_binomial__maximum,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_maximum
% 0.30/0.64  thf(fact_8796_binomial__le__pow2,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_le_pow2
% 0.30/0.64  thf(fact_8797_choose__reduce__nat,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64         => ( ( binomial @ N2 @ K )
% 0.30/0.64            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % choose_reduce_nat
% 0.30/0.64  thf(fact_8798_times__binomial__minus1__eq,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64       => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
% 0.30/0.64          = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % times_binomial_minus1_eq
% 0.30/0.64  thf(fact_8799_Suc__times__gbinomial,axiom,
% 0.30/0.64      ! [K: nat,A: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_gbinomial
% 0.30/0.64  thf(fact_8800_Suc__times__gbinomial,axiom,
% 0.30/0.64      ! [K: nat,A: rat] :
% 0.30/0.64        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_gbinomial
% 0.30/0.64  thf(fact_8801_Suc__times__gbinomial,axiom,
% 0.30/0.64      ! [K: nat,A: real] :
% 0.30/0.64        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % Suc_times_gbinomial
% 0.30/0.64  thf(fact_8802_gbinomial__absorption,axiom,
% 0.30/0.64      ! [K: nat,A: complex] :
% 0.30/0.64        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_absorption
% 0.30/0.64  thf(fact_8803_gbinomial__absorption,axiom,
% 0.30/0.64      ! [K: nat,A: rat] :
% 0.30/0.64        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_absorption
% 0.30/0.64  thf(fact_8804_gbinomial__absorption,axiom,
% 0.30/0.64      ! [K: nat,A: real] :
% 0.30/0.64        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 0.30/0.64        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_absorption
% 0.30/0.64  thf(fact_8805_binomial__altdef__nat,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( binomial @ N2 @ K )
% 0.30/0.64          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_altdef_nat
% 0.30/0.64  thf(fact_8806_gbinomial__trinomial__revision,axiom,
% 0.30/0.64      ! [K: nat,M: nat,A: complex] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.64       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 0.30/0.64          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_trinomial_revision
% 0.30/0.64  thf(fact_8807_gbinomial__trinomial__revision,axiom,
% 0.30/0.64      ! [K: nat,M: nat,A: rat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.64       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 0.30/0.64          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_trinomial_revision
% 0.30/0.64  thf(fact_8808_gbinomial__trinomial__revision,axiom,
% 0.30/0.64      ! [K: nat,M: nat,A: real] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ M )
% 0.30/0.64       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 0.30/0.64          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_trinomial_revision
% 0.30/0.64  thf(fact_8809_pochhammer__product,axiom,
% 0.30/0.64      ! [M: nat,N2: nat,Z2: complex] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( comm_s2602460028002588243omplex @ Z2 @ N2 )
% 0.30/0.64          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product
% 0.30/0.64  thf(fact_8810_pochhammer__product,axiom,
% 0.30/0.64      ! [M: nat,N2: nat,Z2: rat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( comm_s4028243227959126397er_rat @ Z2 @ N2 )
% 0.30/0.64          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product
% 0.30/0.64  thf(fact_8811_pochhammer__product,axiom,
% 0.30/0.64      ! [M: nat,N2: nat,Z2: int] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( comm_s4660882817536571857er_int @ Z2 @ N2 )
% 0.30/0.64          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product
% 0.30/0.64  thf(fact_8812_pochhammer__product,axiom,
% 0.30/0.64      ! [M: nat,N2: nat,Z2: real] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( comm_s7457072308508201937r_real @ Z2 @ N2 )
% 0.30/0.64          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product
% 0.30/0.64  thf(fact_8813_pochhammer__product,axiom,
% 0.30/0.64      ! [M: nat,N2: nat,Z2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.64       => ( ( comm_s4663373288045622133er_nat @ Z2 @ N2 )
% 0.30/0.64          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_product
% 0.30/0.64  thf(fact_8814_binomial__strict__antimono,axiom,
% 0.30/0.64      ! [K: nat,K6: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ K @ K6 )
% 0.30/0.64       => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 0.30/0.64         => ( ( ord_less_eq_nat @ K6 @ N2 )
% 0.30/0.64           => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_strict_antimono
% 0.30/0.64  thf(fact_8815_binomial__strict__mono,axiom,
% 0.30/0.64      ! [K: nat,K6: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ K @ K6 )
% 0.30/0.64       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 0.30/0.64         => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_strict_mono
% 0.30/0.64  thf(fact_8816_binomial__less__binomial__Suc,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.64       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_less_binomial_Suc
% 0.30/0.64  thf(fact_8817_central__binomial__odd,axiom,
% 0.30/0.64      ! [N2: nat] :
% 0.30/0.64        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.64       => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.64          = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % central_binomial_odd
% 0.30/0.64  thf(fact_8818_binomial__addition__formula,axiom,
% 0.30/0.64      ! [N2: nat,K: nat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.64       => ( ( binomial @ N2 @ ( suc @ K ) )
% 0.30/0.64          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_addition_formula
% 0.30/0.64  thf(fact_8819_binomial__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
% 0.30/0.64          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_fact
% 0.30/0.64  thf(fact_8820_binomial__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
% 0.30/0.64          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_fact
% 0.30/0.64  thf(fact_8821_binomial__fact,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
% 0.30/0.64          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % binomial_fact
% 0.30/0.64  thf(fact_8822_fact__binomial,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
% 0.30/0.64          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_binomial
% 0.30/0.64  thf(fact_8823_fact__binomial,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
% 0.30/0.64          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_binomial
% 0.30/0.64  thf(fact_8824_fact__binomial,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( ord_less_eq_nat @ K @ N2 )
% 0.30/0.64       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
% 0.30/0.64          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % fact_binomial
% 0.30/0.64  thf(fact_8825_gbinomial__rec,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_rec
% 0.30/0.64  thf(fact_8826_gbinomial__rec,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_rec
% 0.30/0.64  thf(fact_8827_gbinomial__rec,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_rec
% 0.30/0.64  thf(fact_8828_gbinomial__factors,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_factors
% 0.30/0.64  thf(fact_8829_gbinomial__factors,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_factors
% 0.30/0.64  thf(fact_8830_gbinomial__factors,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 0.30/0.64        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_factors
% 0.30/0.64  thf(fact_8831_pochhammer__absorb__comp,axiom,
% 0.30/0.64      ! [R2: complex,K: nat] :
% 0.30/0.64        ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 0.30/0.64        = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_absorb_comp
% 0.30/0.64  thf(fact_8832_pochhammer__absorb__comp,axiom,
% 0.30/0.64      ! [R2: rat,K: nat] :
% 0.30/0.64        ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 0.30/0.64        = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_absorb_comp
% 0.30/0.64  thf(fact_8833_pochhammer__absorb__comp,axiom,
% 0.30/0.64      ! [R2: code_integer,K: nat] :
% 0.30/0.64        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 0.30/0.64        = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_absorb_comp
% 0.30/0.64  thf(fact_8834_pochhammer__absorb__comp,axiom,
% 0.30/0.64      ! [R2: int,K: nat] :
% 0.30/0.64        ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 0.30/0.64        = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_absorb_comp
% 0.30/0.64  thf(fact_8835_pochhammer__absorb__comp,axiom,
% 0.30/0.64      ! [R2: real,K: nat] :
% 0.30/0.64        ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 0.30/0.64        = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_absorb_comp
% 0.30/0.64  thf(fact_8836_gbinomial__minus,axiom,
% 0.30/0.64      ! [A: complex,K: nat] :
% 0.30/0.64        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 0.30/0.64        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_minus
% 0.30/0.64  thf(fact_8837_gbinomial__minus,axiom,
% 0.30/0.64      ! [A: rat,K: nat] :
% 0.30/0.64        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 0.30/0.64        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_minus
% 0.30/0.64  thf(fact_8838_gbinomial__minus,axiom,
% 0.30/0.64      ! [A: real,K: nat] :
% 0.30/0.64        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 0.30/0.64        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_minus
% 0.30/0.64  thf(fact_8839_gbinomial__reduce__nat,axiom,
% 0.30/0.64      ! [K: nat,A: complex] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64       => ( ( gbinomial_complex @ A @ K )
% 0.30/0.64          = ( 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 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_reduce_nat
% 0.30/0.64  thf(fact_8840_gbinomial__reduce__nat,axiom,
% 0.30/0.64      ! [K: nat,A: real] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64       => ( ( gbinomial_real @ A @ K )
% 0.30/0.64          = ( 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 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_reduce_nat
% 0.30/0.64  thf(fact_8841_gbinomial__reduce__nat,axiom,
% 0.30/0.64      ! [K: nat,A: rat] :
% 0.30/0.64        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.64       => ( ( gbinomial_rat @ A @ K )
% 0.30/0.64          = ( 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 ) ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % gbinomial_reduce_nat
% 0.30/0.64  thf(fact_8842_pochhammer__minus,axiom,
% 0.30/0.64      ! [B: complex,K: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 0.30/0.64        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus
% 0.30/0.64  thf(fact_8843_pochhammer__minus,axiom,
% 0.30/0.64      ! [B: rat,K: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 0.30/0.64        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus
% 0.30/0.64  thf(fact_8844_pochhammer__minus,axiom,
% 0.30/0.64      ! [B: code_integer,K: nat] :
% 0.30/0.64        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 0.30/0.64        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus
% 0.30/0.64  thf(fact_8845_pochhammer__minus,axiom,
% 0.30/0.64      ! [B: int,K: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 0.30/0.64        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus
% 0.30/0.64  thf(fact_8846_pochhammer__minus,axiom,
% 0.30/0.64      ! [B: real,K: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 0.30/0.64        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus
% 0.30/0.64  thf(fact_8847_pochhammer__minus_H,axiom,
% 0.30/0.64      ! [B: complex,K: nat] :
% 0.30/0.64        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 0.30/0.64        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus'
% 0.30/0.64  thf(fact_8848_pochhammer__minus_H,axiom,
% 0.30/0.64      ! [B: rat,K: nat] :
% 0.30/0.64        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 0.30/0.64        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus'
% 0.30/0.64  thf(fact_8849_pochhammer__minus_H,axiom,
% 0.30/0.64      ! [B: code_integer,K: nat] :
% 0.30/0.64        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 0.30/0.64        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus'
% 0.30/0.64  thf(fact_8850_pochhammer__minus_H,axiom,
% 0.30/0.64      ! [B: int,K: nat] :
% 0.30/0.64        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 0.30/0.64        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus'
% 0.30/0.64  thf(fact_8851_pochhammer__minus_H,axiom,
% 0.30/0.64      ! [B: real,K: nat] :
% 0.30/0.64        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 0.30/0.64        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 0.30/0.64  
% 0.30/0.64  % pochhammer_minus'
% 0.30/0.64  thf(fact_8852_gbinomial__sum__up__index,axiom,
% 0.30/0.64      ! [K: nat,N2: nat] :
% 0.30/0.64        ( ( groups2073611262835488442omplex
% 0.30/0.64          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 0.30/0.64          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_sum_up_index
% 0.30/0.65  thf(fact_8853_gbinomial__sum__up__index,axiom,
% 0.30/0.65      ! [K: nat,N2: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat
% 0.30/0.65          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_sum_up_index
% 0.30/0.65  thf(fact_8854_gbinomial__sum__up__index,axiom,
% 0.30/0.65      ! [K: nat,N2: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_sum_up_index
% 0.30/0.65  thf(fact_8855_gbinomial__absorption_H,axiom,
% 0.30/0.65      ! [K: nat,A: complex] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.65       => ( ( gbinomial_complex @ A @ K )
% 0.30/0.65          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_absorption'
% 0.30/0.65  thf(fact_8856_gbinomial__absorption_H,axiom,
% 0.30/0.65      ! [K: nat,A: rat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.65       => ( ( gbinomial_rat @ A @ K )
% 0.30/0.65          = ( 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 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_absorption'
% 0.30/0.65  thf(fact_8857_gbinomial__absorption_H,axiom,
% 0.30/0.65      ! [K: nat,A: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ K )
% 0.30/0.65       => ( ( gbinomial_real @ A @ K )
% 0.30/0.65          = ( 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 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_absorption'
% 0.30/0.65  thf(fact_8858_binomial__code,axiom,
% 0.30/0.65      ( binomial
% 0.30/0.65      = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_code
% 0.30/0.65  thf(fact_8859_gbinomial__code,axiom,
% 0.30/0.65      ( gbinomial_complex
% 0.30/0.65      = ( ^ [A3: complex,K3: nat] :
% 0.30/0.65            ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 0.30/0.65            @ ( divide1717551699836669952omplex
% 0.30/0.65              @ ( set_fo1517530859248394432omplex
% 0.30/0.65                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 0.30/0.65                @ zero_zero_nat
% 0.30/0.65                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.30/0.65                @ one_one_complex )
% 0.30/0.65              @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_code
% 0.30/0.65  thf(fact_8860_gbinomial__code,axiom,
% 0.30/0.65      ( gbinomial_rat
% 0.30/0.65      = ( ^ [A3: rat,K3: nat] :
% 0.30/0.65            ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 0.30/0.65            @ ( divide_divide_rat
% 0.30/0.65              @ ( set_fo1949268297981939178at_rat
% 0.30/0.65                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 0.30/0.65                @ zero_zero_nat
% 0.30/0.65                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.30/0.65                @ one_one_rat )
% 0.30/0.65              @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_code
% 0.30/0.65  thf(fact_8861_gbinomial__code,axiom,
% 0.30/0.65      ( gbinomial_real
% 0.30/0.65      = ( ^ [A3: real,K3: nat] :
% 0.30/0.65            ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 0.30/0.65            @ ( divide_divide_real
% 0.30/0.65              @ ( set_fo3111899725591712190t_real
% 0.30/0.65                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 0.30/0.65                @ zero_zero_nat
% 0.30/0.65                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 0.30/0.65                @ one_one_real )
% 0.30/0.65              @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_code
% 0.30/0.65  thf(fact_8862_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z2: complex,N2: nat] :
% 0.30/0.65        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( groups6464643781859351333omplex
% 0.30/0.65          @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8863_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z2: rat,N2: nat] :
% 0.30/0.65        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8864_pochhammer__times__pochhammer__half,axiom,
% 0.30/0.65      ! [Z2: real,N2: nat] :
% 0.30/0.65        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_times_pochhammer_half
% 0.30/0.65  thf(fact_8865_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s2602460028002588243omplex
% 0.30/0.65      = ( ^ [A3: complex,N: nat] :
% 0.30/0.65            ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
% 0.30/0.65            @ ( set_fo1517530859248394432omplex
% 0.30/0.65              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N @ one_one_nat )
% 0.30/0.65              @ one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_code
% 0.30/0.65  thf(fact_8866_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4028243227959126397er_rat
% 0.30/0.65      = ( ^ [A3: rat,N: nat] :
% 0.30/0.65            ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
% 0.30/0.65            @ ( set_fo1949268297981939178at_rat
% 0.30/0.65              @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N @ one_one_nat )
% 0.30/0.65              @ one_one_rat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_code
% 0.30/0.65  thf(fact_8867_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4660882817536571857er_int
% 0.30/0.65      = ( ^ [A3: int,N: nat] :
% 0.30/0.65            ( if_int @ ( N = zero_zero_nat ) @ one_one_int
% 0.30/0.65            @ ( set_fo2581907887559384638at_int
% 0.30/0.65              @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N @ one_one_nat )
% 0.30/0.65              @ one_one_int ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_code
% 0.30/0.65  thf(fact_8868_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s7457072308508201937r_real
% 0.30/0.65      = ( ^ [A3: real,N: nat] :
% 0.30/0.65            ( if_real @ ( N = zero_zero_nat ) @ one_one_real
% 0.30/0.65            @ ( set_fo3111899725591712190t_real
% 0.30/0.65              @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N @ one_one_nat )
% 0.30/0.65              @ one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_code
% 0.30/0.65  thf(fact_8869_pochhammer__code,axiom,
% 0.30/0.65      ( comm_s4663373288045622133er_nat
% 0.30/0.65      = ( ^ [A3: nat,N: nat] :
% 0.30/0.65            ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
% 0.30/0.65            @ ( set_fo2584398358068434914at_nat
% 0.30/0.65              @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 0.30/0.65              @ zero_zero_nat
% 0.30/0.65              @ ( minus_minus_nat @ N @ one_one_nat )
% 0.30/0.65              @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pochhammer_code
% 0.30/0.65  thf(fact_8870_gbinomial__partial__row__sum,axiom,
% 0.30/0.65      ! [A: complex,M: nat] :
% 0.30/0.65        ( ( groups2073611262835488442omplex
% 0.30/0.65          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_partial_row_sum
% 0.30/0.65  thf(fact_8871_gbinomial__partial__row__sum,axiom,
% 0.30/0.65      ! [A: rat,M: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat
% 0.30/0.65          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_partial_row_sum
% 0.30/0.65  thf(fact_8872_gbinomial__partial__row__sum,axiom,
% 0.30/0.65      ! [A: real,M: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real
% 0.30/0.65          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gbinomial_partial_row_sum
% 0.30/0.65  thf(fact_8873_atMost__iff,axiom,
% 0.30/0.65      ! [I: real,K: real] :
% 0.30/0.65        ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
% 0.30/0.65        = ( ord_less_eq_real @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8874_atMost__iff,axiom,
% 0.30/0.65      ! [I: set_nat,K: set_nat] :
% 0.30/0.65        ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
% 0.30/0.65        = ( ord_less_eq_set_nat @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8875_atMost__iff,axiom,
% 0.30/0.65      ! [I: set_int,K: set_int] :
% 0.30/0.65        ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
% 0.30/0.65        = ( ord_less_eq_set_int @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8876_atMost__iff,axiom,
% 0.30/0.65      ! [I: rat,K: rat] :
% 0.30/0.65        ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
% 0.30/0.65        = ( ord_less_eq_rat @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8877_atMost__iff,axiom,
% 0.30/0.65      ! [I: num,K: num] :
% 0.30/0.65        ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
% 0.30/0.65        = ( ord_less_eq_num @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8878_atMost__iff,axiom,
% 0.30/0.65      ! [I: nat,K: nat] :
% 0.30/0.65        ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
% 0.30/0.65        = ( ord_less_eq_nat @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8879_atMost__iff,axiom,
% 0.30/0.65      ! [I: int,K: int] :
% 0.30/0.65        ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
% 0.30/0.65        = ( ord_less_eq_int @ I @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_iff
% 0.30/0.65  thf(fact_8880_finite__atMost,axiom,
% 0.30/0.65      ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_atMost
% 0.30/0.65  thf(fact_8881_of__nat__prod,axiom,
% 0.30/0.65      ! [F: int > nat,A2: set_int] :
% 0.30/0.65        ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
% 0.30/0.65        = ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_nat_prod
% 0.30/0.65  thf(fact_8882_of__nat__prod,axiom,
% 0.30/0.65      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.65        ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_nat_prod
% 0.30/0.65  thf(fact_8883_of__nat__prod,axiom,
% 0.30/0.65      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.65        ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.30/0.65        = ( groups708209901874060359at_nat
% 0.30/0.65          @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_nat_prod
% 0.30/0.65  thf(fact_8884_of__nat__prod,axiom,
% 0.30/0.65      ! [F: nat > nat,A2: set_nat] :
% 0.30/0.65        ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_nat_prod
% 0.30/0.65  thf(fact_8885_of__int__prod,axiom,
% 0.30/0.65      ! [F: nat > int,A2: set_nat] :
% 0.30/0.65        ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.30/0.65        = ( groups129246275422532515t_real
% 0.30/0.65          @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8886_of__int__prod,axiom,
% 0.30/0.65      ! [F: nat > int,A2: set_nat] :
% 0.30/0.65        ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.30/0.65        = ( groups73079841787564623at_rat
% 0.30/0.65          @ ^ [X: nat] : ( ring_1_of_int_rat @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8887_of__int__prod,axiom,
% 0.30/0.65      ! [F: nat > int,A2: set_nat] :
% 0.30/0.65        ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
% 0.30/0.65        = ( groups705719431365010083at_int
% 0.30/0.65          @ ^ [X: nat] : ( ring_1_of_int_int @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8888_of__int__prod,axiom,
% 0.30/0.65      ! [F: int > int,A2: set_int] :
% 0.30/0.65        ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.30/0.65        = ( groups2316167850115554303t_real
% 0.30/0.65          @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8889_of__int__prod,axiom,
% 0.30/0.65      ! [F: int > int,A2: set_int] :
% 0.30/0.65        ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.30/0.65        = ( groups1072433553688619179nt_rat
% 0.30/0.65          @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8890_of__int__prod,axiom,
% 0.30/0.65      ! [F: int > int,A2: set_int] :
% 0.30/0.65        ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 0.30/0.65        = ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
% 0.30/0.65          @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_int_prod
% 0.30/0.65  thf(fact_8891_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_nat,F: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( ( groups6464643781859351333omplex @ F @ A2 )
% 0.30/0.65            = zero_zero_complex )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8892_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_int,F: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( ( groups7440179247065528705omplex @ F @ A2 )
% 0.30/0.65            = zero_zero_complex )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8893_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_complex,F: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( ( groups3708469109370488835omplex @ F @ A2 )
% 0.30/0.65            = zero_zero_complex )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_complex ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8894_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_nat,F: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( ( groups129246275422532515t_real @ F @ A2 )
% 0.30/0.65            = zero_zero_real )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_real ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8895_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_int,F: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( ( groups2316167850115554303t_real @ F @ A2 )
% 0.30/0.65            = zero_zero_real )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_real ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8896_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_complex,F: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( ( groups766887009212190081x_real @ F @ A2 )
% 0.30/0.65            = zero_zero_real )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_real ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8897_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_nat,F: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( ( groups73079841787564623at_rat @ F @ A2 )
% 0.30/0.65            = zero_zero_rat )
% 0.30/0.65          = ( ? [X: nat] :
% 0.30/0.65                ( ( member_nat @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8898_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_int,F: int > rat] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 0.30/0.65            = zero_zero_rat )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8899_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_complex,F: complex > rat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
% 0.30/0.65            = zero_zero_rat )
% 0.30/0.65          = ( ? [X: complex] :
% 0.30/0.65                ( ( member_complex @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_rat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8900_prod__zero__iff,axiom,
% 0.30/0.65      ! [A2: set_int,F: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 0.30/0.65            = zero_zero_nat )
% 0.30/0.65          = ( ? [X: int] :
% 0.30/0.65                ( ( member_int @ X @ A2 )
% 0.30/0.65                & ( ( F @ X )
% 0.30/0.65                  = zero_zero_nat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_zero_iff
% 0.30/0.65  thf(fact_8901_prod_Oempty,axiom,
% 0.30/0.65      ! [G: real > complex] :
% 0.30/0.65        ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 0.30/0.65        = one_one_complex ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8902_prod_Oempty,axiom,
% 0.30/0.65      ! [G: real > real] :
% 0.30/0.65        ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 0.30/0.65        = one_one_real ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8903_prod_Oempty,axiom,
% 0.30/0.65      ! [G: real > rat] :
% 0.30/0.65        ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 0.30/0.65        = one_one_rat ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8904_prod_Oempty,axiom,
% 0.30/0.65      ! [G: real > nat] :
% 0.30/0.65        ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 0.30/0.65        = one_one_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8905_prod_Oempty,axiom,
% 0.30/0.65      ! [G: real > int] :
% 0.30/0.65        ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 0.30/0.65        = one_one_int ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8906_prod_Oempty,axiom,
% 0.30/0.65      ! [G: nat > complex] :
% 0.30/0.65        ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 0.30/0.65        = one_one_complex ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8907_prod_Oempty,axiom,
% 0.30/0.65      ! [G: nat > real] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 0.30/0.65        = one_one_real ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8908_prod_Oempty,axiom,
% 0.30/0.65      ! [G: nat > rat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 0.30/0.65        = one_one_rat ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8909_prod_Oempty,axiom,
% 0.30/0.65      ! [G: int > complex] :
% 0.30/0.65        ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 0.30/0.65        = one_one_complex ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8910_prod_Oempty,axiom,
% 0.30/0.65      ! [G: int > real] :
% 0.30/0.65        ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 0.30/0.65        = one_one_real ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.empty
% 0.30/0.65  thf(fact_8911_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_nat,G: nat > complex] :
% 0.30/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( groups6464643781859351333omplex @ G @ A2 )
% 0.30/0.65          = one_one_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8912_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_int,G: int > complex] :
% 0.30/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( groups7440179247065528705omplex @ G @ A2 )
% 0.30/0.65          = one_one_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8913_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_complex,G: complex > complex] :
% 0.30/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( groups3708469109370488835omplex @ G @ A2 )
% 0.30/0.65          = one_one_complex ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8914_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_nat,G: nat > real] :
% 0.30/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( groups129246275422532515t_real @ G @ A2 )
% 0.30/0.65          = one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8915_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_int,G: int > real] :
% 0.30/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( groups2316167850115554303t_real @ G @ A2 )
% 0.30/0.65          = one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8916_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_complex,G: complex > real] :
% 0.30/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( groups766887009212190081x_real @ G @ A2 )
% 0.30/0.65          = one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8917_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_nat,G: nat > rat] :
% 0.30/0.65        ( ~ ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( groups73079841787564623at_rat @ G @ A2 )
% 0.30/0.65          = one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8918_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_int,G: int > rat] :
% 0.30/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 0.30/0.65          = one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8919_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_complex,G: complex > rat] :
% 0.30/0.65        ( ~ ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 0.30/0.65          = one_one_rat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8920_prod_Oinfinite,axiom,
% 0.30/0.65      ! [A2: set_int,G: int > nat] :
% 0.30/0.65        ( ~ ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 0.30/0.65          = one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.infinite
% 0.30/0.65  thf(fact_8921_atMost__subset__iff,axiom,
% 0.30/0.65      ! [X3: set_int,Y3: set_int] :
% 0.30/0.65        ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_subset_iff
% 0.30/0.65  thf(fact_8922_atMost__subset__iff,axiom,
% 0.30/0.65      ! [X3: rat,Y3: rat] :
% 0.30/0.65        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_subset_iff
% 0.30/0.65  thf(fact_8923_atMost__subset__iff,axiom,
% 0.30/0.65      ! [X3: num,Y3: num] :
% 0.30/0.65        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_subset_iff
% 0.30/0.65  thf(fact_8924_atMost__subset__iff,axiom,
% 0.30/0.65      ! [X3: nat,Y3: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_subset_iff
% 0.30/0.65  thf(fact_8925_atMost__subset__iff,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y3 ) )
% 0.30/0.65        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_subset_iff
% 0.30/0.65  thf(fact_8926_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,B: nat,F: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8927_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_int,A: int,B: nat,F: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( member_int @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8928_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,B: nat,F: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8929_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,B: int,F: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8930_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,B: int,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8931_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,B: code_integer,F: real > code_integer] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8932_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( member_nat @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8933_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_int,A: int,B: code_integer,F: int > code_integer] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( member_int @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8934_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,B: code_integer,F: complex > code_integer] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_Code_integer @ B @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8935_dvd__prod__eqI,axiom,
% 0.30/0.65      ! [A2: set_nat,A: nat,B: nat,F: nat > nat] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( member_nat @ A @ A2 )
% 0.30/0.65         => ( ( B
% 0.30/0.65              = ( F @ A ) )
% 0.30/0.65           => ( dvd_dvd_nat @ B @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prod_eqI
% 0.30/0.65  thf(fact_8936_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,F: real > nat] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8937_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_int,A: int,F: int > nat] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( member_int @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8938_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,F: complex > nat] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8939_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,F: real > int] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8940_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,F: complex > int] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8941_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_real,A: real,F: real > code_integer] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ( member_real @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8942_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_nat,A: nat,F: nat > code_integer] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( member_nat @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8943_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_int,A: int,F: int > code_integer] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ( member_int @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8944_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_complex,A: complex,F: complex > code_integer] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ( member_complex @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8945_dvd__prodI,axiom,
% 0.30/0.65      ! [A2: set_nat,A: nat,F: nat > nat] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ( member_nat @ A @ A2 )
% 0.30/0.65         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_prodI
% 0.30/0.65  thf(fact_8946_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups713298508707869441omplex
% 0.30/0.65                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups713298508707869441omplex
% 0.30/0.65                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8947_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups6464643781859351333omplex
% 0.30/0.65                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups6464643781859351333omplex
% 0.30/0.65                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8948_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_int,A: int,B: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ S3 )
% 0.30/0.65       => ( ( ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups7440179247065528705omplex
% 0.30/0.65                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups7440179247065528705omplex
% 0.30/0.65                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8949_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_complex,A: complex,B: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.65       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups3708469109370488835omplex
% 0.30/0.65                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups3708469109370488835omplex
% 0.30/0.65                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8950_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups1681761925125756287l_real
% 0.30/0.65                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups1681761925125756287l_real
% 0.30/0.65                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8951_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups129246275422532515t_real
% 0.30/0.65                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups129246275422532515t_real
% 0.30/0.65                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8952_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_int,A: int,B: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ S3 )
% 0.30/0.65       => ( ( ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups2316167850115554303t_real
% 0.30/0.65                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups2316167850115554303t_real
% 0.30/0.65                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8953_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_complex,A: complex,B: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.65       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups766887009212190081x_real
% 0.30/0.65                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups766887009212190081x_real
% 0.30/0.65                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8954_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups4061424788464935467al_rat
% 0.30/0.65                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups4061424788464935467al_rat
% 0.30/0.65                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_rat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8955_prod_Odelta,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups73079841787564623at_rat
% 0.30/0.65                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups73079841787564623at_rat
% 0.30/0.65                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_rat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta
% 0.30/0.65  thf(fact_8956_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups713298508707869441omplex
% 0.30/0.65                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups713298508707869441omplex
% 0.30/0.65                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8957_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups6464643781859351333omplex
% 0.30/0.65                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups6464643781859351333omplex
% 0.30/0.65                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8958_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_int,A: int,B: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ S3 )
% 0.30/0.65       => ( ( ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups7440179247065528705omplex
% 0.30/0.65                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups7440179247065528705omplex
% 0.30/0.65                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8959_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_complex,A: complex,B: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.65       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups3708469109370488835omplex
% 0.30/0.65                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups3708469109370488835omplex
% 0.30/0.65                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_complex ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8960_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups1681761925125756287l_real
% 0.30/0.65                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups1681761925125756287l_real
% 0.30/0.65                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8961_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups129246275422532515t_real
% 0.30/0.65                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups129246275422532515t_real
% 0.30/0.65                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8962_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_int,A: int,B: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ S3 )
% 0.30/0.65       => ( ( ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups2316167850115554303t_real
% 0.30/0.65                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_int @ A @ S3 )
% 0.30/0.65           => ( ( groups2316167850115554303t_real
% 0.30/0.65                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8963_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_complex,A: complex,B: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ S3 )
% 0.30/0.65       => ( ( ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups766887009212190081x_real
% 0.30/0.65                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_complex @ A @ S3 )
% 0.30/0.65           => ( ( groups766887009212190081x_real
% 0.30/0.65                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_real ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8964_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_real,A: real,B: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ S3 )
% 0.30/0.65       => ( ( ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups4061424788464935467al_rat
% 0.30/0.65                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_real @ A @ S3 )
% 0.30/0.65           => ( ( groups4061424788464935467al_rat
% 0.30/0.65                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_rat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8965_prod_Odelta_H,axiom,
% 0.30/0.65      ! [S3: set_nat,A: nat,B: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ( ( ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups73079841787564623at_rat
% 0.30/0.65                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = ( B @ A ) ) )
% 0.30/0.65          & ( ~ ( member_nat @ A @ S3 )
% 0.30/0.65           => ( ( groups73079841787564623at_rat
% 0.30/0.65                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 0.30/0.65                @ S3 )
% 0.30/0.65              = one_one_rat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.delta'
% 0.30/0.65  thf(fact_8966_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_real,X3: real,G: real > complex] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.65         => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_complex @ ( G @ X3 ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8967_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_nat,X3: nat,G: nat > complex] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.65         => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_complex @ ( G @ X3 ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8968_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_int,X3: int,G: int > complex] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.65         => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_complex @ ( G @ X3 ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8969_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_complex,X3: complex,G: complex > complex] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.65         => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_complex @ ( G @ X3 ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8970_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_real,X3: real,G: real > real] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.65         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8971_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_nat,X3: nat,G: nat > real] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8972_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_int,X3: int,G: int > real] :
% 0.30/0.65        ( ( finite_finite_int @ A2 )
% 0.30/0.65       => ( ~ ( member_int @ X3 @ A2 )
% 0.30/0.65         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8973_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_complex,X3: complex,G: complex > real] :
% 0.30/0.65        ( ( finite3207457112153483333omplex @ A2 )
% 0.30/0.65       => ( ~ ( member_complex @ X3 @ A2 )
% 0.30/0.65         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8974_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_real,X3: real,G: real > rat] :
% 0.30/0.65        ( ( finite_finite_real @ A2 )
% 0.30/0.65       => ( ~ ( member_real @ X3 @ A2 )
% 0.30/0.65         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8975_prod_Oinsert,axiom,
% 0.30/0.65      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 0.30/0.65        ( ( finite_finite_nat @ A2 )
% 0.30/0.65       => ( ~ ( member_nat @ X3 @ A2 )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 0.30/0.65            = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.insert
% 0.30/0.65  thf(fact_8976_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: set_int,H2: set_int,H3: set_int] :
% 0.30/0.65        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8977_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: rat,H2: rat,H3: rat] :
% 0.30/0.65        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8978_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: num,H2: num,H3: num] :
% 0.30/0.65        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_num @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8979_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: nat,H2: nat,H3: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8980_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: int,H2: int,H3: int] :
% 0.30/0.65        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_int @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8981_Icc__subset__Iic__iff,axiom,
% 0.30/0.65      ! [L: real,H2: real,H3: real] :
% 0.30/0.65        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 0.30/0.65        = ( ~ ( ord_less_eq_real @ L @ H2 )
% 0.30/0.65          | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Icc_subset_Iic_iff
% 0.30/0.65  thf(fact_8982_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N2: nat] :
% 0.30/0.65        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8983_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N2: nat] :
% 0.30/0.65        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8984_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N2: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8985_sum_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N2: nat] :
% 0.30/0.65        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum.atMost_Suc
% 0.30/0.65  thf(fact_8986_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > complex,N2: nat] :
% 0.30/0.65        ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8987_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N2: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8988_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N2: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8989_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N2: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8990_prod_OlessThan__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N2: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.lessThan_Suc
% 0.30/0.65  thf(fact_8991_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > complex,N2: nat] :
% 0.30/0.65        ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8992_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > real,N2: nat] :
% 0.30/0.65        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8993_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > rat,N2: nat] :
% 0.30/0.65        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8994_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > nat,N2: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8995_prod_OatMost__Suc,axiom,
% 0.30/0.65      ! [G: nat > int,N2: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 0.30/0.65        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.atMost_Suc
% 0.30/0.65  thf(fact_8996_atMost__0,axiom,
% 0.30/0.65      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 0.30/0.65      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_0
% 0.30/0.65  thf(fact_8997_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N2: nat,M: nat,G: nat > complex] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = one_one_complex ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8998_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N2: nat,M: nat,G: nat > real] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = one_one_real ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_8999_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N2: nat,M: nat,G: nat > rat] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = one_one_rat ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_9000_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N2: nat,M: nat,G: nat > nat] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = one_one_nat ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_9001_prod_Ocl__ivl__Suc,axiom,
% 0.30/0.65      ! [N2: nat,M: nat,G: nat > int] :
% 0.30/0.65        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = one_one_int ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 0.30/0.65            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod.cl_ivl_Suc
% 0.30/0.65  thf(fact_9002_atMost__atLeast0,axiom,
% 0.30/0.65      ( set_ord_atMost_nat
% 0.30/0.65      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_atLeast0
% 0.30/0.65  thf(fact_9003_lessThan__Suc__atMost,axiom,
% 0.30/0.65      ! [K: nat] :
% 0.30/0.65        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 0.30/0.65        = ( set_ord_atMost_nat @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % lessThan_Suc_atMost
% 0.30/0.65  thf(fact_9004_atMost__Suc,axiom,
% 0.30/0.65      ! [K: nat] :
% 0.30/0.65        ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 0.30/0.65        = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atMost_Suc
% 0.30/0.65  thf(fact_9005_finite__nat__iff__bounded__le,axiom,
% 0.30/0.65      ( finite_finite_nat
% 0.30/0.65      = ( ^ [S7: set_nat] :
% 0.30/0.65          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_nat_iff_bounded_le
% 0.30/0.65  thf(fact_9006_sum__choose__upper,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N2 ) )
% 0.30/0.65        = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_choose_upper
% 0.30/0.65  thf(fact_9007_sum__choose__lower,axiom,
% 0.30/0.65      ! [R2: nat,N2: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N2 ) )
% 0.30/0.65        = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_choose_lower
% 0.30/0.65  thf(fact_9008_choose__rising__sum_I1_J,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_rising_sum(1)
% 0.30/0.65  thf(fact_9009_choose__rising__sum_I2_J,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 0.30/0.65          @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % choose_rising_sum(2)
% 0.30/0.65  thf(fact_9010_atLeast1__atMost__eq__remove0,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.65        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeast1_atMost_eq_remove0
% 0.30/0.65  thf(fact_9011_fact__eq__fact__times,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.65       => ( ( semiri1408675320244567234ct_nat @ M )
% 0.30/0.65          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 0.30/0.65            @ ( groups708209901874060359at_nat
% 0.30/0.65              @ ^ [X: nat] : X
% 0.30/0.65              @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_eq_fact_times
% 0.30/0.65  thf(fact_9012_sum__choose__diagonal,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65       => ( ( groups3542108847815614940at_nat
% 0.30/0.65            @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 0.30/0.65            @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65          = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_choose_diagonal
% 0.30/0.65  thf(fact_9013_vandermonde,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,R2: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ R2 ) )
% 0.30/0.65        = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % vandermonde
% 0.30/0.65  thf(fact_9014_fact__div__fact,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.65       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
% 0.30/0.65          = ( groups708209901874060359at_nat
% 0.30/0.65            @ ^ [X: nat] : X
% 0.30/0.65            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % fact_div_fact
% 0.30/0.65  thf(fact_9015_binomial,axiom,
% 0.30/0.65      ! [A: nat,B: nat,N2: nat] :
% 0.30/0.65        ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 0.30/0.65        = ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
% 0.30/0.65          @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial
% 0.30/0.65  thf(fact_9016_polynomial__product__nat,axiom,
% 0.30/0.65      ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X3: nat] :
% 0.30/0.65        ( ! [I3: nat] :
% 0.30/0.65            ( ( ord_less_nat @ M @ I3 )
% 0.30/0.65           => ( ( A @ I3 )
% 0.30/0.65              = zero_zero_nat ) )
% 0.30/0.65       => ( ! [J2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ N2 @ J2 )
% 0.30/0.65             => ( ( B @ J2 )
% 0.30/0.65                = zero_zero_nat ) )
% 0.30/0.65         => ( ( times_times_nat
% 0.30/0.65              @ ( groups3542108847815614940at_nat
% 0.30/0.65                @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X3 @ I2 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65              @ ( groups3542108847815614940at_nat
% 0.30/0.65                @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
% 0.30/0.65                @ ( set_ord_atMost_nat @ N2 ) ) )
% 0.30/0.65            = ( groups3542108847815614940at_nat
% 0.30/0.65              @ ^ [R5: nat] :
% 0.30/0.65                  ( times_times_nat
% 0.30/0.65                  @ ( groups3542108847815614940at_nat
% 0.30/0.65                    @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 0.30/0.65                    @ ( set_ord_atMost_nat @ R5 ) )
% 0.30/0.65                  @ ( power_power_nat @ X3 @ R5 ) )
% 0.30/0.65              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % polynomial_product_nat
% 0.30/0.65  thf(fact_9017_binomial__r__part__sum,axiom,
% 0.30/0.65      ! [M: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 0.30/0.65        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_r_part_sum
% 0.30/0.65  thf(fact_9018_cot__less__zero,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.65         => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cot_less_zero
% 0.30/0.65  thf(fact_9019_int__ge__less__than2__def,axiom,
% 0.30/0.65      ( int_ge_less_than2
% 0.30/0.65      = ( ^ [D5: int] :
% 0.30/0.65            ( collec213857154873943460nt_int
% 0.30/0.65            @ ( produc4947309494688390418_int_o
% 0.30/0.65              @ ^ [Z7: int,Z6: int] :
% 0.30/0.65                  ( ( ord_less_eq_int @ D5 @ Z6 )
% 0.30/0.65                  & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_ge_less_than2_def
% 0.30/0.65  thf(fact_9020_of__nat__id,axiom,
% 0.30/0.65      ( semiri1316708129612266289at_nat
% 0.30/0.65      = ( ^ [N: nat] : N ) ) ).
% 0.30/0.65  
% 0.30/0.65  % of_nat_id
% 0.30/0.65  thf(fact_9021_prod__int__plus__eq,axiom,
% 0.30/0.65      ! [I: nat,J: nat] :
% 0.30/0.65        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 0.30/0.65        = ( groups1705073143266064639nt_int
% 0.30/0.65          @ ^ [X: int] : X
% 0.30/0.65          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_int_plus_eq
% 0.30/0.65  thf(fact_9022_cot__gt__zero,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cot_gt_zero
% 0.30/0.65  thf(fact_9023_int__ge__less__than__def,axiom,
% 0.30/0.65      ( int_ge_less_than
% 0.30/0.65      = ( ^ [D5: int] :
% 0.30/0.65            ( collec213857154873943460nt_int
% 0.30/0.65            @ ( produc4947309494688390418_int_o
% 0.30/0.65              @ ^ [Z7: int,Z6: int] :
% 0.30/0.65                  ( ( ord_less_eq_int @ D5 @ Z7 )
% 0.30/0.65                  & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_ge_less_than_def
% 0.30/0.65  thf(fact_9024_arctan__def,axiom,
% 0.30/0.65      ( arctan
% 0.30/0.65      = ( ^ [Y: real] :
% 0.30/0.65            ( the_real
% 0.30/0.65            @ ^ [X: real] :
% 0.30/0.65                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 0.30/0.65                & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 0.30/0.65                & ( ( tan_real @ X )
% 0.30/0.65                  = Y ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arctan_def
% 0.30/0.65  thf(fact_9025_modulo__int__unfold,axiom,
% 0.30/0.65      ! [L: int,K: int,N2: nat,M: nat] :
% 0.30/0.65        ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.65              = zero_zero_int )
% 0.30/0.65            | ( ( sgn_sgn_int @ K )
% 0.30/0.65              = zero_zero_int )
% 0.30/0.65            | ( N2 = zero_zero_nat ) )
% 0.30/0.65         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 0.30/0.65        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.65                = zero_zero_int )
% 0.30/0.65              | ( ( sgn_sgn_int @ K )
% 0.30/0.65                = zero_zero_int )
% 0.30/0.65              | ( N2 = zero_zero_nat ) )
% 0.30/0.65         => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.65                = ( sgn_sgn_int @ L ) )
% 0.30/0.65             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
% 0.30/0.65            & ( ( ( sgn_sgn_int @ K )
% 0.30/0.65               != ( sgn_sgn_int @ L ) )
% 0.30/0.65             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65                = ( times_times_int @ ( sgn_sgn_int @ L )
% 0.30/0.65                  @ ( minus_minus_int
% 0.30/0.65                    @ ( semiri1314217659103216013at_int
% 0.30/0.65                      @ ( times_times_nat @ N2
% 0.30/0.65                        @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65                          @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
% 0.30/0.65                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % modulo_int_unfold
% 0.30/0.65  thf(fact_9026_powr__int,axiom,
% 0.30/0.65      ! [X3: real,I: int] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.65           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 0.30/0.65              = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 0.30/0.65           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 0.30/0.65              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % powr_int
% 0.30/0.65  thf(fact_9027_nat__int,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.65        = N2 ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_int
% 0.30/0.65  thf(fact_9028_nat__numeral,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 0.30/0.65        = ( numeral_numeral_nat @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_numeral
% 0.30/0.65  thf(fact_9029_nat__of__bool,axiom,
% 0.30/0.65      ! [P: $o] :
% 0.30/0.65        ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 0.30/0.65        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_of_bool
% 0.30/0.65  thf(fact_9030_nat__1,axiom,
% 0.30/0.65      ( ( nat2 @ one_one_int )
% 0.30/0.65      = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_1
% 0.30/0.65  thf(fact_9031_nat__0__iff,axiom,
% 0.30/0.65      ! [I: int] :
% 0.30/0.65        ( ( ( nat2 @ I )
% 0.30/0.65          = zero_zero_nat )
% 0.30/0.65        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_0_iff
% 0.30/0.65  thf(fact_9032_nat__le__0,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.30/0.65       => ( ( nat2 @ Z2 )
% 0.30/0.65          = zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_le_0
% 0.30/0.65  thf(fact_9033_zless__nat__conj,axiom,
% 0.30/0.65      ! [W2: int,Z2: int] :
% 0.30/0.65        ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.30/0.65        = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.65          & ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zless_nat_conj
% 0.30/0.65  thf(fact_9034_nat__neg__numeral,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.65        = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_neg_numeral
% 0.30/0.65  thf(fact_9035_nat__zminus__int,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65        = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_zminus_int
% 0.30/0.65  thf(fact_9036_int__nat__eq,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.30/0.65            = Z2 ) )
% 0.30/0.65        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.30/0.65            = zero_zero_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_nat_eq
% 0.30/0.65  thf(fact_9037_sgn__mult__dvd__iff,axiom,
% 0.30/0.65      ! [R2: int,L: int,K: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 0.30/0.65        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65          & ( ( R2 = zero_zero_int )
% 0.30/0.65           => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_mult_dvd_iff
% 0.30/0.65  thf(fact_9038_mult__sgn__dvd__iff,axiom,
% 0.30/0.65      ! [L: int,R2: int,K: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 0.30/0.65        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65          & ( ( R2 = zero_zero_int )
% 0.30/0.65           => ( K = zero_zero_int ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % mult_sgn_dvd_iff
% 0.30/0.65  thf(fact_9039_dvd__sgn__mult__iff,axiom,
% 0.30/0.65      ! [L: int,R2: int,K: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 0.30/0.65        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65          | ( R2 = zero_zero_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_sgn_mult_iff
% 0.30/0.65  thf(fact_9040_dvd__mult__sgn__iff,axiom,
% 0.30/0.65      ! [L: int,K: int,R2: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 0.30/0.65        = ( ( dvd_dvd_int @ L @ K )
% 0.30/0.65          | ( R2 = zero_zero_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_mult_sgn_iff
% 0.30/0.65  thf(fact_9041_zero__less__nat__eq,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 0.30/0.65        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_less_nat_eq
% 0.30/0.65  thf(fact_9042_diff__nat__numeral,axiom,
% 0.30/0.65      ! [V: num,V3: num] :
% 0.30/0.65        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 0.30/0.65        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % diff_nat_numeral
% 0.30/0.65  thf(fact_9043_numeral__power__eq__nat__cancel__iff,axiom,
% 0.30/0.65      ! [X3: num,N2: nat,Y3: int] :
% 0.30/0.65        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 )
% 0.30/0.65          = ( nat2 @ Y3 ) )
% 0.30/0.65        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 0.30/0.65          = Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % numeral_power_eq_nat_cancel_iff
% 0.30/0.65  thf(fact_9044_nat__eq__numeral__power__cancel__iff,axiom,
% 0.30/0.65      ! [Y3: int,X3: num,N2: nat] :
% 0.30/0.65        ( ( ( nat2 @ Y3 )
% 0.30/0.65          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 0.30/0.65        = ( Y3
% 0.30/0.65          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_eq_numeral_power_cancel_iff
% 0.30/0.65  thf(fact_9045_nat__abs__dvd__iff,axiom,
% 0.30/0.65      ! [K: int,N2: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
% 0.30/0.65        = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_abs_dvd_iff
% 0.30/0.65  thf(fact_9046_dvd__nat__abs__iff,axiom,
% 0.30/0.65      ! [N2: nat,K: int] :
% 0.30/0.65        ( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.30/0.65        = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % dvd_nat_abs_iff
% 0.30/0.65  thf(fact_9047_nat__ceiling__le__eq,axiom,
% 0.30/0.65      ! [X3: real,A: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
% 0.30/0.65        = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_ceiling_le_eq
% 0.30/0.65  thf(fact_9048_one__less__nat__eq,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 0.30/0.65        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % one_less_nat_eq
% 0.30/0.65  thf(fact_9049_nat__numeral__diff__1,axiom,
% 0.30/0.65      ! [V: num] :
% 0.30/0.65        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 0.30/0.65        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_numeral_diff_1
% 0.30/0.65  thf(fact_9050_nat__less__numeral__power__cancel__iff,axiom,
% 0.30/0.65      ! [A: int,X3: num,N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 0.30/0.65        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_less_numeral_power_cancel_iff
% 0.30/0.65  thf(fact_9051_numeral__power__less__nat__cancel__iff,axiom,
% 0.30/0.65      ! [X3: num,N2: nat,A: int] :
% 0.30/0.65        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
% 0.30/0.65        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % numeral_power_less_nat_cancel_iff
% 0.30/0.65  thf(fact_9052_nat__le__numeral__power__cancel__iff,axiom,
% 0.30/0.65      ! [A: int,X3: num,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 0.30/0.65        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_le_numeral_power_cancel_iff
% 0.30/0.65  thf(fact_9053_numeral__power__le__nat__cancel__iff,axiom,
% 0.30/0.65      ! [X3: num,N2: nat,A: int] :
% 0.30/0.65        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
% 0.30/0.65        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % numeral_power_le_nat_cancel_iff
% 0.30/0.65  thf(fact_9054_nat__zero__as__int,axiom,
% 0.30/0.65      ( zero_zero_nat
% 0.30/0.65      = ( nat2 @ zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_zero_as_int
% 0.30/0.65  thf(fact_9055_nat__mono,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ X3 @ Y3 )
% 0.30/0.65       => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mono
% 0.30/0.65  thf(fact_9056_int__sgnE,axiom,
% 0.30/0.65      ! [K: int] :
% 0.30/0.65        ~ ! [N3: nat,L4: int] :
% 0.30/0.65            ( K
% 0.30/0.65           != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_sgnE
% 0.30/0.65  thf(fact_9057_ex__nat,axiom,
% 0.30/0.65      ( ( ^ [P2: nat > $o] :
% 0.30/0.65          ? [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.65      = ( ^ [P3: nat > $o] :
% 0.30/0.65          ? [X: int] :
% 0.30/0.65            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.65            & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % ex_nat
% 0.30/0.65  thf(fact_9058_all__nat,axiom,
% 0.30/0.65      ( ( ^ [P2: nat > $o] :
% 0.30/0.65          ! [X6: nat] : ( P2 @ X6 ) )
% 0.30/0.65      = ( ^ [P3: nat > $o] :
% 0.30/0.65          ! [X: int] :
% 0.30/0.65            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 0.30/0.65           => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % all_nat
% 0.30/0.65  thf(fact_9059_eq__nat__nat__iff,axiom,
% 0.30/0.65      ! [Z2: int,Z8: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 0.30/0.65         => ( ( ( nat2 @ Z2 )
% 0.30/0.65              = ( nat2 @ Z8 ) )
% 0.30/0.65            = ( Z2 = Z8 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % eq_nat_nat_iff
% 0.30/0.65  thf(fact_9060_nat__mono__iff,axiom,
% 0.30/0.65      ! [Z2: int,W2: int] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.30/0.65          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mono_iff
% 0.30/0.65  thf(fact_9061_zless__nat__eq__int__zless,axiom,
% 0.30/0.65      ! [M: nat,Z2: int] :
% 0.30/0.65        ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 0.30/0.65        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zless_nat_eq_int_zless
% 0.30/0.65  thf(fact_9062_nat__le__iff,axiom,
% 0.30/0.65      ! [X3: int,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N2 )
% 0.30/0.65        = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_le_iff
% 0.30/0.65  thf(fact_9063_nat__0__le,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 0.30/0.65          = Z2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_0_le
% 0.30/0.65  thf(fact_9064_int__eq__iff,axiom,
% 0.30/0.65      ! [M: nat,Z2: int] :
% 0.30/0.65        ( ( ( semiri1314217659103216013at_int @ M )
% 0.30/0.65          = Z2 )
% 0.30/0.65        = ( ( M
% 0.30/0.65            = ( nat2 @ Z2 ) )
% 0.30/0.65          & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_eq_iff
% 0.30/0.65  thf(fact_9065_nat__int__add,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 0.30/0.65        = ( plus_plus_nat @ A @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_int_add
% 0.30/0.65  thf(fact_9066_nat__abs__mult__distrib,axiom,
% 0.30/0.65      ! [W2: int,Z2: int] :
% 0.30/0.65        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
% 0.30/0.65        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_abs_mult_distrib
% 0.30/0.65  thf(fact_9067_nat__plus__as__int,axiom,
% 0.30/0.65      ( plus_plus_nat
% 0.30/0.65      = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_plus_as_int
% 0.30/0.65  thf(fact_9068_zsgn__def,axiom,
% 0.30/0.65      ( sgn_sgn_int
% 0.30/0.65      = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zsgn_def
% 0.30/0.65  thf(fact_9069_powr__real__of__int,axiom,
% 0.30/0.65      ! [X3: real,N2: int] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 0.30/0.65           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 0.30/0.65              = ( power_power_real @ X3 @ ( nat2 @ N2 ) ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
% 0.30/0.65           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 0.30/0.65              = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % powr_real_of_int
% 0.30/0.65  thf(fact_9070_nat__less__eq__zless,axiom,
% 0.30/0.65      ! [W2: int,Z2: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.30/0.65          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_less_eq_zless
% 0.30/0.65  thf(fact_9071_nat__le__eq__zle,axiom,
% 0.30/0.65      ! [W2: int,Z2: int] :
% 0.30/0.65        ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 0.30/0.65          | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 0.30/0.65       => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 0.30/0.65          = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_le_eq_zle
% 0.30/0.65  thf(fact_9072_nat__eq__iff,axiom,
% 0.30/0.65      ! [W2: int,M: nat] :
% 0.30/0.65        ( ( ( nat2 @ W2 )
% 0.30/0.65          = M )
% 0.30/0.65        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65           => ( W2
% 0.30/0.65              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_eq_iff
% 0.30/0.65  thf(fact_9073_nat__eq__iff2,axiom,
% 0.30/0.65      ! [M: nat,W2: int] :
% 0.30/0.65        ( ( M
% 0.30/0.65          = ( nat2 @ W2 ) )
% 0.30/0.65        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65           => ( W2
% 0.30/0.65              = ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_eq_iff2
% 0.30/0.65  thf(fact_9074_split__nat,axiom,
% 0.30/0.65      ! [P: nat > $o,I: int] :
% 0.30/0.65        ( ( P @ ( nat2 @ I ) )
% 0.30/0.65        = ( ! [N: nat] :
% 0.30/0.65              ( ( I
% 0.30/0.65                = ( semiri1314217659103216013at_int @ N ) )
% 0.30/0.65             => ( P @ N ) )
% 0.30/0.65          & ( ( ord_less_int @ I @ zero_zero_int )
% 0.30/0.65           => ( P @ zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % split_nat
% 0.30/0.65  thf(fact_9075_le__nat__iff,axiom,
% 0.30/0.65      ! [K: int,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
% 0.30/0.65          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % le_nat_iff
% 0.30/0.65  thf(fact_9076_nat__add__distrib,axiom,
% 0.30/0.65      ! [Z2: int,Z8: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 0.30/0.65         => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z8 ) )
% 0.30/0.65            = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_add_distrib
% 0.30/0.65  thf(fact_9077_nat__mult__distrib,axiom,
% 0.30/0.65      ! [Z2: int,Z8: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
% 0.30/0.65          = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_distrib
% 0.30/0.65  thf(fact_9078_Suc__as__int,axiom,
% 0.30/0.65      ( suc
% 0.30/0.65      = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_as_int
% 0.30/0.65  thf(fact_9079_forall__pos__mono__1,axiom,
% 0.30/0.65      ! [P: real > $o,E2: real] :
% 0.30/0.65        ( ! [D4: real,E: real] :
% 0.30/0.65            ( ( ord_less_real @ D4 @ E )
% 0.30/0.65           => ( ( P @ D4 )
% 0.30/0.65             => ( P @ E ) ) )
% 0.30/0.65       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 0.30/0.65         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.65           => ( P @ E2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % forall_pos_mono_1
% 0.30/0.65  thf(fact_9080_forall__pos__mono,axiom,
% 0.30/0.65      ! [P: real > $o,E2: real] :
% 0.30/0.65        ( ! [D4: real,E: real] :
% 0.30/0.65            ( ( ord_less_real @ D4 @ E )
% 0.30/0.65           => ( ( P @ D4 )
% 0.30/0.65             => ( P @ E ) ) )
% 0.30/0.65       => ( ! [N3: nat] :
% 0.30/0.65              ( ( N3 != zero_zero_nat )
% 0.30/0.65             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 0.30/0.65         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.65           => ( P @ E2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % forall_pos_mono
% 0.30/0.65  thf(fact_9081_real__arch__inverse,axiom,
% 0.30/0.65      ! [E2: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ E2 )
% 0.30/0.65        = ( ? [N: nat] :
% 0.30/0.65              ( ( N != zero_zero_nat )
% 0.30/0.65              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.65              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_arch_inverse
% 0.30/0.65  thf(fact_9082_nat__diff__distrib,axiom,
% 0.30/0.65      ! [Z8: int,Z2: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 0.30/0.65       => ( ( ord_less_eq_int @ Z8 @ Z2 )
% 0.30/0.65         => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) )
% 0.30/0.65            = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_diff_distrib
% 0.30/0.65  thf(fact_9083_nat__diff__distrib_H,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.65       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65         => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
% 0.30/0.65            = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_diff_distrib'
% 0.30/0.65  thf(fact_9084_nat__abs__triangle__ineq,axiom,
% 0.30/0.65      ! [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 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_abs_triangle_ineq
% 0.30/0.65  thf(fact_9085_nat__floor__neg,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.65       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.65          = zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_floor_neg
% 0.30/0.65  thf(fact_9086_nat__power__eq,axiom,
% 0.30/0.65      ! [Z2: int,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( nat2 @ ( power_power_int @ Z2 @ N2 ) )
% 0.30/0.65          = ( power_power_nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_power_eq
% 0.30/0.65  thf(fact_9087_ln__inverse,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.65          = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % ln_inverse
% 0.30/0.65  thf(fact_9088_floor__eq3,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 0.30/0.65         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.65            = N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % floor_eq3
% 0.30/0.65  thf(fact_9089_le__nat__floor,axiom,
% 0.30/0.65      ! [X3: nat,A: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
% 0.30/0.65       => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % le_nat_floor
% 0.30/0.65  thf(fact_9090_divide__int__def,axiom,
% 0.30/0.65      ( divide_divide_int
% 0.30/0.65      = ( ^ [K3: int,L2: int] :
% 0.30/0.65            ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 0.30/0.65            @ ( if_int
% 0.30/0.65              @ ( ( sgn_sgn_int @ K3 )
% 0.30/0.65                = ( sgn_sgn_int @ L2 ) )
% 0.30/0.65              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 0.30/0.65              @ ( uminus_uminus_int
% 0.30/0.65                @ ( semiri1314217659103216013at_int
% 0.30/0.65                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 0.30/0.65                    @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65                      @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divide_int_def
% 0.30/0.65  thf(fact_9091_nat__2,axiom,
% 0.30/0.65      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.65      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_2
% 0.30/0.65  thf(fact_9092_Suc__nat__eq__nat__zadd1,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65       => ( ( suc @ ( nat2 @ Z2 ) )
% 0.30/0.65          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_nat_eq_nat_zadd1
% 0.30/0.65  thf(fact_9093_nat__less__iff,axiom,
% 0.30/0.65      ! [W2: int,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 0.30/0.65       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 0.30/0.65          = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_less_iff
% 0.30/0.65  thf(fact_9094_nat__mult__distrib__neg,axiom,
% 0.30/0.65      ! [Z2: int,Z8: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 0.30/0.65       => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
% 0.30/0.65          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_distrib_neg
% 0.30/0.65  thf(fact_9095_nat__abs__int__diff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.65         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.30/0.65            = ( minus_minus_nat @ B @ A ) ) )
% 0.30/0.65        & ( ~ ( ord_less_eq_nat @ A @ B )
% 0.30/0.65         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 0.30/0.65            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_abs_int_diff
% 0.30/0.65  thf(fact_9096_log__inverse,axiom,
% 0.30/0.65      ! [A: real,X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.65       => ( ( A != one_one_real )
% 0.30/0.65         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65           => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
% 0.30/0.65              = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % log_inverse
% 0.30/0.65  thf(fact_9097_floor__eq4,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 0.30/0.65         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 0.30/0.65            = N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % floor_eq4
% 0.30/0.65  thf(fact_9098_diff__nat__eq__if,axiom,
% 0.30/0.65      ! [Z8: int,Z2: int] :
% 0.30/0.65        ( ( ( ord_less_int @ Z8 @ zero_zero_int )
% 0.30/0.65         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 0.30/0.65            = ( nat2 @ Z2 ) ) )
% 0.30/0.65        & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
% 0.30/0.65         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 0.30/0.65            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % diff_nat_eq_if
% 0.30/0.65  thf(fact_9099_nat__dvd__iff,axiom,
% 0.30/0.65      ! [Z2: int,M: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 0.30/0.65        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65           => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 0.30/0.65           => ( M = zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_dvd_iff
% 0.30/0.65  thf(fact_9100_eucl__rel__int__remainderI,axiom,
% 0.30/0.65      ! [R2: int,L: int,K: int,Q4: int] :
% 0.30/0.65        ( ( ( sgn_sgn_int @ R2 )
% 0.30/0.65          = ( sgn_sgn_int @ L ) )
% 0.30/0.65       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 0.30/0.65         => ( ( K
% 0.30/0.65              = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 0.30/0.65           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % eucl_rel_int_remainderI
% 0.30/0.65  thf(fact_9101_plus__inverse__ge__2,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % plus_inverse_ge_2
% 0.30/0.65  thf(fact_9102_real__inv__sqrt__pow2,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 0.30/0.65          = ( inverse_inverse_real @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_inv_sqrt_pow2
% 0.30/0.65  thf(fact_9103_eucl__rel__int_Ocases,axiom,
% 0.30/0.65      ! [A12: int,A23: int,A32: product_prod_int_int] :
% 0.30/0.65        ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 0.30/0.65       => ( ( ( A23 = zero_zero_int )
% 0.30/0.65           => ( A32
% 0.30/0.65             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 0.30/0.65         => ( ! [Q2: int] :
% 0.30/0.65                ( ( A32
% 0.30/0.65                  = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 0.30/0.65               => ( ( A23 != zero_zero_int )
% 0.30/0.65                 => ( A12
% 0.30/0.65                   != ( times_times_int @ Q2 @ A23 ) ) ) )
% 0.30/0.65           => ~ ! [R4: int,Q2: int] :
% 0.30/0.65                  ( ( A32
% 0.30/0.65                    = ( product_Pair_int_int @ Q2 @ R4 ) )
% 0.30/0.65                 => ( ( ( sgn_sgn_int @ R4 )
% 0.30/0.65                      = ( sgn_sgn_int @ A23 ) )
% 0.30/0.65                   => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
% 0.30/0.65                     => ( A12
% 0.30/0.65                       != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % eucl_rel_int.cases
% 0.30/0.65  thf(fact_9104_eucl__rel__int_Osimps,axiom,
% 0.30/0.65      ( eucl_rel_int
% 0.30/0.65      = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 0.30/0.65            ( ? [K3: int] :
% 0.30/0.65                ( ( A1 = K3 )
% 0.30/0.65                & ( A22 = zero_zero_int )
% 0.30/0.65                & ( A33
% 0.30/0.65                  = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 0.30/0.65            | ? [L2: int,K3: int,Q5: int] :
% 0.30/0.65                ( ( A1 = K3 )
% 0.30/0.65                & ( A22 = L2 )
% 0.30/0.65                & ( A33
% 0.30/0.65                  = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 0.30/0.65                & ( L2 != zero_zero_int )
% 0.30/0.65                & ( K3
% 0.30/0.65                  = ( times_times_int @ Q5 @ L2 ) ) )
% 0.30/0.65            | ? [R5: int,L2: int,K3: int,Q5: int] :
% 0.30/0.65                ( ( A1 = K3 )
% 0.30/0.65                & ( A22 = L2 )
% 0.30/0.65                & ( A33
% 0.30/0.65                  = ( product_Pair_int_int @ Q5 @ R5 ) )
% 0.30/0.65                & ( ( sgn_sgn_int @ R5 )
% 0.30/0.65                  = ( sgn_sgn_int @ L2 ) )
% 0.30/0.65                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 0.30/0.65                & ( K3
% 0.30/0.65                  = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % eucl_rel_int.simps
% 0.30/0.65  thf(fact_9105_divide__int__unfold,axiom,
% 0.30/0.65      ! [L: int,K: int,N2: nat,M: nat] :
% 0.30/0.65        ( ( ( ( ( sgn_sgn_int @ L )
% 0.30/0.65              = zero_zero_int )
% 0.30/0.65            | ( ( sgn_sgn_int @ K )
% 0.30/0.65              = zero_zero_int )
% 0.30/0.65            | ( N2 = zero_zero_nat ) )
% 0.30/0.65         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65            = zero_zero_int ) )
% 0.30/0.65        & ( ~ ( ( ( sgn_sgn_int @ L )
% 0.30/0.65                = zero_zero_int )
% 0.30/0.65              | ( ( sgn_sgn_int @ K )
% 0.30/0.65                = zero_zero_int )
% 0.30/0.65              | ( N2 = zero_zero_nat ) )
% 0.30/0.65         => ( ( ( ( sgn_sgn_int @ K )
% 0.30/0.65                = ( sgn_sgn_int @ L ) )
% 0.30/0.65             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
% 0.30/0.65            & ( ( ( sgn_sgn_int @ K )
% 0.30/0.65               != ( sgn_sgn_int @ L ) )
% 0.30/0.65             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 0.30/0.65                = ( uminus_uminus_int
% 0.30/0.65                  @ ( semiri1314217659103216013at_int
% 0.30/0.65                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
% 0.30/0.65                      @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divide_int_unfold
% 0.30/0.65  thf(fact_9106_sinh__ln__real,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.65          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sinh_ln_real
% 0.30/0.65  thf(fact_9107_arcsin__lt__bounded,axiom,
% 0.30/0.65      ! [Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.65       => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.30/0.65         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 0.30/0.65            & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arcsin_lt_bounded
% 0.30/0.65  thf(fact_9108_sinh__real__less__iff,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 0.30/0.65        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sinh_real_less_iff
% 0.30/0.65  thf(fact_9109_sinh__real__neg__iff,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 0.30/0.65        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sinh_real_neg_iff
% 0.30/0.65  thf(fact_9110_sinh__real__pos__iff,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 0.30/0.65        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sinh_real_pos_iff
% 0.30/0.65  thf(fact_9111_sinh__less__cosh__real,axiom,
% 0.30/0.65      ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sinh_less_cosh_real
% 0.30/0.65  thf(fact_9112_cosh__real__pos,axiom,
% 0.30/0.65      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cosh_real_pos
% 0.30/0.65  thf(fact_9113_cosh__real__strict__mono,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.65         => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cosh_real_strict_mono
% 0.30/0.65  thf(fact_9114_cosh__real__nonneg__less__iff,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cosh_real_nonneg_less_iff
% 0.30/0.65  thf(fact_9115_cosh__real__nonpos__less__iff,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 0.30/0.65       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 0.30/0.65         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 0.30/0.65            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cosh_real_nonpos_less_iff
% 0.30/0.65  thf(fact_9116_sgn__real__def,axiom,
% 0.30/0.65      ( sgn_sgn_real
% 0.30/0.65      = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_real_def
% 0.30/0.65  thf(fact_9117_arcsin__less__arcsin,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.65         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.65           => ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arcsin_less_arcsin
% 0.30/0.65  thf(fact_9118_arcsin__less__mono,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.65       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.65         => ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 0.30/0.65            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % arcsin_less_mono
% 0.30/0.65  thf(fact_9119_sgn__power__injE,axiom,
% 0.30/0.65      ! [A: real,N2: nat,X3: real,B: real] :
% 0.30/0.65        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 0.30/0.65          = X3 )
% 0.30/0.65       => ( ( X3
% 0.30/0.65            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
% 0.30/0.65         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65           => ( A = B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_power_injE
% 0.30/0.65  thf(fact_9120_cos__arcsin__nonzero,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.65         => ( ( cos_real @ ( arcsin @ X3 ) )
% 0.30/0.65           != zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cos_arcsin_nonzero
% 0.30/0.65  thf(fact_9121_floor__real__def,axiom,
% 0.30/0.65      ( archim6058952711729229775r_real
% 0.30/0.65      = ( ^ [X: real] :
% 0.30/0.65            ( the_int
% 0.30/0.65            @ ^ [Z6: int] :
% 0.30/0.65                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z6 ) @ X )
% 0.30/0.65                & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % floor_real_def
% 0.30/0.65  thf(fact_9122_cosh__ln__real,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65       => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
% 0.30/0.65          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % cosh_ln_real
% 0.30/0.65  thf(fact_9123_floor__rat__def,axiom,
% 0.30/0.65      ( archim3151403230148437115or_rat
% 0.30/0.65      = ( ^ [X: rat] :
% 0.30/0.65            ( the_int
% 0.30/0.65            @ ^ [Z6: int] :
% 0.30/0.65                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X )
% 0.30/0.65                & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % floor_rat_def
% 0.30/0.65  thf(fact_9124_VEBT_Osize_I3_J,axiom,
% 0.30/0.65      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.65        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.65        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT.size(3)
% 0.30/0.65  thf(fact_9125_sum__power2,axiom,
% 0.30/0.65      ! [K: nat] :
% 0.30/0.65        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 0.30/0.65        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sum_power2
% 0.30/0.65  thf(fact_9126_finite__atLeastLessThan,axiom,
% 0.30/0.65      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_atLeastLessThan
% 0.30/0.65  thf(fact_9127_atLeastLessThan__singleton,axiom,
% 0.30/0.65      ! [M: nat] :
% 0.30/0.65        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 0.30/0.65        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeastLessThan_singleton
% 0.30/0.65  thf(fact_9128_sgn__rat__def,axiom,
% 0.30/0.65      ( sgn_sgn_rat
% 0.30/0.65      = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_rat_def
% 0.30/0.65  thf(fact_9129_abs__rat__def,axiom,
% 0.30/0.65      ( abs_abs_rat
% 0.30/0.65      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % abs_rat_def
% 0.30/0.65  thf(fact_9130_less__eq__rat__def,axiom,
% 0.30/0.65      ( ord_less_eq_rat
% 0.30/0.65      = ( ^ [X: rat,Y: rat] :
% 0.30/0.65            ( ( ord_less_rat @ X @ Y )
% 0.30/0.65            | ( X = Y ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_eq_rat_def
% 0.30/0.65  thf(fact_9131_obtain__pos__sum,axiom,
% 0.30/0.65      ! [R2: rat] :
% 0.30/0.65        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 0.30/0.65       => ~ ! [S2: rat] :
% 0.30/0.65              ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 0.30/0.65             => ! [T6: rat] :
% 0.30/0.65                  ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 0.30/0.65                 => ( R2
% 0.30/0.65                   != ( plus_plus_rat @ S2 @ T6 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % obtain_pos_sum
% 0.30/0.65  thf(fact_9132_ex__nat__less__eq,axiom,
% 0.30/0.65      ! [N2: nat,P: nat > $o] :
% 0.30/0.65        ( ( ? [M2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.65              & ( P @ M2 ) ) )
% 0.30/0.65        = ( ? [X: nat] :
% 0.30/0.65              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65              & ( P @ X ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % ex_nat_less_eq
% 0.30/0.65  thf(fact_9133_all__nat__less__eq,axiom,
% 0.30/0.65      ! [N2: nat,P: nat > $o] :
% 0.30/0.65        ( ( ! [M2: nat] :
% 0.30/0.65              ( ( ord_less_nat @ M2 @ N2 )
% 0.30/0.65             => ( P @ M2 ) ) )
% 0.30/0.65        = ( ! [X: nat] :
% 0.30/0.65              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65             => ( P @ X ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % all_nat_less_eq
% 0.30/0.65  thf(fact_9134_atLeastLessThanSuc__atLeastAtMost,axiom,
% 0.30/0.65      ! [L: nat,U: nat] :
% 0.30/0.65        ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 0.30/0.65        = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeastLessThanSuc_atLeastAtMost
% 0.30/0.65  thf(fact_9135_lessThan__atLeast0,axiom,
% 0.30/0.65      ( set_ord_lessThan_nat
% 0.30/0.65      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % lessThan_atLeast0
% 0.30/0.65  thf(fact_9136_atLeastLessThan0,axiom,
% 0.30/0.65      ! [M: nat] :
% 0.30/0.65        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 0.30/0.65        = bot_bot_set_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeastLessThan0
% 0.30/0.65  thf(fact_9137_atLeast0__lessThan__Suc,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.30/0.65        = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeast0_lessThan_Suc
% 0.30/0.65  thf(fact_9138_subset__eq__atLeast0__lessThan__finite,axiom,
% 0.30/0.65      ! [N7: set_nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65       => ( finite_finite_nat @ N7 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % subset_eq_atLeast0_lessThan_finite
% 0.30/0.65  thf(fact_9139_atLeastLessThanSuc,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 0.30/0.65            = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
% 0.30/0.65        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 0.30/0.65            = bot_bot_set_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeastLessThanSuc
% 0.30/0.65  thf(fact_9140_prod__Suc__Suc__fact,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 0.30/0.65        = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_Suc_Suc_fact
% 0.30/0.65  thf(fact_9141_prod__Suc__fact,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65        = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_Suc_fact
% 0.30/0.65  thf(fact_9142_atLeastLessThan__nat__numeral,axiom,
% 0.30/0.65      ! [M: nat,K: num] :
% 0.30/0.65        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.65         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 0.30/0.65        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 0.30/0.65         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65            = bot_bot_set_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeastLessThan_nat_numeral
% 0.30/0.65  thf(fact_9143_atLeast1__lessThan__eq__remove0,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.65        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % atLeast1_lessThan_eq_remove0
% 0.30/0.65  thf(fact_9144_Chebyshev__sum__upper__nat,axiom,
% 0.30/0.65      ! [N2: nat,A: nat > nat,B: nat > nat] :
% 0.30/0.65        ( ! [I3: nat,J2: nat] :
% 0.30/0.65            ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.65           => ( ( ord_less_nat @ J2 @ N2 )
% 0.30/0.65             => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 0.30/0.65       => ( ! [I3: nat,J2: nat] :
% 0.30/0.65              ( ( ord_less_eq_nat @ I3 @ J2 )
% 0.30/0.65             => ( ( ord_less_nat @ J2 @ N2 )
% 0.30/0.65               => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 0.30/0.65         => ( ord_less_eq_nat
% 0.30/0.65            @ ( times_times_nat @ N2
% 0.30/0.65              @ ( groups3542108847815614940at_nat
% 0.30/0.65                @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
% 0.30/0.65                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 0.30/0.65            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Chebyshev_sum_upper_nat
% 0.30/0.65  thf(fact_9145_VEBT_Osize__gen_I1_J,axiom,
% 0.30/0.65      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 0.30/0.65        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 0.30/0.65        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT.size_gen(1)
% 0.30/0.65  thf(fact_9146_finite__atLeastLessThan__int,axiom,
% 0.30/0.65      ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_atLeastLessThan_int
% 0.30/0.65  thf(fact_9147_finite__atLeastZeroLessThan__int,axiom,
% 0.30/0.65      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_atLeastZeroLessThan_int
% 0.30/0.65  thf(fact_9148_VEBT_Osize__gen_I2_J,axiom,
% 0.30/0.65      ! [X21: $o,X222: $o] :
% 0.30/0.65        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 0.30/0.65        = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT.size_gen(2)
% 0.30/0.65  thf(fact_9149_rat__inverse__code,axiom,
% 0.30/0.65      ! [P4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_inverse_code
% 0.30/0.65  thf(fact_9150_normalize__negative,axiom,
% 0.30/0.65      ! [Q4: int,P4: int] :
% 0.30/0.65        ( ( ord_less_int @ Q4 @ zero_zero_int )
% 0.30/0.65       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_negative
% 0.30/0.65  thf(fact_9151_Cauchy__iff2,axiom,
% 0.30/0.65      ( topolo4055970368930404560y_real
% 0.30/0.65      = ( ^ [X5: nat > real] :
% 0.30/0.65          ! [J3: nat] :
% 0.30/0.65          ? [M8: nat] :
% 0.30/0.65          ! [M2: nat] :
% 0.30/0.65            ( ( ord_less_eq_nat @ M8 @ M2 )
% 0.30/0.65           => ! [N: nat] :
% 0.30/0.65                ( ( ord_less_eq_nat @ M8 @ N )
% 0.30/0.65               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Cauchy_iff2
% 0.30/0.65  thf(fact_9152_vebt__mint_Opelims,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.65        ( ( ( vEBT_vebt_mint @ X3 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
% 0.30/0.65         => ( ! [A4: $o,B4: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.65               => ( ( ( A4
% 0.30/0.65                     => ( Y3
% 0.30/0.65                        = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.65                    & ( ~ A4
% 0.30/0.65                     => ( ( B4
% 0.30/0.65                         => ( Y3
% 0.30/0.65                            = ( some_nat @ one_one_nat ) ) )
% 0.30/0.65                        & ( ~ B4
% 0.30/0.65                         => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.65                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.65           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.65                 => ( ( Y3 = none_nat )
% 0.30/0.65                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.65             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.65                    ( ( X3
% 0.30/0.65                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.65                   => ( ( Y3
% 0.30/0.65                        = ( some_nat @ Mi2 ) )
% 0.30/0.65                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % vebt_mint.pelims
% 0.30/0.65  thf(fact_9153_quotient__of__number_I3_J,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 0.30/0.65        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_number(3)
% 0.30/0.65  thf(fact_9154_normalize__denom__zero,axiom,
% 0.30/0.65      ! [P4: int] :
% 0.30/0.65        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 0.30/0.65        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_denom_zero
% 0.30/0.65  thf(fact_9155_rat__one__code,axiom,
% 0.30/0.65      ( ( quotient_of @ one_one_rat )
% 0.30/0.65      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_one_code
% 0.30/0.65  thf(fact_9156_rat__zero__code,axiom,
% 0.30/0.65      ( ( quotient_of @ zero_zero_rat )
% 0.30/0.65      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_zero_code
% 0.30/0.65  thf(fact_9157_quotient__of__number_I5_J,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 0.30/0.65        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_number(5)
% 0.30/0.65  thf(fact_9158_quotient__of__number_I4_J,axiom,
% 0.30/0.65      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 0.30/0.65      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_number(4)
% 0.30/0.65  thf(fact_9159_rat__divide__code,axiom,
% 0.30/0.65      ! [P4: rat,Q4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int,C3: int] :
% 0.30/0.65              ( produc4245557441103728435nt_int
% 0.30/0.65              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) ) )
% 0.30/0.65              @ ( quotient_of @ Q4 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_divide_code
% 0.30/0.65  thf(fact_9160_rat__times__code,axiom,
% 0.30/0.65      ! [P4: rat,Q4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int,C3: int] :
% 0.30/0.65              ( produc4245557441103728435nt_int
% 0.30/0.65              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C3 @ D5 ) ) )
% 0.30/0.65              @ ( quotient_of @ Q4 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_times_code
% 0.30/0.65  thf(fact_9161_quotient__of__div,axiom,
% 0.30/0.65      ! [R2: rat,N2: int,D: int] :
% 0.30/0.65        ( ( ( quotient_of @ R2 )
% 0.30/0.65          = ( product_Pair_int_int @ N2 @ D ) )
% 0.30/0.65       => ( R2
% 0.30/0.65          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_div
% 0.30/0.65  thf(fact_9162_rat__minus__code,axiom,
% 0.30/0.65      ! [P4: rat,Q4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int,C3: int] :
% 0.30/0.65              ( produc4245557441103728435nt_int
% 0.30/0.65              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
% 0.30/0.65              @ ( quotient_of @ Q4 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_minus_code
% 0.30/0.65  thf(fact_9163_quotient__of__denom__pos,axiom,
% 0.30/0.65      ! [R2: rat,P4: int,Q4: int] :
% 0.30/0.65        ( ( ( quotient_of @ R2 )
% 0.30/0.65          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_denom_pos
% 0.30/0.65  thf(fact_9164_rat__plus__code,axiom,
% 0.30/0.65      ! [P4: rat,Q4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int,C3: int] :
% 0.30/0.65              ( produc4245557441103728435nt_int
% 0.30/0.65              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
% 0.30/0.65              @ ( quotient_of @ Q4 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_plus_code
% 0.30/0.65  thf(fact_9165_normalize__denom__pos,axiom,
% 0.30/0.65      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 0.30/0.65        ( ( ( normalize @ R2 )
% 0.30/0.65          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_denom_pos
% 0.30/0.65  thf(fact_9166_normalize__crossproduct,axiom,
% 0.30/0.65      ! [Q4: int,S: int,P4: int,R2: int] :
% 0.30/0.65        ( ( Q4 != zero_zero_int )
% 0.30/0.65       => ( ( S != zero_zero_int )
% 0.30/0.65         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 0.30/0.65           => ( ( times_times_int @ P4 @ S )
% 0.30/0.65              = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_crossproduct
% 0.30/0.65  thf(fact_9167_rat__uminus__code,axiom,
% 0.30/0.65      ! [P4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_uminus_code
% 0.30/0.65  thf(fact_9168_rat__less__code,axiom,
% 0.30/0.65      ( ord_less_rat
% 0.30/0.65      = ( ^ [P6: rat,Q5: rat] :
% 0.30/0.65            ( produc4947309494688390418_int_o
% 0.30/0.65            @ ^ [A3: int,C3: int] :
% 0.30/0.65                ( produc4947309494688390418_int_o
% 0.30/0.65                @ ^ [B3: int,D5: int] : ( ord_less_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) )
% 0.30/0.65                @ ( quotient_of @ Q5 ) )
% 0.30/0.65            @ ( quotient_of @ P6 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_less_code
% 0.30/0.65  thf(fact_9169_rat__abs__code,axiom,
% 0.30/0.65      ! [P4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 0.30/0.65        = ( produc4245557441103728435nt_int
% 0.30/0.65          @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
% 0.30/0.65          @ ( quotient_of @ P4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_abs_code
% 0.30/0.65  thf(fact_9170_vebt__maxt_Opelims,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 0.30/0.65        ( ( ( vEBT_vebt_maxt @ X3 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
% 0.30/0.65         => ( ! [A4: $o,B4: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 0.30/0.65               => ( ( ( B4
% 0.30/0.65                     => ( Y3
% 0.30/0.65                        = ( some_nat @ one_one_nat ) ) )
% 0.30/0.65                    & ( ~ B4
% 0.30/0.65                     => ( ( A4
% 0.30/0.65                         => ( Y3
% 0.30/0.65                            = ( some_nat @ zero_zero_nat ) ) )
% 0.30/0.65                        & ( ~ A4
% 0.30/0.65                         => ( Y3 = none_nat ) ) ) ) )
% 0.30/0.65                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 0.30/0.65           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 0.30/0.65                 => ( ( Y3 = none_nat )
% 0.30/0.65                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 0.30/0.65             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 0.30/0.65                    ( ( X3
% 0.30/0.65                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 0.30/0.65                   => ( ( Y3
% 0.30/0.65                        = ( some_nat @ Ma2 ) )
% 0.30/0.65                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % vebt_maxt.pelims
% 0.30/0.65  thf(fact_9171_quotient__of__int,axiom,
% 0.30/0.65      ! [A: int] :
% 0.30/0.65        ( ( quotient_of @ ( of_int @ A ) )
% 0.30/0.65        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_int
% 0.30/0.65  thf(fact_9172_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Y3: $o] :
% 0.30/0.65        ( ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.65         => ( ( ( X3
% 0.30/0.65                = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.65             => ( Y3
% 0.30/0.65               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 0.30/0.65           => ( ! [Uv2: $o] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.65                 => ( ~ Y3
% 0.30/0.65                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 0.30/0.65             => ( ! [Uu2: $o] :
% 0.30/0.65                    ( ( X3
% 0.30/0.65                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.65                   => ( ~ Y3
% 0.30/0.65                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 0.30/0.65               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.65                      ( ( X3
% 0.30/0.65                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.65                     => ( Y3
% 0.30/0.65                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 0.30/0.65                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.65                        ( ( X3
% 0.30/0.65                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.30/0.65                       => ( ~ Y3
% 0.30/0.65                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.minNull.pelims(1)
% 0.30/0.65  thf(fact_9173_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT] :
% 0.30/0.65        ( ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.65       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.65         => ( ( ( X3
% 0.30/0.65                = ( vEBT_Leaf @ $false @ $false ) )
% 0.30/0.65             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 0.30/0.65           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 0.30/0.65                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.minNull.pelims(2)
% 0.30/0.65  thf(fact_9174_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT] :
% 0.30/0.65        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 0.30/0.65       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 0.30/0.65         => ( ! [Uv2: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 0.30/0.65               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 0.30/0.65           => ( ! [Uu2: $o] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 0.30/0.65                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 0.30/0.65             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 0.30/0.65                    ( ( X3
% 0.30/0.65                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 0.30/0.65                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.minNull.pelims(3)
% 0.30/0.65  thf(fact_9175_Frct__code__post_I5_J,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 0.30/0.65        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(5)
% 0.30/0.65  thf(fact_9176_divmod__step__integer__def,axiom,
% 0.30/0.65      ( unique4921790084139445826nteger
% 0.30/0.65      = ( ^ [L2: num] :
% 0.30/0.65            ( produc6916734918728496179nteger
% 0.30/0.65            @ ^ [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 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_step_integer_def
% 0.30/0.65  thf(fact_9177_divmod__integer_H__def,axiom,
% 0.30/0.65      ( unique3479559517661332726nteger
% 0.30/0.65      = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_integer'_def
% 0.30/0.65  thf(fact_9178_sgn__integer__code,axiom,
% 0.30/0.65      ( sgn_sgn_Code_integer
% 0.30/0.65      = ( ^ [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 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_integer_code
% 0.30/0.65  thf(fact_9179_exhaustive__integer_H_Ocases,axiom,
% 0.30/0.65      ! [X3: produc8763457246119570046nteger] :
% 0.30/0.65        ~ ! [F2: code_integer > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
% 0.30/0.65            ( X3
% 0.30/0.65           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % exhaustive_integer'.cases
% 0.30/0.65  thf(fact_9180_full__exhaustive__integer_H_Ocases,axiom,
% 0.30/0.65      ! [X3: produc1908205239877642774nteger] :
% 0.30/0.65        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
% 0.30/0.65            ( X3
% 0.30/0.65           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % full_exhaustive_integer'.cases
% 0.30/0.65  thf(fact_9181_zero__natural_Orsp,axiom,
% 0.30/0.65      zero_zero_nat = zero_zero_nat ).
% 0.30/0.65  
% 0.30/0.65  % zero_natural.rsp
% 0.30/0.65  thf(fact_9182_Frct__code__post_I1_J,axiom,
% 0.30/0.65      ! [A: int] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 0.30/0.65        = zero_zero_rat ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(1)
% 0.30/0.65  thf(fact_9183_Frct__code__post_I2_J,axiom,
% 0.30/0.65      ! [A: int] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 0.30/0.65        = zero_zero_rat ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(2)
% 0.30/0.65  thf(fact_9184_Frct__code__post_I8_J,axiom,
% 0.30/0.65      ! [A: int,B: int] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
% 0.30/0.65        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(8)
% 0.30/0.65  thf(fact_9185_Frct__code__post_I7_J,axiom,
% 0.30/0.65      ! [A: int,B: int] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.30/0.65        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(7)
% 0.30/0.65  thf(fact_9186_Frct__code__post_I3_J,axiom,
% 0.30/0.65      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 0.30/0.65      = one_one_rat ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(3)
% 0.30/0.65  thf(fact_9187_Frct__code__post_I4_J,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 0.30/0.65        = ( numeral_numeral_rat @ K ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(4)
% 0.30/0.65  thf(fact_9188_Frct__code__post_I6_J,axiom,
% 0.30/0.65      ! [K: num,L: num] :
% 0.30/0.65        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 0.30/0.65        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Frct_code_post(6)
% 0.30/0.65  thf(fact_9189_integer__of__int__code,axiom,
% 0.30/0.65      ( code_integer_of_int
% 0.30/0.65      = ( ^ [K3: int] :
% 0.30/0.65            ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 0.30/0.65            @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 0.30/0.65              @ ( if_Code_integer
% 0.30/0.65                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 0.30/0.65                  = zero_zero_int )
% 0.30/0.65                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % integer_of_int_code
% 0.30/0.65  thf(fact_9190_pair__leqI2,axiom,
% 0.30/0.65      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ A @ B )
% 0.30/0.65       => ( ( ord_less_eq_nat @ S @ T )
% 0.30/0.65         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pair_leqI2
% 0.30/0.65  thf(fact_9191_card__Collect__less__nat,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( finite_card_nat
% 0.30/0.65          @ ( collect_nat
% 0.30/0.65            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
% 0.30/0.65        = N2 ) ).
% 0.30/0.65  
% 0.30/0.65  % card_Collect_less_nat
% 0.30/0.65  thf(fact_9192_card__atMost,axiom,
% 0.30/0.65      ! [U: nat] :
% 0.30/0.65        ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 0.30/0.65        = ( suc @ U ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_atMost
% 0.30/0.65  thf(fact_9193_card__Collect__le__nat,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( finite_card_nat
% 0.30/0.65          @ ( collect_nat
% 0.30/0.65            @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N2 ) ) )
% 0.30/0.65        = ( suc @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_Collect_le_nat
% 0.30/0.65  thf(fact_9194_card__atLeastAtMost,axiom,
% 0.30/0.65      ! [L: nat,U: nat] :
% 0.30/0.65        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 0.30/0.65        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_atLeastAtMost
% 0.30/0.65  thf(fact_9195_abs__integer__code,axiom,
% 0.30/0.65      ( abs_abs_Code_integer
% 0.30/0.65      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % abs_integer_code
% 0.30/0.65  thf(fact_9196_less__integer__code_I1_J,axiom,
% 0.30/0.65      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 0.30/0.65  
% 0.30/0.65  % less_integer_code(1)
% 0.30/0.65  thf(fact_9197_less__integer_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: int,X3: int] :
% 0.30/0.65        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 0.30/0.65        = ( ord_less_int @ Xa2 @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_integer.abs_eq
% 0.30/0.65  thf(fact_9198_card__less,axiom,
% 0.30/0.65      ! [M7: set_nat,I: nat] :
% 0.30/0.65        ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.65       => ( ( finite_card_nat
% 0.30/0.65            @ ( collect_nat
% 0.30/0.65              @ ^ [K3: nat] :
% 0.30/0.65                  ( ( member_nat @ K3 @ M7 )
% 0.30/0.65                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 0.30/0.65         != zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_less
% 0.30/0.65  thf(fact_9199_card__less__Suc,axiom,
% 0.30/0.65      ! [M7: set_nat,I: nat] :
% 0.30/0.65        ( ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.65       => ( ( suc
% 0.30/0.65            @ ( finite_card_nat
% 0.30/0.65              @ ( collect_nat
% 0.30/0.65                @ ^ [K3: nat] :
% 0.30/0.65                    ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.30/0.65                    & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 0.30/0.65          = ( finite_card_nat
% 0.30/0.65            @ ( collect_nat
% 0.30/0.65              @ ^ [K3: nat] :
% 0.30/0.65                  ( ( member_nat @ K3 @ M7 )
% 0.30/0.65                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_less_Suc
% 0.30/0.65  thf(fact_9200_card__less__Suc2,axiom,
% 0.30/0.65      ! [M7: set_nat,I: nat] :
% 0.30/0.65        ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.65       => ( ( finite_card_nat
% 0.30/0.65            @ ( collect_nat
% 0.30/0.65              @ ^ [K3: nat] :
% 0.30/0.65                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 0.30/0.65                  & ( ord_less_nat @ K3 @ I ) ) ) )
% 0.30/0.65          = ( finite_card_nat
% 0.30/0.65            @ ( collect_nat
% 0.30/0.65              @ ^ [K3: nat] :
% 0.30/0.65                  ( ( member_nat @ K3 @ M7 )
% 0.30/0.65                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_less_Suc2
% 0.30/0.65  thf(fact_9201_subset__card__intvl__is__intvl,axiom,
% 0.30/0.65      ! [A2: set_nat,K: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 0.30/0.65       => ( A2
% 0.30/0.65          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % subset_card_intvl_is_intvl
% 0.30/0.65  thf(fact_9202_subset__eq__atLeast0__lessThan__card,axiom,
% 0.30/0.65      ! [N7: set_nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 0.30/0.65       => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % subset_eq_atLeast0_lessThan_card
% 0.30/0.65  thf(fact_9203_card__sum__le__nat__sum,axiom,
% 0.30/0.65      ! [S3: set_nat] :
% 0.30/0.65        ( ord_less_eq_nat
% 0.30/0.65        @ ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [X: nat] : X
% 0.30/0.65          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
% 0.30/0.65        @ ( groups3542108847815614940at_nat
% 0.30/0.65          @ ^ [X: nat] : X
% 0.30/0.65          @ S3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_sum_le_nat_sum
% 0.30/0.65  thf(fact_9204_card__nth__roots,axiom,
% 0.30/0.65      ! [C: complex,N2: nat] :
% 0.30/0.65        ( ( C != zero_zero_complex )
% 0.30/0.65       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65         => ( ( finite_card_complex
% 0.30/0.65              @ ( collect_complex
% 0.30/0.65                @ ^ [Z6: complex] :
% 0.30/0.65                    ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.65                    = C ) ) )
% 0.30/0.65            = N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_nth_roots
% 0.30/0.65  thf(fact_9205_card__roots__unity__eq,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( finite_card_complex
% 0.30/0.65            @ ( collect_complex
% 0.30/0.65              @ ^ [Z6: complex] :
% 0.30/0.65                  ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.65                  = one_one_complex ) ) )
% 0.30/0.65          = N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card_roots_unity_eq
% 0.30/0.65  thf(fact_9206_pair__leqI1,axiom,
% 0.30/0.65      ! [A: nat,B: nat,S: nat,T: nat] :
% 0.30/0.65        ( ( ord_less_nat @ A @ B )
% 0.30/0.65       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pair_leqI1
% 0.30/0.65  thf(fact_9207_int__of__integer__code,axiom,
% 0.30/0.65      ( code_int_of_integer
% 0.30/0.65      = ( ^ [K3: code_integer] :
% 0.30/0.65            ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 0.30/0.65            @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 0.30/0.65              @ ( produc1553301316500091796er_int
% 0.30/0.65                @ ^ [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 ) )
% 0.30/0.65                @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_of_integer_code
% 0.30/0.65  thf(fact_9208_less__integer_Orep__eq,axiom,
% 0.30/0.65      ( ord_le6747313008572928689nteger
% 0.30/0.65      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_integer.rep_eq
% 0.30/0.65  thf(fact_9209_integer__less__iff,axiom,
% 0.30/0.65      ( ord_le6747313008572928689nteger
% 0.30/0.65      = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % integer_less_iff
% 0.30/0.65  thf(fact_9210_divmod__integer__def,axiom,
% 0.30/0.65      ( code_divmod_integer
% 0.30/0.65      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_integer_def
% 0.30/0.65  thf(fact_9211_nat__of__integer__code,axiom,
% 0.30/0.65      ( code_nat_of_integer
% 0.30/0.65      = ( ^ [K3: code_integer] :
% 0.30/0.65            ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 0.30/0.65            @ ( produc1555791787009142072er_nat
% 0.30/0.65              @ ^ [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 ) )
% 0.30/0.65              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_of_integer_code
% 0.30/0.65  thf(fact_9212_bit__cut__integer__def,axiom,
% 0.30/0.65      ( code_bit_cut_integer
% 0.30/0.65      = ( ^ [K3: code_integer] :
% 0.30/0.65            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 0.30/0.65            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bit_cut_integer_def
% 0.30/0.65  thf(fact_9213_divmod__abs__def,axiom,
% 0.30/0.65      ( code_divmod_abs
% 0.30/0.65      = ( ^ [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 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_abs_def
% 0.30/0.65  thf(fact_9214_nat__of__integer__non__positive,axiom,
% 0.30/0.65      ! [K: code_integer] :
% 0.30/0.65        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 0.30/0.65       => ( ( code_nat_of_integer @ K )
% 0.30/0.65          = zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_of_integer_non_positive
% 0.30/0.65  thf(fact_9215_nat__of__integer__code__post_I1_J,axiom,
% 0.30/0.65      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 0.30/0.65      = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_of_integer_code_post(1)
% 0.30/0.65  thf(fact_9216_divmod__abs__code_I6_J,axiom,
% 0.30/0.65      ! [J: code_integer] :
% 0.30/0.65        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 0.30/0.65        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_abs_code(6)
% 0.30/0.65  thf(fact_9217_divmod__abs__code_I5_J,axiom,
% 0.30/0.65      ! [J: code_integer] :
% 0.30/0.65        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 0.30/0.65        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_abs_code(5)
% 0.30/0.65  thf(fact_9218_bit__cut__integer__code,axiom,
% 0.30/0.65      ( code_bit_cut_integer
% 0.30/0.65      = ( ^ [K3: code_integer] :
% 0.30/0.65            ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 0.30/0.65            @ ( produc9125791028180074456eger_o
% 0.30/0.65              @ ^ [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 ) )
% 0.30/0.65              @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bit_cut_integer_code
% 0.30/0.65  thf(fact_9219_divmod__integer__code,axiom,
% 0.30/0.65      ( code_divmod_integer
% 0.30/0.65      = ( ^ [K3: code_integer,L2: code_integer] :
% 0.30/0.65            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.65            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 0.30/0.65              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.30/0.65                @ ( produc6916734918728496179nteger
% 0.30/0.65                  @ ^ [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 ) ) )
% 0.30/0.65                  @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 0.30/0.65              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.30/0.65                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 0.30/0.65                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 0.30/0.65                    @ ( produc6916734918728496179nteger
% 0.30/0.65                      @ ^ [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 ) ) )
% 0.30/0.65                      @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_integer_code
% 0.30/0.65  thf(fact_9220_nat_Odisc__eq__case_I2_J,axiom,
% 0.30/0.65      ! [Nat: nat] :
% 0.30/0.65        ( ( Nat != zero_zero_nat )
% 0.30/0.65        = ( case_nat_o @ $false
% 0.30/0.65          @ ^ [Uu3: nat] : $true
% 0.30/0.65          @ Nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat.disc_eq_case(2)
% 0.30/0.65  thf(fact_9221_nat_Odisc__eq__case_I1_J,axiom,
% 0.30/0.65      ! [Nat: nat] :
% 0.30/0.65        ( ( Nat = zero_zero_nat )
% 0.30/0.65        = ( case_nat_o @ $true
% 0.30/0.65          @ ^ [Uu3: nat] : $false
% 0.30/0.65          @ Nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat.disc_eq_case(1)
% 0.30/0.65  thf(fact_9222_less__eq__nat_Osimps_I2_J,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 0.30/0.65        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_eq_nat.simps(2)
% 0.30/0.65  thf(fact_9223_diff__Suc,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 0.30/0.65        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.65          @ ^ [K3: nat] : K3
% 0.30/0.65          @ ( minus_minus_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % diff_Suc
% 0.30/0.65  thf(fact_9224_binomial__def,axiom,
% 0.30/0.65      ( binomial
% 0.30/0.65      = ( ^ [N: nat,K3: nat] :
% 0.30/0.65            ( finite_card_set_nat
% 0.30/0.65            @ ( collect_set_nat
% 0.30/0.65              @ ^ [K7: set_nat] :
% 0.30/0.65                  ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 0.30/0.65                  & ( ( finite_card_nat @ K7 )
% 0.30/0.65                    = K3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % binomial_def
% 0.30/0.65  thf(fact_9225_pred__def,axiom,
% 0.30/0.65      ( pred
% 0.30/0.65      = ( case_nat_nat @ zero_zero_nat
% 0.30/0.65        @ ^ [X24: nat] : X24 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pred_def
% 0.30/0.65  thf(fact_9226_wmin__insertI,axiom,
% 0.30/0.65      ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( member8440522571783428010at_nat @ X3 @ XS )
% 0.30/0.65       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
% 0.30/0.65         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 0.30/0.65           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % wmin_insertI
% 0.30/0.65  thf(fact_9227_wmax__insertI,axiom,
% 0.30/0.65      ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( member8440522571783428010at_nat @ Y3 @ YS )
% 0.30/0.65       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
% 0.30/0.65         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 0.30/0.65           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % wmax_insertI
% 0.30/0.65  thf(fact_9228_bezw__0,axiom,
% 0.30/0.65      ! [X3: nat] :
% 0.30/0.65        ( ( bezw @ X3 @ zero_zero_nat )
% 0.30/0.65        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw_0
% 0.30/0.65  thf(fact_9229_push__bit__negative__int__iff,axiom,
% 0.30/0.65      ! [N2: nat,K: int] :
% 0.30/0.65        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.65        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % push_bit_negative_int_iff
% 0.30/0.65  thf(fact_9230_push__bit__of__Suc__0,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % push_bit_of_Suc_0
% 0.30/0.65  thf(fact_9231_wmin__emptyI,axiom,
% 0.30/0.65      ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 0.30/0.65  
% 0.30/0.65  % wmin_emptyI
% 0.30/0.65  thf(fact_9232_wmax__emptyI,axiom,
% 0.30/0.65      ! [X7: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( finite6177210948735845034at_nat @ X7 )
% 0.30/0.65       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).
% 0.30/0.65  
% 0.30/0.65  % wmax_emptyI
% 0.30/0.65  thf(fact_9233_bit__push__bit__iff__int,axiom,
% 0.30/0.65      ! [M: nat,K: int,N2: nat] :
% 0.30/0.65        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
% 0.30/0.65        = ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bit_push_bit_iff_int
% 0.30/0.65  thf(fact_9234_bit__push__bit__iff__nat,axiom,
% 0.30/0.65      ! [M: nat,Q4: nat,N2: nat] :
% 0.30/0.65        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N2 )
% 0.30/0.65        = ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65          & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bit_push_bit_iff_nat
% 0.30/0.65  thf(fact_9235_smax__insertI,axiom,
% 0.30/0.65      ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
% 0.30/0.65       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
% 0.30/0.65         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
% 0.30/0.65           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % smax_insertI
% 0.30/0.65  thf(fact_9236_smin__insertI,axiom,
% 0.30/0.65      ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( member8440522571783428010at_nat @ X3 @ XS )
% 0.30/0.65       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
% 0.30/0.65         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 0.30/0.65           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % smin_insertI
% 0.30/0.65  thf(fact_9237_prod__decode__aux_Osimps,axiom,
% 0.30/0.65      ( nat_prod_decode_aux
% 0.30/0.65      = ( ^ [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 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_decode_aux.simps
% 0.30/0.65  thf(fact_9238_prod__decode__aux_Oelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.65        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 0.30/0.65          & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_decode_aux.elims
% 0.30/0.65  thf(fact_9239_smax__emptyI,axiom,
% 0.30/0.65      ! [Y7: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( finite6177210948735845034at_nat @ Y7 )
% 0.30/0.65       => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 0.30/0.65         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % smax_emptyI
% 0.30/0.65  thf(fact_9240_smin__emptyI,axiom,
% 0.30/0.65      ! [X7: set_Pr1261947904930325089at_nat] :
% 0.30/0.65        ( ( X7 != bot_bo2099793752762293965at_nat )
% 0.30/0.65       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 0.30/0.65  
% 0.30/0.65  % smin_emptyI
% 0.30/0.65  thf(fact_9241_Suc__0__div__numeral,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_0_div_numeral
% 0.30/0.65  thf(fact_9242_Suc__0__mod__numeral,axiom,
% 0.30/0.65      ! [K: num] :
% 0.30/0.65        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_0_mod_numeral
% 0.30/0.65  thf(fact_9243_finite__enumerate,axiom,
% 0.30/0.65      ! [S3: set_nat] :
% 0.30/0.65        ( ( finite_finite_nat @ S3 )
% 0.30/0.65       => ? [R4: nat > nat] :
% 0.30/0.65            ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
% 0.30/0.65            & ! [N6: nat] :
% 0.30/0.65                ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S3 ) )
% 0.30/0.65               => ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % finite_enumerate
% 0.30/0.65  thf(fact_9244_Sup__nat__empty,axiom,
% 0.30/0.65      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 0.30/0.65      = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % Sup_nat_empty
% 0.30/0.65  thf(fact_9245_Inf__nat__def1,axiom,
% 0.30/0.65      ! [K5: set_nat] :
% 0.30/0.65        ( ( K5 != bot_bot_set_nat )
% 0.30/0.65       => ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Inf_nat_def1
% 0.30/0.65  thf(fact_9246_quotient__of__denom__pos_H,axiom,
% 0.30/0.65      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_denom_pos'
% 0.30/0.65  thf(fact_9247_bezw__non__0,axiom,
% 0.30/0.65      ! [Y3: nat,X3: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 0.30/0.65       => ( ( bezw @ X3 @ Y3 )
% 0.30/0.65          = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw_non_0
% 0.30/0.65  thf(fact_9248_bezw_Oelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.65        ( ( ( bezw @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.65          & ( ( Xa2 != zero_zero_nat )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw.elims
% 0.30/0.65  thf(fact_9249_bezw_Osimps,axiom,
% 0.30/0.65      ( bezw
% 0.30/0.65      = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( 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 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw.simps
% 0.30/0.65  thf(fact_9250_rat__sgn__code,axiom,
% 0.30/0.65      ! [P4: rat] :
% 0.30/0.65        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 0.30/0.65        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_sgn_code
% 0.30/0.65  thf(fact_9251_bezw_Opelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 0.30/0.65        ( ( ( bezw @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.65                 => ( Y3
% 0.30/0.65                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 0.30/0.65                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.65                 => ( Y3
% 0.30/0.65                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
% 0.30/0.65             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw.pelims
% 0.30/0.65  thf(fact_9252_prod__decode__aux_Opelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 0.30/0.65        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.65                 => ( Y3
% 0.30/0.65                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 0.30/0.65                & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 0.30/0.65                 => ( Y3
% 0.30/0.65                    = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
% 0.30/0.65             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_decode_aux.pelims
% 0.30/0.65  thf(fact_9253_normalize__def,axiom,
% 0.30/0.65      ( normalize
% 0.30/0.65      = ( ^ [P6: product_prod_int_int] :
% 0.30/0.65            ( 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 ) ) ) )
% 0.30/0.65            @ ( if_Pro3027730157355071871nt_int
% 0.30/0.65              @ ( ( product_snd_int_int @ P6 )
% 0.30/0.65                = zero_zero_int )
% 0.30/0.65              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 0.30/0.65              @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_def
% 0.30/0.65  thf(fact_9254_gcd__1__int,axiom,
% 0.30/0.65      ! [M: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ M @ one_one_int )
% 0.30/0.65        = one_one_int ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_1_int
% 0.30/0.65  thf(fact_9255_gcd__neg2__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_neg2_int
% 0.30/0.65  thf(fact_9256_gcd__neg1__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_neg1_int
% 0.30/0.65  thf(fact_9257_abs__gcd__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( abs_abs_int @ ( gcd_gcd_int @ X3 @ Y3 ) )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % abs_gcd_int
% 0.30/0.65  thf(fact_9258_gcd__abs1__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ ( abs_abs_int @ X3 ) @ Y3 )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_abs1_int
% 0.30/0.65  thf(fact_9259_gcd__abs2__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ X3 @ ( abs_abs_int @ Y3 ) )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_abs2_int
% 0.30/0.65  thf(fact_9260_gcd__pos__int,axiom,
% 0.30/0.65      ! [M: int,N2: int] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
% 0.30/0.65        = ( ( M != zero_zero_int )
% 0.30/0.65          | ( N2 != zero_zero_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_pos_int
% 0.30/0.65  thf(fact_9261_gcd__neg__numeral__1__int,axiom,
% 0.30/0.65      ! [N2: num,X3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X3 )
% 0.30/0.65        = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_neg_numeral_1_int
% 0.30/0.65  thf(fact_9262_gcd__neg__numeral__2__int,axiom,
% 0.30/0.65      ! [X3: int,N2: num] :
% 0.30/0.65        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_neg_numeral_2_int
% 0.30/0.65  thf(fact_9263_gcd__0__int,axiom,
% 0.30/0.65      ! [X3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ X3 @ zero_zero_int )
% 0.30/0.65        = ( abs_abs_int @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_0_int
% 0.30/0.65  thf(fact_9264_gcd__0__left__int,axiom,
% 0.30/0.65      ! [X3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ zero_zero_int @ X3 )
% 0.30/0.65        = ( abs_abs_int @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_0_left_int
% 0.30/0.65  thf(fact_9265_gcd__proj1__if__dvd__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ X3 @ Y3 )
% 0.30/0.65       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 0.30/0.65          = ( abs_abs_int @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_proj1_if_dvd_int
% 0.30/0.65  thf(fact_9266_gcd__proj2__if__dvd__int,axiom,
% 0.30/0.65      ! [Y3: int,X3: int] :
% 0.30/0.65        ( ( dvd_dvd_int @ Y3 @ X3 )
% 0.30/0.65       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 0.30/0.65          = ( abs_abs_int @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_proj2_if_dvd_int
% 0.30/0.65  thf(fact_9267_gcd__idem__int,axiom,
% 0.30/0.65      ! [X3: int] :
% 0.30/0.65        ( ( gcd_gcd_int @ X3 @ X3 )
% 0.30/0.65        = ( abs_abs_int @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_idem_int
% 0.30/0.65  thf(fact_9268_gcd__red__int,axiom,
% 0.30/0.65      ( gcd_gcd_int
% 0.30/0.65      = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_red_int
% 0.30/0.65  thf(fact_9269_gcd__ge__0__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_ge_0_int
% 0.30/0.65  thf(fact_9270_bezout__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int] :
% 0.30/0.65      ? [U3: int,V2: int] :
% 0.30/0.65        ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
% 0.30/0.65        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezout_int
% 0.30/0.65  thf(fact_9271_gcd__mult__distrib__int,axiom,
% 0.30/0.65      ! [K: int,M: int,N2: int] :
% 0.30/0.65        ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
% 0.30/0.65        = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_mult_distrib_int
% 0.30/0.65  thf(fact_9272_gcd__le2__int,axiom,
% 0.30/0.65      ! [B: int,A: int] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.65       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_le2_int
% 0.30/0.65  thf(fact_9273_gcd__le1__int,axiom,
% 0.30/0.65      ! [A: int,B: int] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ A )
% 0.30/0.65       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_le1_int
% 0.30/0.65  thf(fact_9274_gcd__cases__int,axiom,
% 0.30/0.65      ! [X3: int,Y3: int,P: int > $o] :
% 0.30/0.65        ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.65         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
% 0.30/0.65       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.65           => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.65             => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.65         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.30/0.65             => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 0.30/0.65               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
% 0.30/0.65           => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 0.30/0.65               => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 0.30/0.65                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 0.30/0.65             => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_cases_int
% 0.30/0.65  thf(fact_9275_gcd__unique__int,axiom,
% 0.30/0.65      ! [D: int,A: int,B: int] :
% 0.30/0.65        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 0.30/0.65          & ( dvd_dvd_int @ D @ A )
% 0.30/0.65          & ( dvd_dvd_int @ D @ B )
% 0.30/0.65          & ! [E3: int] :
% 0.30/0.65              ( ( ( dvd_dvd_int @ E3 @ A )
% 0.30/0.65                & ( dvd_dvd_int @ E3 @ B ) )
% 0.30/0.65             => ( dvd_dvd_int @ E3 @ D ) ) )
% 0.30/0.65        = ( D
% 0.30/0.65          = ( gcd_gcd_int @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_unique_int
% 0.30/0.65  thf(fact_9276_gcd__non__0__int,axiom,
% 0.30/0.65      ! [Y3: int,X3: int] :
% 0.30/0.65        ( ( ord_less_int @ zero_zero_int @ Y3 )
% 0.30/0.65       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 0.30/0.65          = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_non_0_int
% 0.30/0.65  thf(fact_9277_gcd__code__int,axiom,
% 0.30/0.65      ( gcd_gcd_int
% 0.30/0.65      = ( ^ [K3: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_code_int
% 0.30/0.65  thf(fact_9278_gcd__nat_Oeq__neutr__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65          = zero_zero_nat )
% 0.30/0.65        = ( ( A = zero_zero_nat )
% 0.30/0.65          & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.eq_neutr_iff
% 0.30/0.65  thf(fact_9279_gcd__nat_Oleft__neutral,axiom,
% 0.30/0.65      ! [A: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 0.30/0.65        = A ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.left_neutral
% 0.30/0.65  thf(fact_9280_gcd__nat_Oneutr__eq__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( zero_zero_nat
% 0.30/0.65          = ( gcd_gcd_nat @ A @ B ) )
% 0.30/0.65        = ( ( A = zero_zero_nat )
% 0.30/0.65          & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.neutr_eq_iff
% 0.30/0.65  thf(fact_9281_gcd__nat_Oright__neutral,axiom,
% 0.30/0.65      ! [A: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 0.30/0.65        = A ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.right_neutral
% 0.30/0.65  thf(fact_9282_gcd__0__nat,axiom,
% 0.30/0.65      ! [X3: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ X3 @ zero_zero_nat )
% 0.30/0.65        = X3 ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_0_nat
% 0.30/0.65  thf(fact_9283_gcd__0__left__nat,axiom,
% 0.30/0.65      ! [X3: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ zero_zero_nat @ X3 )
% 0.30/0.65        = X3 ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_0_left_nat
% 0.30/0.65  thf(fact_9284_gcd__1__nat,axiom,
% 0.30/0.65      ! [M: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ M @ one_one_nat )
% 0.30/0.65        = one_one_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_1_nat
% 0.30/0.65  thf(fact_9285_gcd__proj2__if__dvd__nat,axiom,
% 0.30/0.65      ! [Y3: nat,X3: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ Y3 @ X3 )
% 0.30/0.65       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 0.30/0.65          = Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_proj2_if_dvd_nat
% 0.30/0.65  thf(fact_9286_gcd__proj1__if__dvd__nat,axiom,
% 0.30/0.65      ! [X3: nat,Y3: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 0.30/0.65       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 0.30/0.65          = X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_proj1_if_dvd_nat
% 0.30/0.65  thf(fact_9287_gcd__nat_Obounded__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat,C: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.30/0.65        = ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65          & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.bounded_iff
% 0.30/0.65  thf(fact_9288_gcd__nat_Oabsorb2,axiom,
% 0.30/0.65      ! [B: nat,A: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.65       => ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65          = B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb2
% 0.30/0.65  thf(fact_9289_gcd__nat_Oabsorb1,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65       => ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65          = A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb1
% 0.30/0.65  thf(fact_9290_gcd__Suc__0,axiom,
% 0.30/0.65      ! [M: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( suc @ zero_zero_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_Suc_0
% 0.30/0.65  thf(fact_9291_gcd__pos__nat,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
% 0.30/0.65        = ( ( M != zero_zero_nat )
% 0.30/0.65          | ( N2 != zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_pos_nat
% 0.30/0.65  thf(fact_9292_or__negative__int__iff,axiom,
% 0.30/0.65      ! [K: int,L: int] :
% 0.30/0.65        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 0.30/0.65        = ( ( ord_less_int @ K @ zero_zero_int )
% 0.30/0.65          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_negative_int_iff
% 0.30/0.65  thf(fact_9293_gcd__int__int__eq,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 0.30/0.65        = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_int_int_eq
% 0.30/0.65  thf(fact_9294_or__nat__numerals_I4_J,axiom,
% 0.30/0.65      ! [X3: num] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_numerals(4)
% 0.30/0.65  thf(fact_9295_or__nat__numerals_I2_J,axiom,
% 0.30/0.65      ! [Y3: num] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 0.30/0.65        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_numerals(2)
% 0.30/0.65  thf(fact_9296_gcd__nat__abs__right__eq,axiom,
% 0.30/0.65      ! [N2: nat,K: int] :
% 0.30/0.65        ( ( gcd_gcd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 0.30/0.65        = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat_abs_right_eq
% 0.30/0.65  thf(fact_9297_gcd__nat__abs__left__eq,axiom,
% 0.30/0.65      ! [K: int,N2: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
% 0.30/0.65        = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat_abs_left_eq
% 0.30/0.65  thf(fact_9298_or__nat__numerals_I3_J,axiom,
% 0.30/0.65      ! [X3: num] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_numerals(3)
% 0.30/0.65  thf(fact_9299_or__nat__numerals_I1_J,axiom,
% 0.30/0.65      ! [Y3: num] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 0.30/0.65        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_numerals(1)
% 0.30/0.65  thf(fact_9300_gcd__mult__distrib__nat,axiom,
% 0.30/0.65      ! [K: nat,M: nat,N2: nat] :
% 0.30/0.65        ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
% 0.30/0.65        = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_mult_distrib_nat
% 0.30/0.65  thf(fact_9301_gcd__red__nat,axiom,
% 0.30/0.65      ( gcd_gcd_nat
% 0.30/0.65      = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_red_nat
% 0.30/0.65  thf(fact_9302_gcd__unique__nat,axiom,
% 0.30/0.65      ! [D: nat,A: nat,B: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ D @ A )
% 0.30/0.65          & ( dvd_dvd_nat @ D @ B )
% 0.30/0.65          & ! [E3: nat] :
% 0.30/0.65              ( ( ( dvd_dvd_nat @ E3 @ A )
% 0.30/0.65                & ( dvd_dvd_nat @ E3 @ B ) )
% 0.30/0.65             => ( dvd_dvd_nat @ E3 @ D ) ) )
% 0.30/0.65        = ( D
% 0.30/0.65          = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_unique_nat
% 0.30/0.65  thf(fact_9303_gcd__nat_Ostrict__coboundedI2,axiom,
% 0.30/0.65      ! [B: nat,C: nat,A: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.65          & ( B != C ) )
% 0.30/0.65       => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.30/0.65          & ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65           != C ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.strict_coboundedI2
% 0.30/0.65  thf(fact_9304_gcd__nat_Ostrict__coboundedI1,axiom,
% 0.30/0.65      ! [A: nat,C: nat,B: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.65          & ( A != C ) )
% 0.30/0.65       => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 0.30/0.65          & ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65           != C ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.strict_coboundedI1
% 0.30/0.65  thf(fact_9305_gcd__nat_Ostrict__order__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65          & ( A != B ) )
% 0.30/0.65        = ( ( A
% 0.30/0.65            = ( gcd_gcd_nat @ A @ B ) )
% 0.30/0.65          & ( A != B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.strict_order_iff
% 0.30/0.65  thf(fact_9306_gcd__nat_Ostrict__boundedE,axiom,
% 0.30/0.65      ! [A: nat,B: nat,C: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.30/0.65          & ( A
% 0.30/0.65           != ( gcd_gcd_nat @ B @ C ) ) )
% 0.30/0.65       => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65              & ( A != B ) )
% 0.30/0.65           => ~ ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.65                & ( A != C ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.strict_boundedE
% 0.30/0.65  thf(fact_9307_gcd__nat_OcoboundedI2,axiom,
% 0.30/0.65      ! [B: nat,C: nat,A: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ B @ C )
% 0.30/0.65       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.coboundedI2
% 0.30/0.65  thf(fact_9308_gcd__nat_OcoboundedI1,axiom,
% 0.30/0.65      ! [A: nat,C: nat,B: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.65       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.coboundedI1
% 0.30/0.65  thf(fact_9309_gcd__nat_Oabsorb__iff2,axiom,
% 0.30/0.65      ( dvd_dvd_nat
% 0.30/0.65      = ( ^ [B3: nat,A3: nat] :
% 0.30/0.65            ( ( gcd_gcd_nat @ A3 @ B3 )
% 0.30/0.65            = B3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb_iff2
% 0.30/0.65  thf(fact_9310_gcd__nat_Oabsorb__iff1,axiom,
% 0.30/0.65      ( dvd_dvd_nat
% 0.30/0.65      = ( ^ [A3: nat,B3: nat] :
% 0.30/0.65            ( ( gcd_gcd_nat @ A3 @ B3 )
% 0.30/0.65            = A3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb_iff1
% 0.30/0.65  thf(fact_9311_gcd__nat_Ocobounded2,axiom,
% 0.30/0.65      ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.cobounded2
% 0.30/0.65  thf(fact_9312_gcd__nat_Ocobounded1,axiom,
% 0.30/0.65      ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.cobounded1
% 0.30/0.65  thf(fact_9313_gcd__nat_Oorder__iff,axiom,
% 0.30/0.65      ( dvd_dvd_nat
% 0.30/0.65      = ( ^ [A3: nat,B3: nat] :
% 0.30/0.65            ( A3
% 0.30/0.65            = ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.order_iff
% 0.30/0.65  thf(fact_9314_gcd__nat_OboundedI,axiom,
% 0.30/0.65      ! [A: nat,B: nat,C: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65       => ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.65         => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.boundedI
% 0.30/0.65  thf(fact_9315_gcd__nat_OboundedE,axiom,
% 0.30/0.65      ! [A: nat,B: nat,C: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 0.30/0.65       => ~ ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65           => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.boundedE
% 0.30/0.65  thf(fact_9316_gcd__nat_Oabsorb4,axiom,
% 0.30/0.65      ! [B: nat,A: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ B @ A )
% 0.30/0.65          & ( B != A ) )
% 0.30/0.65       => ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65          = B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb4
% 0.30/0.65  thf(fact_9317_gcd__nat_Oabsorb3,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65          & ( A != B ) )
% 0.30/0.65       => ( ( gcd_gcd_nat @ A @ B )
% 0.30/0.65          = A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.absorb3
% 0.30/0.65  thf(fact_9318_gcd__nat_OorderI,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( A
% 0.30/0.65          = ( gcd_gcd_nat @ A @ B ) )
% 0.30/0.65       => ( dvd_dvd_nat @ A @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.orderI
% 0.30/0.65  thf(fact_9319_gcd__nat_OorderE,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ B )
% 0.30/0.65       => ( A
% 0.30/0.65          = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.orderE
% 0.30/0.65  thf(fact_9320_gcd__nat_Omono,axiom,
% 0.30/0.65      ! [A: nat,C: nat,B: nat,D: nat] :
% 0.30/0.65        ( ( dvd_dvd_nat @ A @ C )
% 0.30/0.65       => ( ( dvd_dvd_nat @ B @ D )
% 0.30/0.65         => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.mono
% 0.30/0.65  thf(fact_9321_gcd__le1__nat,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( A != zero_zero_nat )
% 0.30/0.65       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_le1_nat
% 0.30/0.65  thf(fact_9322_gcd__le2__nat,axiom,
% 0.30/0.65      ! [B: nat,A: nat] :
% 0.30/0.65        ( ( B != zero_zero_nat )
% 0.30/0.65       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_le2_nat
% 0.30/0.65  thf(fact_9323_gcd__diff1__nat,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N2 @ M )
% 0.30/0.65       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
% 0.30/0.65          = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_diff1_nat
% 0.30/0.65  thf(fact_9324_gcd__diff2__nat,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.65       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
% 0.30/0.65          = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_diff2_nat
% 0.30/0.65  thf(fact_9325_gcd__nat_Oelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: nat] :
% 0.30/0.65        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.65           => ( Y3 = X3 ) )
% 0.30/0.65          & ( ( Xa2 != zero_zero_nat )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.elims
% 0.30/0.65  thf(fact_9326_gcd__nat_Osimps,axiom,
% 0.30/0.65      ( gcd_gcd_nat
% 0.30/0.65      = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.simps
% 0.30/0.65  thf(fact_9327_gcd__non__0__nat,axiom,
% 0.30/0.65      ! [Y3: nat,X3: nat] :
% 0.30/0.65        ( ( Y3 != zero_zero_nat )
% 0.30/0.65       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 0.30/0.65          = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_non_0_nat
% 0.30/0.65  thf(fact_9328_gcd__integer_Orep__eq,axiom,
% 0.30/0.65      ! [X3: code_integer,Xa2: code_integer] :
% 0.30/0.65        ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X3 @ Xa2 ) )
% 0.30/0.65        = ( gcd_gcd_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_integer.rep_eq
% 0.30/0.65  thf(fact_9329_gcd__integer_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: int,X3: int] :
% 0.30/0.65        ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 0.30/0.65        = ( code_integer_of_int @ ( gcd_gcd_int @ Xa2 @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_integer.abs_eq
% 0.30/0.65  thf(fact_9330_bezout__nat,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( A != zero_zero_nat )
% 0.30/0.65       => ? [X4: nat,Y5: nat] :
% 0.30/0.65            ( ( times_times_nat @ A @ X4 )
% 0.30/0.65            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezout_nat
% 0.30/0.65  thf(fact_9331_bezout__gcd__nat_H,axiom,
% 0.30/0.65      ! [B: nat,A: nat] :
% 0.30/0.65      ? [X4: nat,Y5: nat] :
% 0.30/0.65        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y5 ) @ ( times_times_nat @ A @ X4 ) )
% 0.30/0.65          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
% 0.30/0.65            = ( gcd_gcd_nat @ A @ B ) ) )
% 0.30/0.65        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y5 ) @ ( times_times_nat @ B @ X4 ) )
% 0.30/0.65          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
% 0.30/0.65            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezout_gcd_nat'
% 0.30/0.65  thf(fact_9332_gcd__code__integer,axiom,
% 0.30/0.65      ( gcd_gcd_Code_integer
% 0.30/0.65      = ( ^ [K3: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_code_integer
% 0.30/0.65  thf(fact_9333_gcd__int__def,axiom,
% 0.30/0.65      ( gcd_gcd_int
% 0.30/0.65      = ( ^ [X: int,Y: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X ) ) @ ( nat2 @ ( abs_abs_int @ Y ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_int_def
% 0.30/0.65  thf(fact_9334_bezw__aux,axiom,
% 0.30/0.65      ! [X3: nat,Y3: nat] :
% 0.30/0.65        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
% 0.30/0.65        = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % bezw_aux
% 0.30/0.65  thf(fact_9335_OR__upper,axiom,
% 0.30/0.65      ! [X3: int,N2: nat,Y3: int] :
% 0.30/0.65        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 0.30/0.65       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.65         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 0.30/0.65           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % OR_upper
% 0.30/0.65  thf(fact_9336_or__Suc__0__eq,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_Suc_0_eq
% 0.30/0.65  thf(fact_9337_Suc__0__or__eq,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 0.30/0.65        = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_0_or_eq
% 0.30/0.65  thf(fact_9338_or__nat__rec,axiom,
% 0.30/0.65      ( bit_se1412395901928357646or_nat
% 0.30/0.65      = ( ^ [M2: nat,N: nat] :
% 0.30/0.65            ( plus_plus_nat
% 0.30/0.65            @ ( zero_n2687167440665602831ol_nat
% 0.30/0.65              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
% 0.30/0.65                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.65            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_rec
% 0.30/0.65  thf(fact_9339_gcd__nat_Opelims,axiom,
% 0.30/0.65      ! [X3: nat,Xa2: nat,Y3: nat] :
% 0.30/0.65        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 0.30/0.65                 => ( Y3 = X3 ) )
% 0.30/0.65                & ( ( Xa2 != zero_zero_nat )
% 0.30/0.65                 => ( Y3
% 0.30/0.65                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
% 0.30/0.65             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.pelims
% 0.30/0.65  thf(fact_9340_gcd__nat_Oidem,axiom,
% 0.30/0.65      ! [A: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ A @ A )
% 0.30/0.65        = A ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.idem
% 0.30/0.65  thf(fact_9341_gcd__nat_Oleft__idem,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
% 0.30/0.65        = ( gcd_gcd_nat @ A @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.left_idem
% 0.30/0.65  thf(fact_9342_gcd__nat_Oright__idem,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
% 0.30/0.65        = ( gcd_gcd_nat @ A @ B ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.right_idem
% 0.30/0.65  thf(fact_9343_gcd__idem__nat,axiom,
% 0.30/0.65      ! [X3: nat] :
% 0.30/0.65        ( ( gcd_gcd_nat @ X3 @ X3 )
% 0.30/0.65        = X3 ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_idem_nat
% 0.30/0.65  thf(fact_9344_or__nat__unfold,axiom,
% 0.30/0.65      ( bit_se1412395901928357646or_nat
% 0.30/0.65      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % or_nat_unfold
% 0.30/0.65  thf(fact_9345_max__Suc__Suc,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.65        = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_Suc_Suc
% 0.30/0.65  thf(fact_9346_max__0R,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_max_nat @ N2 @ zero_zero_nat )
% 0.30/0.65        = N2 ) ).
% 0.30/0.65  
% 0.30/0.65  % max_0R
% 0.30/0.65  thf(fact_9347_max__0L,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_max_nat @ zero_zero_nat @ N2 )
% 0.30/0.65        = N2 ) ).
% 0.30/0.65  
% 0.30/0.65  % max_0L
% 0.30/0.65  thf(fact_9348_max__nat_Oright__neutral,axiom,
% 0.30/0.65      ! [A: nat] :
% 0.30/0.65        ( ( ord_max_nat @ A @ zero_zero_nat )
% 0.30/0.65        = A ) ).
% 0.30/0.65  
% 0.30/0.65  % max_nat.right_neutral
% 0.30/0.65  thf(fact_9349_max__nat_Oneutr__eq__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( zero_zero_nat
% 0.30/0.65          = ( ord_max_nat @ A @ B ) )
% 0.30/0.65        = ( ( A = zero_zero_nat )
% 0.30/0.65          & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_nat.neutr_eq_iff
% 0.30/0.65  thf(fact_9350_max__nat_Oleft__neutral,axiom,
% 0.30/0.65      ! [A: nat] :
% 0.30/0.65        ( ( ord_max_nat @ zero_zero_nat @ A )
% 0.30/0.65        = A ) ).
% 0.30/0.65  
% 0.30/0.65  % max_nat.left_neutral
% 0.30/0.65  thf(fact_9351_max__nat_Oeq__neutr__iff,axiom,
% 0.30/0.65      ! [A: nat,B: nat] :
% 0.30/0.65        ( ( ( ord_max_nat @ A @ B )
% 0.30/0.65          = zero_zero_nat )
% 0.30/0.65        = ( ( A = zero_zero_nat )
% 0.30/0.65          & ( B = zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_nat.eq_neutr_iff
% 0.30/0.65  thf(fact_9352_max__Suc__numeral,axiom,
% 0.30/0.65      ! [N2: nat,K: num] :
% 0.30/0.65        ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65        = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_Suc_numeral
% 0.30/0.65  thf(fact_9353_max__numeral__Suc,axiom,
% 0.30/0.65      ! [K: num,N2: nat] :
% 0.30/0.65        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.30/0.65        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_numeral_Suc
% 0.30/0.65  thf(fact_9354_nat__add__max__right,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
% 0.30/0.65        = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_add_max_right
% 0.30/0.65  thf(fact_9355_nat__add__max__left,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
% 0.30/0.65        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N2 @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_add_max_left
% 0.30/0.65  thf(fact_9356_nat__mult__max__left,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
% 0.30/0.65        = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_max_left
% 0.30/0.65  thf(fact_9357_nat__mult__max__right,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
% 0.30/0.65        = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_max_right
% 0.30/0.65  thf(fact_9358_nat__minus__add__max,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
% 0.30/0.65        = ( ord_max_nat @ N2 @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_minus_add_max
% 0.30/0.65  thf(fact_9359_max__Suc2,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_max_nat @ M @ ( suc @ N2 ) )
% 0.30/0.65        = ( case_nat_nat @ ( suc @ N2 )
% 0.30/0.65          @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N2 ) )
% 0.30/0.65          @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_Suc2
% 0.30/0.65  thf(fact_9360_max__Suc1,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( ord_max_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65        = ( case_nat_nat @ ( suc @ N2 )
% 0.30/0.65          @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N2 @ M4 ) )
% 0.30/0.65          @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_Suc1
% 0.30/0.65  thf(fact_9361_divmod__integer__eq__cases,axiom,
% 0.30/0.65      ( code_divmod_integer
% 0.30/0.65      = ( ^ [K3: code_integer,L2: code_integer] :
% 0.30/0.65            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 0.30/0.65            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 0.30/0.65              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 0.30/0.65                @ ( if_Pro6119634080678213985nteger
% 0.30/0.65                  @ ( ( sgn_sgn_Code_integer @ K3 )
% 0.30/0.65                    = ( sgn_sgn_Code_integer @ L2 ) )
% 0.30/0.65                  @ ( code_divmod_abs @ K3 @ L2 )
% 0.30/0.65                  @ ( produc6916734918728496179nteger
% 0.30/0.65                    @ ^ [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 ) ) )
% 0.30/0.65                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % divmod_integer_eq_cases
% 0.30/0.65  thf(fact_9362_card_Ocomp__fun__commute__on,axiom,
% 0.30/0.65      ( ( comp_nat_nat_nat @ suc @ suc )
% 0.30/0.65      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 0.30/0.65  
% 0.30/0.65  % card.comp_fun_commute_on
% 0.30/0.65  thf(fact_9363_root__powr__inverse,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ( root @ N2 @ X3 )
% 0.30/0.65            = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % root_powr_inverse
% 0.30/0.65  thf(fact_9364_drop__bit__negative__int__iff,axiom,
% 0.30/0.65      ! [N2: nat,K: int] :
% 0.30/0.65        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
% 0.30/0.65        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.65  
% 0.30/0.65  % drop_bit_negative_int_iff
% 0.30/0.65  thf(fact_9365_real__root__Suc__0,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
% 0.30/0.65        = X3 ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_Suc_0
% 0.30/0.65  thf(fact_9366_root__0,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65        ( ( root @ zero_zero_nat @ X3 )
% 0.30/0.65        = zero_zero_real ) ).
% 0.30/0.65  
% 0.30/0.65  % root_0
% 0.30/0.65  thf(fact_9367_real__root__eq__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ( root @ N2 @ X3 )
% 0.30/0.65            = ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( X3 = Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_eq_iff
% 0.30/0.65  thf(fact_9368_drop__bit__of__Suc__0,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 0.30/0.65        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % drop_bit_of_Suc_0
% 0.30/0.65  thf(fact_9369_drop__bit__Suc__minus__bit0,axiom,
% 0.30/0.65      ! [N2: nat,K: num] :
% 0.30/0.65        ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 0.30/0.65        = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % drop_bit_Suc_minus_bit0
% 0.30/0.65  thf(fact_9370_real__root__eq__0__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ( root @ N2 @ X3 )
% 0.30/0.65            = zero_zero_real )
% 0.30/0.65          = ( X3 = zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_eq_0_iff
% 0.30/0.65  thf(fact_9371_real__root__less__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_less_iff
% 0.30/0.65  thf(fact_9372_real__root__le__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_le_iff
% 0.30/0.65  thf(fact_9373_real__root__eq__1__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ( root @ N2 @ X3 )
% 0.30/0.65            = one_one_real )
% 0.30/0.65          = ( X3 = one_one_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_eq_1_iff
% 0.30/0.65  thf(fact_9374_real__root__one,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( root @ N2 @ one_one_real )
% 0.30/0.65          = one_one_real ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_one
% 0.30/0.65  thf(fact_9375_real__root__lt__0__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
% 0.30/0.65          = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_lt_0_iff
% 0.30/0.65  thf(fact_9376_real__root__gt__0__iff,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_gt_0_iff
% 0.30/0.65  thf(fact_9377_real__root__le__0__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
% 0.30/0.65          = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_le_0_iff
% 0.30/0.65  thf(fact_9378_real__root__ge__0__iff,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_ge_0_iff
% 0.30/0.65  thf(fact_9379_real__root__lt__1__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ one_one_real )
% 0.30/0.65          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_lt_1_iff
% 0.30/0.65  thf(fact_9380_real__root__gt__1__iff,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_gt_1_iff
% 0.30/0.65  thf(fact_9381_real__root__le__1__iff,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ one_one_real )
% 0.30/0.65          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_le_1_iff
% 0.30/0.65  thf(fact_9382_real__root__ge__1__iff,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y3 ) )
% 0.30/0.65          = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_ge_1_iff
% 0.30/0.65  thf(fact_9383_drop__bit__Suc__minus__bit1,axiom,
% 0.30/0.65      ! [N2: nat,K: num] :
% 0.30/0.65        ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 0.30/0.65        = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % drop_bit_Suc_minus_bit1
% 0.30/0.65  thf(fact_9384_real__root__pow__pos2,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
% 0.30/0.65            = X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_pow_pos2
% 0.30/0.65  thf(fact_9385_real__root__less__mono,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.65         => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_less_mono
% 0.30/0.65  thf(fact_9386_real__root__le__mono,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 0.30/0.65         => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_le_mono
% 0.30/0.65  thf(fact_9387_real__root__power,axiom,
% 0.30/0.65      ! [N2: nat,X3: real,K: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( root @ N2 @ ( power_power_real @ X3 @ K ) )
% 0.30/0.65          = ( power_power_real @ ( root @ N2 @ X3 ) @ K ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_power
% 0.30/0.65  thf(fact_9388_real__root__abs,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( root @ N2 @ ( abs_abs_real @ X3 ) )
% 0.30/0.65          = ( abs_abs_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_abs
% 0.30/0.65  thf(fact_9389_sgn__root,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( sgn_sgn_real @ ( root @ N2 @ X3 ) )
% 0.30/0.65          = ( sgn_sgn_real @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_root
% 0.30/0.65  thf(fact_9390_real__root__gt__zero,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_gt_zero
% 0.30/0.65  thf(fact_9391_real__root__strict__decreasing,axiom,
% 0.30/0.65      ! [N2: nat,N7: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_nat @ N2 @ N7 )
% 0.30/0.65         => ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.65           => ( ord_less_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_strict_decreasing
% 0.30/0.65  thf(fact_9392_root__abs__power,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y3 @ N2 ) ) )
% 0.30/0.65          = ( abs_abs_real @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % root_abs_power
% 0.30/0.65  thf(fact_9393_real__root__pos__pos,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_pos_pos
% 0.30/0.65  thf(fact_9394_real__root__strict__increasing,axiom,
% 0.30/0.65      ! [N2: nat,N7: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_nat @ N2 @ N7 )
% 0.30/0.65         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65           => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.65             => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_strict_increasing
% 0.30/0.65  thf(fact_9395_real__root__decreasing,axiom,
% 0.30/0.65      ! [N2: nat,N7: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.30/0.65         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 0.30/0.65           => ( ord_less_eq_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_decreasing
% 0.30/0.65  thf(fact_9396_real__root__pow__pos,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
% 0.30/0.65            = X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_pow_pos
% 0.30/0.65  thf(fact_9397_real__root__pos__unique,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 0.30/0.65         => ( ( ( power_power_real @ Y3 @ N2 )
% 0.30/0.65              = X3 )
% 0.30/0.65           => ( ( root @ N2 @ X3 )
% 0.30/0.65              = Y3 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_pos_unique
% 0.30/0.65  thf(fact_9398_real__root__power__cancel,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65         => ( ( root @ N2 @ ( power_power_real @ X3 @ N2 ) )
% 0.30/0.65            = X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_power_cancel
% 0.30/0.65  thf(fact_9399_real__root__increasing,axiom,
% 0.30/0.65      ! [N2: nat,N7: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_eq_nat @ N2 @ N7 )
% 0.30/0.65         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.65           => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 0.30/0.65             => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % real_root_increasing
% 0.30/0.65  thf(fact_9400_root__sgn__power,axiom,
% 0.30/0.65      ! [N2: nat,Y3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N2 ) ) )
% 0.30/0.65          = Y3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % root_sgn_power
% 0.30/0.65  thf(fact_9401_sgn__power__root,axiom,
% 0.30/0.65      ! [N2: nat,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X3 ) ) @ N2 ) )
% 0.30/0.65          = X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sgn_power_root
% 0.30/0.65  thf(fact_9402_ln__root,axiom,
% 0.30/0.65      ! [N2: nat,B: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.65         => ( ( ln_ln_real @ ( root @ N2 @ B ) )
% 0.30/0.65            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % ln_root
% 0.30/0.65  thf(fact_9403_log__root,axiom,
% 0.30/0.65      ! [N2: nat,A: real,B: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ A )
% 0.30/0.65         => ( ( log @ B @ ( root @ N2 @ A ) )
% 0.30/0.65            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % log_root
% 0.30/0.65  thf(fact_9404_log__base__root,axiom,
% 0.30/0.65      ! [N2: nat,B: real,X3: real] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( ord_less_real @ zero_zero_real @ B )
% 0.30/0.65         => ( ( log @ ( root @ N2 @ B ) @ X3 )
% 0.30/0.65            = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % log_base_root
% 0.30/0.65  thf(fact_9405_split__root,axiom,
% 0.30/0.65      ! [P: real > $o,N2: nat,X3: real] :
% 0.30/0.65        ( ( P @ ( root @ N2 @ X3 ) )
% 0.30/0.65        = ( ( ( N2 = zero_zero_nat )
% 0.30/0.65           => ( P @ zero_zero_real ) )
% 0.30/0.65          & ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65           => ! [Y: real] :
% 0.30/0.65                ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.30/0.65                  = X3 )
% 0.30/0.65               => ( P @ Y ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % split_root
% 0.30/0.65  thf(fact_9406_Suc__funpow,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( compow_nat_nat @ N2 @ suc )
% 0.30/0.65        = ( plus_plus_nat @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Suc_funpow
% 0.30/0.65  thf(fact_9407_min__Suc__Suc,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 0.30/0.65        = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_Suc_Suc
% 0.30/0.65  thf(fact_9408_min__0R,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ N2 @ zero_zero_nat )
% 0.30/0.65        = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % min_0R
% 0.30/0.65  thf(fact_9409_min__0L,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ zero_zero_nat @ N2 )
% 0.30/0.65        = zero_zero_nat ) ).
% 0.30/0.65  
% 0.30/0.65  % min_0L
% 0.30/0.65  thf(fact_9410_min__Suc__numeral,axiom,
% 0.30/0.65      ! [N2: nat,K: num] :
% 0.30/0.65        ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 0.30/0.65        = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_Suc_numeral
% 0.30/0.65  thf(fact_9411_min__numeral__Suc,axiom,
% 0.30/0.65      ! [K: num,N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 0.30/0.65        = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_numeral_Suc
% 0.30/0.65  thf(fact_9412_min__diff,axiom,
% 0.30/0.65      ! [M: nat,I: nat,N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
% 0.30/0.65        = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_diff
% 0.30/0.65  thf(fact_9413_nat__mult__min__left,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q4 )
% 0.30/0.65        = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_min_left
% 0.30/0.65  thf(fact_9414_nat__mult__min__right,axiom,
% 0.30/0.65      ! [M: nat,N2: nat,Q4: nat] :
% 0.30/0.65        ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q4 ) )
% 0.30/0.65        = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat_mult_min_right
% 0.30/0.65  thf(fact_9415_inf__nat__def,axiom,
% 0.30/0.65      inf_inf_nat = ord_min_nat ).
% 0.30/0.65  
% 0.30/0.65  % inf_nat_def
% 0.30/0.65  thf(fact_9416_min__Suc2,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_min_nat @ M @ ( suc @ N2 ) )
% 0.30/0.65        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.65          @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N2 ) )
% 0.30/0.65          @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_Suc2
% 0.30/0.65  thf(fact_9417_min__Suc1,axiom,
% 0.30/0.65      ! [N2: nat,M: nat] :
% 0.30/0.65        ( ( ord_min_nat @ ( suc @ N2 ) @ M )
% 0.30/0.65        = ( case_nat_nat @ zero_zero_nat
% 0.30/0.65          @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N2 @ M4 ) )
% 0.30/0.65          @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % min_Suc1
% 0.30/0.65  thf(fact_9418_inf__int__def,axiom,
% 0.30/0.65      inf_inf_int = ord_min_int ).
% 0.30/0.65  
% 0.30/0.65  % inf_int_def
% 0.30/0.65  thf(fact_9419_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.65      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 0.30/0.65      @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 0.30/0.65      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 0.30/0.65  
% 0.30/0.65  % max_nat.semilattice_neutr_order_axioms
% 0.30/0.65  thf(fact_9420_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 0.30/0.65      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 0.30/0.65      @ ^ [M2: nat,N: nat] :
% 0.30/0.65          ( ( dvd_dvd_nat @ M2 @ N )
% 0.30/0.65          & ( M2 != N ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % gcd_nat.semilattice_neutr_order_axioms
% 0.30/0.65  thf(fact_9421_times__int_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( abs_Integ
% 0.30/0.65          @ ( produc27273713700761075at_nat
% 0.30/0.65            @ ^ [X: nat,Y: nat] :
% 0.30/0.65                ( produc2626176000494625587at_nat
% 0.30/0.65                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
% 0.30/0.65            @ Xa2
% 0.30/0.65            @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % times_int.abs_eq
% 0.30/0.65  thf(fact_9422_int_Oabs__induct,axiom,
% 0.30/0.65      ! [P: int > $o,X3: int] :
% 0.30/0.65        ( ! [Y5: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y5 ) )
% 0.30/0.65       => ( P @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int.abs_induct
% 0.30/0.65  thf(fact_9423_eq__Abs__Integ,axiom,
% 0.30/0.65      ! [Z2: int] :
% 0.30/0.65        ~ ! [X4: nat,Y5: nat] :
% 0.30/0.65            ( Z2
% 0.30/0.65           != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % eq_Abs_Integ
% 0.30/0.65  thf(fact_9424_nat_Oabs__eq,axiom,
% 0.30/0.65      ! [X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( nat2 @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat.abs_eq
% 0.30/0.65  thf(fact_9425_zero__int__def,axiom,
% 0.30/0.65      ( zero_zero_int
% 0.30/0.65      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_int_def
% 0.30/0.65  thf(fact_9426_int__def,axiom,
% 0.30/0.65      ( semiri1314217659103216013at_int
% 0.30/0.65      = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % int_def
% 0.30/0.65  thf(fact_9427_uminus__int_Oabs__eq,axiom,
% 0.30/0.65      ! [X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( abs_Integ
% 0.30/0.65          @ ( produc2626176000494625587at_nat
% 0.30/0.65            @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 0.30/0.65            @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % uminus_int.abs_eq
% 0.30/0.65  thf(fact_9428_one__int__def,axiom,
% 0.30/0.65      ( one_one_int
% 0.30/0.65      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % one_int_def
% 0.30/0.65  thf(fact_9429_less__int_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( produc8739625826339149834_nat_o
% 0.30/0.65          @ ^ [X: nat,Y: nat] :
% 0.30/0.65              ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.65          @ Xa2
% 0.30/0.65          @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_int.abs_eq
% 0.30/0.65  thf(fact_9430_less__eq__int_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( produc8739625826339149834_nat_o
% 0.30/0.65          @ ^ [X: nat,Y: nat] :
% 0.30/0.65              ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 0.30/0.65          @ Xa2
% 0.30/0.65          @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_eq_int.abs_eq
% 0.30/0.65  thf(fact_9431_plus__int_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( abs_Integ
% 0.30/0.65          @ ( produc27273713700761075at_nat
% 0.30/0.65            @ ^ [X: nat,Y: nat] :
% 0.30/0.65                ( produc2626176000494625587at_nat
% 0.30/0.65                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 0.30/0.65            @ Xa2
% 0.30/0.65            @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % plus_int.abs_eq
% 0.30/0.65  thf(fact_9432_minus__int_Oabs__eq,axiom,
% 0.30/0.65      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 0.30/0.65        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 0.30/0.65        = ( abs_Integ
% 0.30/0.65          @ ( produc27273713700761075at_nat
% 0.30/0.65            @ ^ [X: nat,Y: nat] :
% 0.30/0.65                ( produc2626176000494625587at_nat
% 0.30/0.65                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 0.30/0.65            @ Xa2
% 0.30/0.65            @ X3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % minus_int.abs_eq
% 0.30/0.65  thf(fact_9433_max__rpair__set,axiom,
% 0.30/0.65      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 0.30/0.65  
% 0.30/0.65  % max_rpair_set
% 0.30/0.65  thf(fact_9434_min__rpair__set,axiom,
% 0.30/0.65      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 0.30/0.65  
% 0.30/0.65  % min_rpair_set
% 0.30/0.65  thf(fact_9435_less__eq__int_Orep__eq,axiom,
% 0.30/0.65      ( ord_less_eq_int
% 0.30/0.65      = ( ^ [X: int,Xa4: int] :
% 0.30/0.65            ( produc8739625826339149834_nat_o
% 0.30/0.65            @ ^ [Y: nat,Z6: nat] :
% 0.30/0.65                ( produc6081775807080527818_nat_o
% 0.30/0.65                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
% 0.30/0.65            @ ( rep_Integ @ X )
% 0.30/0.65            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_eq_int.rep_eq
% 0.30/0.65  thf(fact_9436_less__int_Orep__eq,axiom,
% 0.30/0.65      ( ord_less_int
% 0.30/0.65      = ( ^ [X: int,Xa4: int] :
% 0.30/0.65            ( produc8739625826339149834_nat_o
% 0.30/0.65            @ ^ [Y: nat,Z6: nat] :
% 0.30/0.65                ( produc6081775807080527818_nat_o
% 0.30/0.65                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
% 0.30/0.65            @ ( rep_Integ @ X )
% 0.30/0.65            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_int.rep_eq
% 0.30/0.65  thf(fact_9437_prod__encode__def,axiom,
% 0.30/0.65      ( nat_prod_encode
% 0.30/0.65      = ( produc6842872674320459806at_nat
% 0.30/0.65        @ ^ [M2: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N ) ) @ M2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_encode_def
% 0.30/0.65  thf(fact_9438_le__prod__encode__2,axiom,
% 0.30/0.65      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % le_prod_encode_2
% 0.30/0.65  thf(fact_9439_le__prod__encode__1,axiom,
% 0.30/0.65      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % le_prod_encode_1
% 0.30/0.65  thf(fact_9440_nat_Orep__eq,axiom,
% 0.30/0.65      ( nat2
% 0.30/0.65      = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % nat.rep_eq
% 0.30/0.65  thf(fact_9441_prod__encode__prod__decode__aux,axiom,
% 0.30/0.65      ! [K: nat,M: nat] :
% 0.30/0.65        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 0.30/0.65        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 0.30/0.65  
% 0.30/0.65  % prod_encode_prod_decode_aux
% 0.30/0.65  thf(fact_9442_uminus__int__def,axiom,
% 0.30/0.65      ( uminus_uminus_int
% 0.30/0.65      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 0.30/0.65        @ ( produc2626176000494625587at_nat
% 0.30/0.65          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % uminus_int_def
% 0.30/0.65  thf(fact_9443_times__int__def,axiom,
% 0.30/0.65      ( times_times_int
% 0.30/0.65      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.65        @ ( produc27273713700761075at_nat
% 0.30/0.65          @ ^ [X: nat,Y: nat] :
% 0.30/0.65              ( produc2626176000494625587at_nat
% 0.30/0.65              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % times_int_def
% 0.30/0.65  thf(fact_9444_minus__int__def,axiom,
% 0.30/0.65      ( minus_minus_int
% 0.30/0.65      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.65        @ ( produc27273713700761075at_nat
% 0.30/0.65          @ ^ [X: nat,Y: nat] :
% 0.30/0.65              ( produc2626176000494625587at_nat
% 0.30/0.65              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % minus_int_def
% 0.30/0.65  thf(fact_9445_plus__int__def,axiom,
% 0.30/0.65      ( plus_plus_int
% 0.30/0.65      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 0.30/0.65        @ ( produc27273713700761075at_nat
% 0.30/0.65          @ ^ [X: nat,Y: nat] :
% 0.30/0.65              ( produc2626176000494625587at_nat
% 0.30/0.65              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % plus_int_def
% 0.30/0.65  thf(fact_9446_Gcd__remove0__nat,axiom,
% 0.30/0.65      ! [M7: set_nat] :
% 0.30/0.65        ( ( finite_finite_nat @ M7 )
% 0.30/0.65       => ( ( gcd_Gcd_nat @ M7 )
% 0.30/0.65          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_remove0_nat
% 0.30/0.65  thf(fact_9447_Gcd__dvd__nat,axiom,
% 0.30/0.65      ! [A: nat,A2: set_nat] :
% 0.30/0.65        ( ( member_nat @ A @ A2 )
% 0.30/0.65       => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_dvd_nat
% 0.30/0.65  thf(fact_9448_Gcd__greatest__nat,axiom,
% 0.30/0.65      ! [A2: set_nat,A: nat] :
% 0.30/0.65        ( ! [B4: nat] :
% 0.30/0.65            ( ( member_nat @ B4 @ A2 )
% 0.30/0.65           => ( dvd_dvd_nat @ A @ B4 ) )
% 0.30/0.65       => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_greatest_nat
% 0.30/0.65  thf(fact_9449_Gcd__nat__eq__one,axiom,
% 0.30/0.65      ! [N7: set_nat] :
% 0.30/0.65        ( ( member_nat @ one_one_nat @ N7 )
% 0.30/0.65       => ( ( gcd_Gcd_nat @ N7 )
% 0.30/0.65          = one_one_nat ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_nat_eq_one
% 0.30/0.65  thf(fact_9450_Gcd__in,axiom,
% 0.30/0.65      ! [A2: set_nat] :
% 0.30/0.65        ( ! [A4: nat,B4: nat] :
% 0.30/0.65            ( ( member_nat @ A4 @ A2 )
% 0.30/0.65           => ( ( member_nat @ B4 @ A2 )
% 0.30/0.65             => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A2 ) ) )
% 0.30/0.65       => ( ( A2 != bot_bot_set_nat )
% 0.30/0.65         => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_in
% 0.30/0.65  thf(fact_9451_sub__BitM__One__eq,axiom,
% 0.30/0.65      ! [N2: num] :
% 0.30/0.65        ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 0.30/0.65        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % sub_BitM_One_eq
% 0.30/0.65  thf(fact_9452_num__of__nat_Osimps_I2_J,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65         => ( ( num_of_nat @ ( suc @ N2 ) )
% 0.30/0.65            = ( inc @ ( num_of_nat @ N2 ) ) ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65         => ( ( num_of_nat @ ( suc @ N2 ) )
% 0.30/0.65            = one ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % num_of_nat.simps(2)
% 0.30/0.65  thf(fact_9453_pred__nat__def,axiom,
% 0.30/0.65      ( pred_nat
% 0.30/0.65      = ( collec3392354462482085612at_nat
% 0.30/0.65        @ ( produc6081775807080527818_nat_o
% 0.30/0.65          @ ^ [M2: nat,N: nat] :
% 0.30/0.65              ( N
% 0.30/0.65              = ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % pred_nat_def
% 0.30/0.65  thf(fact_9454_abs__Gcd__eq,axiom,
% 0.30/0.65      ! [K5: set_int] :
% 0.30/0.65        ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
% 0.30/0.65        = ( gcd_Gcd_int @ K5 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % abs_Gcd_eq
% 0.30/0.65  thf(fact_9455_Gcd__dvd__int,axiom,
% 0.30/0.65      ! [A: int,A2: set_int] :
% 0.30/0.65        ( ( member_int @ A @ A2 )
% 0.30/0.65       => ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_dvd_int
% 0.30/0.65  thf(fact_9456_Gcd__greatest__int,axiom,
% 0.30/0.65      ! [A2: set_int,A: int] :
% 0.30/0.65        ( ! [B4: int] :
% 0.30/0.65            ( ( member_int @ B4 @ A2 )
% 0.30/0.65           => ( dvd_dvd_int @ A @ B4 ) )
% 0.30/0.65       => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_greatest_int
% 0.30/0.65  thf(fact_9457_Gcd__int__greater__eq__0,axiom,
% 0.30/0.65      ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Gcd_int_greater_eq_0
% 0.30/0.65  thf(fact_9458_num__of__nat_Osimps_I1_J,axiom,
% 0.30/0.65      ( ( num_of_nat @ zero_zero_nat )
% 0.30/0.65      = one ) ).
% 0.30/0.65  
% 0.30/0.65  % num_of_nat.simps(1)
% 0.30/0.65  thf(fact_9459_numeral__num__of__nat,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
% 0.30/0.65          = N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % numeral_num_of_nat
% 0.30/0.65  thf(fact_9460_num__of__nat__One,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_less_eq_nat @ N2 @ one_one_nat )
% 0.30/0.65       => ( ( num_of_nat @ N2 )
% 0.30/0.65          = one ) ) ).
% 0.30/0.65  
% 0.30/0.65  % num_of_nat_One
% 0.30/0.65  thf(fact_9461_num__of__nat__double,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65       => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
% 0.30/0.65          = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % num_of_nat_double
% 0.30/0.65  thf(fact_9462_num__of__nat__plus__distrib,axiom,
% 0.30/0.65      ! [M: nat,N2: nat] :
% 0.30/0.65        ( ( ord_less_nat @ zero_zero_nat @ M )
% 0.30/0.65       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.65         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
% 0.30/0.65            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % num_of_nat_plus_distrib
% 0.30/0.65  thf(fact_9463_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.65        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( X3
% 0.30/0.65                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65           => ( Xa2 = one_one_nat ) )
% 0.30/0.65         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65               => ( ( Deg2 = Xa2 )
% 0.30/0.65                  & ! [X4: vEBT_VEBT] :
% 0.30/0.65                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                     => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                  & ( case_o184042715313410164at_nat
% 0.30/0.65                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                      & ! [X: vEBT_VEBT] :
% 0.30/0.65                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                    @ ( produc6081775807080527818_nat_o
% 0.30/0.65                      @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                          & ! [I2: nat] :
% 0.30/0.65                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                          & ( ( Mi3 = Ma3 )
% 0.30/0.65                           => ! [X: vEBT_VEBT] :
% 0.30/0.65                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                          & ( ( Mi3 != Ma3 )
% 0.30/0.65                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                              & ! [X: nat] :
% 0.30/0.65                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                   => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                    @ Mima ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.elims(3)
% 0.30/0.65  thf(fact_9464_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 0.30/0.65      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 0.30/0.65        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 0.30/0.65        = ( ( Deg = Deg4 )
% 0.30/0.65          & ! [X: vEBT_VEBT] :
% 0.30/0.65              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.65             => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65          & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 0.30/0.65            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65          & ( case_o184042715313410164at_nat
% 0.30/0.65            @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
% 0.30/0.65              & ! [X: vEBT_VEBT] :
% 0.30/0.65                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.65                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65            @ ( produc6081775807080527818_nat_o
% 0.30/0.65              @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.65                  & ! [I2: nat] :
% 0.30/0.65                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
% 0.30/0.65                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 0.30/0.65                  & ( ( Mi3 = Ma3 )
% 0.30/0.65                   => ! [X: vEBT_VEBT] :
% 0.30/0.65                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 0.30/0.65                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                  & ( ( Mi3 != Ma3 )
% 0.30/0.65                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 0.30/0.65                      & ! [X: nat] :
% 0.30/0.65                          ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 0.30/0.65                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 0.30/0.65                           => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65            @ Mima2 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.simps(2)
% 0.30/0.65  thf(fact_9465_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.65        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( X3
% 0.30/0.65                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65           => ( Y3
% 0.30/0.65              = ( Xa2 != one_one_nat ) ) )
% 0.30/0.65         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65               => ( Y3
% 0.30/0.65                  = ( ~ ( ( Deg2 = Xa2 )
% 0.30/0.65                        & ! [X: vEBT_VEBT] :
% 0.30/0.65                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                        & ( case_o184042715313410164at_nat
% 0.30/0.65                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                            & ! [X: vEBT_VEBT] :
% 0.30/0.65                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                          @ ( produc6081775807080527818_nat_o
% 0.30/0.65                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                & ! [I2: nat] :
% 0.30/0.65                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                                & ( ( Mi3 = Ma3 )
% 0.30/0.65                                 => ! [X: vEBT_VEBT] :
% 0.30/0.65                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                                & ( ( Mi3 != Ma3 )
% 0.30/0.65                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                                    & ! [X: nat] :
% 0.30/0.65                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                          @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.elims(1)
% 0.30/0.65  thf(fact_9466_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.65        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65       => ( ( ? [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( X3
% 0.30/0.65                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65           => ( Xa2 != one_one_nat ) )
% 0.30/0.65         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65               => ~ ( ( Deg2 = Xa2 )
% 0.30/0.65                    & ! [X2: vEBT_VEBT] :
% 0.30/0.65                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                    & ( case_o184042715313410164at_nat
% 0.30/0.65                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                        & ! [X: vEBT_VEBT] :
% 0.30/0.65                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                      @ ( produc6081775807080527818_nat_o
% 0.30/0.65                        @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                            & ! [I2: nat] :
% 0.30/0.65                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                            & ( ( Mi3 = Ma3 )
% 0.30/0.65                             => ! [X: vEBT_VEBT] :
% 0.30/0.65                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                            & ( ( Mi3 != Ma3 )
% 0.30/0.65                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                                & ! [X: nat] :
% 0.30/0.65                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                     => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                      @ Mima ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.elims(2)
% 0.30/0.65  thf(fact_9467_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 0.30/0.65        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65          = Y3 )
% 0.30/0.65       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65               => ( ( Y3
% 0.30/0.65                    = ( Xa2 = one_one_nat ) )
% 0.30/0.65                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 0.30/0.65           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65                 => ( ( Y3
% 0.30/0.65                      = ( ( Deg2 = Xa2 )
% 0.30/0.65                        & ! [X: vEBT_VEBT] :
% 0.30/0.65                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                        & ( case_o184042715313410164at_nat
% 0.30/0.65                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                            & ! [X: vEBT_VEBT] :
% 0.30/0.65                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                          @ ( produc6081775807080527818_nat_o
% 0.30/0.65                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                & ! [I2: nat] :
% 0.30/0.65                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                                & ( ( Mi3 = Ma3 )
% 0.30/0.65                                 => ! [X: vEBT_VEBT] :
% 0.30/0.65                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                                & ( ( Mi3 != Ma3 )
% 0.30/0.65                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                                    & ! [X: nat] :
% 0.30/0.65                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                          @ Mima ) ) )
% 0.30/0.65                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.pelims(1)
% 0.30/0.65  thf(fact_9468_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.65        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.65                 => ( Xa2 != one_one_nat ) ) )
% 0.30/0.65           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.65                   => ~ ( ( Deg2 = Xa2 )
% 0.30/0.65                        & ! [X2: vEBT_VEBT] :
% 0.30/0.65                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                        & ( case_o184042715313410164at_nat
% 0.30/0.65                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                            & ! [X: vEBT_VEBT] :
% 0.30/0.65                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                          @ ( produc6081775807080527818_nat_o
% 0.30/0.65                            @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                & ! [I2: nat] :
% 0.30/0.65                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                                & ( ( Mi3 = Ma3 )
% 0.30/0.65                                 => ! [X: vEBT_VEBT] :
% 0.30/0.65                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                                & ( ( Mi3 != Ma3 )
% 0.30/0.65                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                                    & ! [X: nat] :
% 0.30/0.65                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                         => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                          @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.pelims(2)
% 0.30/0.65  thf(fact_9469_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 0.30/0.65      ! [X3: vEBT_VEBT,Xa2: nat] :
% 0.30/0.65        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 0.30/0.65       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 0.30/0.65         => ( ! [Uu2: $o,Uv2: $o] :
% 0.30/0.65                ( ( X3
% 0.30/0.65                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 0.30/0.65               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 0.30/0.65                 => ( Xa2 = one_one_nat ) ) )
% 0.30/0.65           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 0.30/0.65                  ( ( X3
% 0.30/0.65                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 0.30/0.65                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 0.30/0.65                   => ( ( Deg2 = Xa2 )
% 0.30/0.65                      & ! [X4: vEBT_VEBT] :
% 0.30/0.65                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 0.30/0.65                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 0.30/0.65                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                      & ( case_o184042715313410164at_nat
% 0.30/0.65                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 0.30/0.65                          & ! [X: vEBT_VEBT] :
% 0.30/0.65                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                        @ ( produc6081775807080527818_nat_o
% 0.30/0.65                          @ ^ [Mi3: nat,Ma3: nat] :
% 0.30/0.65                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 0.30/0.65                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                              & ! [I2: nat] :
% 0.30/0.65                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.65                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 0.30/0.65                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 0.30/0.65                              & ( ( Mi3 = Ma3 )
% 0.30/0.65                               => ! [X: vEBT_VEBT] :
% 0.30/0.65                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 0.30/0.65                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 0.30/0.65                              & ( ( Mi3 != Ma3 )
% 0.30/0.65                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 0.30/0.65                                  & ! [X: nat] :
% 0.30/0.65                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 0.30/0.65                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 0.30/0.65                                       => ( ( ord_less_nat @ Mi3 @ X )
% 0.30/0.65                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 0.30/0.65                        @ Mima ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % VEBT_internal.valid'.pelims(3)
% 0.30/0.65  thf(fact_9470_GreatestI__nat,axiom,
% 0.30/0.65      ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.65        ( ( P @ K )
% 0.30/0.65       => ( ! [Y5: nat] :
% 0.30/0.65              ( ( P @ Y5 )
% 0.30/0.65             => ( ord_less_eq_nat @ Y5 @ B ) )
% 0.30/0.65         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % GreatestI_nat
% 0.30/0.65  thf(fact_9471_Greatest__le__nat,axiom,
% 0.30/0.65      ! [P: nat > $o,K: nat,B: nat] :
% 0.30/0.65        ( ( P @ K )
% 0.30/0.65       => ( ! [Y5: nat] :
% 0.30/0.65              ( ( P @ Y5 )
% 0.30/0.65             => ( ord_less_eq_nat @ Y5 @ B ) )
% 0.30/0.65         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Greatest_le_nat
% 0.30/0.65  thf(fact_9472_GreatestI__ex__nat,axiom,
% 0.30/0.65      ! [P: nat > $o,B: nat] :
% 0.30/0.65        ( ? [X_12: nat] : ( P @ X_12 )
% 0.30/0.65       => ( ! [Y5: nat] :
% 0.30/0.65              ( ( P @ Y5 )
% 0.30/0.65             => ( ord_less_eq_nat @ Y5 @ B ) )
% 0.30/0.65         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % GreatestI_ex_nat
% 0.30/0.65  thf(fact_9473_take__bit__num__simps_I1_J,axiom,
% 0.30/0.65      ! [M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 0.30/0.65        = none_num ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_simps(1)
% 0.30/0.65  thf(fact_9474_take__bit__num__simps_I2_J,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
% 0.30/0.65        = ( some_num @ one ) ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_simps(2)
% 0.30/0.65  thf(fact_9475_take__bit__num__simps_I3_J,axiom,
% 0.30/0.65      ! [N2: nat,M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
% 0.30/0.65        = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.65          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.30/0.65          @ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_simps(3)
% 0.30/0.65  thf(fact_9476_take__bit__num__simps_I4_J,axiom,
% 0.30/0.65      ! [N2: nat,M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
% 0.30/0.65        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_simps(4)
% 0.30/0.65  thf(fact_9477_take__bit__num__simps_I6_J,axiom,
% 0.30/0.65      ! [R2: num,M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 0.30/0.65        = ( case_o6005452278849405969um_num @ none_num
% 0.30/0.65          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.30/0.65          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_simps(6)
% 0.30/0.65  thf(fact_9478_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 0.30/0.65      ! [N2: nat,M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
% 0.30/0.65        = ( case_nat_option_num @ none_num
% 0.30/0.65          @ ^ [N: nat] :
% 0.30/0.65              ( case_o6005452278849405969um_num @ none_num
% 0.30/0.65              @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.30/0.65              @ ( bit_take_bit_num @ N @ M ) )
% 0.30/0.65          @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Code_Abstract_Nat.take_bit_num_code(2)
% 0.30/0.65  thf(fact_9479_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 0.30/0.65      ! [N2: nat] :
% 0.30/0.65        ( ( bit_take_bit_num @ N2 @ one )
% 0.30/0.65        = ( case_nat_option_num @ none_num
% 0.30/0.65          @ ^ [N: nat] : ( some_num @ one )
% 0.30/0.65          @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Code_Abstract_Nat.take_bit_num_code(1)
% 0.30/0.65  thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 0.30/0.65      ! [N2: nat,M: num] :
% 0.30/0.65        ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
% 0.30/0.65        = ( case_nat_option_num @ none_num
% 0.30/0.65          @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
% 0.30/0.65          @ N2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Code_Abstract_Nat.take_bit_num_code(3)
% 0.30/0.65  thf(fact_9481_take__bit__num__def,axiom,
% 0.30/0.65      ( bit_take_bit_num
% 0.30/0.65      = ( ^ [N: nat,M2: num] :
% 0.30/0.65            ( if_option_num
% 0.30/0.65            @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) )
% 0.30/0.65              = zero_zero_nat )
% 0.30/0.65            @ none_num
% 0.30/0.65            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % take_bit_num_def
% 0.30/0.65  thf(fact_9482_Bit__Operations_Otake__bit__num__code,axiom,
% 0.30/0.65      ( bit_take_bit_num
% 0.30/0.65      = ( ^ [N: nat,M2: num] :
% 0.30/0.65            ( produc478579273971653890on_num
% 0.30/0.65            @ ^ [A3: nat,X: num] :
% 0.30/0.65                ( case_nat_option_num @ none_num
% 0.30/0.65                @ ^ [O: nat] :
% 0.30/0.65                    ( case_num_option_num @ ( some_num @ one )
% 0.30/0.65                    @ ^ [P6: num] :
% 0.30/0.65                        ( case_o6005452278849405969um_num @ none_num
% 0.30/0.65                        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 0.30/0.65                        @ ( bit_take_bit_num @ O @ P6 ) )
% 0.30/0.65                    @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 0.30/0.65                    @ X )
% 0.30/0.65                @ A3 )
% 0.30/0.65            @ ( product_Pair_nat_num @ N @ M2 ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Bit_Operations.take_bit_num_code
% 0.30/0.65  thf(fact_9483_Rats__eq__int__div__nat,axiom,
% 0.30/0.65      ( field_5140801741446780682s_real
% 0.30/0.65      = ( collect_real
% 0.30/0.65        @ ^ [Uu3: real] :
% 0.30/0.65          ? [I2: int,N: nat] :
% 0.30/0.65            ( ( Uu3
% 0.30/0.65              = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 0.30/0.65            & ( N != zero_zero_nat ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Rats_eq_int_div_nat
% 0.30/0.65  thf(fact_9484_Rats__dense__in__real,axiom,
% 0.30/0.65      ! [X3: real,Y3: real] :
% 0.30/0.65        ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.65       => ? [X4: real] :
% 0.30/0.65            ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.30/0.65            & ( ord_less_real @ X3 @ X4 )
% 0.30/0.65            & ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Rats_dense_in_real
% 0.30/0.65  thf(fact_9485_Rats__no__bot__less,axiom,
% 0.30/0.65      ! [X3: real] :
% 0.30/0.65      ? [X4: real] :
% 0.30/0.65        ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 0.30/0.65        & ( ord_less_real @ X4 @ X3 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Rats_no_bot_less
% 0.30/0.65  thf(fact_9486_rat__floor__lemma,axiom,
% 0.30/0.65      ! [A: int,B: int] :
% 0.30/0.65        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 0.30/0.65        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % rat_floor_lemma
% 0.30/0.65  thf(fact_9487_image__minus__const__atLeastLessThan__nat,axiom,
% 0.30/0.65      ! [C: nat,Y3: nat,X3: nat] :
% 0.30/0.65        ( ( ( ord_less_nat @ C @ Y3 )
% 0.30/0.65         => ( ( image_nat_nat
% 0.30/0.65              @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.30/0.65              @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.65            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 0.30/0.65        & ( ~ ( ord_less_nat @ C @ Y3 )
% 0.30/0.65         => ( ( ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.65             => ( ( image_nat_nat
% 0.30/0.65                  @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.30/0.65                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.65                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 0.30/0.65            & ( ~ ( ord_less_nat @ X3 @ Y3 )
% 0.30/0.65             => ( ( image_nat_nat
% 0.30/0.65                  @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 0.30/0.65                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 0.30/0.65                = bot_bot_set_nat ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % image_minus_const_atLeastLessThan_nat
% 0.30/0.65  thf(fact_9488_image__Suc__atLeastAtMost,axiom,
% 0.30/0.65      ! [I: nat,J: nat] :
% 0.30/0.65        ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 0.30/0.65        = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % image_Suc_atLeastAtMost
% 0.30/0.65  thf(fact_9489_image__Suc__atLeastLessThan,axiom,
% 0.30/0.65      ! [I: nat,J: nat] :
% 0.30/0.65        ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 0.30/0.65        = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % image_Suc_atLeastLessThan
% 0.30/0.65  thf(fact_9490_less__rat,axiom,
% 0.30/0.65      ! [B: int,D: int,A: int,C: int] :
% 0.30/0.65        ( ( B != zero_zero_int )
% 0.30/0.65       => ( ( D != zero_zero_int )
% 0.30/0.65         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 0.30/0.65            = ( 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 ) ) ) ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % less_rat
% 0.30/0.65  thf(fact_9491_zero__notin__Suc__image,axiom,
% 0.30/0.65      ! [A2: set_nat] :
% 0.30/0.65        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % zero_notin_Suc_image
% 0.30/0.65  thf(fact_9492_Rat__induct__pos,axiom,
% 0.30/0.65      ! [P: rat > $o,Q4: rat] :
% 0.30/0.65        ( ! [A4: int,B4: int] :
% 0.30/0.65            ( ( ord_less_int @ zero_zero_int @ B4 )
% 0.30/0.65           => ( P @ ( fract @ A4 @ B4 ) ) )
% 0.30/0.65       => ( P @ Q4 ) ) ).
% 0.30/0.65  
% 0.30/0.65  % Rat_induct_pos
% 0.30/0.65  thf(fact_9493_quotient__of__eq,axiom,
% 0.30/0.65      ! [A: int,B: int,P4: int,Q4: int] :
% 0.30/0.65        ( ( ( quotient_of @ ( fract @ A @ B ) )
% 0.30/0.65          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65       => ( ( fract @ P4 @ Q4 )
% 0.30/0.65          = ( fract @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % quotient_of_eq
% 0.30/0.65  thf(fact_9494_normalize__eq,axiom,
% 0.30/0.65      ! [A: int,B: int,P4: int,Q4: int] :
% 0.30/0.65        ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
% 0.30/0.65          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 0.30/0.65       => ( ( fract @ P4 @ Q4 )
% 0.30/0.65          = ( fract @ A @ B ) ) ) ).
% 0.30/0.65  
% 0.30/0.65  % normalize_eq
% 0.30/0.65  thf(fact_9495_quotient__of__Fract,axiom,
% 0.30/0.65      ! [A: int,B: int] :
% 0.30/0.65        ( ( quotient_of @ ( fract @ A @ B ) )
% 0.30/0.66        = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % quotient_of_Fract
% 0.30/0.66  thf(fact_9496_image__Suc__lessThan,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66        = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % image_Suc_lessThan
% 0.30/0.66  thf(fact_9497_image__Suc__atMost,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
% 0.30/0.66        = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % image_Suc_atMost
% 0.30/0.66  thf(fact_9498_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.30/0.66        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast0_atMost_Suc_eq_insert_0
% 0.30/0.66  thf(fact_9499_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 0.30/0.66        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast0_lessThan_Suc_eq_insert_0
% 0.30/0.66  thf(fact_9500_lessThan__Suc__eq__insert__0,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
% 0.30/0.66        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % lessThan_Suc_eq_insert_0
% 0.30/0.66  thf(fact_9501_atMost__Suc__eq__insert__0,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
% 0.30/0.66        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atMost_Suc_eq_insert_0
% 0.30/0.66  thf(fact_9502_Fract__less__zero__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.30/0.66          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Fract_less_zero_iff
% 0.30/0.66  thf(fact_9503_zero__less__Fract__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.30/0.66          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_less_Fract_iff
% 0.30/0.66  thf(fact_9504_one__less__Fract__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.30/0.66          = ( ord_less_int @ B @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_less_Fract_iff
% 0.30/0.66  thf(fact_9505_Fract__less__one__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.30/0.66          = ( ord_less_int @ A @ B ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Fract_less_one_iff
% 0.30/0.66  thf(fact_9506_zero__le__Fract__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 0.30/0.66          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zero_le_Fract_iff
% 0.30/0.66  thf(fact_9507_Fract__le__zero__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 0.30/0.66          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Fract_le_zero_iff
% 0.30/0.66  thf(fact_9508_Fract__le__one__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 0.30/0.66          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Fract_le_one_iff
% 0.30/0.66  thf(fact_9509_one__le__Fract__iff,axiom,
% 0.30/0.66      ! [B: int,A: int] :
% 0.30/0.66        ( ( ord_less_int @ zero_zero_int @ B )
% 0.30/0.66       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 0.30/0.66          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % one_le_Fract_iff
% 0.30/0.66  thf(fact_9510_Gcd__abs__eq,axiom,
% 0.30/0.66      ! [K5: set_int] :
% 0.30/0.66        ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 0.30/0.66        = ( gcd_Gcd_int @ K5 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Gcd_abs_eq
% 0.30/0.66  thf(fact_9511_Gcd__int__eq,axiom,
% 0.30/0.66      ! [N7: set_nat] :
% 0.30/0.66        ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N7 ) )
% 0.30/0.66        = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N7 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Gcd_int_eq
% 0.30/0.66  thf(fact_9512_Gcd__nat__abs__eq,axiom,
% 0.30/0.66      ! [K5: set_int] :
% 0.30/0.66        ( ( gcd_Gcd_nat
% 0.30/0.66          @ ( image_int_nat
% 0.30/0.66            @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
% 0.30/0.66            @ K5 ) )
% 0.30/0.66        = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Gcd_nat_abs_eq
% 0.30/0.66  thf(fact_9513_finite__int__iff__bounded,axiom,
% 0.30/0.66      ( finite_finite_int
% 0.30/0.66      = ( ^ [S7: set_int] :
% 0.30/0.66          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_int_iff_bounded
% 0.30/0.66  thf(fact_9514_finite__int__iff__bounded__le,axiom,
% 0.30/0.66      ( finite_finite_int
% 0.30/0.66      = ( ^ [S7: set_int] :
% 0.30/0.66          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_int_iff_bounded_le
% 0.30/0.66  thf(fact_9515_infinite__int__iff__infinite__nat__abs,axiom,
% 0.30/0.66      ! [S3: set_int] :
% 0.30/0.66        ( ( ~ ( finite_finite_int @ S3 ) )
% 0.30/0.66        = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % infinite_int_iff_infinite_nat_abs
% 0.30/0.66  thf(fact_9516_Gcd__int__def,axiom,
% 0.30/0.66      ( gcd_Gcd_int
% 0.30/0.66      = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Gcd_int_def
% 0.30/0.66  thf(fact_9517_range__mult,axiom,
% 0.30/0.66      ! [A: real] :
% 0.30/0.66        ( ( ( A = zero_zero_real )
% 0.30/0.66         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.66            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 0.30/0.66        & ( ( A != zero_zero_real )
% 0.30/0.66         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 0.30/0.66            = top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % range_mult
% 0.30/0.66  thf(fact_9518_range__abs__Nats,axiom,
% 0.30/0.66      ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 0.30/0.66      = semiring_1_Nats_int ) ).
% 0.30/0.66  
% 0.30/0.66  % range_abs_Nats
% 0.30/0.66  thf(fact_9519_infinite__UNIV__int,axiom,
% 0.30/0.66      ~ ( finite_finite_int @ top_top_set_int ) ).
% 0.30/0.66  
% 0.30/0.66  % infinite_UNIV_int
% 0.30/0.66  thf(fact_9520_nat__not__finite,axiom,
% 0.30/0.66      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_not_finite
% 0.30/0.66  thf(fact_9521_infinite__UNIV__nat,axiom,
% 0.30/0.66      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % infinite_UNIV_nat
% 0.30/0.66  thf(fact_9522_int__in__range__abs,axiom,
% 0.30/0.66      ! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % int_in_range_abs
% 0.30/0.66  thf(fact_9523_UNIV__nat__eq,axiom,
% 0.30/0.66      ( top_top_set_nat
% 0.30/0.66      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % UNIV_nat_eq
% 0.30/0.66  thf(fact_9524_range__mod,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( image_nat_nat
% 0.30/0.66            @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N2 )
% 0.30/0.66            @ top_top_set_nat )
% 0.30/0.66          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % range_mod
% 0.30/0.66  thf(fact_9525_root__def,axiom,
% 0.30/0.66      ( root
% 0.30/0.66      = ( ^ [N: nat,X: real] :
% 0.30/0.66            ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 0.30/0.66            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 0.30/0.66              @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 0.30/0.66              @ X ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % root_def
% 0.30/0.66  thf(fact_9526_DERIV__even__real__root,axiom,
% 0.30/0.66      ! [N2: nat,X3: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66         => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.66           => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_even_real_root
% 0.30/0.66  thf(fact_9527_DERIV__real__root__generic,axiom,
% 0.30/0.66      ! [N2: nat,X3: real,D6: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( X3 != zero_zero_real )
% 0.30/0.66         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66               => ( D6
% 0.30/0.66                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 0.30/0.66           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66               => ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.66                 => ( D6
% 0.30/0.66                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 0.30/0.66             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66                 => ( D6
% 0.30/0.66                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_real_root_generic
% 0.30/0.66  thf(fact_9528_DERIV__neg__dec__right,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_neg_dec_right
% 0.30/0.66  thf(fact_9529_DERIV__pos__inc__right,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pos_inc_right
% 0.30/0.66  thf(fact_9530_DERIV__pos__inc__left,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pos_inc_left
% 0.30/0.66  thf(fact_9531_DERIV__neg__dec__left,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_neg_dec_left
% 0.30/0.66  thf(fact_9532_DERIV__neg__imp__decreasing,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.66             => ( ( ord_less_eq_real @ X4 @ B )
% 0.30/0.66               => ? [Y6: real] :
% 0.30/0.66                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                    & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
% 0.30/0.66         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_neg_imp_decreasing
% 0.30/0.66  thf(fact_9533_DERIV__pos__imp__increasing,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.66             => ( ( ord_less_eq_real @ X4 @ B )
% 0.30/0.66               => ? [Y6: real] :
% 0.30/0.66                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                    & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
% 0.30/0.66         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pos_imp_increasing
% 0.30/0.66  thf(fact_9534_has__real__derivative__pos__inc__right,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
% 0.30/0.66                   => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % has_real_derivative_pos_inc_right
% 0.30/0.66  thf(fact_9535_has__real__derivative__neg__dec__right,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 0.30/0.66       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
% 0.30/0.66                   => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % has_real_derivative_neg_dec_right
% 0.30/0.66  thf(fact_9536_has__real__derivative__neg__dec__left,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 0.30/0.66       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
% 0.30/0.66                   => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % has_real_derivative_neg_dec_left
% 0.30/0.66  thf(fact_9537_has__real__derivative__pos__inc__left,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66         => ? [D4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ D4 )
% 0.30/0.66              & ! [H4: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 0.30/0.66                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
% 0.30/0.66                   => ( ( ord_less_real @ H4 @ D4 )
% 0.30/0.66                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % has_real_derivative_pos_inc_left
% 0.30/0.66  thf(fact_9538_MVT2,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,F5: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.66             => ( ( ord_less_eq_real @ X4 @ B )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66         => ? [Z3: real] :
% 0.30/0.66              ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66              & ( ord_less_real @ Z3 @ B )
% 0.30/0.66              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % MVT2
% 0.30/0.66  thf(fact_9539_DERIV__local__const,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66         => ( ! [Y5: real] :
% 0.30/0.66                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 0.30/0.66               => ( ( F @ X3 )
% 0.30/0.66                  = ( F @ Y5 ) ) )
% 0.30/0.66           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_local_const
% 0.30/0.66  thf(fact_9540_DERIV__ln,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_ln
% 0.30/0.66  thf(fact_9541_DERIV__local__max,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66         => ( ! [Y5: real] :
% 0.30/0.66                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 0.30/0.66               => ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 0.30/0.66           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_local_max
% 0.30/0.66  thf(fact_9542_DERIV__local__min,axiom,
% 0.30/0.66      ! [F: real > real,L: real,X3: real,D: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66         => ( ! [Y5: real] :
% 0.30/0.66                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 0.30/0.66               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 0.30/0.66           => ( L = zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_local_min
% 0.30/0.66  thf(fact_9543_DERIV__ln__divide,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_ln_divide
% 0.30/0.66  thf(fact_9544_DERIV__pow,axiom,
% 0.30/0.66      ! [N2: nat,X3: real,S: set_real] :
% 0.30/0.66        ( has_fi5821293074295781190e_real
% 0.30/0.66        @ ^ [X: real] : ( power_power_real @ X @ N2 )
% 0.30/0.66        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 0.30/0.66        @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pow
% 0.30/0.66  thf(fact_9545_has__real__derivative__powr,axiom,
% 0.30/0.66      ! [Z2: real,R2: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real
% 0.30/0.66          @ ^ [Z6: real] : ( powr_real @ Z6 @ R2 )
% 0.30/0.66          @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 0.30/0.66          @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % has_real_derivative_powr
% 0.30/0.66  thf(fact_9546_DERIV__log,axiom,
% 0.30/0.66      ! [X3: real,B: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_log
% 0.30/0.66  thf(fact_9547_DERIV__fun__powr,axiom,
% 0.30/0.66      ! [G: real > real,M: real,X3: real,R2: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 0.30/0.66         => ( has_fi5821293074295781190e_real
% 0.30/0.66            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 0.30/0.66            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 0.30/0.66            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_fun_powr
% 0.30/0.66  thf(fact_9548_DERIV__powr,axiom,
% 0.30/0.66      ! [G: real > real,M: real,X3: real,F: real > real,R2: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 0.30/0.66         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 0.30/0.66           => ( has_fi5821293074295781190e_real
% 0.30/0.66              @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 0.30/0.66              @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
% 0.30/0.66              @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_powr
% 0.30/0.66  thf(fact_9549_arcosh__real__has__field__derivative,axiom,
% 0.30/0.66      ! [X3: real,A2: set_real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arcosh_real_has_field_derivative
% 0.30/0.66  thf(fact_9550_DERIV__real__sqrt,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_real_sqrt
% 0.30/0.66  thf(fact_9551_DERIV__real__sqrt__generic,axiom,
% 0.30/0.66      ! [X3: real,D6: real] :
% 0.30/0.66        ( ( X3 != zero_zero_real )
% 0.30/0.66       => ( ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66           => ( D6
% 0.30/0.66              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66         => ( ( ( ord_less_real @ X3 @ zero_zero_real )
% 0.30/0.66             => ( D6
% 0.30/0.66                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 0.30/0.66           => ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_real_sqrt_generic
% 0.30/0.66  thf(fact_9552_artanh__real__has__field__derivative,axiom,
% 0.30/0.66      ! [X3: real,A2: set_real] :
% 0.30/0.66        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.66       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % artanh_real_has_field_derivative
% 0.30/0.66  thf(fact_9553_DERIV__real__root,axiom,
% 0.30/0.66      ! [N2: nat,X3: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 0.30/0.66         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_real_root
% 0.30/0.66  thf(fact_9554_DERIV__arcsin,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_arcsin
% 0.30/0.66  thf(fact_9555_Maclaurin__all__le__objl,axiom,
% 0.30/0.66      ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
% 0.30/0.66        ( ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66            = F )
% 0.30/0.66          & ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.30/0.66       => ? [T6: real] :
% 0.30/0.66            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.66            & ( ( F @ X3 )
% 0.30/0.66              = ( plus_plus_real
% 0.30/0.66                @ ( groups6591440286371151544t_real
% 0.30/0.66                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_all_le_objl
% 0.30/0.66  thf(fact_9556_Maclaurin__all__le,axiom,
% 0.30/0.66      ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
% 0.30/0.66        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66          = F )
% 0.30/0.66       => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66         => ? [T6: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.66              & ( ( F @ X3 )
% 0.30/0.66                = ( plus_plus_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_all_le
% 0.30/0.66  thf(fact_9557_DERIV__odd__real__root,axiom,
% 0.30/0.66      ! [N2: nat,X3: real] :
% 0.30/0.66        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66       => ( ( X3 != zero_zero_real )
% 0.30/0.66         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_odd_real_root
% 0.30/0.66  thf(fact_9558_Maclaurin__minus,axiom,
% 0.30/0.66      ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ H2 @ zero_zero_real )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66         => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66              = F )
% 0.30/0.66           => ( ! [M3: nat,T6: real] :
% 0.30/0.66                  ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                    & ( ord_less_eq_real @ H2 @ T6 )
% 0.30/0.66                    & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66             => ? [T6: real] :
% 0.30/0.66                  ( ( ord_less_real @ H2 @ T6 )
% 0.30/0.66                  & ( ord_less_real @ T6 @ zero_zero_real )
% 0.30/0.66                  & ( ( F @ H2 )
% 0.30/0.66                    = ( plus_plus_real
% 0.30/0.66                      @ ( groups6591440286371151544t_real
% 0.30/0.66                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.30/0.66                        @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_minus
% 0.30/0.66  thf(fact_9559_Maclaurin2,axiom,
% 0.30/0.66      ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.66       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66            = F )
% 0.30/0.66         => ( ! [M3: nat,T6: real] :
% 0.30/0.66                ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.66                  & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66           => ? [T6: real] :
% 0.30/0.66                ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.30/0.66                & ( ord_less_eq_real @ T6 @ H2 )
% 0.30/0.66                & ( ( F @ H2 )
% 0.30/0.66                  = ( plus_plus_real
% 0.30/0.66                    @ ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin2
% 0.30/0.66  thf(fact_9560_Maclaurin,axiom,
% 0.30/0.66      ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ H2 )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66         => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66              = F )
% 0.30/0.66           => ( ! [M3: nat,T6: real] :
% 0.30/0.66                  ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                    & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.66                    & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66             => ? [T6: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ T6 )
% 0.30/0.66                  & ( ord_less_real @ T6 @ H2 )
% 0.30/0.66                  & ( ( F @ H2 )
% 0.30/0.66                    = ( plus_plus_real
% 0.30/0.66                      @ ( groups6591440286371151544t_real
% 0.30/0.66                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 0.30/0.66                        @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin
% 0.30/0.66  thf(fact_9561_Maclaurin__all__lt,axiom,
% 0.30/0.66      ! [Diff: nat > real > real,F: real > real,N2: nat,X3: real] :
% 0.30/0.66        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66          = F )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66         => ( ( X3 != zero_zero_real )
% 0.30/0.66           => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66             => ? [T6: real] :
% 0.30/0.66                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 0.30/0.66                  & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.66                  & ( ( F @ X3 )
% 0.30/0.66                    = ( plus_plus_real
% 0.30/0.66                      @ ( groups6591440286371151544t_real
% 0.30/0.66                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.66                        @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_all_lt
% 0.30/0.66  thf(fact_9562_Maclaurin__bi__le,axiom,
% 0.30/0.66      ! [Diff: nat > real > real,F: real > real,N2: nat,X3: real] :
% 0.30/0.66        ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66          = F )
% 0.30/0.66       => ( ! [M3: nat,T6: real] :
% 0.30/0.66              ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) ) )
% 0.30/0.66             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66         => ? [T6: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 0.30/0.66              & ( ( F @ X3 )
% 0.30/0.66                = ( plus_plus_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_bi_le
% 0.30/0.66  thf(fact_9563_Taylor,axiom,
% 0.30/0.66      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X3: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66            = F )
% 0.30/0.66         => ( ! [M3: nat,T6: real] :
% 0.30/0.66                ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                  & ( ord_less_eq_real @ A @ T6 )
% 0.30/0.66                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66           => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.66             => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.66               => ( ( ord_less_eq_real @ A @ X3 )
% 0.30/0.66                 => ( ( ord_less_eq_real @ X3 @ B )
% 0.30/0.66                   => ( ( X3 != C )
% 0.30/0.66                     => ? [T6: real] :
% 0.30/0.66                          ( ( ( ord_less_real @ X3 @ C )
% 0.30/0.66                           => ( ( ord_less_real @ X3 @ T6 )
% 0.30/0.66                              & ( ord_less_real @ T6 @ C ) ) )
% 0.30/0.66                          & ( ~ ( ord_less_real @ X3 @ C )
% 0.30/0.66                           => ( ( ord_less_real @ C @ T6 )
% 0.30/0.66                              & ( ord_less_real @ T6 @ X3 ) ) )
% 0.30/0.66                          & ( ( F @ X3 )
% 0.30/0.66                            = ( plus_plus_real
% 0.30/0.66                              @ ( groups6591440286371151544t_real
% 0.30/0.66                                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M2 ) )
% 0.30/0.66                                @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Taylor
% 0.30/0.66  thf(fact_9564_Taylor__up,axiom,
% 0.30/0.66      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66            = F )
% 0.30/0.66         => ( ! [M3: nat,T6: real] :
% 0.30/0.66                ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                  & ( ord_less_eq_real @ A @ T6 )
% 0.30/0.66                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66           => ( ( ord_less_eq_real @ A @ C )
% 0.30/0.66             => ( ( ord_less_real @ C @ B )
% 0.30/0.66               => ? [T6: real] :
% 0.30/0.66                    ( ( ord_less_real @ C @ T6 )
% 0.30/0.66                    & ( ord_less_real @ T6 @ B )
% 0.30/0.66                    & ( ( F @ B )
% 0.30/0.66                      = ( plus_plus_real
% 0.30/0.66                        @ ( groups6591440286371151544t_real
% 0.30/0.66                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
% 0.30/0.66                          @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Taylor_up
% 0.30/0.66  thf(fact_9565_Taylor__down,axiom,
% 0.30/0.66      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( ( Diff @ zero_zero_nat )
% 0.30/0.66            = F )
% 0.30/0.66         => ( ! [M3: nat,T6: real] :
% 0.30/0.66                ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66                  & ( ord_less_eq_real @ A @ T6 )
% 0.30/0.66                  & ( ord_less_eq_real @ T6 @ B ) )
% 0.30/0.66               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66           => ( ( ord_less_real @ A @ C )
% 0.30/0.66             => ( ( ord_less_eq_real @ C @ B )
% 0.30/0.66               => ? [T6: real] :
% 0.30/0.66                    ( ( ord_less_real @ A @ T6 )
% 0.30/0.66                    & ( ord_less_real @ T6 @ C )
% 0.30/0.66                    & ( ( F @ A )
% 0.30/0.66                      = ( plus_plus_real
% 0.30/0.66                        @ ( groups6591440286371151544t_real
% 0.30/0.66                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
% 0.30/0.66                          @ ( set_ord_lessThan_nat @ N2 ) )
% 0.30/0.66                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Taylor_down
% 0.30/0.66  thf(fact_9566_Maclaurin__lemma2,axiom,
% 0.30/0.66      ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 0.30/0.66        ( ! [M3: nat,T6: real] :
% 0.30/0.66            ( ( ( ord_less_nat @ M3 @ N2 )
% 0.30/0.66              & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 0.30/0.66              & ( ord_less_eq_real @ T6 @ H2 ) )
% 0.30/0.66           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 0.30/0.66       => ( ( N2
% 0.30/0.66            = ( suc @ K ) )
% 0.30/0.66         => ! [M6: nat,T7: real] :
% 0.30/0.66              ( ( ( ord_less_nat @ M6 @ N2 )
% 0.30/0.66                & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 0.30/0.66                & ( ord_less_eq_real @ T7 @ H2 ) )
% 0.30/0.66             => ( has_fi5821293074295781190e_real
% 0.30/0.66                @ ^ [U2: real] :
% 0.30/0.66                    ( minus_minus_real @ ( Diff @ M6 @ U2 )
% 0.30/0.66                    @ ( plus_plus_real
% 0.30/0.66                      @ ( groups6591440286371151544t_real
% 0.30/0.66                        @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M6 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
% 0.30/0.66                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M6 ) ) )
% 0.30/0.66                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M6 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M6 ) ) ) ) ) )
% 0.30/0.66                @ ( minus_minus_real @ ( Diff @ ( suc @ M6 ) @ T7 )
% 0.30/0.66                  @ ( plus_plus_real
% 0.30/0.66                    @ ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M6 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) )
% 0.30/0.66                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) )
% 0.30/0.66                @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Maclaurin_lemma2
% 0.30/0.66  thf(fact_9567_DERIV__arctan__series,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.66       => ( has_fi5821293074295781190e_real
% 0.30/0.66          @ ^ [X8: real] :
% 0.30/0.66              ( suminf_real
% 0.30/0.66              @ ^ [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 @ X8 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 0.30/0.66          @ ( suminf_real
% 0.30/0.66            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 0.30/0.66          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_arctan_series
% 0.30/0.66  thf(fact_9568_DERIV__arccos,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_arccos
% 0.30/0.66  thf(fact_9569_DERIV__power__series_H,axiom,
% 0.30/0.66      ! [R: real,F: nat > real,X0: real] :
% 0.30/0.66        ( ! [X4: real] :
% 0.30/0.66            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.30/0.66           => ( summable_real
% 0.30/0.66              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X4 @ N ) ) ) )
% 0.30/0.66       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ R )
% 0.30/0.66           => ( has_fi5821293074295781190e_real
% 0.30/0.66              @ ^ [X: real] :
% 0.30/0.66                  ( suminf_real
% 0.30/0.66                  @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ ( suc @ N ) ) ) )
% 0.30/0.66              @ ( suminf_real
% 0.30/0.66                @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
% 0.30/0.66              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_power_series'
% 0.30/0.66  thf(fact_9570_arccos__less__arccos,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ Y3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 0.30/0.66           => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_less_arccos
% 0.30/0.66  thf(fact_9571_arccos__less__mono,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.66       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
% 0.30/0.66            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_less_mono
% 0.30/0.66  thf(fact_9572_DERIV__isconst3,axiom,
% 0.30/0.66      ! [A: real,B: real,X3: real,Y3: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66         => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66           => ( ! [X4: real] :
% 0.30/0.66                  ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.30/0.66             => ( ( F @ X3 )
% 0.30/0.66                = ( F @ Y3 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_isconst3
% 0.30/0.66  thf(fact_9573_arccos__lt__bounded,axiom,
% 0.30/0.66      ! [Y3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 0.30/0.66       => ( ( ord_less_real @ Y3 @ one_one_real )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 0.30/0.66            & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % arccos_lt_bounded
% 0.30/0.66  thf(fact_9574_sin__arccos__nonzero,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( ( sin_real @ ( arccos @ X3 ) )
% 0.30/0.66           != zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sin_arccos_nonzero
% 0.30/0.66  thf(fact_9575_finite__greaterThanLessThan,axiom,
% 0.30/0.66      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_greaterThanLessThan
% 0.30/0.66  thf(fact_9576_finite__greaterThanLessThan__int,axiom,
% 0.30/0.66      ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_greaterThanLessThan_int
% 0.30/0.66  thf(fact_9577_card__greaterThanLessThan,axiom,
% 0.30/0.66      ! [L: nat,U: nat] :
% 0.30/0.66        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 0.30/0.66        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % card_greaterThanLessThan
% 0.30/0.66  thf(fact_9578_atLeastSucLessThan__greaterThanLessThan,axiom,
% 0.30/0.66      ! [L: nat,U: nat] :
% 0.30/0.66        ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 0.30/0.66        = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastSucLessThan_greaterThanLessThan
% 0.30/0.66  thf(fact_9579_LIM__fun__less__zero,axiom,
% 0.30/0.66      ! [F: real > real,L: real,C: real] :
% 0.30/0.66        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ L @ zero_zero_real )
% 0.30/0.66         => ? [R4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.30/0.66              & ! [X2: real] :
% 0.30/0.66                  ( ( ( X2 != C )
% 0.30/0.66                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 0.30/0.66                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIM_fun_less_zero
% 0.30/0.66  thf(fact_9580_LIM__fun__not__zero,axiom,
% 0.30/0.66      ! [F: real > real,L: real,C: real] :
% 0.30/0.66        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66       => ( ( L != zero_zero_real )
% 0.30/0.66         => ? [R4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.30/0.66              & ! [X2: real] :
% 0.30/0.66                  ( ( ( X2 != C )
% 0.30/0.66                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 0.30/0.66                 => ( ( F @ X2 )
% 0.30/0.66                   != zero_zero_real ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIM_fun_not_zero
% 0.30/0.66  thf(fact_9581_LIM__fun__gt__zero,axiom,
% 0.30/0.66      ! [F: real > real,L: real,C: real] :
% 0.30/0.66        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 0.30/0.66       => ( ( ord_less_real @ zero_zero_real @ L )
% 0.30/0.66         => ? [R4: real] :
% 0.30/0.66              ( ( ord_less_real @ zero_zero_real @ R4 )
% 0.30/0.66              & ! [X2: real] :
% 0.30/0.66                  ( ( ( X2 != C )
% 0.30/0.66                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 0.30/0.66                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIM_fun_gt_zero
% 0.30/0.66  thf(fact_9582_isCont__inverse__function2,axiom,
% 0.30/0.66      ! [A: real,X3: real,B: real,G: real > real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ B )
% 0.30/0.66         => ( ! [Z3: real] :
% 0.30/0.66                ( ( ord_less_eq_real @ A @ Z3 )
% 0.30/0.66               => ( ( ord_less_eq_real @ Z3 @ B )
% 0.30/0.66                 => ( ( G @ ( F @ Z3 ) )
% 0.30/0.66                    = Z3 ) ) )
% 0.30/0.66           => ( ! [Z3: real] :
% 0.30/0.66                  ( ( ord_less_eq_real @ A @ Z3 )
% 0.30/0.66                 => ( ( ord_less_eq_real @ Z3 @ B )
% 0.30/0.66                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 0.30/0.66             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_inverse_function2
% 0.30/0.66  thf(fact_9583_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.66      ! [N2: nat,J: nat,I: nat] :
% 0.30/0.66        ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 0.30/0.66       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
% 0.30/0.66          = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nth_sorted_list_of_set_greaterThanLessThan
% 0.30/0.66  thf(fact_9584_isCont__arcosh,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_arcosh
% 0.30/0.66  thf(fact_9585_DERIV__inverse__function,axiom,
% 0.30/0.66      ! [F: real > real,D6: real,G: real > real,X3: real,A: real,B: real] :
% 0.30/0.66        ( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
% 0.30/0.66       => ( ( D6 != zero_zero_real )
% 0.30/0.66         => ( ( ord_less_real @ A @ X3 )
% 0.30/0.66           => ( ( ord_less_real @ X3 @ B )
% 0.30/0.66             => ( ! [Y5: real] :
% 0.30/0.66                    ( ( ord_less_real @ A @ Y5 )
% 0.30/0.66                   => ( ( ord_less_real @ Y5 @ B )
% 0.30/0.66                     => ( ( F @ ( G @ Y5 ) )
% 0.30/0.66                        = Y5 ) ) )
% 0.30/0.66               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_inverse_function
% 0.30/0.66  thf(fact_9586_isCont__arccos,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_arccos
% 0.30/0.66  thf(fact_9587_isCont__arcsin,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_arcsin
% 0.30/0.66  thf(fact_9588_LIM__less__bound,axiom,
% 0.30/0.66      ! [B: real,X3: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ B @ X3 )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X3 ) )
% 0.30/0.66             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 0.30/0.66         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
% 0.30/0.66           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIM_less_bound
% 0.30/0.66  thf(fact_9589_isCont__artanh,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_artanh
% 0.30/0.66  thf(fact_9590_isCont__inverse__function,axiom,
% 0.30/0.66      ! [D: real,X3: real,G: real > real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ D )
% 0.30/0.66       => ( ! [Z3: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
% 0.30/0.66             => ( ( G @ ( F @ Z3 ) )
% 0.30/0.66                = Z3 ) )
% 0.30/0.66         => ( ! [Z3: real] :
% 0.30/0.66                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
% 0.30/0.66               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
% 0.30/0.66           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % isCont_inverse_function
% 0.30/0.66  thf(fact_9591_GMVT_H,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [Z3: real] :
% 0.30/0.66              ( ( ord_less_eq_real @ A @ Z3 )
% 0.30/0.66             => ( ( ord_less_eq_real @ Z3 @ B )
% 0.30/0.66               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 0.30/0.66         => ( ! [Z3: real] :
% 0.30/0.66                ( ( ord_less_eq_real @ A @ Z3 )
% 0.30/0.66               => ( ( ord_less_eq_real @ Z3 @ B )
% 0.30/0.66                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
% 0.30/0.66           => ( ! [Z3: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66                 => ( ( ord_less_real @ Z3 @ B )
% 0.30/0.66                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 0.30/0.66             => ( ! [Z3: real] :
% 0.30/0.66                    ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66                   => ( ( ord_less_real @ Z3 @ B )
% 0.30/0.66                     => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 0.30/0.66               => ? [C2: real] :
% 0.30/0.66                    ( ( ord_less_real @ A @ C2 )
% 0.30/0.66                    & ( ord_less_real @ C2 @ B )
% 0.30/0.66                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 0.30/0.66                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % GMVT'
% 0.30/0.66  thf(fact_9592_summable__Leibniz_I2_J,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 0.30/0.66           => ! [N6: nat] :
% 0.30/0.66                ( member_real
% 0.30/0.66                @ ( suminf_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.30/0.66                @ ( set_or1222579329274155063t_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz(2)
% 0.30/0.66  thf(fact_9593_summable__Leibniz_I3_J,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 0.30/0.66           => ! [N6: nat] :
% 0.30/0.66                ( member_real
% 0.30/0.66                @ ( suminf_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.30/0.66                @ ( set_or1222579329274155063t_real
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 0.30/0.66                  @ ( groups6591440286371151544t_real
% 0.30/0.66                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz(3)
% 0.30/0.66  thf(fact_9594_summable__Leibniz_H_I5_J,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66           => ( filterlim_nat_real
% 0.30/0.66              @ ^ [N: nat] :
% 0.30/0.66                  ( groups6591440286371151544t_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.30/0.66              @ ( topolo2815343760600316023s_real
% 0.30/0.66                @ ( suminf_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.30/0.66              @ at_top_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz'(5)
% 0.30/0.66  thf(fact_9595_trivial__limit__sequentially,axiom,
% 0.30/0.66      at_top_nat != bot_bot_filter_nat ).
% 0.30/0.66  
% 0.30/0.66  % trivial_limit_sequentially
% 0.30/0.66  thf(fact_9596_filterlim__Suc,axiom,
% 0.30/0.66      filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 0.30/0.66  
% 0.30/0.66  % filterlim_Suc
% 0.30/0.66  thf(fact_9597_mult__nat__left__at__top,axiom,
% 0.30/0.66      ! [C: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.66       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_nat_left_at_top
% 0.30/0.66  thf(fact_9598_mult__nat__right__at__top,axiom,
% 0.30/0.66      ! [C: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ C )
% 0.30/0.66       => ( filterlim_nat_nat
% 0.30/0.66          @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 0.30/0.66          @ at_top_nat
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mult_nat_right_at_top
% 0.30/0.66  thf(fact_9599_nested__sequence__unique,axiom,
% 0.30/0.66      ! [F: nat > real,G: nat > real] :
% 0.30/0.66        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 0.30/0.66           => ( ( filterlim_nat_real
% 0.30/0.66                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 0.30/0.66                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66                @ at_top_nat )
% 0.30/0.66             => ? [L4: real] :
% 0.30/0.66                  ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
% 0.30/0.66                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 0.30/0.66                  & ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G @ N6 ) )
% 0.30/0.66                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nested_sequence_unique
% 0.30/0.66  thf(fact_9600_LIMSEQ__inverse__zero,axiom,
% 0.30/0.66      ! [X7: nat > real] :
% 0.30/0.66        ( ! [R4: real] :
% 0.30/0.66          ? [N8: nat] :
% 0.30/0.66          ! [N3: nat] :
% 0.30/0.66            ( ( ord_less_eq_nat @ N8 @ N3 )
% 0.30/0.66           => ( ord_less_real @ R4 @ ( X7 @ N3 ) ) )
% 0.30/0.66       => ( filterlim_nat_real
% 0.30/0.66          @ ^ [N: nat] : ( inverse_inverse_real @ ( X7 @ N ) )
% 0.30/0.66          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_zero
% 0.30/0.66  thf(fact_9601_LIMSEQ__root__const,axiom,
% 0.30/0.66      ! [C: real] :
% 0.30/0.66        ( ( ord_less_real @ zero_zero_real @ C )
% 0.30/0.66       => ( filterlim_nat_real
% 0.30/0.66          @ ^ [N: nat] : ( root @ N @ C )
% 0.30/0.66          @ ( topolo2815343760600316023s_real @ one_one_real )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_root_const
% 0.30/0.66  thf(fact_9602_LIMSEQ__inverse__real__of__nat,axiom,
% 0.30/0.66      ( filterlim_nat_real
% 0.30/0.66      @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 0.30/0.66      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66      @ at_top_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_real_of_nat
% 0.30/0.66  thf(fact_9603_LIMSEQ__inverse__real__of__nat__add,axiom,
% 0.30/0.66      ! [R2: real] :
% 0.30/0.66        ( filterlim_nat_real
% 0.30/0.66        @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
% 0.30/0.66        @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66        @ at_top_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_real_of_nat_add
% 0.30/0.66  thf(fact_9604_increasing__LIMSEQ,axiom,
% 0.30/0.66      ! [F: nat > real,L: real] :
% 0.30/0.66        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 0.30/0.66         => ( ! [E: real] :
% 0.30/0.66                ( ( ord_less_real @ zero_zero_real @ E )
% 0.30/0.66               => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
% 0.30/0.66           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % increasing_LIMSEQ
% 0.30/0.66  thf(fact_9605_LIMSEQ__realpow__zero,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ord_less_real @ X3 @ one_one_real )
% 0.30/0.66         => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_realpow_zero
% 0.30/0.66  thf(fact_9606_LIMSEQ__divide__realpow__zero,axiom,
% 0.30/0.66      ! [X3: real,A: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( filterlim_nat_real
% 0.30/0.66          @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N ) )
% 0.30/0.66          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_divide_realpow_zero
% 0.30/0.66  thf(fact_9607_LIMSEQ__abs__realpow__zero2,axiom,
% 0.30/0.66      ! [C: real] :
% 0.30/0.66        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.66       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_abs_realpow_zero2
% 0.30/0.66  thf(fact_9608_LIMSEQ__abs__realpow__zero,axiom,
% 0.30/0.66      ! [C: real] :
% 0.30/0.66        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 0.30/0.66       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_abs_realpow_zero
% 0.30/0.66  thf(fact_9609_LIMSEQ__inverse__realpow__zero,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ X3 )
% 0.30/0.66       => ( filterlim_nat_real
% 0.30/0.66          @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N ) )
% 0.30/0.66          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_realpow_zero
% 0.30/0.66  thf(fact_9610_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 0.30/0.66      ! [R2: real] :
% 0.30/0.66        ( filterlim_nat_real
% 0.30/0.66        @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
% 0.30/0.66        @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66        @ at_top_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_real_of_nat_add_minus
% 0.30/0.66  thf(fact_9611_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 0.30/0.66      ! [R2: real] :
% 0.30/0.66        ( filterlim_nat_real
% 0.30/0.66        @ ^ [N: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
% 0.30/0.66        @ ( topolo2815343760600316023s_real @ R2 )
% 0.30/0.66        @ at_top_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % LIMSEQ_inverse_real_of_nat_add_minus_mult
% 0.30/0.66  thf(fact_9612_summable,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66           => ( summable_real
% 0.30/0.66              @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable
% 0.30/0.66  thf(fact_9613_zeroseq__arctan__series,axiom,
% 0.30/0.66      ! [X3: real] :
% 0.30/0.66        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 0.30/0.66       => ( filterlim_nat_real
% 0.30/0.66          @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 0.30/0.66          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % zeroseq_arctan_series
% 0.30/0.66  thf(fact_9614_summable__Leibniz_H_I3_J,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66           => ( filterlim_nat_real
% 0.30/0.66              @ ^ [N: nat] :
% 0.30/0.66                  ( groups6591440286371151544t_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.66              @ ( topolo2815343760600316023s_real
% 0.30/0.66                @ ( suminf_real
% 0.30/0.66                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.30/0.66              @ at_top_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz'(3)
% 0.30/0.66  thf(fact_9615_summable__Leibniz_H_I2_J,axiom,
% 0.30/0.66      ! [A: nat > real,N2: nat] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66           => ( ord_less_eq_real
% 0.30/0.66              @ ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 0.30/0.66              @ ( suminf_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz'(2)
% 0.30/0.66  thf(fact_9616_sums__alternating__upper__lower,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66           => ? [L4: real] :
% 0.30/0.66                ( ! [N6: nat] :
% 0.30/0.66                    ( ord_less_eq_real
% 0.30/0.66                    @ ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 0.30/0.66                    @ L4 )
% 0.30/0.66                & ( filterlim_nat_real
% 0.30/0.66                  @ ^ [N: nat] :
% 0.30/0.66                      ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 0.30/0.66                  @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.66                  @ at_top_nat )
% 0.30/0.66                & ! [N6: nat] :
% 0.30/0.66                    ( ord_less_eq_real @ L4
% 0.30/0.66                    @ ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 0.30/0.66                & ( filterlim_nat_real
% 0.30/0.66                  @ ^ [N: nat] :
% 0.30/0.66                      ( groups6591440286371151544t_real
% 0.30/0.66                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.30/0.66                  @ ( topolo2815343760600316023s_real @ L4 )
% 0.30/0.66                  @ at_top_nat ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sums_alternating_upper_lower
% 0.30/0.66  thf(fact_9617_summable__Leibniz_I5_J,axiom,
% 0.30/0.66      ! [A: nat > real] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ( topolo6980174941875973593q_real @ A )
% 0.30/0.66         => ( filterlim_nat_real
% 0.30/0.66            @ ^ [N: nat] :
% 0.30/0.66                ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 0.30/0.66            @ ( topolo2815343760600316023s_real
% 0.30/0.66              @ ( suminf_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 0.30/0.66            @ at_top_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz(5)
% 0.30/0.66  thf(fact_9618_summable__Leibniz_H_I4_J,axiom,
% 0.30/0.66      ! [A: nat > real,N2: nat] :
% 0.30/0.66        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 0.30/0.66       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 0.30/0.66         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 0.30/0.66           => ( ord_less_eq_real
% 0.30/0.66              @ ( suminf_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 0.30/0.66              @ ( groups6591440286371151544t_real
% 0.30/0.66                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 0.30/0.66                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % summable_Leibniz'(4)
% 0.30/0.66  thf(fact_9619_eventually__sequentially__Suc,axiom,
% 0.30/0.66      ! [P: nat > $o] :
% 0.30/0.66        ( ( eventually_nat
% 0.30/0.66          @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 0.30/0.66          @ at_top_nat )
% 0.30/0.66        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_sequentially_Suc
% 0.30/0.66  thf(fact_9620_eventually__sequentially__seg,axiom,
% 0.30/0.66      ! [P: nat > $o,K: nat] :
% 0.30/0.66        ( ( eventually_nat
% 0.30/0.66          @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
% 0.30/0.66          @ at_top_nat )
% 0.30/0.66        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_sequentially_seg
% 0.30/0.66  thf(fact_9621_eventually__sequentiallyI,axiom,
% 0.30/0.66      ! [C: nat,P: nat > $o] :
% 0.30/0.66        ( ! [X4: nat] :
% 0.30/0.66            ( ( ord_less_eq_nat @ C @ X4 )
% 0.30/0.66           => ( P @ X4 ) )
% 0.30/0.66       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_sequentiallyI
% 0.30/0.66  thf(fact_9622_eventually__sequentially,axiom,
% 0.30/0.66      ! [P: nat > $o] :
% 0.30/0.66        ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.66        = ( ? [N5: nat] :
% 0.30/0.66            ! [N: nat] :
% 0.30/0.66              ( ( ord_less_eq_nat @ N5 @ N )
% 0.30/0.66             => ( P @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_sequentially
% 0.30/0.66  thf(fact_9623_le__sequentially,axiom,
% 0.30/0.66      ! [F3: filter_nat] :
% 0.30/0.66        ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
% 0.30/0.66        = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % le_sequentially
% 0.30/0.66  thf(fact_9624_sequentially__offset,axiom,
% 0.30/0.66      ! [P: nat > $o,K: nat] :
% 0.30/0.66        ( ( eventually_nat @ P @ at_top_nat )
% 0.30/0.66       => ( eventually_nat
% 0.30/0.66          @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
% 0.30/0.66          @ at_top_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sequentially_offset
% 0.30/0.66  thf(fact_9625_DERIV__neg__imp__decreasing__at__top,axiom,
% 0.30/0.66      ! [B: real,F: real > real,Flim: real] :
% 0.30/0.66        ( ! [X4: real] :
% 0.30/0.66            ( ( ord_less_eq_real @ B @ X4 )
% 0.30/0.66           => ? [Y6: real] :
% 0.30/0.66                ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                & ( ord_less_real @ Y6 @ zero_zero_real ) ) )
% 0.30/0.66       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 0.30/0.66         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_neg_imp_decreasing_at_top
% 0.30/0.66  thf(fact_9626_finite__greaterThanAtMost,axiom,
% 0.30/0.66      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_greaterThanAtMost
% 0.30/0.66  thf(fact_9627_atLeastSucAtMost__greaterThanAtMost,axiom,
% 0.30/0.66      ! [L: nat,U: nat] :
% 0.30/0.66        ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 0.30/0.66        = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastSucAtMost_greaterThanAtMost
% 0.30/0.66  thf(fact_9628_eventually__at__left__real,axiom,
% 0.30/0.66      ! [B: real,A: real] :
% 0.30/0.66        ( ( ord_less_real @ B @ A )
% 0.30/0.66       => ( eventually_real
% 0.30/0.66          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B @ A ) )
% 0.30/0.66          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_at_left_real
% 0.30/0.66  thf(fact_9629_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.66      ! [N2: nat,J: nat,I: nat] :
% 0.30/0.66        ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
% 0.30/0.66       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
% 0.30/0.66          = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nth_sorted_list_of_set_greaterThanAtMost
% 0.30/0.66  thf(fact_9630_filterlim__pow__at__bot__even,axiom,
% 0.30/0.66      ! [N2: nat,F: real > real,F3: filter_real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 0.30/0.66         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66           => ( filterlim_real_real
% 0.30/0.66              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
% 0.30/0.66              @ at_top_real
% 0.30/0.66              @ F3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % filterlim_pow_at_bot_even
% 0.30/0.66  thf(fact_9631_finite__greaterThanAtMost__int,axiom,
% 0.30/0.66      ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_greaterThanAtMost_int
% 0.30/0.66  thf(fact_9632_DERIV__pos__imp__increasing__at__bot,axiom,
% 0.30/0.66      ! [B: real,F: real > real,Flim: real] :
% 0.30/0.66        ( ! [X4: real] :
% 0.30/0.66            ( ( ord_less_eq_real @ X4 @ B )
% 0.30/0.66           => ? [Y6: real] :
% 0.30/0.66                ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                & ( ord_less_real @ zero_zero_real @ Y6 ) ) )
% 0.30/0.66       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 0.30/0.66         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pos_imp_increasing_at_bot
% 0.30/0.66  thf(fact_9633_filterlim__pow__at__bot__odd,axiom,
% 0.30/0.66      ! [N2: nat,F: real > real,F3: filter_real] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 0.30/0.66         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 0.30/0.66           => ( filterlim_real_real
% 0.30/0.66              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
% 0.30/0.66              @ at_bot_real
% 0.30/0.66              @ F3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % filterlim_pow_at_bot_odd
% 0.30/0.66  thf(fact_9634_GMVT,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,G: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.66                & ( ord_less_eq_real @ X4 @ B ) )
% 0.30/0.66             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 0.30/0.66         => ( ! [X4: real] :
% 0.30/0.66                ( ( ( ord_less_real @ A @ X4 )
% 0.30/0.66                  & ( ord_less_real @ X4 @ B ) )
% 0.30/0.66               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.30/0.66           => ( ! [X4: real] :
% 0.30/0.66                  ( ( ( ord_less_eq_real @ A @ X4 )
% 0.30/0.66                    & ( ord_less_eq_real @ X4 @ B ) )
% 0.30/0.66                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 0.30/0.66             => ( ! [X4: real] :
% 0.30/0.66                    ( ( ( ord_less_real @ A @ X4 )
% 0.30/0.66                      & ( ord_less_real @ X4 @ B ) )
% 0.30/0.66                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 0.30/0.66               => ? [G_c: real,F_c: real,C2: real] :
% 0.30/0.66                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.30/0.66                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 0.30/0.66                    & ( ord_less_real @ A @ C2 )
% 0.30/0.66                    & ( ord_less_real @ C2 @ B )
% 0.30/0.66                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 0.30/0.66                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % GMVT
% 0.30/0.66  thf(fact_9635_MVT,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66         => ( ! [X4: real] :
% 0.30/0.66                ( ( ord_less_real @ A @ X4 )
% 0.30/0.66               => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66           => ? [L4: real,Z3: real] :
% 0.30/0.66                ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66                & ( ord_less_real @ Z3 @ B )
% 0.30/0.66                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
% 0.30/0.66                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % MVT
% 0.30/0.66  thf(fact_9636_Rolle__deriv,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,F5: real > real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( ( F @ A )
% 0.30/0.66            = ( F @ B ) )
% 0.30/0.66         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66           => ( ! [X4: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ X4 )
% 0.30/0.66                 => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                   => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66             => ? [Z3: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66                  & ( ord_less_real @ Z3 @ B )
% 0.30/0.66                  & ( ( F5 @ Z3 )
% 0.30/0.66                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Rolle_deriv
% 0.30/0.66  thf(fact_9637_mvt,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,F5: real > real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66         => ( ! [X4: real] :
% 0.30/0.66                ( ( ord_less_real @ A @ X4 )
% 0.30/0.66               => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66           => ~ ! [Xi: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ Xi )
% 0.30/0.66                 => ( ( ord_less_real @ Xi @ B )
% 0.30/0.66                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 0.30/0.66                     != ( F5 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mvt
% 0.30/0.66  thf(fact_9638_DERIV__pos__imp__increasing__open,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ord_less_real @ A @ X4 )
% 0.30/0.66             => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66               => ? [Y6: real] :
% 0.30/0.66                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                    & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
% 0.30/0.66         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_pos_imp_increasing_open
% 0.30/0.66  thf(fact_9639_DERIV__neg__imp__decreasing__open,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ! [X4: real] :
% 0.30/0.66              ( ( ord_less_real @ A @ X4 )
% 0.30/0.66             => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66               => ? [Y6: real] :
% 0.30/0.66                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66                    & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
% 0.30/0.66         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_neg_imp_decreasing_open
% 0.30/0.66  thf(fact_9640_DERIV__isconst__end,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66         => ( ! [X4: real] :
% 0.30/0.66                ( ( ord_less_real @ A @ X4 )
% 0.30/0.66               => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66           => ( ( F @ B )
% 0.30/0.66              = ( F @ A ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_isconst_end
% 0.30/0.66  thf(fact_9641_DERIV__isconst2,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real,X3: real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66         => ( ! [X4: real] :
% 0.30/0.66                ( ( ord_less_real @ A @ X4 )
% 0.30/0.66               => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66           => ( ( ord_less_eq_real @ A @ X3 )
% 0.30/0.66             => ( ( ord_less_eq_real @ X3 @ B )
% 0.30/0.66               => ( ( F @ X3 )
% 0.30/0.66                  = ( F @ A ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % DERIV_isconst2
% 0.30/0.66  thf(fact_9642_Rolle,axiom,
% 0.30/0.66      ! [A: real,B: real,F: real > real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( ( ( F @ A )
% 0.30/0.66            = ( F @ B ) )
% 0.30/0.66         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 0.30/0.66           => ( ! [X4: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ X4 )
% 0.30/0.66                 => ( ( ord_less_real @ X4 @ B )
% 0.30/0.66                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 0.30/0.66             => ? [Z3: real] :
% 0.30/0.66                  ( ( ord_less_real @ A @ Z3 )
% 0.30/0.66                  & ( ord_less_real @ Z3 @ B )
% 0.30/0.66                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Rolle
% 0.30/0.66  thf(fact_9643_INT__greaterThan__UNIV,axiom,
% 0.30/0.66      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 0.30/0.66      = bot_bot_set_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % INT_greaterThan_UNIV
% 0.30/0.66  thf(fact_9644_greaterThan__0,axiom,
% 0.30/0.66      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 0.30/0.66      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThan_0
% 0.30/0.66  thf(fact_9645_eventually__at__right__real,axiom,
% 0.30/0.66      ! [A: real,B: real] :
% 0.30/0.66        ( ( ord_less_real @ A @ B )
% 0.30/0.66       => ( eventually_real
% 0.30/0.66          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 0.30/0.66          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_at_right_real
% 0.30/0.66  thf(fact_9646_greaterThan__Suc,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 0.30/0.66        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThan_Suc
% 0.30/0.66  thf(fact_9647_atLeast__0,axiom,
% 0.30/0.66      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 0.30/0.66      = top_top_set_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast_0
% 0.30/0.66  thf(fact_9648_atLeast__Suc__greaterThan,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.30/0.66        = ( set_or1210151606488870762an_nat @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast_Suc_greaterThan
% 0.30/0.66  thf(fact_9649_atLeast__Suc,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 0.30/0.66        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast_Suc
% 0.30/0.66  thf(fact_9650_Gcd__eq__Max,axiom,
% 0.30/0.66      ! [M7: set_nat] :
% 0.30/0.66        ( ( finite_finite_nat @ M7 )
% 0.30/0.66       => ( ( M7 != bot_bot_set_nat )
% 0.30/0.66         => ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 0.30/0.66           => ( ( gcd_Gcd_nat @ M7 )
% 0.30/0.66              = ( lattic8265883725875713057ax_nat
% 0.30/0.66                @ ( comple7806235888213564991et_nat
% 0.30/0.66                  @ ( image_nat_set_nat
% 0.30/0.66                    @ ^ [M2: nat] :
% 0.30/0.66                        ( collect_nat
% 0.30/0.66                        @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
% 0.30/0.66                    @ M7 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Gcd_eq_Max
% 0.30/0.66  thf(fact_9651_Max__divisors__self__nat,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( N2 != zero_zero_nat )
% 0.30/0.66       => ( ( lattic8265883725875713057ax_nat
% 0.30/0.66            @ ( collect_nat
% 0.30/0.66              @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N2 ) ) )
% 0.30/0.66          = N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Max_divisors_self_nat
% 0.30/0.66  thf(fact_9652_bdd__above__nat,axiom,
% 0.30/0.66      condit2214826472909112428ve_nat = finite_finite_nat ).
% 0.30/0.66  
% 0.30/0.66  % bdd_above_nat
% 0.30/0.66  thf(fact_9653_card__le__Suc__Max,axiom,
% 0.30/0.66      ! [S3: set_nat] :
% 0.30/0.66        ( ( finite_finite_nat @ S3 )
% 0.30/0.66       => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % card_le_Suc_Max
% 0.30/0.66  thf(fact_9654_Sup__nat__def,axiom,
% 0.30/0.66      ( complete_Sup_Sup_nat
% 0.30/0.66      = ( ^ [X5: set_nat] : ( if_nat @ ( X5 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Sup_nat_def
% 0.30/0.66  thf(fact_9655_divide__nat__def,axiom,
% 0.30/0.66      ( divide_divide_nat
% 0.30/0.66      = ( ^ [M2: nat,N: nat] :
% 0.30/0.66            ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 0.30/0.66            @ ( lattic8265883725875713057ax_nat
% 0.30/0.66              @ ( collect_nat
% 0.30/0.66                @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % divide_nat_def
% 0.30/0.66  thf(fact_9656_gcd__is__Max__divisors__nat,axiom,
% 0.30/0.66      ! [N2: nat,M: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( ( gcd_gcd_nat @ M @ N2 )
% 0.30/0.66          = ( lattic8265883725875713057ax_nat
% 0.30/0.66            @ ( collect_nat
% 0.30/0.66              @ ^ [D5: nat] :
% 0.30/0.66                  ( ( dvd_dvd_nat @ D5 @ M )
% 0.30/0.66                  & ( dvd_dvd_nat @ D5 @ N2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gcd_is_Max_divisors_nat
% 0.30/0.66  thf(fact_9657_Max__divisors__self__int,axiom,
% 0.30/0.66      ! [N2: int] :
% 0.30/0.66        ( ( N2 != zero_zero_int )
% 0.30/0.66       => ( ( lattic8263393255366662781ax_int
% 0.30/0.66            @ ( collect_int
% 0.30/0.66              @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ N2 ) ) )
% 0.30/0.66          = ( abs_abs_int @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Max_divisors_self_int
% 0.30/0.66  thf(fact_9658_gcd__is__Max__divisors__int,axiom,
% 0.30/0.66      ! [N2: int,M: int] :
% 0.30/0.66        ( ( N2 != zero_zero_int )
% 0.30/0.66       => ( ( gcd_gcd_int @ M @ N2 )
% 0.30/0.66          = ( lattic8263393255366662781ax_int
% 0.30/0.66            @ ( collect_int
% 0.30/0.66              @ ^ [D5: int] :
% 0.30/0.66                  ( ( dvd_dvd_int @ D5 @ M )
% 0.30/0.66                  & ( dvd_dvd_int @ D5 @ N2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % gcd_is_Max_divisors_int
% 0.30/0.66  thf(fact_9659_uniformity__complex__def,axiom,
% 0.30/0.66      ( topolo896644834953643431omplex
% 0.30/0.66      = ( comple8358262395181532106omplex
% 0.30/0.66        @ ( image_5971271580939081552omplex
% 0.30/0.66          @ ^ [E3: real] :
% 0.30/0.66              ( princi3496590319149328850omplex
% 0.30/0.66              @ ( collec8663557070575231912omplex
% 0.30/0.66                @ ( produc6771430404735790350plex_o
% 0.30/0.66                  @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 0.30/0.66          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % uniformity_complex_def
% 0.30/0.66  thf(fact_9660_uniformity__real__def,axiom,
% 0.30/0.66      ( topolo1511823702728130853y_real
% 0.30/0.66      = ( comple2936214249959783750l_real
% 0.30/0.66        @ ( image_2178119161166701260l_real
% 0.30/0.66          @ ^ [E3: real] :
% 0.30/0.66              ( princi6114159922880469582l_real
% 0.30/0.66              @ ( collec3799799289383736868l_real
% 0.30/0.66                @ ( produc5414030515140494994real_o
% 0.30/0.66                  @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 0.30/0.66          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % uniformity_real_def
% 0.30/0.66  thf(fact_9661_eventually__prod__sequentially,axiom,
% 0.30/0.66      ! [P: product_prod_nat_nat > $o] :
% 0.30/0.66        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 0.30/0.66        = ( ? [N5: nat] :
% 0.30/0.66            ! [M2: nat] :
% 0.30/0.66              ( ( ord_less_eq_nat @ N5 @ M2 )
% 0.30/0.66             => ! [N: nat] :
% 0.30/0.66                  ( ( ord_less_eq_nat @ N5 @ N )
% 0.30/0.66                 => ( P @ ( product_Pair_nat_nat @ N @ M2 ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % eventually_prod_sequentially
% 0.30/0.66  thf(fact_9662_mono__Suc,axiom,
% 0.30/0.66      order_mono_nat_nat @ suc ).
% 0.30/0.66  
% 0.30/0.66  % mono_Suc
% 0.30/0.66  thf(fact_9663_mono__times__nat,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mono_times_nat
% 0.30/0.66  thf(fact_9664_mono__ge2__power__minus__self,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 0.30/0.66       => ( order_mono_nat_nat
% 0.30/0.66          @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % mono_ge2_power_minus_self
% 0.30/0.66  thf(fact_9665_less__eq,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 0.30/0.66        = ( ord_less_nat @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_eq
% 0.30/0.66  thf(fact_9666_inj__sgn__power,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( inj_on_real_real
% 0.30/0.66          @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 0.30/0.66          @ top_top_set_real ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inj_sgn_power
% 0.30/0.66  thf(fact_9667_log__inj,axiom,
% 0.30/0.66      ! [B: real] :
% 0.30/0.66        ( ( ord_less_real @ one_one_real @ B )
% 0.30/0.66       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % log_inj
% 0.30/0.66  thf(fact_9668_inj__Suc,axiom,
% 0.30/0.66      ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).
% 0.30/0.66  
% 0.30/0.66  % inj_Suc
% 0.30/0.66  thf(fact_9669_inj__on__diff__nat,axiom,
% 0.30/0.66      ! [N7: set_nat,K: nat] :
% 0.30/0.66        ( ! [N3: nat] :
% 0.30/0.66            ( ( member_nat @ N3 @ N7 )
% 0.30/0.66           => ( ord_less_eq_nat @ K @ N3 ) )
% 0.30/0.66       => ( inj_on_nat_nat
% 0.30/0.66          @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
% 0.30/0.66          @ N7 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % inj_on_diff_nat
% 0.30/0.66  thf(fact_9670_inj__on__set__encode,axiom,
% 0.30/0.66      inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % inj_on_set_encode
% 0.30/0.66  thf(fact_9671_pred__nat__trancl__eq__le,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 0.30/0.66        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pred_nat_trancl_eq_le
% 0.30/0.66  thf(fact_9672_measure__function__int,axiom,
% 0.30/0.66      fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 0.30/0.66  
% 0.30/0.66  % measure_function_int
% 0.30/0.66  thf(fact_9673_powr__real__of__int_H,axiom,
% 0.30/0.66      ! [X3: real,N2: int] :
% 0.30/0.66        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 0.30/0.66       => ( ( ( X3 != zero_zero_real )
% 0.30/0.66            | ( ord_less_int @ zero_zero_int @ N2 ) )
% 0.30/0.66         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 0.30/0.66            = ( power_int_real @ X3 @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % powr_real_of_int'
% 0.30/0.66  thf(fact_9674_atLeastLessThan__add__Un,axiom,
% 0.30/0.66      ! [I: nat,J: nat,K: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.66       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.66          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastLessThan_add_Un
% 0.30/0.66  thf(fact_9675_min__weak__def,axiom,
% 0.30/0.66      ( fun_min_weak
% 0.30/0.66      = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % min_weak_def
% 0.30/0.66  thf(fact_9676_sup__int__def,axiom,
% 0.30/0.66      sup_sup_int = ord_max_int ).
% 0.30/0.66  
% 0.30/0.66  % sup_int_def
% 0.30/0.66  thf(fact_9677_sup__nat__def,axiom,
% 0.30/0.66      sup_sup_nat = ord_max_nat ).
% 0.30/0.66  
% 0.30/0.66  % sup_nat_def
% 0.30/0.66  thf(fact_9678_max__weak__def,axiom,
% 0.30/0.66      ( fun_max_weak
% 0.30/0.66      = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % max_weak_def
% 0.30/0.66  thf(fact_9679_pos__deriv__imp__strict__mono,axiom,
% 0.30/0.66      ! [F: real > real,F5: real > real] :
% 0.30/0.66        ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 0.30/0.66       => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
% 0.30/0.66         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pos_deriv_imp_strict_mono
% 0.30/0.66  thf(fact_9680_of__nat__eq__id,axiom,
% 0.30/0.66      semiri1316708129612266289at_nat = id_nat ).
% 0.30/0.66  
% 0.30/0.66  % of_nat_eq_id
% 0.30/0.66  thf(fact_9681_infinite__enumerate,axiom,
% 0.30/0.66      ! [S3: set_nat] :
% 0.30/0.66        ( ~ ( finite_finite_nat @ S3 )
% 0.30/0.66       => ? [R4: nat > nat] :
% 0.30/0.66            ( ( order_5726023648592871131at_nat @ R4 )
% 0.30/0.66            & ! [N6: nat] : ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % infinite_enumerate
% 0.30/0.66  thf(fact_9682_strict__mono__imp__increasing,axiom,
% 0.30/0.66      ! [F: nat > nat,N2: nat] :
% 0.30/0.66        ( ( order_5726023648592871131at_nat @ F )
% 0.30/0.66       => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % strict_mono_imp_increasing
% 0.30/0.66  thf(fact_9683_less__int__def,axiom,
% 0.30/0.66      ( ord_less_int
% 0.30/0.66      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.66        @ ( produc8739625826339149834_nat_o
% 0.30/0.66          @ ^ [X: nat,Y: nat] :
% 0.30/0.66              ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_int_def
% 0.30/0.66  thf(fact_9684_nat__def,axiom,
% 0.30/0.66      ( nat2
% 0.30/0.66      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nat_def
% 0.30/0.66  thf(fact_9685_less__eq__int__def,axiom,
% 0.30/0.66      ( ord_less_eq_int
% 0.30/0.66      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 0.30/0.66        @ ( produc8739625826339149834_nat_o
% 0.30/0.66          @ ^ [X: nat,Y: nat] :
% 0.30/0.66              ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_eq_int_def
% 0.30/0.66  thf(fact_9686_positive__rat,axiom,
% 0.30/0.66      ! [A: int,B: int] :
% 0.30/0.66        ( ( positive @ ( fract @ A @ B ) )
% 0.30/0.66        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % positive_rat
% 0.30/0.66  thf(fact_9687_less__rat__def,axiom,
% 0.30/0.66      ( ord_less_rat
% 0.30/0.66      = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % less_rat_def
% 0.30/0.66  thf(fact_9688_Rat_Opositive_Orep__eq,axiom,
% 0.30/0.66      ( positive
% 0.30/0.66      = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Rat.positive.rep_eq
% 0.30/0.66  thf(fact_9689_Rat_Opositive__def,axiom,
% 0.30/0.66      ( positive
% 0.30/0.66      = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 0.30/0.66        @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Rat.positive_def
% 0.30/0.66  thf(fact_9690_sorted__list__of__set__greaterThanAtMost,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 0.30/0.66       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 0.30/0.66          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_list_of_set_greaterThanAtMost
% 0.30/0.66  thf(fact_9691_sorted__list__of__set__greaterThanLessThan,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 0.30/0.66       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 0.30/0.66          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_list_of_set_greaterThanLessThan
% 0.30/0.66  thf(fact_9692_sorted__list__of__set__lessThan__Suc,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 0.30/0.66        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_list_of_set_lessThan_Suc
% 0.30/0.66  thf(fact_9693_sorted__list__of__set__atMost__Suc,axiom,
% 0.30/0.66      ! [K: nat] :
% 0.30/0.66        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 0.30/0.66        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_list_of_set_atMost_Suc
% 0.30/0.66  thf(fact_9694_list__encode_Oelims,axiom,
% 0.30/0.66      ! [X3: list_nat,Y3: nat] :
% 0.30/0.66        ( ( ( nat_list_encode @ X3 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( X3 = nil_nat )
% 0.30/0.66           => ( Y3 != zero_zero_nat ) )
% 0.30/0.66         => ~ ! [X4: nat,Xs3: list_nat] :
% 0.30/0.66                ( ( X3
% 0.30/0.66                  = ( cons_nat @ X4 @ Xs3 ) )
% 0.30/0.66               => ( Y3
% 0.30/0.66                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % list_encode.elims
% 0.30/0.66  thf(fact_9695_list__encode_Osimps_I1_J,axiom,
% 0.30/0.66      ( ( nat_list_encode @ nil_nat )
% 0.30/0.66      = zero_zero_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % list_encode.simps(1)
% 0.30/0.66  thf(fact_9696_list__encode_Osimps_I2_J,axiom,
% 0.30/0.66      ! [X3: nat,Xs: list_nat] :
% 0.30/0.66        ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs ) )
% 0.30/0.66        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % list_encode.simps(2)
% 0.30/0.66  thf(fact_9697_upto__aux__rec,axiom,
% 0.30/0.66      ( upto_aux
% 0.30/0.66      = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_aux_rec
% 0.30/0.66  thf(fact_9698_upto_Opsimps,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 0.30/0.66       => ( ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66           => ( ( upto @ I @ J )
% 0.30/0.66              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ I @ J )
% 0.30/0.66           => ( ( upto @ I @ J )
% 0.30/0.66              = nil_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto.psimps
% 0.30/0.66  thf(fact_9699_upto_Opelims,axiom,
% 0.30/0.66      ! [X3: int,Xa2: int,Y3: list_int] :
% 0.30/0.66        ( ( ( upto @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 0.30/0.66         => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.66                 => ( Y3
% 0.30/0.66                    = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.66                & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.66                 => ( Y3 = nil_int ) ) )
% 0.30/0.66             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto.pelims
% 0.30/0.66  thf(fact_9700_upto__empty,axiom,
% 0.30/0.66      ! [J: int,I: int] :
% 0.30/0.66        ( ( ord_less_int @ J @ I )
% 0.30/0.66       => ( ( upto @ I @ J )
% 0.30/0.66          = nil_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_empty
% 0.30/0.66  thf(fact_9701_upto__Nil2,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( nil_int
% 0.30/0.66          = ( upto @ I @ J ) )
% 0.30/0.66        = ( ord_less_int @ J @ I ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_Nil2
% 0.30/0.66  thf(fact_9702_upto__Nil,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( ( upto @ I @ J )
% 0.30/0.66          = nil_int )
% 0.30/0.66        = ( ord_less_int @ J @ I ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_Nil
% 0.30/0.66  thf(fact_9703_upto__single,axiom,
% 0.30/0.66      ! [I: int] :
% 0.30/0.66        ( ( upto @ I @ I )
% 0.30/0.66        = ( cons_int @ I @ nil_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_single
% 0.30/0.66  thf(fact_9704_nth__upto,axiom,
% 0.30/0.66      ! [I: int,K: nat,J: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 0.30/0.66       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 0.30/0.66          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nth_upto
% 0.30/0.66  thf(fact_9705_length__upto,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 0.30/0.66        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % length_upto
% 0.30/0.66  thf(fact_9706_upto__rec__numeral_I1_J,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66            = nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec_numeral(1)
% 0.30/0.66  thf(fact_9707_upto__rec__numeral_I2_J,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66            = nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec_numeral(2)
% 0.30/0.66  thf(fact_9708_upto__rec__numeral_I3_J,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66            = ( 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 @ N2 ) ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 0.30/0.66            = nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec_numeral(3)
% 0.30/0.66  thf(fact_9709_upto__rec__numeral_I4_J,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66            = ( 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 @ N2 ) ) ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66            = nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec_numeral(4)
% 0.30/0.66  thf(fact_9710_atLeastAtMost__upto,axiom,
% 0.30/0.66      ( set_or1266510415728281911st_int
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ J3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastAtMost_upto
% 0.30/0.66  thf(fact_9711_distinct__upto,axiom,
% 0.30/0.66      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 0.30/0.66  
% 0.30/0.66  % distinct_upto
% 0.30/0.66  thf(fact_9712_upto__aux__def,axiom,
% 0.30/0.66      ( upto_aux
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( append_int @ ( upto @ I2 @ J3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_aux_def
% 0.30/0.66  thf(fact_9713_upto__code,axiom,
% 0.30/0.66      ( upto
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( upto_aux @ I2 @ J3 @ nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_code
% 0.30/0.66  thf(fact_9714_upto__split2,axiom,
% 0.30/0.66      ! [I: int,J: int,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66         => ( ( upto @ I @ K )
% 0.30/0.66            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_split2
% 0.30/0.66  thf(fact_9715_upto__split1,axiom,
% 0.30/0.66      ! [I: int,J: int,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66         => ( ( upto @ I @ K )
% 0.30/0.66            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_split1
% 0.30/0.66  thf(fact_9716_atLeastLessThan__upto,axiom,
% 0.30/0.66      ( set_or4662586982721622107an_int
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastLessThan_upto
% 0.30/0.66  thf(fact_9717_greaterThanAtMost__upto,axiom,
% 0.30/0.66      ( set_or6656581121297822940st_int
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThanAtMost_upto
% 0.30/0.66  thf(fact_9718_upto__rec1,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66       => ( ( upto @ I @ J )
% 0.30/0.66          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec1
% 0.30/0.66  thf(fact_9719_upto_Osimps,axiom,
% 0.30/0.66      ( upto
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto.simps
% 0.30/0.66  thf(fact_9720_upto_Oelims,axiom,
% 0.30/0.66      ! [X3: int,Xa2: int,Y3: list_int] :
% 0.30/0.66        ( ( ( upto @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.66           => ( Y3
% 0.30/0.66              = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 0.30/0.66          & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 0.30/0.66           => ( Y3 = nil_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto.elims
% 0.30/0.66  thf(fact_9721_upto__rec2,axiom,
% 0.30/0.66      ! [I: int,J: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66       => ( ( upto @ I @ J )
% 0.30/0.66          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_rec2
% 0.30/0.66  thf(fact_9722_greaterThanLessThan__upto,axiom,
% 0.30/0.66      ( set_or5832277885323065728an_int
% 0.30/0.66      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThanLessThan_upto
% 0.30/0.66  thf(fact_9723_upto__split3,axiom,
% 0.30/0.66      ! [I: int,J: int,K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ I @ J )
% 0.30/0.66       => ( ( ord_less_eq_int @ J @ K )
% 0.30/0.66         => ( ( upto @ I @ K )
% 0.30/0.66            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upto_split3
% 0.30/0.66  thf(fact_9724_list__encode_Opelims,axiom,
% 0.30/0.66      ! [X3: list_nat,Y3: nat] :
% 0.30/0.66        ( ( ( nat_list_encode @ X3 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
% 0.30/0.66         => ( ( ( X3 = nil_nat )
% 0.30/0.66             => ( ( Y3 = zero_zero_nat )
% 0.30/0.66               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 0.30/0.66           => ~ ! [X4: nat,Xs3: list_nat] :
% 0.30/0.66                  ( ( X3
% 0.30/0.66                    = ( cons_nat @ X4 @ Xs3 ) )
% 0.30/0.66                 => ( ( Y3
% 0.30/0.66                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 0.30/0.66                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % list_encode.pelims
% 0.30/0.66  thf(fact_9725_remdups__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( remdups_nat @ ( upt @ M @ N2 ) )
% 0.30/0.66        = ( upt @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % remdups_upt
% 0.30/0.66  thf(fact_9726_hd__upt,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_nat @ I @ J )
% 0.30/0.66       => ( ( hd_nat @ ( upt @ I @ J ) )
% 0.30/0.66          = I ) ) ).
% 0.30/0.66  
% 0.30/0.66  % hd_upt
% 0.30/0.66  thf(fact_9727_drop__upt,axiom,
% 0.30/0.66      ! [M: nat,I: nat,J: nat] :
% 0.30/0.66        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 0.30/0.66        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 0.30/0.66  
% 0.30/0.66  % drop_upt
% 0.30/0.66  thf(fact_9728_length__upt,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 0.30/0.66        = ( minus_minus_nat @ J @ I ) ) ).
% 0.30/0.66  
% 0.30/0.66  % length_upt
% 0.30/0.66  thf(fact_9729_take__upt,axiom,
% 0.30/0.66      ! [I: nat,M: nat,N2: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
% 0.30/0.66       => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
% 0.30/0.66          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % take_upt
% 0.30/0.66  thf(fact_9730_upt__conv__Nil,axiom,
% 0.30/0.66      ! [J: nat,I: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ J @ I )
% 0.30/0.66       => ( ( upt @ I @ J )
% 0.30/0.66          = nil_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_conv_Nil
% 0.30/0.66  thf(fact_9731_sorted__list__of__set__range,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
% 0.30/0.66        = ( upt @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_list_of_set_range
% 0.30/0.66  thf(fact_9732_upt__eq__Nil__conv,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ( upt @ I @ J )
% 0.30/0.66          = nil_nat )
% 0.30/0.66        = ( ( J = zero_zero_nat )
% 0.30/0.66          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_eq_Nil_conv
% 0.30/0.66  thf(fact_9733_nth__upt,axiom,
% 0.30/0.66      ! [I: nat,K: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 0.30/0.66       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 0.30/0.66          = ( plus_plus_nat @ I @ K ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % nth_upt
% 0.30/0.66  thf(fact_9734_upt__rec__numeral,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.66         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.66            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.66         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 0.30/0.66            = nil_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_rec_numeral
% 0.30/0.66  thf(fact_9735_upt__0,axiom,
% 0.30/0.66      ! [I: nat] :
% 0.30/0.66        ( ( upt @ I @ zero_zero_nat )
% 0.30/0.66        = nil_nat ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_0
% 0.30/0.66  thf(fact_9736_upt__conv__Cons__Cons,axiom,
% 0.30/0.66      ! [M: nat,N2: nat,Ns: list_nat,Q4: nat] :
% 0.30/0.66        ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
% 0.30/0.66          = ( upt @ M @ Q4 ) )
% 0.30/0.66        = ( ( cons_nat @ N2 @ Ns )
% 0.30/0.66          = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_conv_Cons_Cons
% 0.30/0.66  thf(fact_9737_map__add__upt,axiom,
% 0.30/0.66      ! [N2: nat,M: nat] :
% 0.30/0.66        ( ( map_nat_nat
% 0.30/0.66          @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N2 )
% 0.30/0.66          @ ( upt @ zero_zero_nat @ M ) )
% 0.30/0.66        = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % map_add_upt
% 0.30/0.66  thf(fact_9738_map__Suc__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
% 0.30/0.66        = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % map_Suc_upt
% 0.30/0.66  thf(fact_9739_atMost__upto,axiom,
% 0.30/0.66      ( set_ord_atMost_nat
% 0.30/0.66      = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atMost_upto
% 0.30/0.66  thf(fact_9740_greaterThanAtMost__upt,axiom,
% 0.30/0.66      ( set_or6659071591806873216st_nat
% 0.30/0.66      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThanAtMost_upt
% 0.30/0.66  thf(fact_9741_atLeast__upt,axiom,
% 0.30/0.66      ( set_ord_lessThan_nat
% 0.30/0.66      = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeast_upt
% 0.30/0.66  thf(fact_9742_atLeastAtMost__upt,axiom,
% 0.30/0.66      ( set_or1269000886237332187st_nat
% 0.30/0.66      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastAtMost_upt
% 0.30/0.66  thf(fact_9743_atLeastLessThan__upt,axiom,
% 0.30/0.66      ( set_or4665077453230672383an_nat
% 0.30/0.66      = ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % atLeastLessThan_upt
% 0.30/0.66  thf(fact_9744_greaterThanLessThan__upt,axiom,
% 0.30/0.66      ( set_or5834768355832116004an_nat
% 0.30/0.66      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % greaterThanLessThan_upt
% 0.30/0.66  thf(fact_9745_distinct__upt,axiom,
% 0.30/0.66      ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% 0.30/0.66  
% 0.30/0.66  % distinct_upt
% 0.30/0.66  thf(fact_9746_upt__conv__Cons,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_nat @ I @ J )
% 0.30/0.66       => ( ( upt @ I @ J )
% 0.30/0.66          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_conv_Cons
% 0.30/0.66  thf(fact_9747_map__decr__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( map_nat_nat
% 0.30/0.66          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
% 0.30/0.66          @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 0.30/0.66        = ( upt @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % map_decr_upt
% 0.30/0.66  thf(fact_9748_upt__add__eq__append,axiom,
% 0.30/0.66      ! [I: nat,J: nat,K: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.66       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 0.30/0.66          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_add_eq_append
% 0.30/0.66  thf(fact_9749_upt__eq__Cons__conv,axiom,
% 0.30/0.66      ! [I: nat,J: nat,X3: nat,Xs: list_nat] :
% 0.30/0.66        ( ( ( upt @ I @ J )
% 0.30/0.66          = ( cons_nat @ X3 @ Xs ) )
% 0.30/0.66        = ( ( ord_less_nat @ I @ J )
% 0.30/0.66          & ( I = X3 )
% 0.30/0.66          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 0.30/0.66            = Xs ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_eq_Cons_conv
% 0.30/0.66  thf(fact_9750_upt__rec,axiom,
% 0.30/0.66      ( upt
% 0.30/0.66      = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_rec
% 0.30/0.66  thf(fact_9751_upt__Suc,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.66         => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.66            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 0.30/0.66        & ( ~ ( ord_less_eq_nat @ I @ J )
% 0.30/0.66         => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.66            = nil_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_Suc
% 0.30/0.66  thf(fact_9752_upt__Suc__append,axiom,
% 0.30/0.66      ! [I: nat,J: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ I @ J )
% 0.30/0.66       => ( ( upt @ I @ ( suc @ J ) )
% 0.30/0.66          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % upt_Suc_append
% 0.30/0.66  thf(fact_9753_sum__list__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ M @ N2 )
% 0.30/0.66       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
% 0.30/0.66          = ( groups3542108847815614940at_nat
% 0.30/0.66            @ ^ [X: nat] : X
% 0.30/0.66            @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sum_list_upt
% 0.30/0.66  thf(fact_9754_card__length__sum__list__rec,axiom,
% 0.30/0.66      ! [M: nat,N7: nat] :
% 0.30/0.66        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 0.30/0.66       => ( ( finite_card_list_nat
% 0.30/0.66            @ ( collect_list_nat
% 0.30/0.66              @ ^ [L2: list_nat] :
% 0.30/0.66                  ( ( ( size_size_list_nat @ L2 )
% 0.30/0.66                    = M )
% 0.30/0.66                  & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.66                    = N7 ) ) ) )
% 0.30/0.66          = ( plus_plus_nat
% 0.30/0.66            @ ( finite_card_list_nat
% 0.30/0.66              @ ( collect_list_nat
% 0.30/0.66                @ ^ [L2: list_nat] :
% 0.30/0.66                    ( ( ( size_size_list_nat @ L2 )
% 0.30/0.66                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 0.30/0.66                    & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.66                      = N7 ) ) ) )
% 0.30/0.66            @ ( finite_card_list_nat
% 0.30/0.66              @ ( collect_list_nat
% 0.30/0.66                @ ^ [L2: list_nat] :
% 0.30/0.66                    ( ( ( size_size_list_nat @ L2 )
% 0.30/0.66                      = M )
% 0.30/0.66                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 0.30/0.66                      = N7 ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % card_length_sum_list_rec
% 0.30/0.66  thf(fact_9755_card__length__sum__list,axiom,
% 0.30/0.66      ! [M: nat,N7: nat] :
% 0.30/0.66        ( ( finite_card_list_nat
% 0.30/0.66          @ ( collect_list_nat
% 0.30/0.66            @ ^ [L2: list_nat] :
% 0.30/0.66                ( ( ( size_size_list_nat @ L2 )
% 0.30/0.66                  = M )
% 0.30/0.66                & ( ( groups4561878855575611511st_nat @ L2 )
% 0.30/0.66                  = N7 ) ) ) )
% 0.30/0.66        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % card_length_sum_list
% 0.30/0.66  thf(fact_9756_tl__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] :
% 0.30/0.66        ( ( tl_nat @ ( upt @ M @ N2 ) )
% 0.30/0.66        = ( upt @ ( suc @ M ) @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % tl_upt
% 0.30/0.66  thf(fact_9757_sorted__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_upt
% 0.30/0.66  thf(fact_9758_sorted__wrt__upt,axiom,
% 0.30/0.66      ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_wrt_upt
% 0.30/0.66  thf(fact_9759_sorted__wrt__less__idx,axiom,
% 0.30/0.66      ! [Ns: list_nat,I: nat] :
% 0.30/0.66        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 0.30/0.66       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 0.30/0.66         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_wrt_less_idx
% 0.30/0.66  thf(fact_9760_sorted__upto,axiom,
% 0.30/0.66      ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_upto
% 0.30/0.66  thf(fact_9761_sorted__wrt__upto,axiom,
% 0.30/0.66      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 0.30/0.66  
% 0.30/0.66  % sorted_wrt_upto
% 0.30/0.66  thf(fact_9762_pairs__le__eq__Sigma,axiom,
% 0.30/0.66      ! [M: nat] :
% 0.30/0.66        ( ( collec3392354462482085612at_nat
% 0.30/0.66          @ ( produc6081775807080527818_nat_o
% 0.30/0.66            @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 0.30/0.66        = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 0.30/0.66          @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % pairs_le_eq_Sigma
% 0.30/0.66  thf(fact_9763_natLess__def,axiom,
% 0.30/0.66      ( bNF_Ca8459412986667044542atLess
% 0.30/0.66      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % natLess_def
% 0.30/0.66  thf(fact_9764_vimage__Suc__insert__Suc,axiom,
% 0.30/0.66      ! [N2: nat,A2: set_nat] :
% 0.30/0.66        ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N2 ) @ A2 ) )
% 0.30/0.66        = ( insert_nat @ N2 @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % vimage_Suc_insert_Suc
% 0.30/0.66  thf(fact_9765_vimage__Suc__insert__0,axiom,
% 0.30/0.66      ! [A2: set_nat] :
% 0.30/0.66        ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 0.30/0.66        = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % vimage_Suc_insert_0
% 0.30/0.66  thf(fact_9766_finite__vimage__Suc__iff,axiom,
% 0.30/0.66      ! [F3: set_nat] :
% 0.30/0.66        ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
% 0.30/0.66        = ( finite_finite_nat @ F3 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % finite_vimage_Suc_iff
% 0.30/0.66  thf(fact_9767_set__decode__div__2,axiom,
% 0.30/0.66      ! [X3: nat] :
% 0.30/0.66        ( ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 0.30/0.66        = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % set_decode_div_2
% 0.30/0.66  thf(fact_9768_set__encode__vimage__Suc,axiom,
% 0.30/0.66      ! [A2: set_nat] :
% 0.30/0.66        ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 0.30/0.66        = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % set_encode_vimage_Suc
% 0.30/0.66  thf(fact_9769_Restr__natLeq,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.30/0.66          @ ( produc457027306803732586at_nat
% 0.30/0.66            @ ( collect_nat
% 0.30/0.66              @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) )
% 0.30/0.66            @ ^ [Uu3: nat] :
% 0.30/0.66                ( collect_nat
% 0.30/0.66                @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) )
% 0.30/0.66        = ( collec3392354462482085612at_nat
% 0.30/0.66          @ ( produc6081775807080527818_nat_o
% 0.30/0.66            @ ^ [X: nat,Y: nat] :
% 0.30/0.66                ( ( ord_less_nat @ X @ N2 )
% 0.30/0.66                & ( ord_less_nat @ Y @ N2 )
% 0.30/0.66                & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Restr_natLeq
% 0.30/0.66  thf(fact_9770_natLeq__def,axiom,
% 0.30/0.66      ( bNF_Ca8665028551170535155natLeq
% 0.30/0.66      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % natLeq_def
% 0.30/0.66  thf(fact_9771_Restr__natLeq2,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 0.30/0.66          @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
% 0.30/0.66            @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
% 0.30/0.66        = ( collec3392354462482085612at_nat
% 0.30/0.66          @ ( produc6081775807080527818_nat_o
% 0.30/0.66            @ ^ [X: nat,Y: nat] :
% 0.30/0.66                ( ( ord_less_nat @ X @ N2 )
% 0.30/0.66                & ( ord_less_nat @ Y @ N2 )
% 0.30/0.66                & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Restr_natLeq2
% 0.30/0.66  thf(fact_9772_natLeq__underS__less,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
% 0.30/0.66        = ( collect_nat
% 0.30/0.66          @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % natLeq_underS_less
% 0.30/0.66  thf(fact_9773_Arg__bounded,axiom,
% 0.30/0.66      ! [Z2: complex] :
% 0.30/0.66        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 0.30/0.66        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_bounded
% 0.30/0.66  thf(fact_9774_Arg__correct,axiom,
% 0.30/0.66      ! [Z2: complex] :
% 0.30/0.66        ( ( Z2 != zero_zero_complex )
% 0.30/0.66       => ( ( ( sgn_sgn_complex @ Z2 )
% 0.30/0.66            = ( cis @ ( arg @ Z2 ) ) )
% 0.30/0.66          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 0.30/0.66          & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_correct
% 0.30/0.66  thf(fact_9775_Field__natLeq__on,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( field_nat
% 0.30/0.66          @ ( collec3392354462482085612at_nat
% 0.30/0.66            @ ( produc6081775807080527818_nat_o
% 0.30/0.66              @ ^ [X: nat,Y: nat] :
% 0.30/0.66                  ( ( ord_less_nat @ X @ N2 )
% 0.30/0.66                  & ( ord_less_nat @ Y @ N2 )
% 0.30/0.66                  & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 0.30/0.66        = ( collect_nat
% 0.30/0.66          @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Field_natLeq_on
% 0.30/0.66  thf(fact_9776_cis__Arg__unique,axiom,
% 0.30/0.66      ! [Z2: complex,X3: real] :
% 0.30/0.66        ( ( ( sgn_sgn_complex @ Z2 )
% 0.30/0.66          = ( cis @ X3 ) )
% 0.30/0.66       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 0.30/0.66         => ( ( ord_less_eq_real @ X3 @ pi )
% 0.30/0.66           => ( ( arg @ Z2 )
% 0.30/0.66              = X3 ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % cis_Arg_unique
% 0.30/0.66  thf(fact_9777_bij__betw__roots__unity,axiom,
% 0.30/0.66      ! [N2: nat] :
% 0.30/0.66        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66       => ( bij_betw_nat_complex
% 0.30/0.66          @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 0.30/0.66          @ ( set_ord_lessThan_nat @ N2 )
% 0.30/0.66          @ ( collect_complex
% 0.30/0.66            @ ^ [Z6: complex] :
% 0.30/0.66                ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.66                = one_one_complex ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bij_betw_roots_unity
% 0.30/0.66  thf(fact_9778_bij__betw__nth__root__unity,axiom,
% 0.30/0.66      ! [C: complex,N2: nat] :
% 0.30/0.66        ( ( C != zero_zero_complex )
% 0.30/0.66       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 0.30/0.66         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
% 0.30/0.66            @ ( collect_complex
% 0.30/0.66              @ ^ [Z6: complex] :
% 0.30/0.66                  ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.66                  = one_one_complex ) )
% 0.30/0.66            @ ( collect_complex
% 0.30/0.66              @ ^ [Z6: complex] :
% 0.30/0.66                  ( ( power_power_complex @ Z6 @ N2 )
% 0.30/0.66                  = C ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bij_betw_nth_root_unity
% 0.30/0.66  thf(fact_9779_bij__betw__Suc,axiom,
% 0.30/0.66      ! [M7: set_nat,N7: set_nat] :
% 0.30/0.66        ( ( bij_betw_nat_nat @ suc @ M7 @ N7 )
% 0.30/0.66        = ( ( image_nat_nat @ suc @ M7 )
% 0.30/0.66          = N7 ) ) ).
% 0.30/0.66  
% 0.30/0.66  % bij_betw_Suc
% 0.30/0.66  thf(fact_9780_Arg__def,axiom,
% 0.30/0.66      ( arg
% 0.30/0.66      = ( ^ [Z6: complex] :
% 0.30/0.66            ( if_real @ ( Z6 = zero_zero_complex ) @ zero_zero_real
% 0.30/0.66            @ ( fChoice_real
% 0.30/0.66              @ ^ [A3: real] :
% 0.30/0.66                  ( ( ( sgn_sgn_complex @ Z6 )
% 0.30/0.66                    = ( cis @ A3 ) )
% 0.30/0.66                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 0.30/0.66                  & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % Arg_def
% 0.30/0.66  thf(fact_9781_not__negative__int__iff,axiom,
% 0.30/0.66      ! [K: int] :
% 0.30/0.66        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 0.30/0.66        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 0.30/0.66  
% 0.30/0.66  % not_negative_int_iff
% 0.30/0.66  thf(fact_9782_not__nonnegative__int__iff,axiom,
% 0.30/0.66      ! [K: int] :
% 0.30/0.66        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 0.30/0.66        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % not_nonnegative_int_iff
% 0.30/0.66  thf(fact_9783_wf__int__ge__less__than2,axiom,
% 0.30/0.66      ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).
% 0.30/0.66  
% 0.30/0.66  % wf_int_ge_less_than2
% 0.30/0.66  thf(fact_9784_wf__int__ge__less__than,axiom,
% 0.30/0.66      ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).
% 0.30/0.66  
% 0.30/0.66  % wf_int_ge_less_than
% 0.30/0.66  thf(fact_9785_wf__less,axiom,
% 0.30/0.66      wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 0.30/0.66  
% 0.30/0.66  % wf_less
% 0.30/0.66  thf(fact_9786_and__not__num__eq__None__iff,axiom,
% 0.30/0.66      ! [M: num,N2: num] :
% 0.30/0.66        ( ( ( bit_and_not_num @ M @ N2 )
% 0.30/0.66          = none_num )
% 0.30/0.66        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 0.30/0.66          = zero_zero_int ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_not_num_eq_None_iff
% 0.30/0.66  thf(fact_9787_and__not__num_Osimps_I1_J,axiom,
% 0.30/0.66      ( ( bit_and_not_num @ one @ one )
% 0.30/0.66      = none_num ) ).
% 0.30/0.66  
% 0.30/0.66  % and_not_num.simps(1)
% 0.30/0.66  thf(fact_9788_and__not__num_Osimps_I3_J,axiom,
% 0.30/0.66      ! [N2: num] :
% 0.30/0.66        ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
% 0.30/0.66        = none_num ) ).
% 0.30/0.66  
% 0.30/0.66  % and_not_num.simps(3)
% 0.30/0.66  thf(fact_9789_and__not__num_Oelims,axiom,
% 0.30/0.66      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.66        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( X3 = one )
% 0.30/0.66           => ( ( Xa2 = one )
% 0.30/0.66             => ( Y3 != none_num ) ) )
% 0.30/0.66         => ( ( ( X3 = one )
% 0.30/0.66             => ( ? [N3: num] :
% 0.30/0.66                    ( Xa2
% 0.30/0.66                    = ( bit0 @ N3 ) )
% 0.30/0.66               => ( Y3
% 0.30/0.66                 != ( some_num @ one ) ) ) )
% 0.30/0.66           => ( ( ( X3 = one )
% 0.30/0.66               => ( ? [N3: num] :
% 0.30/0.66                      ( Xa2
% 0.30/0.66                      = ( bit1 @ N3 ) )
% 0.30/0.66                 => ( Y3 != none_num ) ) )
% 0.30/0.66             => ( ! [M3: num] :
% 0.30/0.66                    ( ( X3
% 0.30/0.66                      = ( bit0 @ M3 ) )
% 0.30/0.66                   => ( ( Xa2 = one )
% 0.30/0.66                     => ( Y3
% 0.30/0.66                       != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 0.30/0.66               => ( ! [M3: num] :
% 0.30/0.66                      ( ( X3
% 0.30/0.66                        = ( bit0 @ M3 ) )
% 0.30/0.66                     => ! [N3: num] :
% 0.30/0.66                          ( ( Xa2
% 0.30/0.66                            = ( bit0 @ N3 ) )
% 0.30/0.66                         => ( Y3
% 0.30/0.66                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                 => ( ! [M3: num] :
% 0.30/0.66                        ( ( X3
% 0.30/0.66                          = ( bit0 @ M3 ) )
% 0.30/0.66                       => ! [N3: num] :
% 0.30/0.66                            ( ( Xa2
% 0.30/0.66                              = ( bit1 @ N3 ) )
% 0.30/0.66                           => ( Y3
% 0.30/0.66                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                   => ( ! [M3: num] :
% 0.30/0.66                          ( ( X3
% 0.30/0.66                            = ( bit1 @ M3 ) )
% 0.30/0.66                         => ( ( Xa2 = one )
% 0.30/0.66                           => ( Y3
% 0.30/0.66                             != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 0.30/0.66                     => ( ! [M3: num] :
% 0.30/0.66                            ( ( X3
% 0.30/0.66                              = ( bit1 @ M3 ) )
% 0.30/0.66                           => ! [N3: num] :
% 0.30/0.66                                ( ( Xa2
% 0.30/0.66                                  = ( bit0 @ N3 ) )
% 0.30/0.66                               => ( Y3
% 0.30/0.66                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66                                    @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                       => ~ ! [M3: num] :
% 0.30/0.66                              ( ( X3
% 0.30/0.66                                = ( bit1 @ M3 ) )
% 0.30/0.66                             => ! [N3: num] :
% 0.30/0.66                                  ( ( Xa2
% 0.30/0.66                                    = ( bit1 @ N3 ) )
% 0.30/0.66                                 => ( Y3
% 0.30/0.66                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_not_num.elims
% 0.30/0.66  thf(fact_9790_and__num_Oelims,axiom,
% 0.30/0.66      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.66        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( X3 = one )
% 0.30/0.66           => ( ( Xa2 = one )
% 0.30/0.66             => ( Y3
% 0.30/0.66               != ( some_num @ one ) ) ) )
% 0.30/0.66         => ( ( ( X3 = one )
% 0.30/0.66             => ( ? [N3: num] :
% 0.30/0.66                    ( Xa2
% 0.30/0.66                    = ( bit0 @ N3 ) )
% 0.30/0.66               => ( Y3 != none_num ) ) )
% 0.30/0.66           => ( ( ( X3 = one )
% 0.30/0.66               => ( ? [N3: num] :
% 0.30/0.66                      ( Xa2
% 0.30/0.66                      = ( bit1 @ N3 ) )
% 0.30/0.66                 => ( Y3
% 0.30/0.66                   != ( some_num @ one ) ) ) )
% 0.30/0.66             => ( ( ? [M3: num] :
% 0.30/0.66                      ( X3
% 0.30/0.66                      = ( bit0 @ M3 ) )
% 0.30/0.66                 => ( ( Xa2 = one )
% 0.30/0.66                   => ( Y3 != none_num ) ) )
% 0.30/0.66               => ( ! [M3: num] :
% 0.30/0.66                      ( ( X3
% 0.30/0.66                        = ( bit0 @ M3 ) )
% 0.30/0.66                     => ! [N3: num] :
% 0.30/0.66                          ( ( Xa2
% 0.30/0.66                            = ( bit0 @ N3 ) )
% 0.30/0.66                         => ( Y3
% 0.30/0.66                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                 => ( ! [M3: num] :
% 0.30/0.66                        ( ( X3
% 0.30/0.66                          = ( bit0 @ M3 ) )
% 0.30/0.66                       => ! [N3: num] :
% 0.30/0.66                            ( ( Xa2
% 0.30/0.66                              = ( bit1 @ N3 ) )
% 0.30/0.66                           => ( Y3
% 0.30/0.66                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                   => ( ( ? [M3: num] :
% 0.30/0.66                            ( X3
% 0.30/0.66                            = ( bit1 @ M3 ) )
% 0.30/0.66                       => ( ( Xa2 = one )
% 0.30/0.66                         => ( Y3
% 0.30/0.66                           != ( some_num @ one ) ) ) )
% 0.30/0.66                     => ( ! [M3: num] :
% 0.30/0.66                            ( ( X3
% 0.30/0.66                              = ( bit1 @ M3 ) )
% 0.30/0.66                           => ! [N3: num] :
% 0.30/0.66                                ( ( Xa2
% 0.30/0.66                                  = ( bit0 @ N3 ) )
% 0.30/0.66                               => ( Y3
% 0.30/0.66                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                       => ~ ! [M3: num] :
% 0.30/0.66                              ( ( X3
% 0.30/0.66                                = ( bit1 @ M3 ) )
% 0.30/0.66                             => ! [N3: num] :
% 0.30/0.66                                  ( ( Xa2
% 0.30/0.66                                    = ( bit1 @ N3 ) )
% 0.30/0.66                                 => ( Y3
% 0.30/0.66                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 0.30/0.66                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 0.30/0.66                                      @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % and_num.elims
% 0.30/0.66  thf(fact_9791_and__num_Osimps_I4_J,axiom,
% 0.30/0.66      ! [M: num] :
% 0.30/0.66        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 0.30/0.66        = none_num ) ).
% 0.30/0.66  
% 0.30/0.66  % and_num.simps(4)
% 0.30/0.66  thf(fact_9792_and__num_Osimps_I2_J,axiom,
% 0.30/0.66      ! [N2: num] :
% 0.30/0.66        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
% 0.30/0.66        = none_num ) ).
% 0.30/0.66  
% 0.30/0.66  % and_num.simps(2)
% 0.30/0.66  thf(fact_9793_xor__num_Oelims,axiom,
% 0.30/0.66      ! [X3: num,Xa2: num,Y3: option_num] :
% 0.30/0.66        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 0.30/0.66          = Y3 )
% 0.30/0.66       => ( ( ( X3 = one )
% 0.30/0.66           => ( ( Xa2 = one )
% 0.30/0.66             => ( Y3 != none_num ) ) )
% 0.30/0.66         => ( ( ( X3 = one )
% 0.30/0.66             => ! [N3: num] :
% 0.30/0.66                  ( ( Xa2
% 0.30/0.66                    = ( bit0 @ N3 ) )
% 0.30/0.66                 => ( Y3
% 0.30/0.66                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 0.30/0.66           => ( ( ( X3 = one )
% 0.30/0.66               => ! [N3: num] :
% 0.30/0.66                    ( ( Xa2
% 0.30/0.66                      = ( bit1 @ N3 ) )
% 0.30/0.66                   => ( Y3
% 0.30/0.66                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 0.30/0.66             => ( ! [M3: num] :
% 0.30/0.66                    ( ( X3
% 0.30/0.66                      = ( bit0 @ M3 ) )
% 0.30/0.66                   => ( ( Xa2 = one )
% 0.30/0.66                     => ( Y3
% 0.30/0.66                       != ( some_num @ ( bit1 @ M3 ) ) ) ) )
% 0.30/0.66               => ( ! [M3: num] :
% 0.30/0.66                      ( ( X3
% 0.30/0.66                        = ( bit0 @ M3 ) )
% 0.30/0.66                     => ! [N3: num] :
% 0.30/0.66                          ( ( Xa2
% 0.30/0.66                            = ( bit0 @ N3 ) )
% 0.30/0.66                         => ( Y3
% 0.30/0.66                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) )
% 0.30/0.66                 => ( ! [M3: num] :
% 0.30/0.66                        ( ( X3
% 0.30/0.66                          = ( bit0 @ M3 ) )
% 0.30/0.66                       => ! [N3: num] :
% 0.30/0.66                            ( ( Xa2
% 0.30/0.66                              = ( bit1 @ N3 ) )
% 0.30/0.66                           => ( Y3
% 0.30/0.66                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
% 0.30/0.66                   => ( ! [M3: num] :
% 0.30/0.66                          ( ( X3
% 0.30/0.66                            = ( bit1 @ M3 ) )
% 0.30/0.66                         => ( ( Xa2 = one )
% 0.30/0.66                           => ( Y3
% 0.30/0.66                             != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 0.30/0.66                     => ( ! [M3: num] :
% 0.30/0.66                            ( ( X3
% 0.30/0.66                              = ( bit1 @ M3 ) )
% 0.30/0.66                           => ! [N3: num] :
% 0.30/0.66                                ( ( Xa2
% 0.30/0.66                                  = ( bit0 @ N3 ) )
% 0.30/0.66                               => ( Y3
% 0.30/0.66                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
% 0.30/0.66                       => ~ ! [M3: num] :
% 0.30/0.66                              ( ( X3
% 0.30/0.66                                = ( bit1 @ M3 ) )
% 0.30/0.66                             => ! [N3: num] :
% 0.30/0.66                                  ( ( Xa2
% 0.30/0.66                                    = ( bit1 @ N3 ) )
% 0.30/0.66                                 => ( Y3
% 0.30/0.66                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  % xor_num.elims
% 0.30/0.66  thf(fact_9794_xor__num_Osimps_I1_J,axiom,
% 0.30/0.66      ( ( bit_un2480387367778600638or_num @ one @ one )
% 0.30/0.66      = none_num ) ).
% 0.30/0.66  
% 0.30/0.66  % xor_num.simps(1)
% 0.30/0.66  
% 0.30/0.66  % Helper facts (34)
% 0.30/0.66  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( if_int @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 0.30/0.66      ! [X3: int,Y3: int] :
% 0.30/0.66        ( ( if_int @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.66      ! [X3: nat,Y3: nat] :
% 0.30/0.66        ( ( if_nat @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 0.30/0.66      ! [X3: nat,Y3: nat] :
% 0.30/0.66        ( ( if_nat @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.66      ! [X3: rat,Y3: rat] :
% 0.30/0.66        ( ( if_rat @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 0.30/0.66      ! [X3: rat,Y3: rat] :
% 0.30/0.66        ( ( if_rat @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( if_real @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 0.30/0.66      ! [X3: real,Y3: real] :
% 0.30/0.66        ( ( if_real @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 0.30/0.66      ! [P: real > $o] :
% 0.30/0.66        ( ( P @ ( fChoice_real @ P ) )
% 0.30/0.66        = ( ? [X5: real] : ( P @ X5 ) ) ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.66      ! [X3: complex,Y3: complex] :
% 0.30/0.66        ( ( if_complex @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 0.30/0.66      ! [X3: complex,Y3: complex] :
% 0.30/0.66        ( ( if_complex @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.66      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.66        ( ( if_Extended_enat @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 0.30/0.66      ! [X3: extended_enat,Y3: extended_enat] :
% 0.30/0.66        ( ( if_Extended_enat @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.66      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.66        ( ( if_Code_integer @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 0.30/0.66      ! [X3: code_integer,Y3: code_integer] :
% 0.30/0.66        ( ( if_Code_integer @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: set_int,Y3: set_int] :
% 0.30/0.66        ( ( if_set_int @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: set_int,Y3: set_int] :
% 0.30/0.66        ( ( if_set_int @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: list_int,Y3: list_int] :
% 0.30/0.66        ( ( if_list_int @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: list_int,Y3: list_int] :
% 0.30/0.66        ( ( if_list_int @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.66      ! [X3: list_nat,Y3: list_nat] :
% 0.30/0.66        ( ( if_list_nat @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 0.30/0.66      ! [X3: list_nat,Y3: list_nat] :
% 0.30/0.66        ( ( if_list_nat @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: int > int,Y3: int > int] :
% 0.30/0.66        ( ( if_int_int @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: int > int,Y3: int > int] :
% 0.30/0.66        ( ( if_int_int @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.66      ! [X3: option_num,Y3: option_num] :
% 0.30/0.66        ( ( if_option_num @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 0.30/0.66      ! [X3: option_num,Y3: option_num] :
% 0.30/0.66        ( ( if_option_num @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.66        ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 0.30/0.66      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 0.30/0.66        ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.66      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.66        ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 0.30/0.66      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 0.30/0.66        ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.66      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.66        ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 0.30/0.66      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 0.30/0.66        ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.66      ! [P: $o] :
% 0.30/0.66        ( ( P = $true )
% 0.30/0.66        | ( P = $false ) ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.66      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.66        ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
% 0.30/0.66        = Y3 ) ).
% 0.30/0.66  
% 0.30/0.66  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 0.30/0.66      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 0.30/0.66        ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
% 0.30/0.66        = X3 ) ).
% 0.30/0.66  
% 0.30/0.66  % Conjectures (2)
% 0.30/0.66  thf(conj_0,hypothesis,
% 0.30/0.66      ! [X2: nat] :
% 0.30/0.66        ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X2 )
% 0.30/0.66       => thesis ) ).
% 0.30/0.66  
% 0.30/0.66  thf(conj_1,conjecture,
% 0.30/0.66      thesis ).
% 0.30/0.66  
% 0.30/0.66  %------------------------------------------------------------------------------
% 0.30/0.66  ------------------------
% 6.43/2.58  ---- Embedded in THF ---
% 6.43/2.58  %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 6.43/2.58  %%% Generated on Thu Feb 26 16:00:53 EST 2026
% 6.43/2.58  %%% using '$$dhol' embedding, version 1.3.0.
% 6.43/2.58  %%% Logic specification used:
% 6.43/2.58  %%% thf(spec, logic, $$dhol).
% 6.43/2.58  
% 6.43/2.58  % SZS output start ListOfTHF for /export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2DTF22391.p
% See solution above
% 6.73/2.63  ------------------------
% 9.15/5.06  ---- Cleaned THF ---
% 9.15/5.06  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,
% 9.15/5.06      produc1319942482725812455at_nat: $tType ).
% 9.15/5.06  
% 9.15/5.06  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,
% 9.15/5.06      produc5542196010084753463at_nat: $tType ).
% 9.15/5.06  
% 9.15/5.06  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,
% 9.15/5.06      list_P7828571989066258726nteger: $tType ).
% 9.15/5.06  
% 9.15/5.06  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,
% 9.15/5.06      set_Pr1281608226676607948nteger: $tType ).
% 9.15/5.06  
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% 9.15/5.07      comm_s4028243227959126397er_rat: rat > nat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer_001t__Real__Oreal,type,
% 9.15/5.07      comm_s7457072308508201937r_real: real > nat > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Code____Numeral__Ointeger,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Complex__Ocomplex,type,
% 9.15/5.07      semiri5044797733671781792omplex: nat > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Int__Oint,type,
% 9.15/5.07      semiri1406184849735516958ct_int: nat > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Nat__Onat,type,
% 9.15/5.07      semiri1408675320244567234ct_nat: nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Rat__Orat,type,
% 9.15/5.07      semiri773545260158071498ct_rat: nat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Factorial_Osemiring__char__0__class_Ofact_001t__Real__Oreal,type,
% 9.15/5.07      semiri2265585572941072030t_real: nat > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Rat__Orat,type,
% 9.15/5.07      inverse_inverse_rat: rat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Fields_Oinverse__class_Oinverse_001t__Real__Oreal,type,
% 9.15/5.07      inverse_inverse_real: real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oat__bot_001t__Real__Oreal,type,
% 9.15/5.07      at_bot_real: filter_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oat__top_001t__Nat__Onat,type,
% 9.15/5.07      at_top_nat: filter_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
% 9.15/5.07      at_top_real: filter_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oeventually_001t__Nat__Onat,type,
% 9.15/5.07      eventually_nat: ( nat > $o ) > filter_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oeventually_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oeventually_001t__Real__Oreal,type,
% 9.15/5.07      eventually_real: ( real > $o ) > filter_real > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      filterlim_nat_nat: ( nat > nat ) > filter_nat > filter_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Ofilterlim_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.15/5.07      filterlim_nat_real: ( nat > real ) > filter_real > filter_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Ofilterlim_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      filterlim_real_real: ( real > real ) > filter_real > filter_real > $o ).
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% 9.15/5.07  thf(sy_c_Filter_Oprincipal_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Filter_Oprod__filter_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ocard_001t__Complex__Ocomplex,type,
% 9.15/5.07      finite_card_complex: set_complex > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ocard_001t__List__Olist_It__Nat__Onat_J,type,
% 9.15/5.07      finite_card_list_nat: set_list_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
% 9.15/5.07      finite_card_nat: set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ocard_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      finite_card_set_nat: set_set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001_Eo,type,
% 9.15/5.07      finite_finite_o: set_o > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Complex__Ocomplex,type,
% 9.15/5.07      finite3207457112153483333omplex: set_complex > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Int__Oint,type,
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% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_I_Eo_J,type,
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% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Complex__Ocomplex_J,type,
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% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Int__Oint_J,type,
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% 9.15/5.07      finite_finite_nat: set_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Num__Onum,type,
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% 9.15/5.07  
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Real__Oreal,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Complex__Ocomplex_J,type,
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% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
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% 9.15/5.07  thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Complex__Ocomplex,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Fun_Obij__betw_001t__Nat__Onat_001t__Nat__Onat,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Fun_Ocomp_001t__Int__Oint_001t__Nat__Onat_001t__Int__Oint,type,
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% 9.15/5.07  thf(sy_c_Fun_Oinj__on_001t__Real__Oreal_001t__Real__Oreal,type,
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% 9.15/5.07      bezw: nat > nat > product_prod_int_int ).
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% 9.15/5.07  thf(sy_c_GCD_Obezw__rel,type,
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% 9.15/5.07      gcd_gcd_int: int > int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_GCD_Ogcd__class_Ogcd_001t__Nat__Onat,type,
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% 9.15/5.07  thf(sy_c_GCD_Ogcd__nat__rel,type,
% 9.15/5.07      gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
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% 9.15/5.07  thf(sy_c_Groups_Oabs__class_Oabs_001t__Rat__Orat,type,
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% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Int__Oint_001t__Real__Oreal,type,
% 9.15/5.07      lattic2675449441010098035t_real: ( int > real ) > set_int > int ).
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% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Num__Onum,type,
% 9.15/5.07      lattic4004264746738138117at_num: ( nat > num ) > set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.15/5.07      lattic6811802900495863747at_rat: ( nat > rat ) > set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.15/5.07      lattic488527866317076247t_real: ( nat > real ) > set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.15/5.07      lattic5055836439445974935al_nat: ( real > nat ) > set_real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Num__Onum,type,
% 9.15/5.07      lattic1613168225601753569al_num: ( real > num ) > set_real > real ).
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% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Rat__Orat,type,
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% 9.15/5.07  thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      lattic8440615504127631091l_real: ( real > real ) > set_real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oappend_001t__Int__Oint,type,
% 9.15/5.07      append_int: list_int > list_int > list_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oappend_001t__Nat__Onat,type,
% 9.15/5.07      append_nat: list_nat > list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001_Eo,type,
% 9.15/5.07      count_list_o: list_o > $o > nat ).
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% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Complex__Ocomplex,type,
% 9.15/5.07      count_list_complex: list_complex > complex > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Int__Oint,type,
% 9.15/5.07      count_list_int: list_int > int > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Nat__Onat,type,
% 9.15/5.07      count_list_nat: list_nat > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Real__Oreal,type,
% 9.15/5.07      count_list_real: list_real > real > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      count_list_set_nat: list_set_nat > set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ocount__list_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat ).
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% 9.15/5.07  thf(sy_c_List_Odistinct_001t__Int__Oint,type,
% 9.15/5.07      distinct_int: list_int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
% 9.15/5.07      distinct_nat: list_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Odrop_001t__Nat__Onat,type,
% 9.15/5.07      drop_nat: nat > list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oenumerate_001_Eo,type,
% 9.15/5.07      enumerate_o: nat > list_o > list_P7333126701944960589_nat_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
% 9.15/5.07      enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
% 9.15/5.07      enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oenumerate_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001_Eo,type,
% 9.15/5.07      find_o: ( $o > $o ) > list_o > option_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001t__Int__Oint,type,
% 9.15/5.07      find_int: ( int > $o ) > list_int > option_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001t__Nat__Onat,type,
% 9.15/5.07      find_nat: ( nat > $o ) > list_nat > option_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001t__Num__Onum,type,
% 9.15/5.07      find_num: ( num > $o ) > list_num > option_num ).
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      find_set_nat: ( set_nat > $o ) > list_set_nat > option_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ofind_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      find_VEBT_VEBT: ( vEBT_VEBT > $o ) > list_VEBT_VEBT > option_VEBT_VEBT ).
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% 9.15/5.07  thf(sy_c_List_Ogen__length_001_Eo,type,
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% 9.15/5.07  thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
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% 9.15/5.07  thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
% 9.15/5.07      cons_nat: nat > list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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% 9.15/5.07  thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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% 9.15/5.07  
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
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% 9.15/5.07      set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.15/5.07      set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.15/5.07      set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
% 9.15/5.07      set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.15/5.07      set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Real__Oreal,type,
% 9.15/5.07      set_real2: list_real > set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      set_set_nat2: list_set_nat > set_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      size_list_VEBT_VEBT: ( vEBT_VEBT > nat ) > list_VEBT_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
% 9.15/5.07      tl_nat: list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > ( code_integer > option6357759511663192854e_term ) > list_C878401137130745250e_term ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > ( int > option6357759511663192854e_term ) > list_i8448526496819171953e_term ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > ( produc6241069584506657477e_term > option6357759511663192854e_term ) > list_P1316552470764441098e_term ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > ( produc8551481072490612790e_term > option6357759511663192854e_term ) > list_P1743416141875011707e_term ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001_Eo,type,
% 9.15/5.07      list_update_o: list_o > nat > $o > list_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Int__Oint,type,
% 9.15/5.07      list_update_int: list_int > nat > int > list_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
% 9.15/5.07      list_update_nat: list_nat > nat > nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
% 9.15/5.07      list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
% 9.15/5.07      list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
% 9.15/5.07      list_update_real: list_real > nat > real > list_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olist__update_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olistrel1_001_Eo,type,
% 9.15/5.07      listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
% 9.15/5.07      listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
% 9.15/5.07      listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olistrel1_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Olistrel1p_001t__Int__Oint,type,
% 9.15/5.07      listrel1p_int: ( int > int > $o ) > list_int > list_int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001_Eo,type,
% 9.15/5.07      nth_o: list_o > nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Int__Oint,type,
% 9.15/5.07      nth_int: list_int > nat > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
% 9.15/5.07      nth_list_nat: list_list_nat > nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Nat__Onat,type,
% 9.15/5.07      nth_nat: list_nat > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Num__Onum,type,
% 9.15/5.07      nth_num: list_num > nat > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
% 9.15/5.07      nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
% 9.15/5.07      nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
% 9.15/5.07      nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.15/5.07      nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
% 9.15/5.07      nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
% 9.15/5.07      nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.15/5.07      nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
% 9.15/5.07      nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
% 9.15/5.07      nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
% 9.15/5.07      nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
% 9.15/5.07      nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Real__Oreal,type,
% 9.15/5.07      nth_real: list_real > nat > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      nth_set_nat: list_set_nat > nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001_Eo_001_Eo,type,
% 9.15/5.07      product_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001_Eo_001t__Int__Oint,type,
% 9.15/5.07      product_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
% 9.15/5.07      product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
% 9.15/5.07      product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.15/5.07      product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.15/5.07      produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.15/5.07      produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
% 9.15/5.07      remdups_nat: list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001_Eo,type,
% 9.15/5.07      replicate_o: nat > $o > list_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__Int__Oint,type,
% 9.15/5.07      replicate_int: nat > int > list_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__Nat__Onat,type,
% 9.15/5.07      replicate_nat: nat > nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__Real__Oreal,type,
% 9.15/5.07      replicate_real: nat > real > list_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      replicate_set_nat: nat > set_nat > list_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oreplicate_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Osorted__wrt_001t__Int__Oint,type,
% 9.15/5.07      sorted_wrt_int: ( int > int > $o ) > list_int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
% 9.15/5.07      sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Otake_001t__Nat__Onat,type,
% 9.15/5.07      take_nat: nat > list_nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oupt,type,
% 9.15/5.07      upt: nat > nat > list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oupto,type,
% 9.15/5.07      upto: int > int > list_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oupto__aux,type,
% 9.15/5.07      upto_aux: int > int > list_int > list_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Oupto__rel,type,
% 9.15/5.07      upto_rel: product_prod_int_int > product_prod_int_int > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
% 9.15/5.07      zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001_Eo_001t__Int__Oint,type,
% 9.15/5.07      zip_o_int: list_o > list_int > list_P3795440434834930179_o_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001_Eo_001t__Nat__Onat,type,
% 9.15/5.07      zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
% 9.15/5.07      zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Int__Oint,type,
% 9.15/5.07      zip_int_int: list_int > list_int > list_P5707943133018811711nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
% 9.15/5.07      zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Nat__Onat_001_Eo,type,
% 9.15/5.07      zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
% 9.15/5.07      zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
% 9.15/5.07      zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.15/5.07      zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
% 9.15/5.07      zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
% 9.15/5.07      zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
% 9.15/5.07      zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
% 9.15/5.07      zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_OSuc,type,
% 9.15/5.07      suc: nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
% 9.15/5.07      case_nat_o: $o > ( nat > $o ) > nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
% 9.15/5.07      case_nat_nat: nat > ( nat > nat ) > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.15/5.07      case_nat_option_num: option_num > ( nat > option_num ) > nat > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Onat_Opred,type,
% 9.15/5.07      pred: nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
% 9.15/5.07      semiring_1_Nats_int: set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Code____Numeral__Ointeger,type,
% 9.15/5.07      semiri4939895301339042750nteger: nat > code_integer ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex,type,
% 9.15/5.07      semiri8010041392384452111omplex: nat > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Extended____Nat__Oenat,type,
% 9.15/5.07      semiri4216267220026989637d_enat: nat > extended_enat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
% 9.15/5.07      semiri1314217659103216013at_int: nat > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
% 9.15/5.07      semiri1316708129612266289at_nat: nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Rat__Orat,type,
% 9.15/5.07      semiri681578069525770553at_rat: nat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal,type,
% 9.15/5.07      semiri5074537144036343181t_real: nat > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Complex__Ocomplex,type,
% 9.15/5.07      semiri2816024913162550771omplex: ( complex > complex ) > nat > complex > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Int__Oint,type,
% 9.15/5.07      semiri8420488043553186161ux_int: ( int > int ) > nat > int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Nat__Onat,type,
% 9.15/5.07      semiri8422978514062236437ux_nat: ( nat > nat ) > nat > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Rat__Orat,type,
% 9.15/5.07      semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
% 9.15/5.07      semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
% 9.15/5.07      size_size_list_o: list_o > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Complex__Ocomplex_J,type,
% 9.15/5.07      size_s3451745648224563538omplex: list_complex > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
% 9.15/5.07      size_size_list_int: list_int > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.15/5.07      size_s3023201423986296836st_nat: list_list_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
% 9.15/5.07      size_size_list_nat: list_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Num__Onum_J,type,
% 9.15/5.07      size_size_list_num: list_num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
% 9.15/5.07      size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
% 9.15/5.07      size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
% 9.15/5.07      size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.15/5.07      size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
% 9.15/5.07      size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
% 9.15/5.07      size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.15/5.07      size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
% 9.15/5.07      size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
% 9.15/5.07      size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
% 9.15/5.07      size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
% 9.15/5.07      size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
% 9.15/5.07      size_size_list_real: list_real > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.15/5.07      size_s3254054031482475050et_nat: list_set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
% 9.15/5.07      size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
% 9.15/5.07      size_size_num: num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.15/5.07      size_size_option_nat: option_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.15/5.07      size_size_option_num: option_num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
% 9.15/5.07      size_s170228958280169651at_nat: option4927543243414619207at_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      size_size_VEBT_VEBT: vEBT_VEBT > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Olist__encode,type,
% 9.15/5.07      nat_list_encode: list_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
% 9.15/5.07      nat_list_encode_rel: list_nat > list_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
% 9.15/5.07      nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
% 9.15/5.07      nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Oprod__encode,type,
% 9.15/5.07      nat_prod_encode: product_prod_nat_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Oset__decode,type,
% 9.15/5.07      nat_set_decode: nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Oset__encode,type,
% 9.15/5.07      nat_set_encode: set_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Nat__Bijection_Otriangle,type,
% 9.15/5.07      nat_triangle: nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_NthRoot_Oroot,type,
% 9.15/5.07      root: nat > real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_NthRoot_Osqrt,type,
% 9.15/5.07      sqrt: real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_OBitM,type,
% 9.15/5.07      bitM: num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oinc,type,
% 9.15/5.07      inc: num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Code____Numeral__Ointeger,type,
% 9.15/5.07      neg_nu7757733837767384882nteger: code_integer > code_integer ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex,type,
% 9.15/5.07      neg_nu6511756317524482435omplex: complex > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
% 9.15/5.07      neg_nu3811975205180677377ec_int: int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Rat__Orat,type,
% 9.15/5.07      neg_nu3179335615603231917ec_rat: rat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Real__Oreal,type,
% 9.15/5.07      neg_nu6075765906172075777c_real: real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Oneg__numeral__class_Osub_001t__Int__Oint,type,
% 9.15/5.07      neg_numeral_sub_int: num > num > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum_OBit0,type,
% 9.15/5.07      bit0: num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum_OBit1,type,
% 9.15/5.07      bit1: num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum_OOne,type,
% 9.15/5.07      one: num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum_Ocase__num_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.15/5.07      case_num_option_num: option_num > ( num > option_num ) > ( num > option_num ) > num > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum_Osize__num,type,
% 9.15/5.07      size_num: num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onum__of__nat,type,
% 9.15/5.07      num_of_nat: nat > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Code____Numeral__Ointeger,type,
% 9.15/5.07      numera6620942414471956472nteger: num > code_integer ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex,type,
% 9.15/5.07      numera6690914467698888265omplex: num > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Extended____Nat__Oenat,type,
% 9.15/5.07      numera1916890842035813515d_enat: num > extended_enat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Int__Oint,type,
% 9.15/5.07      numeral_numeral_int: num > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
% 9.15/5.07      numeral_numeral_nat: num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Rat__Orat,type,
% 9.15/5.07      numeral_numeral_rat: num > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal,type,
% 9.15/5.07      numeral_numeral_real: num > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Num_Opred__numeral,type,
% 9.15/5.07      pred_numeral: num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Int__Oint,type,
% 9.15/5.07      none_int: option_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
% 9.15/5.07      none_nat: option_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
% 9.15/5.07      none_num: option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      none_P5556105721700978146at_nat: option4927543243414619207at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Real__Oreal,type,
% 9.15/5.07      none_real: option_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      none_set_nat: option_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_ONone_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      none_VEBT_VEBT: option_VEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001_Eo,type,
% 9.15/5.07      some_o: $o > option_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001t__Int__Oint,type,
% 9.15/5.07      some_int: int > option_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
% 9.15/5.07      some_nat: nat > option_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001t__Num__Onum,type,
% 9.15/5.07      some_num: num > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_OSome_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      case_o184042715313410164at_nat: $o > ( product_prod_nat_nat > $o ) > option4927543243414619207at_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Ocase__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.15/5.07      case_option_num_num: num > ( num > num ) > option_num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Ocase__option_001t__Option__Ooption_It__Num__Onum_J_001t__Num__Onum,type,
% 9.15/5.07      case_o6005452278849405969um_num: option_num > ( num > option_num ) > option_num > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
% 9.15/5.07      map_option_num_num: ( num > num ) > option_num > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Osize__option_001t__Nat__Onat,type,
% 9.15/5.07      size_option_nat: ( nat > nat ) > option_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Osize__option_001t__Num__Onum,type,
% 9.15/5.07      size_option_num: ( num > nat ) > option_num > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Osize__option_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      size_o8335143837870341156at_nat: ( product_prod_nat_nat > nat ) > option4927543243414619207at_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Othe_001t__Nat__Onat,type,
% 9.15/5.07      the_nat: option_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
% 9.15/5.07      the_num: option_num > num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.07      the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
% 9.15/5.07      order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo1403522918969695512_int_o: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo3000040243691356879eger_o: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo8662317086119403298_int_o: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J,type,
% 9.15/5.07      bot_bot_int_int_o: int > int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Int__Oint_M_Eo_J,type,
% 9.15/5.07      bot_bot_int_o: int > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
% 9.15/5.07      bot_bot_list_nat_o: list_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J,type,
% 9.15/5.07      bot_bot_nat_nat_o: nat > nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Nat__Onat_M_Eo_J,type,
% 9.15/5.07      bot_bot_nat_o: nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
% 9.15/5.07      bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Real__Oreal_M_Eo_J,type,
% 9.15/5.07      bot_bot_real_o: real > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
% 9.15/5.07      bot_bot_set_nat_o: set_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
% 9.15/5.07      bot_bo4199563552545308370d_enat: extended_enat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Filter__Ofilter_It__Nat__Onat_J,type,
% 9.15/5.07      bot_bot_filter_nat: filter_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
% 9.15/5.07      bot_bot_nat: nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Complex__Ocomplex_J,type,
% 9.15/5.07      bot_bot_set_complex: set_complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
% 9.15/5.07      bot_bot_set_int: set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
% 9.15/5.07      bot_bot_set_list_nat: set_list_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      bot_bot_set_nat: set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Num__Onum_J,type,
% 9.15/5.07      bot_bot_set_num: set_num ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger ).
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% 9.15/5.07  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,
% 9.15/5.07      bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Rat__Orat_J,type,
% 9.15/5.07      bot_bot_set_rat: set_rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
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% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
% 9.15/5.07      bot_bot_set_set_int: set_set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
% 9.15/5.07      bot_bot_set_set_nat: set_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__VEBT____Definitions__OVEBT_J,type,
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% 9.15/5.07      top_top_set_int: set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      top_top_set_nat: set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.15/5.07      top_top_set_real: set_real ).
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% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Code____Numeral__Ointeger,type,
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% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Extended____Nat__Oenat,type,
% 9.15/5.07      power_8040749407984259932d_enat: extended_enat > nat > extended_enat ).
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% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
% 9.15/5.07      power_power_int: int > nat > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
% 9.15/5.07      power_power_nat: nat > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
% 9.15/5.07      power_power_rat: rat > nat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  
% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  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,
% 9.15/5.07      produc3994169339658061776at_nat: ( product_prod_nat_nat > product_prod_nat_nat > $o ) > produc6121120109295599847at_nat > produc5491161045314408544at_nat ).
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% 9.15/5.07  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,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
% 9.15/5.07      product_Pair_o_o: $o > $o > product_prod_o_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001_Eo_001t__Int__Oint,type,
% 9.15/5.07      product_Pair_o_int: $o > int > product_prod_o_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
% 9.15/5.07      product_Pair_o_nat: $o > nat > product_prod_o_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
% 9.15/5.07      product_Pair_o_real: $o > real > product_prod_o_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
% 9.15/5.07      produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001_Eo,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
% 9.15/5.07      produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
% 9.15/5.07      produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__List__Olist_It__VEBT____Definitions__OVEBT_J_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001_Eo,type,
% 9.15/5.07      product_Pair_nat_o: nat > $o > product_prod_nat_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
% 9.15/5.07      product_Pair_nat_int: nat > int > product_prod_nat_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Nat__Onat_J,type,
% 9.15/5.07      produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Option__Ooption_It__Num__Onum_J_001t__Option__Ooption_It__Num__Onum_J,type,
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001_Eo,type,
% 9.15/5.07      product_Pair_real_o: real > $o > product_prod_real_o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Int__Oint,type,
% 9.15/5.07      produc3179012173361985393al_int: real > int > produc8786904178792722361al_int ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Nat__Onat,type,
% 9.15/5.07      produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      produc4511245868158468465l_real: real > real > produc2422161461964618553l_real ).
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat ).
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% 9.15/5.07  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,
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% 9.15/5.07  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,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OPair_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
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% 9.15/5.07  thf(sy_c_Product__Type_OSigma_001t__Nat__Onat_001t__Nat__Onat,type,
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% 9.15/5.07      produc8604463032469472703et_int: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int ) > produc8763457246119570046nteger > set_int ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc3558942015123893603et_nat: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat ) > produc8763457246119570046nteger > set_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc815715089573277247t_real: ( ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real ) > produc8763457246119570046nteger > set_real ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc2558449545302689196_int_o: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc7773217078559923341nt_int > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc5122537100556696953nt_int: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > produc7773217078559923341nt_int ) > produc7773217078559923341nt_int > produc7773217078559923341nt_int ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc8289552606927098482et_nat: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat ) > produc7773217078559923341nt_int > set_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc8709739885379107790t_real: ( ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real ) > produc7773217078559923341nt_int > set_real ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc6253627499356882019eger_o: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ) > produc1908205239877642774nteger > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc6512950862096126219nteger: ( ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > produc1908205239877642774nteger ) > produc1908205239877642774nteger > produc1908205239877642774nteger ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc1573362020775583542_int_o: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o ) > produc2285326912895808259nt_int > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc8492565224438309093nt_int: ( ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > produc2285326912895808259nt_int ) > produc2285326912895808259nt_int > produc2285326912895808259nt_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Int__Oint,type,
% 9.15/5.07      produc1553301316500091796er_int: ( code_integer > code_integer > int ) > produc8923325533196201883nteger > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Nat__Onat,type,
% 9.15/5.07      produc1555791787009142072er_nat: ( code_integer > code_integer > nat ) > produc8923325533196201883nteger > nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc9125791028180074456eger_o: ( code_integer > code_integer > produc6271795597528267376eger_o ) > produc8923325533196201883nteger > produc6271795597528267376eger_o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Complex__Ocomplex_001t__Complex__Ocomplex_001_Eo,type,
% 9.15/5.07      produc6771430404735790350plex_o: ( complex > complex > $o ) > produc4411394909380815293omplex > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001_Eo,type,
% 9.15/5.07      produc4947309494688390418_int_o: ( int > int > $o ) > product_prod_int_int > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc4245557441103728435nt_int: ( int > int > product_prod_int_int ) > product_prod_int_int > product_prod_int_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Int__Oint_J,type,
% 9.15/5.07      produc73460835934605544et_int: ( int > int > set_int ) > product_prod_int_int > set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      produc4251311855443802252et_nat: ( int > int > set_nat ) > product_prod_int_int > set_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc1656060378719767003at_nat: ( int > int > set_Pr1261947904930325089at_nat ) > product_prod_int_int > set_Pr1261947904930325089at_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Int__Oint_001t__Int__Oint_001t__Set__Oset_It__Real__Oreal_J,type,
% 9.15/5.07      produc6452406959799940328t_real: ( int > int > set_real ) > product_prod_int_int > set_real ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc5233655623923918146et_nat: ( int > int > set_set_nat ) > product_prod_int_int > set_set_nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc8739625826339149834_nat_o: ( nat > nat > product_prod_nat_nat > $o ) > product_prod_nat_nat > product_prod_nat_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc27273713700761075at_nat: ( nat > nat > product_prod_nat_nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001_Eo,type,
% 9.15/5.07      produc6081775807080527818_nat_o: ( nat > nat > $o ) > product_prod_nat_nat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Complex__Ocomplex,type,
% 9.15/5.07      produc1917071388513777916omplex: ( nat > nat > complex ) > product_prod_nat_nat > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Extended____Nat__Oenat,type,
% 9.15/5.07      produc2676513652042109336d_enat: ( nat > nat > extended_enat ) > product_prod_nat_nat > extended_enat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Int__Oint,type,
% 9.15/5.07      produc6840382203811409530at_int: ( nat > nat > int ) > product_prod_nat_nat > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      produc6842872674320459806at_nat: ( nat > nat > nat ) > product_prod_nat_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  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,
% 9.15/5.07      produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Rat__Orat,type,
% 9.15/5.07      produc6207742614233964070at_rat: ( nat > nat > rat ) > product_prod_nat_nat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Real__Oreal,type,
% 9.15/5.07      produc1703576794950452218t_real: ( nat > nat > real ) > product_prod_nat_nat > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
% 9.15/5.07      produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Real__Oreal_001t__Real__Oreal_001_Eo,type,
% 9.15/5.07      produc5414030515140494994real_o: ( real > real > $o ) > produc2422161461964618553l_real > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Int__Oint,type,
% 9.15/5.07      product_fst_int_int: product_prod_int_int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      product_fst_nat_nat: product_prod_nat_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
% 9.15/5.07      product_snd_int_int: product_prod_int_int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
% 9.15/5.07      product_snd_nat_nat: product_prod_nat_nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_OFract,type,
% 9.15/5.07      fract: int > int > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_OFrct,type,
% 9.15/5.07      frct: product_prod_int_int > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_ORep__Rat,type,
% 9.15/5.07      rep_Rat: rat > product_prod_int_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
% 9.15/5.07      field_5140801741446780682s_real: set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_Onormalize,type,
% 9.15/5.07      normalize: product_prod_int_int > product_prod_int_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_Oof__int,type,
% 9.15/5.07      of_int: int > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_Opositive,type,
% 9.15/5.07      positive: rat > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rat_Oquotient__of,type,
% 9.15/5.07      quotient_of: rat > product_prod_int_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
% 9.15/5.07      real_V3694042436643373181omplex: complex > complex > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
% 9.15/5.07      real_V975177566351809787t_real: real > real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Complex__Ocomplex,type,
% 9.15/5.07      real_V1022390504157884413omplex: complex > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm_001t__Real__Oreal,type,
% 9.15/5.07      real_V7735802525324610683m_real: real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
% 9.15/5.07      real_V4546457046886955230omplex: real > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
% 9.15/5.07      real_V1803761363581548252l_real: real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Relation_OField_001t__Nat__Onat,type,
% 9.15/5.07      field_nat: set_Pr1261947904930325089at_nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
% 9.15/5.07      divide6298287555418463151nteger: code_integer > code_integer > code_integer ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
% 9.15/5.07      divide1717551699836669952omplex: complex > complex > complex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint,type,
% 9.15/5.07      divide_divide_int: int > int > int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat,type,
% 9.15/5.07      divide_divide_nat: nat > nat > nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Rat__Orat,type,
% 9.15/5.07      divide_divide_rat: rat > rat > rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
% 9.15/5.07      divide_divide_real: real > real > real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
% 9.15/5.07      dvd_dvd_Code_integer: code_integer > code_integer > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
% 9.15/5.07      dvd_dvd_complex: complex > complex > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
% 9.15/5.07      dvd_dvd_int: int > int > $o ).
% 9.15/5.07  
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% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Num__Onum,type,
% 9.15/5.07      set_ord_atMost_num: num > set_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Rat__Orat,type,
% 9.15/5.07      set_ord_atMost_rat: rat > set_rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Real__Oreal,type,
% 9.15/5.07      set_ord_atMost_real: real > set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Int__Oint_J,type,
% 9.15/5.07      set_or58775011639299419et_int: set_int > set_set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OatMost_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      set_or4236626031148496127et_nat: set_nat > set_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Int__Oint,type,
% 9.15/5.07      set_or6656581121297822940st_int: int > int > set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost_001t__Nat__Onat,type,
% 9.15/5.07      set_or6659071591806873216st_nat: nat > nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Int__Oint,type,
% 9.15/5.07      set_or5832277885323065728an_int: int > int > set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat,type,
% 9.15/5.07      set_or5834768355832116004an_nat: nat > nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Real__Oreal,type,
% 9.15/5.07      set_or1633881224788618240n_real: real > real > set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Nat__Onat,type,
% 9.15/5.07      set_or1210151606488870762an_nat: nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OgreaterThan_001t__Real__Oreal,type,
% 9.15/5.07      set_or5849166863359141190n_real: real > set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Int__Oint,type,
% 9.15/5.07      set_ord_lessThan_int: int > set_int ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
% 9.15/5.07      set_ord_lessThan_nat: nat > set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Num__Onum,type,
% 9.15/5.07      set_ord_lessThan_num: num > set_num ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
% 9.15/5.07      set_ord_lessThan_rat: rat > set_rat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Real__Oreal,type,
% 9.15/5.07      set_or5984915006950818249n_real: real > set_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Set__Oset_It__Nat__Onat_J,type,
% 9.15/5.07      set_or890127255671739683et_nat: set_nat > set_set_nat ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Ocontinuous__on_001t__Real__Oreal_001t__Real__Oreal,type,
% 9.15/5.07      topolo5044208981011980120l_real: set_real > ( real > real ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
% 9.15/5.07      topolo4899668324122417113eq_int: ( nat > int ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
% 9.15/5.07      topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
% 9.15/5.07      topolo1459490580787246023eq_num: ( nat > num ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
% 9.15/5.07      topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Real__Oreal,type,
% 9.15/5.07      topolo6980174941875973593q_real: ( nat > real ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
% 9.15/5.07      topolo3100542954746470799et_int: ( nat > set_int ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within_001t__Real__Oreal,type,
% 9.15/5.07      topolo2177554685111907308n_real: real > set_real > filter_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds_001t__Real__Oreal,type,
% 9.15/5.07      topolo2815343760600316023s_real: real > filter_real ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy_001t__Real__Oreal,type,
% 9.15/5.07      topolo4055970368930404560y_real: ( nat > real ) > $o ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
% 9.15/5.07      topolo896644834953643431omplex: filter6041513312241820739omplex ).
% 9.15/5.07  
% 9.15/5.07  thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
% 9.15/5.08      topolo1511823702728130853y_real: filter2146258269922977983l_real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oarccos,type,
% 9.15/5.08      arccos: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oarcosh_001t__Real__Oreal,type,
% 9.15/5.08      arcosh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oarcsin,type,
% 9.15/5.08      arcsin: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oarctan,type,
% 9.15/5.08      arctan: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oarsinh_001t__Real__Oreal,type,
% 9.15/5.08      arsinh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oartanh_001t__Real__Oreal,type,
% 9.15/5.08      artanh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Ocos_001t__Complex__Ocomplex,type,
% 9.15/5.08      cos_complex: complex > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Ocos_001t__Real__Oreal,type,
% 9.15/5.08      cos_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Ocos__coeff,type,
% 9.15/5.08      cos_coeff: nat > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Ocosh_001t__Real__Oreal,type,
% 9.15/5.08      cosh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
% 9.15/5.08      cot_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Odiffs_001t__Complex__Ocomplex,type,
% 9.15/5.08      diffs_complex: ( nat > complex ) > nat > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Odiffs_001t__Int__Oint,type,
% 9.15/5.08      diffs_int: ( nat > int ) > nat > int ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Odiffs_001t__Rat__Orat,type,
% 9.15/5.08      diffs_rat: ( nat > rat ) > nat > rat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Odiffs_001t__Real__Oreal,type,
% 9.15/5.08      diffs_real: ( nat > real ) > nat > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oexp_001t__Complex__Ocomplex,type,
% 9.15/5.08      exp_complex: complex > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oexp_001t__Real__Oreal,type,
% 9.15/5.08      exp_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
% 9.15/5.08      ln_ln_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Olog,type,
% 9.15/5.08      log: real > real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Opi,type,
% 9.15/5.08      pi: real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Opowr_001t__Real__Oreal,type,
% 9.15/5.08      powr_real: real > real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Osin_001t__Complex__Ocomplex,type,
% 9.15/5.08      sin_complex: complex > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Osin_001t__Real__Oreal,type,
% 9.15/5.08      sin_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Osin__coeff,type,
% 9.15/5.08      sin_coeff: nat > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Osinh_001t__Real__Oreal,type,
% 9.15/5.08      sinh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Otan_001t__Complex__Ocomplex,type,
% 9.15/5.08      tan_complex: complex > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Otan_001t__Real__Oreal,type,
% 9.15/5.08      tan_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Otanh_001t__Complex__Ocomplex,type,
% 9.15/5.08      tanh_complex: complex > complex ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transcendental_Otanh_001t__Real__Oreal,type,
% 9.15/5.08      tanh_real: real > real ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transitive__Closure_Ortrancl_001t__Nat__Onat,type,
% 9.15/5.08      transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Transitive__Closure_Otrancl_001t__Nat__Onat,type,
% 9.15/5.08      transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
% 9.15/5.08      vEBT_Leaf: $o > $o > vEBT_VEBT ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
% 9.15/5.08      vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
% 9.15/5.08      vEBT_size_VEBT: vEBT_VEBT > nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
% 9.15/5.08      vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
% 9.15/5.08      vEBT_VEBT_high: nat > nat > nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
% 9.15/5.08      vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
% 9.15/5.08      vEBT_VEBT_low: nat > nat > nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
% 9.15/5.08      vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
% 9.15/5.08      vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
% 9.15/5.08      vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
% 9.15/5.08      vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
% 9.15/5.08      vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
% 9.15/5.08      vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
% 9.15/5.08      vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_Oset__vebt,type,
% 9.15/5.08      vEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
% 9.15/5.08      vEBT_vebt_buildup: nat > vEBT_VEBT ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
% 9.15/5.08      vEBT_v4011308405150292612up_rel: nat > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_OVEBT__internal_Obit__concat,type,
% 9.15/5.08      vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
% 9.15/5.08      vEBT_VEBT_minNull: vEBT_VEBT > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
% 9.15/5.08      vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
% 9.15/5.08      vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_Ovebt__member,type,
% 9.15/5.08      vEBT_vebt_member: vEBT_VEBT > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
% 9.15/5.08      vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oadd,type,
% 9.15/5.08      vEBT_VEBT_add: option_nat > option_nat > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ogreater,type,
% 9.15/5.08      vEBT_VEBT_greater: option_nat > option_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Oless,type,
% 9.15/5.08      vEBT_VEBT_less: option_nat > option_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Olesseq,type,
% 9.15/5.08      vEBT_VEBT_lesseq: option_nat > option_nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omax__in__set,type,
% 9.15/5.08      vEBT_VEBT_max_in_set: set_nat > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omin__in__set,type,
% 9.15/5.08      vEBT_VEBT_min_in_set: set_nat > nat > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Omul,type,
% 9.15/5.08      vEBT_VEBT_mul: option_nat > option_nat > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Nat__Onat,type,
% 9.15/5.08      vEBT_V4262088993061758097ft_nat: ( nat > nat > nat ) > option_nat > option_nat > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
% 9.15/5.08      vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
% 9.15/5.08      vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_OVEBT__internal_Opower,type,
% 9.15/5.08      vEBT_VEBT_power: option_nat > option_nat > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_Ovebt__maxt,type,
% 9.15/5.08      vEBT_vebt_maxt: vEBT_VEBT > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_Ovebt__maxt__rel,type,
% 9.15/5.08      vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_Ovebt__mint,type,
% 9.15/5.08      vEBT_vebt_mint: vEBT_VEBT > option_nat ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_VEBT__MinMax_Ovebt__mint__rel,type,
% 9.15/5.08      vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o ).
% 9.15/5.08  
% 9.15/5.08  thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
% 9.15/5.08      accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).
% 9.15/5.08  
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% 9.15/5.08     => ! [X2: vEBT_VEBT] :
% 9.15/5.08          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.15/5.08         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_5__C5_C_I1_J,axiom,
% 9.15/5.08      ! [X2: vEBT_VEBT] :
% 9.15/5.08        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
% 9.15/5.08       => ( vEBT_invar_vebt @ X2 @ n ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_6_in__children__def,axiom,
% 9.15/5.08      ( vEBT_V5917875025757280293ildren
% 9.15/5.08      = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_7_both__member__options__equiv__member,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 9.15/5.08          = ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_8_valid__member__both__member__options,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
% 9.15/5.08         => ( vEBT_vebt_member @ T @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_9_intind,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,P: nat > $o,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( P @ X3 )
% 9.15/5.08         => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_10_intind,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,P: int > $o,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( P @ X3 )
% 9.15/5.08         => ( P @ ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_11_intind,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( P @ X3 )
% 9.15/5.08         => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_12_assms_I2_J,axiom,
% 9.15/5.08      mi != ma ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_13_list__update__id,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,I: nat] :
% 9.15/5.08        ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ I ) )
% 9.15/5.08        = Xs ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_14_list__update__id,axiom,
% 9.15/5.08      ! [Xs: list_nat,I: nat] :
% 9.15/5.08        ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
% 9.15/5.08        = Xs ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_15_list__update__id,axiom,
% 9.15/5.08      ! [Xs: list_int,I: nat] :
% 9.15/5.08        ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
% 9.15/5.08        = Xs ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_16_nth__list__update__neq,axiom,
% 9.15/5.08      ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( I != J )
% 9.15/5.08       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 9.15/5.08          = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_17_nth__list__update__neq,axiom,
% 9.15/5.08      ! [I: nat,J: nat,Xs: list_nat,X3: nat] :
% 9.15/5.08        ( ( I != J )
% 9.15/5.08       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 9.15/5.08          = ( nth_nat @ Xs @ J ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_18_nth__list__update__neq,axiom,
% 9.15/5.08      ! [I: nat,J: nat,Xs: list_int,X3: int] :
% 9.15/5.08        ( ( I != J )
% 9.15/5.08       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 9.15/5.08          = ( nth_int @ Xs @ J ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_19_min__Null__member,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,X3: nat] :
% 9.15/5.08        ( ( vEBT_VEBT_minNull @ T )
% 9.15/5.08       => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_20__C5_C_I2_J,axiom,
% 9.15/5.08      vEBT_invar_vebt @ summary @ m ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_21_bit__split__inv,axiom,
% 9.15/5.08      ! [X3: nat,D: nat] :
% 9.15/5.08        ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
% 9.15/5.08        = X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 9.15/5.08      ~ ! [Maxs: nat] :
% 9.15/5.08          ( ( some_nat @ Maxs )
% 9.15/5.08         != ( vEBT_vebt_maxt @ summary ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_23_member__valid__both__member__options,axiom,
% 9.15/5.08      ! [Tree: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ Tree @ N2 )
% 9.15/5.08       => ( ( vEBT_vebt_member @ Tree @ X3 )
% 9.15/5.08         => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
% 9.15/5.08            | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_24_maxt__member,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_maxt @ T )
% 9.15/5.08            = ( some_nat @ Maxi ) )
% 9.15/5.08         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_25_member__correct,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_vebt_member @ T @ X3 )
% 9.15/5.08          = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_26__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
% 9.15/5.08      vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_27_nth__replicate,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_28_nth__replicate,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_29_nth__replicate,axiom,
% 9.15/5.08      ! [I: nat,N2: nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ N2 )
% 9.15/5.08       => ( ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_30__C5_C_I8_J,axiom,
% 9.15/5.08      ord_less_eq_nat @ mi @ ma ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_31_greater__shift,axiom,
% 9.15/5.08      ( ord_less_nat
% 9.15/5.08      = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_32_less__shift,axiom,
% 9.15/5.08      ( ord_less_nat
% 9.15/5.08      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_33__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
% 9.15/5.08      ord_less_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_34_mint__member,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_mint @ T )
% 9.15/5.08            = ( some_nat @ Maxi ) )
% 9.15/5.08         => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_35_maxt__corr__help,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_maxt @ T )
% 9.15/5.08            = ( some_nat @ Maxi ) )
% 9.15/5.08         => ( ( vEBT_vebt_member @ T @ X3 )
% 9.15/5.08           => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_36_maxt__corr,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_maxt @ T )
% 9.15/5.08            = ( some_nat @ X3 ) )
% 9.15/5.08         => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_37_maxt__sound,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 9.15/5.08         => ( ( vEBT_vebt_maxt @ T )
% 9.15/5.08            = ( some_nat @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_38_buildup__nothing__in__leaf,axiom,
% 9.15/5.08      ! [N2: nat,X3: nat] :
% 9.15/5.08        ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_39_buildup__nothing__in__min__max,axiom,
% 9.15/5.08      ! [N2: nat,X3: nat] :
% 9.15/5.08        ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_40_option_Oinject,axiom,
% 9.15/5.08      ! [X22: nat,Y2: nat] :
% 9.15/5.08        ( ( ( some_nat @ X22 )
% 9.15/5.08          = ( some_nat @ Y2 ) )
% 9.15/5.08        = ( X22 = Y2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_41_option_Oinject,axiom,
% 9.15/5.08      ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
% 9.15/5.08        ( ( ( some_P7363390416028606310at_nat @ X22 )
% 9.15/5.08          = ( some_P7363390416028606310at_nat @ Y2 ) )
% 9.15/5.08        = ( X22 = Y2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_42_option_Oinject,axiom,
% 9.15/5.08      ! [X22: num,Y2: num] :
% 9.15/5.08        ( ( ( some_num @ X22 )
% 9.15/5.08          = ( some_num @ Y2 ) )
% 9.15/5.08        = ( X22 = Y2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_43_valid__eq,axiom,
% 9.15/5.08      vEBT_VEBT_valid = vEBT_invar_vebt ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_44_valid__eq1,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,D: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ D )
% 9.15/5.08       => ( vEBT_VEBT_valid @ T @ D ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_45_valid__eq2,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,D: nat] :
% 9.15/5.08        ( ( vEBT_VEBT_valid @ T @ D )
% 9.15/5.08       => ( vEBT_invar_vebt @ T @ D ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_46_inthall,axiom,
% 9.15/5.08      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: product_prod_nat_nat] :
% 9.15/5.08            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_47_inthall,axiom,
% 9.15/5.08      ! [Xs: list_real,P: real > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: real] :
% 9.15/5.08            ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_48_inthall,axiom,
% 9.15/5.08      ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: set_nat] :
% 9.15/5.08            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_49_inthall,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: vEBT_VEBT] :
% 9.15/5.08            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_50_inthall,axiom,
% 9.15/5.08      ! [Xs: list_o,P: $o > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: $o] :
% 9.15/5.08            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_51_inthall,axiom,
% 9.15/5.08      ! [Xs: list_nat,P: nat > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: nat] :
% 9.15/5.08            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_52_inthall,axiom,
% 9.15/5.08      ! [Xs: list_int,P: int > $o,N2: nat] :
% 9.15/5.08        ( ! [X4: int] :
% 9.15/5.08            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.15/5.08           => ( P @ X4 ) )
% 9.15/5.08       => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08         => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_53_deg__not__0,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_54__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
% 9.15/5.08      ord_less_eq_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_55_valid__tree__deg__neq__0,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT] :
% 9.15/5.08        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_56_valid__0__not,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT] :
% 9.15/5.08        ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_57_min__in__set__def,axiom,
% 9.15/5.08      ( vEBT_VEBT_min_in_set
% 9.15/5.08      = ( ^ [Xs2: set_nat,X: nat] :
% 9.15/5.08            ( ( member_nat @ X @ Xs2 )
% 9.15/5.08            & ! [Y: nat] :
% 9.15/5.08                ( ( member_nat @ Y @ Xs2 )
% 9.15/5.08               => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_58_max__in__set__def,axiom,
% 9.15/5.08      ( vEBT_VEBT_max_in_set
% 9.15/5.08      = ( ^ [Xs2: set_nat,X: nat] :
% 9.15/5.08            ( ( member_nat @ X @ Xs2 )
% 9.15/5.08            & ! [Y: nat] :
% 9.15/5.08                ( ( member_nat @ Y @ Xs2 )
% 9.15/5.08               => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_59_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.15/5.08        ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_60_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: real,P: real > $o] :
% 9.15/5.08        ( ( member_real @ A @ ( collect_real @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_61_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: list_nat,P: list_nat > $o] :
% 9.15/5.08        ( ( member_list_nat @ A @ ( collect_list_nat @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_62_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: set_nat,P: set_nat > $o] :
% 9.15/5.08        ( ( member_set_nat @ A @ ( collect_set_nat @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_63_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: nat,P: nat > $o] :
% 9.15/5.08        ( ( member_nat @ A @ ( collect_nat @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_64_mem__Collect__eq,axiom,
% 9.15/5.08      ! [A: int,P: int > $o] :
% 9.15/5.08        ( ( member_int @ A @ ( collect_int @ P ) )
% 9.15/5.08        = ( P @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_65_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.08        ( ( collec3392354462482085612at_nat
% 9.15/5.08          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_66_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_real] :
% 9.15/5.08        ( ( collect_real
% 9.15/5.08          @ ^ [X: real] : ( member_real @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_67_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_list_nat] :
% 9.15/5.08        ( ( collect_list_nat
% 9.15/5.08          @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_68_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_set_nat] :
% 9.15/5.08        ( ( collect_set_nat
% 9.15/5.08          @ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_69_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_nat] :
% 9.15/5.08        ( ( collect_nat
% 9.15/5.08          @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_70_Collect__mem__eq,axiom,
% 9.15/5.08      ! [A2: set_int] :
% 9.15/5.08        ( ( collect_int
% 9.15/5.08          @ ^ [X: int] : ( member_int @ X @ A2 ) )
% 9.15/5.08        = A2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_71_Collect__cong,axiom,
% 9.15/5.08      ! [P: real > $o,Q: real > $o] :
% 9.15/5.08        ( ! [X4: real] :
% 9.15/5.08            ( ( P @ X4 )
% 9.15/5.08            = ( Q @ X4 ) )
% 9.15/5.08       => ( ( collect_real @ P )
% 9.15/5.08          = ( collect_real @ Q ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_72_Collect__cong,axiom,
% 9.15/5.08      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.15/5.08        ( ! [X4: list_nat] :
% 9.15/5.08            ( ( P @ X4 )
% 9.15/5.08            = ( Q @ X4 ) )
% 9.15/5.08       => ( ( collect_list_nat @ P )
% 9.15/5.08          = ( collect_list_nat @ Q ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_73_Collect__cong,axiom,
% 9.15/5.08      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.15/5.08        ( ! [X4: set_nat] :
% 9.15/5.08            ( ( P @ X4 )
% 9.15/5.08            = ( Q @ X4 ) )
% 9.15/5.08       => ( ( collect_set_nat @ P )
% 9.15/5.08          = ( collect_set_nat @ Q ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_74_Collect__cong,axiom,
% 9.15/5.08      ! [P: nat > $o,Q: nat > $o] :
% 9.15/5.08        ( ! [X4: nat] :
% 9.15/5.08            ( ( P @ X4 )
% 9.15/5.08            = ( Q @ X4 ) )
% 9.15/5.08       => ( ( collect_nat @ P )
% 9.15/5.08          = ( collect_nat @ Q ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_75_Collect__cong,axiom,
% 9.15/5.08      ! [P: int > $o,Q: int > $o] :
% 9.15/5.08        ( ! [X4: int] :
% 9.15/5.08            ( ( P @ X4 )
% 9.15/5.08            = ( Q @ X4 ) )
% 9.15/5.08       => ( ( collect_int @ P )
% 9.15/5.08          = ( collect_int @ Q ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_76_buildup__gives__valid,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.08       => ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N2 ) @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_77_set__vebt__set__vebt_H__valid,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_set_vebt @ T )
% 9.15/5.08          = ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_78_mint__corr__help,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_mint @ T )
% 9.15/5.08            = ( some_nat @ Mini ) )
% 9.15/5.08         => ( ( vEBT_vebt_member @ T @ X3 )
% 9.15/5.08           => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_79_replicate__eq__replicate,axiom,
% 9.15/5.08      ! [M: nat,X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
% 9.15/5.08        ( ( ( replicate_VEBT_VEBT @ M @ X3 )
% 9.15/5.08          = ( replicate_VEBT_VEBT @ N2 @ Y3 ) )
% 9.15/5.08        = ( ( M = N2 )
% 9.15/5.08          & ( ( M != zero_zero_nat )
% 9.15/5.08           => ( X3 = Y3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_80_replicate__eq__replicate,axiom,
% 9.15/5.08      ! [M: nat,X3: int,N2: nat,Y3: int] :
% 9.15/5.08        ( ( ( replicate_int @ M @ X3 )
% 9.15/5.08          = ( replicate_int @ N2 @ Y3 ) )
% 9.15/5.08        = ( ( M = N2 )
% 9.15/5.08          & ( ( M != zero_zero_nat )
% 9.15/5.08           => ( X3 = Y3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_81_replicate__eq__replicate,axiom,
% 9.15/5.08      ! [M: nat,X3: nat,N2: nat,Y3: nat] :
% 9.15/5.08        ( ( ( replicate_nat @ M @ X3 )
% 9.15/5.08          = ( replicate_nat @ N2 @ Y3 ) )
% 9.15/5.08        = ( ( M = N2 )
% 9.15/5.08          & ( ( M != zero_zero_nat )
% 9.15/5.08           => ( X3 = Y3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_82_length__replicate,axiom,
% 9.15/5.08      ! [N2: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_83_length__replicate,axiom,
% 9.15/5.08      ! [N2: nat,X3: $o] :
% 9.15/5.08        ( ( size_size_list_o @ ( replicate_o @ N2 @ X3 ) )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_84_length__replicate,axiom,
% 9.15/5.08      ! [N2: nat,X3: nat] :
% 9.15/5.08        ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X3 ) )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_85_length__replicate,axiom,
% 9.15/5.08      ! [N2: nat,X3: int] :
% 9.15/5.08        ( ( size_size_list_int @ ( replicate_int @ N2 @ X3 ) )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_86_length__list__update,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) )
% 9.15/5.08        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_87_length__list__update,axiom,
% 9.15/5.08      ! [Xs: list_o,I: nat,X3: $o] :
% 9.15/5.08        ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X3 ) )
% 9.15/5.08        = ( size_size_list_o @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_88_length__list__update,axiom,
% 9.15/5.08      ! [Xs: list_nat,I: nat,X3: nat] :
% 9.15/5.08        ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X3 ) )
% 9.15/5.08        = ( size_size_list_nat @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_89_length__list__update,axiom,
% 9.15/5.08      ! [Xs: list_int,I: nat,X3: int] :
% 9.15/5.08        ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X3 ) )
% 9.15/5.08        = ( size_size_list_int @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_90_mint__sound,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
% 9.15/5.08         => ( ( vEBT_vebt_mint @ T )
% 9.15/5.08            = ( some_nat @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_91_mint__corr,axiom,
% 9.15/5.08      ! [T: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.08        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.08       => ( ( ( vEBT_vebt_mint @ T )
% 9.15/5.08            = ( some_nat @ X3 ) )
% 9.15/5.08         => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_92_Ball__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.15/5.08        ( ( ! [X: vEBT_VEBT] :
% 9.15/5.08              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          | ( N2 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_93_Ball__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: int,P: int > $o] :
% 9.15/5.08        ( ( ! [X: int] :
% 9.15/5.08              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          | ( N2 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_94_Ball__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: nat,P: nat > $o] :
% 9.15/5.08        ( ( ! [X: nat] :
% 9.15/5.08              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          | ( N2 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_95_Bex__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.15/5.08        ( ( ? [X: vEBT_VEBT] :
% 9.15/5.08              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
% 9.15/5.08              & ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_96_Bex__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: int,P: int > $o] :
% 9.15/5.08        ( ( ? [X: int] :
% 9.15/5.08              ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
% 9.15/5.08              & ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_97_Bex__set__replicate,axiom,
% 9.15/5.08      ! [N2: nat,A: nat,P: nat > $o] :
% 9.15/5.08        ( ( ? [X: nat] :
% 9.15/5.08              ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
% 9.15/5.08              & ( P @ X ) ) )
% 9.15/5.08        = ( ( P @ A )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_98_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: product_prod_nat_nat,N2: nat,Y3: product_prod_nat_nat] :
% 9.15/5.08        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_99_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: real,N2: nat,Y3: real] :
% 9.15/5.08        ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_100_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: set_nat,N2: nat,Y3: set_nat] :
% 9.15/5.08        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_101_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
% 9.15/5.08        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_102_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: int,N2: nat,Y3: int] :
% 9.15/5.08        ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_103_in__set__replicate,axiom,
% 9.15/5.08      ! [X3: nat,N2: nat,Y3: nat] :
% 9.15/5.08        ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y3 ) ) )
% 9.15/5.08        = ( ( X3 = Y3 )
% 9.15/5.08          & ( N2 != zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_104_list__update__beyond,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
% 9.15/5.08       => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_105_list__update__beyond,axiom,
% 9.15/5.08      ! [Xs: list_o,I: nat,X3: $o] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
% 9.15/5.08       => ( ( list_update_o @ Xs @ I @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_106_list__update__beyond,axiom,
% 9.15/5.08      ! [Xs: list_nat,I: nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
% 9.15/5.08       => ( ( list_update_nat @ Xs @ I @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_107_list__update__beyond,axiom,
% 9.15/5.08      ! [Xs: list_int,I: nat,X3: int] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
% 9.15/5.08       => ( ( list_update_int @ Xs @ I @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_108__C5_C_I4_J,axiom,
% 9.15/5.08      ( m
% 9.15/5.08      = ( suc @ n ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_109_nth__list__update__eq,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_110_nth__list__update__eq,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_o,X3: $o] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_111_nth__list__update__eq,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_112_nth__list__update__eq,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_int,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ I )
% 9.15/5.08          = X3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_113_lesseq__shift,axiom,
% 9.15/5.08      ( ord_less_eq_nat
% 9.15/5.08      = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_114_set__swap,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_VEBT_VEBT,J: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( ( ord_less_nat @ J @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08         => ( ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ ( nth_VEBT_VEBT @ Xs @ J ) ) @ J @ ( nth_VEBT_VEBT @ Xs @ I ) ) )
% 9.15/5.08            = ( set_VEBT_VEBT2 @ Xs ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_115_set__swap,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_o,J: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
% 9.15/5.08         => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
% 9.15/5.08            = ( set_o2 @ Xs ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_116_set__swap,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_nat,J: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08         => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
% 9.15/5.08            = ( set_nat2 @ Xs ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_117_set__swap,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_int,J: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( ( ord_less_nat @ J @ ( size_size_list_int @ Xs ) )
% 9.15/5.08         => ( ( set_int2 @ ( list_update_int @ ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ J ) ) @ J @ ( nth_int @ Xs @ I ) ) )
% 9.15/5.08            = ( set_int2 @ Xs ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_118_Ex__list__of__length,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08      ? [Xs3: list_VEBT_VEBT] :
% 9.15/5.08        ( ( size_s6755466524823107622T_VEBT @ Xs3 )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_119_Ex__list__of__length,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08      ? [Xs3: list_o] :
% 9.15/5.08        ( ( size_size_list_o @ Xs3 )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_120_Ex__list__of__length,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08      ? [Xs3: list_nat] :
% 9.15/5.08        ( ( size_size_list_nat @ Xs3 )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_121_Ex__list__of__length,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08      ? [Xs3: list_int] :
% 9.15/5.08        ( ( size_size_list_int @ Xs3 )
% 9.15/5.08        = N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_122_neq__if__length__neq,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.15/5.08        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.08         != ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.08       => ( Xs != Ys ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_123_neq__if__length__neq,axiom,
% 9.15/5.08      ! [Xs: list_o,Ys: list_o] :
% 9.15/5.08        ( ( ( size_size_list_o @ Xs )
% 9.15/5.08         != ( size_size_list_o @ Ys ) )
% 9.15/5.08       => ( Xs != Ys ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_124_neq__if__length__neq,axiom,
% 9.15/5.08      ! [Xs: list_nat,Ys: list_nat] :
% 9.15/5.08        ( ( ( size_size_list_nat @ Xs )
% 9.15/5.08         != ( size_size_list_nat @ Ys ) )
% 9.15/5.08       => ( Xs != Ys ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_125_neq__if__length__neq,axiom,
% 9.15/5.08      ! [Xs: list_int,Ys: list_int] :
% 9.15/5.08        ( ( ( size_size_list_int @ Xs )
% 9.15/5.08         != ( size_size_list_int @ Ys ) )
% 9.15/5.08       => ( Xs != Ys ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_126_option_Osel,axiom,
% 9.15/5.08      ! [X22: nat] :
% 9.15/5.08        ( ( the_nat @ ( some_nat @ X22 ) )
% 9.15/5.08        = X22 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_127_option_Osel,axiom,
% 9.15/5.08      ! [X22: product_prod_nat_nat] :
% 9.15/5.08        ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.15/5.08        = X22 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_128_option_Osel,axiom,
% 9.15/5.08      ! [X22: num] :
% 9.15/5.08        ( ( the_num @ ( some_num @ X22 ) )
% 9.15/5.08        = X22 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_129_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.08        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_130_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: real,Xs: list_real] :
% 9.15/5.08        ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_131_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: set_nat,Xs: list_set_nat] :
% 9.15/5.08        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_132_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.15/5.08        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_133_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: $o,Xs: list_o] :
% 9.15/5.08        ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_134_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: nat,Xs: list_nat] :
% 9.15/5.08        ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_135_length__pos__if__in__set,axiom,
% 9.15/5.08      ! [X3: int,Xs: list_int] :
% 9.15/5.08        ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.08       => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_136_length__induct,axiom,
% 9.15/5.08      ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.15/5.08        ( ! [Xs3: list_VEBT_VEBT] :
% 9.15/5.08            ( ! [Ys2: list_VEBT_VEBT] :
% 9.15/5.08                ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
% 9.15/5.08               => ( P @ Ys2 ) )
% 9.15/5.08           => ( P @ Xs3 ) )
% 9.15/5.08       => ( P @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_137_length__induct,axiom,
% 9.15/5.08      ! [P: list_o > $o,Xs: list_o] :
% 9.15/5.08        ( ! [Xs3: list_o] :
% 9.15/5.08            ( ! [Ys2: list_o] :
% 9.15/5.08                ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
% 9.15/5.08               => ( P @ Ys2 ) )
% 9.15/5.08           => ( P @ Xs3 ) )
% 9.15/5.08       => ( P @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_138_length__induct,axiom,
% 9.15/5.08      ! [P: list_nat > $o,Xs: list_nat] :
% 9.15/5.08        ( ! [Xs3: list_nat] :
% 9.15/5.08            ( ! [Ys2: list_nat] :
% 9.15/5.08                ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
% 9.15/5.08               => ( P @ Ys2 ) )
% 9.15/5.08           => ( P @ Xs3 ) )
% 9.15/5.08       => ( P @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_139_length__induct,axiom,
% 9.15/5.08      ! [P: list_int > $o,Xs: list_int] :
% 9.15/5.08        ( ! [Xs3: list_int] :
% 9.15/5.08            ( ! [Ys2: list_int] :
% 9.15/5.08                ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
% 9.15/5.08               => ( P @ Ys2 ) )
% 9.15/5.08           => ( P @ Xs3 ) )
% 9.15/5.08       => ( P @ Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_140_list__eq__iff__nth__eq,axiom,
% 9.15/5.08      ( ( ^ [Y4: list_VEBT_VEBT,Z: list_VEBT_VEBT] : ( Y4 = Z ) )
% 9.15/5.08      = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
% 9.15/5.08            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.15/5.08              = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
% 9.15/5.08            & ! [I2: nat] :
% 9.15/5.08                ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
% 9.15/5.08               => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
% 9.15/5.08                  = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_141_list__eq__iff__nth__eq,axiom,
% 9.15/5.08      ( ( ^ [Y4: list_o,Z: list_o] : ( Y4 = Z ) )
% 9.15/5.08      = ( ^ [Xs2: list_o,Ys3: list_o] :
% 9.15/5.08            ( ( ( size_size_list_o @ Xs2 )
% 9.15/5.08              = ( size_size_list_o @ Ys3 ) )
% 9.15/5.08            & ! [I2: nat] :
% 9.15/5.08                ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
% 9.15/5.08               => ( ( nth_o @ Xs2 @ I2 )
% 9.15/5.08                  = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_142_list__eq__iff__nth__eq,axiom,
% 9.15/5.08      ( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
% 9.15/5.08      = ( ^ [Xs2: list_nat,Ys3: list_nat] :
% 9.15/5.08            ( ( ( size_size_list_nat @ Xs2 )
% 9.15/5.08              = ( size_size_list_nat @ Ys3 ) )
% 9.15/5.08            & ! [I2: nat] :
% 9.15/5.08                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
% 9.15/5.08               => ( ( nth_nat @ Xs2 @ I2 )
% 9.15/5.08                  = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_143_list__eq__iff__nth__eq,axiom,
% 9.15/5.08      ( ( ^ [Y4: list_int,Z: list_int] : ( Y4 = Z ) )
% 9.15/5.08      = ( ^ [Xs2: list_int,Ys3: list_int] :
% 9.15/5.08            ( ( ( size_size_list_int @ Xs2 )
% 9.15/5.08              = ( size_size_list_int @ Ys3 ) )
% 9.15/5.08            & ! [I2: nat] :
% 9.15/5.08                ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
% 9.15/5.08               => ( ( nth_int @ Xs2 @ I2 )
% 9.15/5.08                  = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_144_Skolem__list__nth,axiom,
% 9.15/5.08      ! [K: nat,P: nat > vEBT_VEBT > $o] :
% 9.15/5.08        ( ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08             => ? [X5: vEBT_VEBT] : ( P @ I2 @ X5 ) ) )
% 9.15/5.08        = ( ? [Xs2: list_VEBT_VEBT] :
% 9.15/5.08              ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.15/5.08                = K )
% 9.15/5.08              & ! [I2: nat] :
% 9.15/5.08                  ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08                 => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_145_Skolem__list__nth,axiom,
% 9.15/5.08      ! [K: nat,P: nat > $o > $o] :
% 9.15/5.08        ( ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08             => ? [X5: $o] : ( P @ I2 @ X5 ) ) )
% 9.15/5.08        = ( ? [Xs2: list_o] :
% 9.15/5.08              ( ( ( size_size_list_o @ Xs2 )
% 9.15/5.08                = K )
% 9.15/5.08              & ! [I2: nat] :
% 9.15/5.08                  ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08                 => ( P @ I2 @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_146_Skolem__list__nth,axiom,
% 9.15/5.08      ! [K: nat,P: nat > nat > $o] :
% 9.15/5.08        ( ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08             => ? [X5: nat] : ( P @ I2 @ X5 ) ) )
% 9.15/5.08        = ( ? [Xs2: list_nat] :
% 9.15/5.08              ( ( ( size_size_list_nat @ Xs2 )
% 9.15/5.08                = K )
% 9.15/5.08              & ! [I2: nat] :
% 9.15/5.08                  ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08                 => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_147_Skolem__list__nth,axiom,
% 9.15/5.08      ! [K: nat,P: nat > int > $o] :
% 9.15/5.08        ( ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08             => ? [X5: int] : ( P @ I2 @ X5 ) ) )
% 9.15/5.08        = ( ? [Xs2: list_int] :
% 9.15/5.08              ( ( ( size_size_list_int @ Xs2 )
% 9.15/5.08                = K )
% 9.15/5.08              & ! [I2: nat] :
% 9.15/5.08                  ( ( ord_less_nat @ I2 @ K )
% 9.15/5.08                 => ( P @ I2 @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_148_nth__equalityI,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.15/5.08        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.08          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.08       => ( ! [I3: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08             => ( ( nth_VEBT_VEBT @ Xs @ I3 )
% 9.15/5.08                = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
% 9.15/5.08         => ( Xs = Ys ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_149_nth__equalityI,axiom,
% 9.15/5.08      ! [Xs: list_o,Ys: list_o] :
% 9.15/5.08        ( ( ( size_size_list_o @ Xs )
% 9.15/5.08          = ( size_size_list_o @ Ys ) )
% 9.15/5.08       => ( ! [I3: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08             => ( ( nth_o @ Xs @ I3 )
% 9.15/5.08                = ( nth_o @ Ys @ I3 ) ) )
% 9.15/5.08         => ( Xs = Ys ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_150_nth__equalityI,axiom,
% 9.15/5.08      ! [Xs: list_nat,Ys: list_nat] :
% 9.15/5.08        ( ( ( size_size_list_nat @ Xs )
% 9.15/5.08          = ( size_size_list_nat @ Ys ) )
% 9.15/5.08       => ( ! [I3: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08             => ( ( nth_nat @ Xs @ I3 )
% 9.15/5.08                = ( nth_nat @ Ys @ I3 ) ) )
% 9.15/5.08         => ( Xs = Ys ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_151_nth__equalityI,axiom,
% 9.15/5.08      ! [Xs: list_int,Ys: list_int] :
% 9.15/5.08        ( ( ( size_size_list_int @ Xs )
% 9.15/5.08          = ( size_size_list_int @ Ys ) )
% 9.15/5.08       => ( ! [I3: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08             => ( ( nth_int @ Xs @ I3 )
% 9.15/5.08                = ( nth_int @ Ys @ I3 ) ) )
% 9.15/5.08         => ( Xs = Ys ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_152_replicate__length__same,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.08        ( ! [X4: vEBT_VEBT] :
% 9.15/5.08            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08           => ( X4 = X3 ) )
% 9.15/5.08       => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_153_replicate__length__same,axiom,
% 9.15/5.08      ! [Xs: list_o,X3: $o] :
% 9.15/5.08        ( ! [X4: $o] :
% 9.15/5.08            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.15/5.08           => ( X4 = X3 ) )
% 9.15/5.08       => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_154_replicate__length__same,axiom,
% 9.15/5.08      ! [Xs: list_nat,X3: nat] :
% 9.15/5.08        ( ! [X4: nat] :
% 9.15/5.08            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.15/5.08           => ( X4 = X3 ) )
% 9.15/5.08       => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_155_replicate__length__same,axiom,
% 9.15/5.08      ! [Xs: list_int,X3: int] :
% 9.15/5.08        ( ! [X4: int] :
% 9.15/5.08            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.15/5.08           => ( X4 = X3 ) )
% 9.15/5.08       => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X3 )
% 9.15/5.08          = Xs ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_156_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_P6011104703257516679at_nat,N2: nat,X3: product_prod_nat_nat] :
% 9.15/5.08        ( ( ( size_s5460976970255530739at_nat @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: product_prod_nat_nat] :
% 9.15/5.08              ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replic4235873036481779905at_nat @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_157_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_real,N2: nat,X3: real] :
% 9.15/5.08        ( ( ( size_size_list_real @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: real] :
% 9.15/5.08              ( ( member_real @ Y5 @ ( set_real2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_real @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_158_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_set_nat,N2: nat,X3: set_nat] :
% 9.15/5.08        ( ( ( size_s3254054031482475050et_nat @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: set_nat] :
% 9.15/5.08              ( ( member_set_nat @ Y5 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_set_nat @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_159_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,N2: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: vEBT_VEBT] :
% 9.15/5.08              ( ( member_VEBT_VEBT @ Y5 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_VEBT_VEBT @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_160_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_o,N2: nat,X3: $o] :
% 9.15/5.08        ( ( ( size_size_list_o @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: $o] :
% 9.15/5.08              ( ( member_o @ Y5 @ ( set_o2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_o @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_161_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_nat,N2: nat,X3: nat] :
% 9.15/5.08        ( ( ( size_size_list_nat @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: nat] :
% 9.15/5.08              ( ( member_nat @ Y5 @ ( set_nat2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_nat @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_162_replicate__eqI,axiom,
% 9.15/5.08      ! [Xs: list_int,N2: nat,X3: int] :
% 9.15/5.08        ( ( ( size_size_list_int @ Xs )
% 9.15/5.08          = N2 )
% 9.15/5.08       => ( ! [Y5: int] :
% 9.15/5.08              ( ( member_int @ Y5 @ ( set_int2 @ Xs ) )
% 9.15/5.08             => ( Y5 = X3 ) )
% 9.15/5.08         => ( Xs
% 9.15/5.08            = ( replicate_int @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_163_all__set__conv__all__nth,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.15/5.08        ( ( ! [X: vEBT_VEBT] :
% 9.15/5.08              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08             => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_164_all__set__conv__all__nth,axiom,
% 9.15/5.08      ! [Xs: list_o,P: $o > $o] :
% 9.15/5.08        ( ( ! [X: $o] :
% 9.15/5.08              ( ( member_o @ X @ ( set_o2 @ Xs ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08             => ( P @ ( nth_o @ Xs @ I2 ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_165_all__set__conv__all__nth,axiom,
% 9.15/5.08      ! [Xs: list_nat,P: nat > $o] :
% 9.15/5.08        ( ( ! [X: nat] :
% 9.15/5.08              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08             => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_166_all__set__conv__all__nth,axiom,
% 9.15/5.08      ! [Xs: list_int,P: int > $o] :
% 9.15/5.08        ( ( ! [X: int] :
% 9.15/5.08              ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.15/5.08             => ( P @ X ) ) )
% 9.15/5.08        = ( ! [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08             => ( P @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_167_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_168_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_real,P: real > $o,X3: real] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_real @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_169_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_set_nat,P: set_nat > $o,X3: set_nat] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_set_nat @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_170_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_171_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_o,P: $o > $o,X3: $o] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_o @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_172_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_nat,P: nat > $o,X3: nat] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_nat @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_173_all__nth__imp__all__set,axiom,
% 9.15/5.08      ! [Xs: list_int,P: int > $o,X3: int] :
% 9.15/5.08        ( ! [I3: nat] :
% 9.15/5.08            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08           => ( P @ ( nth_int @ Xs @ I3 ) ) )
% 9.15/5.08       => ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.08         => ( P @ X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_174_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.08        ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.08              & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_175_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: real,Xs: list_real] :
% 9.15/5.08        ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
% 9.15/5.08              & ( ( nth_real @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_176_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: set_nat,Xs: list_set_nat] :
% 9.15/5.08        ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.15/5.08              & ( ( nth_set_nat @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_177_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.15/5.08        ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08              & ( ( nth_VEBT_VEBT @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_178_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: $o,Xs: list_o] :
% 9.15/5.08        ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08              & ( ( nth_o @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_179_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: nat,Xs: list_nat] :
% 9.15/5.08        ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08              & ( ( nth_nat @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_180_in__set__conv__nth,axiom,
% 9.15/5.08      ! [X3: int,Xs: list_int] :
% 9.15/5.08        ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.08        = ( ? [I2: nat] :
% 9.15/5.08              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08              & ( ( nth_int @ Xs @ I2 )
% 9.15/5.08                = X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_181_list__ball__nth,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( ! [X4: vEBT_VEBT] :
% 9.15/5.08              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.08             => ( P @ X4 ) )
% 9.15/5.08         => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_182_list__ball__nth,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_o,P: $o > $o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( ! [X4: $o] :
% 9.15/5.08              ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
% 9.15/5.08             => ( P @ X4 ) )
% 9.15/5.08         => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_183_list__ball__nth,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_nat,P: nat > $o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( ! [X4: nat] :
% 9.15/5.08              ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
% 9.15/5.08             => ( P @ X4 ) )
% 9.15/5.08         => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_184_list__ball__nth,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_int,P: int > $o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( ! [X4: int] :
% 9.15/5.08              ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
% 9.15/5.08             => ( P @ X4 ) )
% 9.15/5.08         => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_185_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.08       => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_186_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_real] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 9.15/5.08       => ( member_real @ ( nth_real @ Xs @ N2 ) @ ( set_real2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_187_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_set_nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.15/5.08       => ( member_set_nat @ ( nth_set_nat @ Xs @ N2 ) @ ( set_set_nat2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_188_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_VEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( member_VEBT_VEBT @ ( nth_VEBT_VEBT @ Xs @ N2 ) @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_189_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_190_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( member_nat @ ( nth_nat @ Xs @ N2 ) @ ( set_nat2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_191_nth__mem,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_int] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( member_int @ ( nth_int @ Xs @ N2 ) @ ( set_int2 @ Xs ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_192_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.08       => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_193_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_real,X3: real] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
% 9.15/5.08       => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_194_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_set_nat,X3: set_nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
% 9.15/5.08       => ( member_set_nat @ X3 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_195_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_196_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_o,X3: $o] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_197_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_198_set__update__memI,axiom,
% 9.15/5.08      ! [N2: nat,Xs: list_int,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_199_list__update__same__conv,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
% 9.15/5.08            = Xs )
% 9.15/5.08          = ( ( nth_VEBT_VEBT @ Xs @ I )
% 9.15/5.08            = X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_200_list__update__same__conv,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_o,X3: $o] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( ( ( list_update_o @ Xs @ I @ X3 )
% 9.15/5.08            = Xs )
% 9.15/5.08          = ( ( nth_o @ Xs @ I )
% 9.15/5.08            = X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_201_list__update__same__conv,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( ( ( list_update_nat @ Xs @ I @ X3 )
% 9.15/5.08            = Xs )
% 9.15/5.08          = ( ( nth_nat @ Xs @ I )
% 9.15/5.08            = X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_202_list__update__same__conv,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_int,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( ( ( list_update_int @ Xs @ I @ X3 )
% 9.15/5.08            = Xs )
% 9.15/5.08          = ( ( nth_int @ Xs @ I )
% 9.15/5.08            = X3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_203_nth__list__update,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.08       => ( ( ( I = J )
% 9.15/5.08           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = X3 ) )
% 9.15/5.08          & ( ( I != J )
% 9.15/5.08           => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_204_nth__list__update,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_o,X3: $o,J: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.08       => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ J )
% 9.15/5.08          = ( ( ( I = J )
% 9.15/5.08             => X3 )
% 9.15/5.08            & ( ( I != J )
% 9.15/5.08             => ( nth_o @ Xs @ J ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_205_nth__list__update,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_nat,J: nat,X3: nat] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.08       => ( ( ( I = J )
% 9.15/5.08           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = X3 ) )
% 9.15/5.08          & ( ( I != J )
% 9.15/5.08           => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = ( nth_nat @ Xs @ J ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_206_nth__list__update,axiom,
% 9.15/5.08      ! [I: nat,Xs: list_int,J: nat,X3: int] :
% 9.15/5.08        ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
% 9.15/5.08       => ( ( ( I = J )
% 9.15/5.08           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = X3 ) )
% 9.15/5.08          & ( ( I != J )
% 9.15/5.08           => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
% 9.15/5.08              = ( nth_int @ Xs @ J ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_207_le0,axiom,
% 9.15/5.08      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_208_bot__nat__0_Oextremum,axiom,
% 9.15/5.08      ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_209_less__nat__zero__code,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_210_neq0__conv,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( N2 != zero_zero_nat )
% 9.15/5.08        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_211_bot__nat__0_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( A != zero_zero_nat )
% 9.15/5.08        = ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_212_not__gr__zero,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 9.15/5.08        = ( N2 = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_213_le__zero__eq,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 9.15/5.08        = ( N2 = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_214__C5_C_I5_J,axiom,
% 9.15/5.08      ( deg
% 9.15/5.08      = ( plus_plus_nat @ n @ m ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_215_ex__least__nat__le,axiom,
% 9.15/5.08      ! [P: nat > $o,N2: nat] :
% 9.15/5.08        ( ( P @ N2 )
% 9.15/5.08       => ( ~ ( P @ zero_zero_nat )
% 9.15/5.08         => ? [K2: nat] :
% 9.15/5.08              ( ( ord_less_eq_nat @ K2 @ N2 )
% 9.15/5.08              & ! [I4: nat] :
% 9.15/5.08                  ( ( ord_less_nat @ I4 @ K2 )
% 9.15/5.08                 => ~ ( P @ I4 ) )
% 9.15/5.08              & ( P @ K2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_216_buildup__gives__empty,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
% 9.15/5.08        = bot_bot_set_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_217_dual__order_Orefl,axiom,
% 9.15/5.08      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_218_dual__order_Orefl,axiom,
% 9.15/5.08      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_219_dual__order_Orefl,axiom,
% 9.15/5.08      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_220_dual__order_Orefl,axiom,
% 9.15/5.08      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_221_dual__order_Orefl,axiom,
% 9.15/5.08      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_222_even__odd__cases,axiom,
% 9.15/5.08      ! [X3: nat] :
% 9.15/5.08        ( ! [N3: nat] :
% 9.15/5.08            ( X3
% 9.15/5.08           != ( plus_plus_nat @ N3 @ N3 ) )
% 9.15/5.08       => ~ ! [N3: nat] :
% 9.15/5.08              ( X3
% 9.15/5.08             != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_223_add__shift,axiom,
% 9.15/5.08      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.15/5.08          = Z2 )
% 9.15/5.08        = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.15/5.08          = ( some_nat @ Z2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_224_order__refl,axiom,
% 9.15/5.08      ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_225_order__refl,axiom,
% 9.15/5.08      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_226_order__refl,axiom,
% 9.15/5.08      ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_227_order__refl,axiom,
% 9.15/5.08      ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_228_order__refl,axiom,
% 9.15/5.08      ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_229_add__left__cancel,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ A @ B )
% 9.15/5.08          = ( plus_plus_real @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_230_add__left__cancel,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ A @ B )
% 9.15/5.08          = ( plus_plus_rat @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_231_add__left__cancel,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ A @ B )
% 9.15/5.08          = ( plus_plus_nat @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_232_add__left__cancel,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ A @ B )
% 9.15/5.08          = ( plus_plus_int @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_233_add__right__cancel,axiom,
% 9.15/5.08      ! [B: real,A: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ B @ A )
% 9.15/5.08          = ( plus_plus_real @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_234_add__right__cancel,axiom,
% 9.15/5.08      ! [B: rat,A: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ B @ A )
% 9.15/5.08          = ( plus_plus_rat @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_235_add__right__cancel,axiom,
% 9.15/5.08      ! [B: nat,A: nat,C: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ B @ A )
% 9.15/5.08          = ( plus_plus_nat @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_236_add__right__cancel,axiom,
% 9.15/5.08      ! [B: int,A: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ B @ A )
% 9.15/5.08          = ( plus_plus_int @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_237_nat_Oinject,axiom,
% 9.15/5.08      ! [X22: nat,Y2: nat] :
% 9.15/5.08        ( ( ( suc @ X22 )
% 9.15/5.08          = ( suc @ Y2 ) )
% 9.15/5.08        = ( X22 = Y2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_238_old_Onat_Oinject,axiom,
% 9.15/5.08      ! [Nat: nat,Nat2: nat] :
% 9.15/5.08        ( ( ( suc @ Nat )
% 9.15/5.08          = ( suc @ Nat2 ) )
% 9.15/5.08        = ( Nat = Nat2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_239_add__le__cancel__left,axiom,
% 9.15/5.08      ! [C: real,A: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.15/5.08        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_240_add__le__cancel__left,axiom,
% 9.15/5.08      ! [C: rat,A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.15/5.08        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_241_add__le__cancel__left,axiom,
% 9.15/5.08      ! [C: nat,A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.15/5.08        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_242_add__le__cancel__left,axiom,
% 9.15/5.08      ! [C: int,A: int,B: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.15/5.08        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_243_add__le__cancel__right,axiom,
% 9.15/5.08      ! [A: real,C: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.15/5.08        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_244_add__le__cancel__right,axiom,
% 9.15/5.08      ! [A: rat,C: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.15/5.08        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_245_add__le__cancel__right,axiom,
% 9.15/5.08      ! [A: nat,C: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.15/5.08        = ( ord_less_eq_nat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_246_add__le__cancel__right,axiom,
% 9.15/5.08      ! [A: int,C: int,B: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.15/5.08        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_247_add_Oright__neutral,axiom,
% 9.15/5.08      ! [A: complex] :
% 9.15/5.08        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_248_add_Oright__neutral,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_249_add_Oright__neutral,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_250_add_Oright__neutral,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_251_add_Oright__neutral,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_252_double__zero__sym,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( zero_zero_real
% 9.15/5.08          = ( plus_plus_real @ A @ A ) )
% 9.15/5.08        = ( A = zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_253_double__zero__sym,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( zero_zero_rat
% 9.15/5.08          = ( plus_plus_rat @ A @ A ) )
% 9.15/5.08        = ( A = zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_254_double__zero__sym,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( zero_zero_int
% 9.15/5.08          = ( plus_plus_int @ A @ A ) )
% 9.15/5.08        = ( A = zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_255_add__cancel__left__left,axiom,
% 9.15/5.08      ! [B: complex,A: complex] :
% 9.15/5.08        ( ( ( plus_plus_complex @ B @ A )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_complex ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_256_add__cancel__left__left,axiom,
% 9.15/5.08      ! [B: real,A: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ B @ A )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_257_add__cancel__left__left,axiom,
% 9.15/5.08      ! [B: rat,A: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ B @ A )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_258_add__cancel__left__left,axiom,
% 9.15/5.08      ! [B: nat,A: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ B @ A )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_259_add__cancel__left__left,axiom,
% 9.15/5.08      ! [B: int,A: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ B @ A )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_260_add__cancel__left__right,axiom,
% 9.15/5.08      ! [A: complex,B: complex] :
% 9.15/5.08        ( ( ( plus_plus_complex @ A @ B )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_complex ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_261_add__cancel__left__right,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ A @ B )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_262_add__cancel__left__right,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ A @ B )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_263_add__cancel__left__right,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ A @ B )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_264_add__cancel__left__right,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ A @ B )
% 9.15/5.08          = A )
% 9.15/5.08        = ( B = zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_265_add__cancel__right__left,axiom,
% 9.15/5.08      ! [A: complex,B: complex] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_complex @ B @ A ) )
% 9.15/5.08        = ( B = zero_zero_complex ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_266_add__cancel__right__left,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_real @ B @ A ) )
% 9.15/5.08        = ( B = zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_267_add__cancel__right__left,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_rat @ B @ A ) )
% 9.15/5.08        = ( B = zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_268_add__cancel__right__left,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_nat @ B @ A ) )
% 9.15/5.08        = ( B = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_269_add__cancel__right__left,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_int @ B @ A ) )
% 9.15/5.08        = ( B = zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_270_add__cancel__right__right,axiom,
% 9.15/5.08      ! [A: complex,B: complex] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_complex @ A @ B ) )
% 9.15/5.08        = ( B = zero_zero_complex ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_271_add__cancel__right__right,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_real @ A @ B ) )
% 9.15/5.08        = ( B = zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_272_add__cancel__right__right,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_rat @ A @ B ) )
% 9.15/5.08        = ( B = zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_273_add__cancel__right__right,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_nat @ A @ B ) )
% 9.15/5.08        = ( B = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_274_add__cancel__right__right,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( A
% 9.15/5.08          = ( plus_plus_int @ A @ B ) )
% 9.15/5.08        = ( B = zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_275_add__eq__0__iff__both__eq__0,axiom,
% 9.15/5.08      ! [X3: nat,Y3: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.15/5.08          = zero_zero_nat )
% 9.15/5.08        = ( ( X3 = zero_zero_nat )
% 9.15/5.08          & ( Y3 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_276_zero__eq__add__iff__both__eq__0,axiom,
% 9.15/5.08      ! [X3: nat,Y3: nat] :
% 9.15/5.08        ( ( zero_zero_nat
% 9.15/5.08          = ( plus_plus_nat @ X3 @ Y3 ) )
% 9.15/5.08        = ( ( X3 = zero_zero_nat )
% 9.15/5.08          & ( Y3 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_277_add__0,axiom,
% 9.15/5.08      ! [A: complex] :
% 9.15/5.08        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_278_add__0,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_279_add__0,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_280_add__0,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_281_add__0,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.15/5.08        = A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_282_add__less__cancel__left,axiom,
% 9.15/5.08      ! [C: real,A: real,B: real] :
% 9.15/5.08        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.15/5.08        = ( ord_less_real @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_283_add__less__cancel__left,axiom,
% 9.15/5.08      ! [C: rat,A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.15/5.08        = ( ord_less_rat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_284_add__less__cancel__left,axiom,
% 9.15/5.08      ! [C: nat,A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.15/5.08        = ( ord_less_nat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_285_add__less__cancel__left,axiom,
% 9.15/5.08      ! [C: int,A: int,B: int] :
% 9.15/5.08        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.15/5.08        = ( ord_less_int @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_286_add__less__cancel__right,axiom,
% 9.15/5.08      ! [A: real,C: real,B: real] :
% 9.15/5.08        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.15/5.08        = ( ord_less_real @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_287_add__less__cancel__right,axiom,
% 9.15/5.08      ! [A: rat,C: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.15/5.08        = ( ord_less_rat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_288_add__less__cancel__right,axiom,
% 9.15/5.08      ! [A: nat,C: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.15/5.08        = ( ord_less_nat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_289_add__less__cancel__right,axiom,
% 9.15/5.08      ! [A: int,C: int,B: int] :
% 9.15/5.08        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.15/5.08        = ( ord_less_int @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_290_lessI,axiom,
% 9.15/5.08      ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_291_Suc__mono,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.08       => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_292_Suc__less__eq,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 9.15/5.08        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_293_Suc__le__mono,axiom,
% 9.15/5.08      ! [N2: nat,M: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
% 9.15/5.08        = ( ord_less_eq_nat @ N2 @ M ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_294_add__Suc__right,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
% 9.15/5.08        = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_295_add__is__0,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ M @ N2 )
% 9.15/5.08          = zero_zero_nat )
% 9.15/5.08        = ( ( M = zero_zero_nat )
% 9.15/5.08          & ( N2 = zero_zero_nat ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_296_Nat_Oadd__0__right,axiom,
% 9.15/5.08      ! [M: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ M @ zero_zero_nat )
% 9.15/5.08        = M ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_297_nat__add__left__cancel__less,axiom,
% 9.15/5.08      ! [K: nat,M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 9.15/5.08        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_298_nat__add__left__cancel__le,axiom,
% 9.15/5.08      ! [K: nat,M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 9.15/5.08        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_299_add__le__same__cancel1,axiom,
% 9.15/5.08      ! [B: real,A: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_300_add__le__same__cancel1,axiom,
% 9.15/5.08      ! [B: rat,A: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_301_add__le__same__cancel1,axiom,
% 9.15/5.08      ! [B: nat,A: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_302_add__le__same__cancel1,axiom,
% 9.15/5.08      ! [B: int,A: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_303_add__le__same__cancel2,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_304_add__le__same__cancel2,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_305_add__le__same__cancel2,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_306_add__le__same__cancel2,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_307_le__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.15/5.08        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_308_le__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.15/5.08        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_309_le__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.15/5.08        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_310_le__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.15/5.08        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_311_le__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.15/5.08        = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_312_le__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.15/5.08        = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_313_le__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.15/5.08        = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_314_le__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.15/5.08        = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_315_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.15/5.08        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_316_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.15/5.08        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_317_double__add__le__zero__iff__single__add__le__zero,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.15/5.08        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_318_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.15/5.08        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_319_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.15/5.08        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_320_zero__le__double__add__iff__zero__le__single__add,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.15/5.08        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_321_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ A ) )
% 9.15/5.08        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_322_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
% 9.15/5.08        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_323_zero__less__double__add__iff__zero__less__single__add,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
% 9.15/5.08        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_324_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.15/5.08      ! [A: real] :
% 9.15/5.08        ( ( ord_less_real @ ( plus_plus_real @ A @ A ) @ zero_zero_real )
% 9.15/5.08        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_325_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.15/5.08      ! [A: rat] :
% 9.15/5.08        ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
% 9.15/5.08        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_326_double__add__less__zero__iff__single__add__less__zero,axiom,
% 9.15/5.08      ! [A: int] :
% 9.15/5.08        ( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
% 9.15/5.08        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_327_less__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.15/5.08        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_328_less__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.15/5.08        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_329_less__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.15/5.08        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_330_less__add__same__cancel2,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.15/5.08        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_331_less__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.15/5.08        = ( ord_less_real @ zero_zero_real @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_332_less__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.15/5.08        = ( ord_less_rat @ zero_zero_rat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_333_less__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.15/5.08        = ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_334_less__add__same__cancel1,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.15/5.08        = ( ord_less_int @ zero_zero_int @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_335_add__less__same__cancel2,axiom,
% 9.15/5.08      ! [A: real,B: real] :
% 9.15/5.08        ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_336_add__less__same__cancel2,axiom,
% 9.15/5.08      ! [A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_337_add__less__same__cancel2,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_338_add__less__same__cancel2,axiom,
% 9.15/5.08      ! [A: int,B: int] :
% 9.15/5.08        ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.15/5.08        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_339_add__less__same__cancel1,axiom,
% 9.15/5.08      ! [B: real,A: real] :
% 9.15/5.08        ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_340_add__less__same__cancel1,axiom,
% 9.15/5.08      ! [B: rat,A: rat] :
% 9.15/5.08        ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_341_add__less__same__cancel1,axiom,
% 9.15/5.08      ! [B: nat,A: nat] :
% 9.15/5.08        ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_342_add__less__same__cancel1,axiom,
% 9.15/5.08      ! [B: int,A: int] :
% 9.15/5.08        ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.15/5.08        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_343_zero__less__Suc,axiom,
% 9.15/5.08      ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_344_less__Suc0,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.15/5.08        = ( N2 = zero_zero_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_345_add__gr__0,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
% 9.15/5.08        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.15/5.08          | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_346_one__is__add,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ( suc @ zero_zero_nat )
% 9.15/5.08          = ( plus_plus_nat @ M @ N2 ) )
% 9.15/5.08        = ( ( ( M
% 9.15/5.08              = ( suc @ zero_zero_nat ) )
% 9.15/5.08            & ( N2 = zero_zero_nat ) )
% 9.15/5.08          | ( ( M = zero_zero_nat )
% 9.15/5.08            & ( N2
% 9.15/5.08              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_347_add__is__1,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ M @ N2 )
% 9.15/5.08          = ( suc @ zero_zero_nat ) )
% 9.15/5.08        = ( ( ( M
% 9.15/5.08              = ( suc @ zero_zero_nat ) )
% 9.15/5.08            & ( N2 = zero_zero_nat ) )
% 9.15/5.08          | ( ( M = zero_zero_nat )
% 9.15/5.08            & ( N2
% 9.15/5.08              = ( suc @ zero_zero_nat ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_348_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_350_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_351_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_352_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.15/5.08      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ( plus_plus_real @ I @ K )
% 9.15/5.08          = ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_353_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.15/5.08      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ( plus_plus_rat @ I @ K )
% 9.15/5.08          = ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_354_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.15/5.08      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ( plus_plus_nat @ I @ K )
% 9.15/5.08          = ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_355_add__mono__thms__linordered__semiring_I4_J,axiom,
% 9.15/5.08      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ( plus_plus_int @ I @ K )
% 9.15/5.08          = ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_356_nat__arith_Osuc1,axiom,
% 9.15/5.08      ! [A2: nat,K: nat,A: nat] :
% 9.15/5.08        ( ( A2
% 9.15/5.08          = ( plus_plus_nat @ K @ A ) )
% 9.15/5.08       => ( ( suc @ A2 )
% 9.15/5.08          = ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_357_add__Suc,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 9.15/5.08        = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_358_group__cancel_Oadd1,axiom,
% 9.15/5.08      ! [A2: real,K: real,A: real,B: real] :
% 9.15/5.08        ( ( A2
% 9.15/5.08          = ( plus_plus_real @ K @ A ) )
% 9.15/5.08       => ( ( plus_plus_real @ A2 @ B )
% 9.15/5.08          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_359_group__cancel_Oadd1,axiom,
% 9.15/5.08      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.15/5.08        ( ( A2
% 9.15/5.08          = ( plus_plus_rat @ K @ A ) )
% 9.15/5.08       => ( ( plus_plus_rat @ A2 @ B )
% 9.15/5.08          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_360_group__cancel_Oadd1,axiom,
% 9.15/5.08      ! [A2: nat,K: nat,A: nat,B: nat] :
% 9.15/5.08        ( ( A2
% 9.15/5.08          = ( plus_plus_nat @ K @ A ) )
% 9.15/5.08       => ( ( plus_plus_nat @ A2 @ B )
% 9.15/5.08          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_361_group__cancel_Oadd1,axiom,
% 9.15/5.08      ! [A2: int,K: int,A: int,B: int] :
% 9.15/5.08        ( ( A2
% 9.15/5.08          = ( plus_plus_int @ K @ A ) )
% 9.15/5.08       => ( ( plus_plus_int @ A2 @ B )
% 9.15/5.08          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_362_group__cancel_Oadd2,axiom,
% 9.15/5.08      ! [B2: real,K: real,B: real,A: real] :
% 9.15/5.08        ( ( B2
% 9.15/5.08          = ( plus_plus_real @ K @ B ) )
% 9.15/5.08       => ( ( plus_plus_real @ A @ B2 )
% 9.15/5.08          = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_363_group__cancel_Oadd2,axiom,
% 9.15/5.08      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.15/5.08        ( ( B2
% 9.15/5.08          = ( plus_plus_rat @ K @ B ) )
% 9.15/5.08       => ( ( plus_plus_rat @ A @ B2 )
% 9.15/5.08          = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_364_group__cancel_Oadd2,axiom,
% 9.15/5.08      ! [B2: nat,K: nat,B: nat,A: nat] :
% 9.15/5.08        ( ( B2
% 9.15/5.08          = ( plus_plus_nat @ K @ B ) )
% 9.15/5.08       => ( ( plus_plus_nat @ A @ B2 )
% 9.15/5.08          = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_365_group__cancel_Oadd2,axiom,
% 9.15/5.08      ! [B2: int,K: int,B: int,A: int] :
% 9.15/5.08        ( ( B2
% 9.15/5.08          = ( plus_plus_int @ K @ B ) )
% 9.15/5.08       => ( ( plus_plus_int @ A @ B2 )
% 9.15/5.08          = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_366_add_Oassoc,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_367_add_Oassoc,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_368_add_Oassoc,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_369_add_Oassoc,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.15/5.08        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_370_add_Oleft__cancel,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ A @ B )
% 9.15/5.08          = ( plus_plus_real @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_371_add_Oleft__cancel,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ A @ B )
% 9.15/5.08          = ( plus_plus_rat @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_372_add_Oleft__cancel,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ A @ B )
% 9.15/5.08          = ( plus_plus_int @ A @ C ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_373_add_Oright__cancel,axiom,
% 9.15/5.08      ! [B: real,A: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ B @ A )
% 9.15/5.08          = ( plus_plus_real @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_374_add_Oright__cancel,axiom,
% 9.15/5.08      ! [B: rat,A: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ B @ A )
% 9.15/5.08          = ( plus_plus_rat @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_375_add_Oright__cancel,axiom,
% 9.15/5.08      ! [B: int,A: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ B @ A )
% 9.15/5.08          = ( plus_plus_int @ C @ A ) )
% 9.15/5.08        = ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_376_add_Ocommute,axiom,
% 9.15/5.08      ( plus_plus_real
% 9.15/5.08      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_377_add_Ocommute,axiom,
% 9.15/5.08      ( plus_plus_rat
% 9.15/5.08      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_378_add_Ocommute,axiom,
% 9.15/5.08      ( plus_plus_nat
% 9.15/5.08      = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_379_add_Ocommute,axiom,
% 9.15/5.08      ( plus_plus_int
% 9.15/5.08      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_380_add_Oleft__commute,axiom,
% 9.15/5.08      ! [B: real,A: real,C: real] :
% 9.15/5.08        ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
% 9.15/5.08        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_381_add_Oleft__commute,axiom,
% 9.15/5.08      ! [B: rat,A: rat,C: rat] :
% 9.15/5.08        ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
% 9.15/5.08        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_382_add_Oleft__commute,axiom,
% 9.15/5.08      ! [B: nat,A: nat,C: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
% 9.15/5.08        = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_383_add_Oleft__commute,axiom,
% 9.15/5.08      ! [B: int,A: int,C: int] :
% 9.15/5.08        ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 9.15/5.08        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_384_add__left__imp__eq,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ A @ B )
% 9.15/5.08          = ( plus_plus_real @ A @ C ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_385_add__left__imp__eq,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ A @ B )
% 9.15/5.08          = ( plus_plus_rat @ A @ C ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_386_add__left__imp__eq,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ A @ B )
% 9.15/5.08          = ( plus_plus_nat @ A @ C ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_387_add__left__imp__eq,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ A @ B )
% 9.15/5.08          = ( plus_plus_int @ A @ C ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_388_add__right__imp__eq,axiom,
% 9.15/5.08      ! [B: real,A: real,C: real] :
% 9.15/5.08        ( ( ( plus_plus_real @ B @ A )
% 9.15/5.08          = ( plus_plus_real @ C @ A ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_389_add__right__imp__eq,axiom,
% 9.15/5.08      ! [B: rat,A: rat,C: rat] :
% 9.15/5.08        ( ( ( plus_plus_rat @ B @ A )
% 9.15/5.08          = ( plus_plus_rat @ C @ A ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_390_add__right__imp__eq,axiom,
% 9.15/5.08      ! [B: nat,A: nat,C: nat] :
% 9.15/5.08        ( ( ( plus_plus_nat @ B @ A )
% 9.15/5.08          = ( plus_plus_nat @ C @ A ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_391_add__right__imp__eq,axiom,
% 9.15/5.08      ! [B: int,A: int,C: int] :
% 9.15/5.08        ( ( ( plus_plus_int @ B @ A )
% 9.15/5.08          = ( plus_plus_int @ C @ A ) )
% 9.15/5.08       => ( B = C ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_392_Suc__inject,axiom,
% 9.15/5.08      ! [X3: nat,Y3: nat] :
% 9.15/5.08        ( ( ( suc @ X3 )
% 9.15/5.08          = ( suc @ Y3 ) )
% 9.15/5.08       => ( X3 = Y3 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_393_n__not__Suc__n,axiom,
% 9.15/5.08      ! [N2: nat] :
% 9.15/5.08        ( N2
% 9.15/5.08       != ( suc @ N2 ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_394_add__Suc__shift,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
% 9.15/5.08        = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_395_less__imp__Suc__add,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.08       => ? [K2: nat] :
% 9.15/5.08            ( N2
% 9.15/5.08            = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_396_less__iff__Suc__add,axiom,
% 9.15/5.08      ( ord_less_nat
% 9.15/5.08      = ( ^ [M2: nat,N: nat] :
% 9.15/5.08          ? [K3: nat] :
% 9.15/5.08            ( N
% 9.15/5.08            = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_397_less__add__Suc2,axiom,
% 9.15/5.08      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_398_less__add__Suc1,axiom,
% 9.15/5.08      ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_399_less__natE,axiom,
% 9.15/5.08      ! [M: nat,N2: nat] :
% 9.15/5.08        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.08       => ~ ! [Q2: nat] :
% 9.15/5.08              ( N2
% 9.15/5.08             != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_400_bot_Oextremum,axiom,
% 9.15/5.08      ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_401_bot_Oextremum,axiom,
% 9.15/5.08      ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_402_bot_Oextremum,axiom,
% 9.15/5.08      ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_403_bot_Oextremum,axiom,
% 9.15/5.08      ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_404_bot_Oextremum,axiom,
% 9.15/5.08      ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_405_bot_Oextremum__unique,axiom,
% 9.15/5.08      ! [A: set_Pr1261947904930325089at_nat] :
% 9.15/5.08        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.15/5.08        = ( A = bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_406_bot_Oextremum__unique,axiom,
% 9.15/5.08      ! [A: set_real] :
% 9.15/5.08        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.15/5.08        = ( A = bot_bot_set_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_407_bot_Oextremum__unique,axiom,
% 9.15/5.08      ! [A: set_nat] :
% 9.15/5.08        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.15/5.08        = ( A = bot_bot_set_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_408_bot_Oextremum__unique,axiom,
% 9.15/5.08      ! [A: set_int] :
% 9.15/5.08        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.15/5.08        = ( A = bot_bot_set_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_409_bot_Oextremum__unique,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.15/5.08        = ( A = bot_bot_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_410_bot_Oextremum__uniqueI,axiom,
% 9.15/5.08      ! [A: set_Pr1261947904930325089at_nat] :
% 9.15/5.08        ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.15/5.08       => ( A = bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_411_bot_Oextremum__uniqueI,axiom,
% 9.15/5.08      ! [A: set_real] :
% 9.15/5.08        ( ( ord_less_eq_set_real @ A @ bot_bot_set_real )
% 9.15/5.08       => ( A = bot_bot_set_real ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_412_bot_Oextremum__uniqueI,axiom,
% 9.15/5.08      ! [A: set_nat] :
% 9.15/5.08        ( ( ord_less_eq_set_nat @ A @ bot_bot_set_nat )
% 9.15/5.08       => ( A = bot_bot_set_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_413_bot_Oextremum__uniqueI,axiom,
% 9.15/5.08      ! [A: set_int] :
% 9.15/5.08        ( ( ord_less_eq_set_int @ A @ bot_bot_set_int )
% 9.15/5.08       => ( A = bot_bot_set_int ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_414_bot_Oextremum__uniqueI,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ bot_bot_nat )
% 9.15/5.08       => ( A = bot_bot_nat ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_415_bot_Oextremum__strict,axiom,
% 9.15/5.08      ! [A: set_Pr1261947904930325089at_nat] :
% 9.15/5.08        ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_416_bot_Oextremum__strict,axiom,
% 9.15/5.08      ! [A: set_real] :
% 9.15/5.08        ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_417_bot_Oextremum__strict,axiom,
% 9.15/5.08      ! [A: set_nat] :
% 9.15/5.08        ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_418_bot_Oextremum__strict,axiom,
% 9.15/5.08      ! [A: set_int] :
% 9.15/5.08        ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_419_bot_Oextremum__strict,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ~ ( ord_less_nat @ A @ bot_bot_nat ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_420_bot_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: set_Pr1261947904930325089at_nat] :
% 9.15/5.08        ( ( A != bot_bo2099793752762293965at_nat )
% 9.15/5.08        = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_421_bot_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: set_real] :
% 9.15/5.08        ( ( A != bot_bot_set_real )
% 9.15/5.08        = ( ord_less_set_real @ bot_bot_set_real @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_422_bot_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: set_nat] :
% 9.15/5.08        ( ( A != bot_bot_set_nat )
% 9.15/5.08        = ( ord_less_set_nat @ bot_bot_set_nat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_423_bot_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: set_int] :
% 9.15/5.08        ( ( A != bot_bot_set_int )
% 9.15/5.08        = ( ord_less_set_int @ bot_bot_set_int @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_424_bot_Onot__eq__extremum,axiom,
% 9.15/5.08      ! [A: nat] :
% 9.15/5.08        ( ( A != bot_bot_nat )
% 9.15/5.08        = ( ord_less_nat @ bot_bot_nat @ A ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_425_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.15/5.08      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.08        ( ( ( ord_less_eq_real @ I @ J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_426_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.15/5.08      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.08        ( ( ( ord_less_eq_rat @ I @ J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_427_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.15/5.08      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.08        ( ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_428_add__mono__thms__linordered__semiring_I3_J,axiom,
% 9.15/5.08      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.08        ( ( ( ord_less_eq_int @ I @ J )
% 9.15/5.08          & ( K = L ) )
% 9.15/5.08       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_429_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.15/5.08      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( ord_less_eq_real @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_430_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.15/5.08      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( ord_less_eq_rat @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_431_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.15/5.08      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( ord_less_eq_nat @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_432_add__mono__thms__linordered__semiring_I2_J,axiom,
% 9.15/5.08      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.08        ( ( ( I = J )
% 9.15/5.08          & ( ord_less_eq_int @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_433_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.15/5.08      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.08        ( ( ( ord_less_eq_real @ I @ J )
% 9.15/5.08          & ( ord_less_eq_real @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_434_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.15/5.08      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.08        ( ( ( ord_less_eq_rat @ I @ J )
% 9.15/5.08          & ( ord_less_eq_rat @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_435_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.15/5.08      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.08        ( ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.08          & ( ord_less_eq_nat @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_436_add__mono__thms__linordered__semiring_I1_J,axiom,
% 9.15/5.08      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.08        ( ( ( ord_less_eq_int @ I @ J )
% 9.15/5.08          & ( ord_less_eq_int @ K @ L ) )
% 9.15/5.08       => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_437_add__mono,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real,D: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.08       => ( ( ord_less_eq_real @ C @ D )
% 9.15/5.08         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_438_add__mono,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.08       => ( ( ord_less_eq_rat @ C @ D )
% 9.15/5.08         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_439_add__mono,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.08       => ( ( ord_less_eq_nat @ C @ D )
% 9.15/5.08         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_440_add__mono,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int,D: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.08       => ( ( ord_less_eq_int @ C @ D )
% 9.15/5.08         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_441_add__left__mono,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.08       => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_442_add__left__mono,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.08       => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_443_add__left__mono,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.08       => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_444_add__left__mono,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.08       => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_445_less__eqE,axiom,
% 9.15/5.08      ! [A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.08       => ~ ! [C2: nat] :
% 9.15/5.08              ( B
% 9.15/5.08             != ( plus_plus_nat @ A @ C2 ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_446_add__right__mono,axiom,
% 9.15/5.08      ! [A: real,B: real,C: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.08       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_447_add__right__mono,axiom,
% 9.15/5.08      ! [A: rat,B: rat,C: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.08       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_448_add__right__mono,axiom,
% 9.15/5.08      ! [A: nat,B: nat,C: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.08       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_449_add__right__mono,axiom,
% 9.15/5.08      ! [A: int,B: int,C: int] :
% 9.15/5.08        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.08       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_450_le__iff__add,axiom,
% 9.15/5.08      ( ord_less_eq_nat
% 9.15/5.08      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.08          ? [C3: nat] :
% 9.15/5.08            ( B3
% 9.15/5.08            = ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_451_add__le__imp__le__left,axiom,
% 9.15/5.08      ! [C: real,A: real,B: real] :
% 9.15/5.08        ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.15/5.08       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_452_add__le__imp__le__left,axiom,
% 9.15/5.08      ! [C: rat,A: rat,B: rat] :
% 9.15/5.08        ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.15/5.08       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.15/5.08  
% 9.15/5.08  thf(fact_453_add__le__imp__le__left,axiom,
% 9.15/5.08      ! [C: nat,A: nat,B: nat] :
% 9.15/5.08        ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.15/5.09       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_454_add__le__imp__le__left,axiom,
% 9.15/5.09      ! [C: int,A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.15/5.09       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_455_add__le__imp__le__right,axiom,
% 9.15/5.09      ! [A: real,C: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.15/5.09       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_456_add__le__imp__le__right,axiom,
% 9.15/5.09      ! [A: rat,C: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.15/5.09       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_457_add__le__imp__le__right,axiom,
% 9.15/5.09      ! [A: nat,C: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.15/5.09       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_458_add__le__imp__le__right,axiom,
% 9.15/5.09      ! [A: int,C: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.15/5.09       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_459_comm__monoid__add__class_Oadd__0,axiom,
% 9.15/5.09      ! [A: complex] :
% 9.15/5.09        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_460_comm__monoid__add__class_Oadd__0,axiom,
% 9.15/5.09      ! [A: real] :
% 9.15/5.09        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_461_comm__monoid__add__class_Oadd__0,axiom,
% 9.15/5.09      ! [A: rat] :
% 9.15/5.09        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_462_comm__monoid__add__class_Oadd__0,axiom,
% 9.15/5.09      ! [A: nat] :
% 9.15/5.09        ( ( plus_plus_nat @ zero_zero_nat @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_463_comm__monoid__add__class_Oadd__0,axiom,
% 9.15/5.09      ! [A: int] :
% 9.15/5.09        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_464_add_Ocomm__neutral,axiom,
% 9.15/5.09      ! [A: complex] :
% 9.15/5.09        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_465_add_Ocomm__neutral,axiom,
% 9.15/5.09      ! [A: real] :
% 9.15/5.09        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_466_add_Ocomm__neutral,axiom,
% 9.15/5.09      ! [A: rat] :
% 9.15/5.09        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_467_add_Ocomm__neutral,axiom,
% 9.15/5.09      ! [A: nat] :
% 9.15/5.09        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_468_add_Ocomm__neutral,axiom,
% 9.15/5.09      ! [A: int] :
% 9.15/5.09        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_469_add_Ogroup__left__neutral,axiom,
% 9.15/5.09      ! [A: complex] :
% 9.15/5.09        ( ( plus_plus_complex @ zero_zero_complex @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_470_add_Ogroup__left__neutral,axiom,
% 9.15/5.09      ! [A: real] :
% 9.15/5.09        ( ( plus_plus_real @ zero_zero_real @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_471_add_Ogroup__left__neutral,axiom,
% 9.15/5.09      ! [A: rat] :
% 9.15/5.09        ( ( plus_plus_rat @ zero_zero_rat @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_472_add_Ogroup__left__neutral,axiom,
% 9.15/5.09      ! [A: int] :
% 9.15/5.09        ( ( plus_plus_int @ zero_zero_int @ A )
% 9.15/5.09        = A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_473_add__mono__thms__linordered__field_I5_J,axiom,
% 9.15/5.09      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.09        ( ( ( ord_less_real @ I @ J )
% 9.15/5.09          & ( ord_less_real @ K @ L ) )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_474_add__mono__thms__linordered__field_I5_J,axiom,
% 9.15/5.09      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.09        ( ( ( ord_less_rat @ I @ J )
% 9.15/5.09          & ( ord_less_rat @ K @ L ) )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_475_add__mono__thms__linordered__field_I5_J,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ( ord_less_nat @ I @ J )
% 9.15/5.09          & ( ord_less_nat @ K @ L ) )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_476_add__mono__thms__linordered__field_I5_J,axiom,
% 9.15/5.09      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.09        ( ( ( ord_less_int @ I @ J )
% 9.15/5.09          & ( ord_less_int @ K @ L ) )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_477_add__mono__thms__linordered__field_I2_J,axiom,
% 9.15/5.09      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.09        ( ( ( I = J )
% 9.15/5.09          & ( ord_less_real @ K @ L ) )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_478_add__mono__thms__linordered__field_I2_J,axiom,
% 9.15/5.09      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.09        ( ( ( I = J )
% 9.15/5.09          & ( ord_less_rat @ K @ L ) )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_479_add__mono__thms__linordered__field_I2_J,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ( I = J )
% 9.15/5.09          & ( ord_less_nat @ K @ L ) )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_480_add__mono__thms__linordered__field_I2_J,axiom,
% 9.15/5.09      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.09        ( ( ( I = J )
% 9.15/5.09          & ( ord_less_int @ K @ L ) )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_481_add__mono__thms__linordered__field_I1_J,axiom,
% 9.15/5.09      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.09        ( ( ( ord_less_real @ I @ J )
% 9.15/5.09          & ( K = L ) )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_482_add__mono__thms__linordered__field_I1_J,axiom,
% 9.15/5.09      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.09        ( ( ( ord_less_rat @ I @ J )
% 9.15/5.09          & ( K = L ) )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_483_add__mono__thms__linordered__field_I1_J,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ( ord_less_nat @ I @ J )
% 9.15/5.09          & ( K = L ) )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_484_add__mono__thms__linordered__field_I1_J,axiom,
% 9.15/5.09      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.09        ( ( ( ord_less_int @ I @ J )
% 9.15/5.09          & ( K = L ) )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_485_add__strict__mono,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real,D: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ( ord_less_real @ C @ D )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_486_add__strict__mono,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_rat @ C @ D )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_487_add__strict__mono,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_nat @ C @ D )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_488_add__strict__mono,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int,D: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_int @ C @ D )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_489_add__strict__left__mono,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_490_add__strict__left__mono,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_491_add__strict__left__mono,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_492_add__strict__left__mono,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_493_add__strict__right__mono,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_494_add__strict__right__mono,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_495_add__strict__right__mono,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_496_add__strict__right__mono,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_497_add__less__imp__less__left,axiom,
% 9.15/5.09      ! [C: real,A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.15/5.09       => ( ord_less_real @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_498_add__less__imp__less__left,axiom,
% 9.15/5.09      ! [C: rat,A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.15/5.09       => ( ord_less_rat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_499_add__less__imp__less__left,axiom,
% 9.15/5.09      ! [C: nat,A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.15/5.09       => ( ord_less_nat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_500_add__less__imp__less__left,axiom,
% 9.15/5.09      ! [C: int,A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.15/5.09       => ( ord_less_int @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_501_add__less__imp__less__right,axiom,
% 9.15/5.09      ! [A: real,C: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.15/5.09       => ( ord_less_real @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_502_add__less__imp__less__right,axiom,
% 9.15/5.09      ! [A: rat,C: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.15/5.09       => ( ord_less_rat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_503_add__less__imp__less__right,axiom,
% 9.15/5.09      ! [A: nat,C: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.15/5.09       => ( ord_less_nat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_504_add__less__imp__less__right,axiom,
% 9.15/5.09      ! [A: int,C: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.15/5.09       => ( ord_less_int @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_505_nat_Odistinct_I1_J,axiom,
% 9.15/5.09      ! [X22: nat] :
% 9.15/5.09        ( zero_zero_nat
% 9.15/5.09       != ( suc @ X22 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_506_old_Onat_Odistinct_I2_J,axiom,
% 9.15/5.09      ! [Nat2: nat] :
% 9.15/5.09        ( ( suc @ Nat2 )
% 9.15/5.09       != zero_zero_nat ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_507_old_Onat_Odistinct_I1_J,axiom,
% 9.15/5.09      ! [Nat2: nat] :
% 9.15/5.09        ( zero_zero_nat
% 9.15/5.09       != ( suc @ Nat2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_508_nat_OdiscI,axiom,
% 9.15/5.09      ! [Nat: nat,X22: nat] :
% 9.15/5.09        ( ( Nat
% 9.15/5.09          = ( suc @ X22 ) )
% 9.15/5.09       => ( Nat != zero_zero_nat ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_509_old_Onat_Oexhaust,axiom,
% 9.15/5.09      ! [Y3: nat] :
% 9.15/5.09        ( ( Y3 != zero_zero_nat )
% 9.15/5.09       => ~ ! [Nat3: nat] :
% 9.15/5.09              ( Y3
% 9.15/5.09             != ( suc @ Nat3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_510_nat__induct,axiom,
% 9.15/5.09      ! [P: nat > $o,N2: nat] :
% 9.15/5.09        ( ( P @ zero_zero_nat )
% 9.15/5.09       => ( ! [N3: nat] :
% 9.15/5.09              ( ( P @ N3 )
% 9.15/5.09             => ( P @ ( suc @ N3 ) ) )
% 9.15/5.09         => ( P @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_511_diff__induct,axiom,
% 9.15/5.09      ! [P: nat > nat > $o,M: nat,N2: nat] :
% 9.15/5.09        ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
% 9.15/5.09       => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( P @ X4 @ Y5 )
% 9.15/5.09               => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
% 9.15/5.09           => ( P @ M @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_512_zero__induct,axiom,
% 9.15/5.09      ! [P: nat > $o,K: nat] :
% 9.15/5.09        ( ( P @ K )
% 9.15/5.09       => ( ! [N3: nat] :
% 9.15/5.09              ( ( P @ ( suc @ N3 ) )
% 9.15/5.09             => ( P @ N3 ) )
% 9.15/5.09         => ( P @ zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_513_Suc__neq__Zero,axiom,
% 9.15/5.09      ! [M: nat] :
% 9.15/5.09        ( ( suc @ M )
% 9.15/5.09       != zero_zero_nat ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_514_Zero__neq__Suc,axiom,
% 9.15/5.09      ! [M: nat] :
% 9.15/5.09        ( zero_zero_nat
% 9.15/5.09       != ( suc @ M ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_515_Zero__not__Suc,axiom,
% 9.15/5.09      ! [M: nat] :
% 9.15/5.09        ( zero_zero_nat
% 9.15/5.09       != ( suc @ M ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_516_not0__implies__Suc,axiom,
% 9.15/5.09      ! [N2: nat] :
% 9.15/5.09        ( ( N2 != zero_zero_nat )
% 9.15/5.09       => ? [M3: nat] :
% 9.15/5.09            ( N2
% 9.15/5.09            = ( suc @ M3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_517_plus__nat_Oadd__0,axiom,
% 9.15/5.09      ! [N2: nat] :
% 9.15/5.09        ( ( plus_plus_nat @ zero_zero_nat @ N2 )
% 9.15/5.09        = N2 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_518_add__eq__self__zero,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ( plus_plus_nat @ M @ N2 )
% 9.15/5.09          = M )
% 9.15/5.09       => ( N2 = zero_zero_nat ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_519_Nat_OlessE,axiom,
% 9.15/5.09      ! [I: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ K )
% 9.15/5.09       => ( ( K
% 9.15/5.09           != ( suc @ I ) )
% 9.15/5.09         => ~ ! [J2: nat] :
% 9.15/5.09                ( ( ord_less_nat @ I @ J2 )
% 9.15/5.09               => ( K
% 9.15/5.09                 != ( suc @ J2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_520_Suc__lessD,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( suc @ M ) @ N2 )
% 9.15/5.09       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_521_Suc__lessE,axiom,
% 9.15/5.09      ! [I: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( suc @ I ) @ K )
% 9.15/5.09       => ~ ! [J2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I @ J2 )
% 9.15/5.09             => ( K
% 9.15/5.09               != ( suc @ J2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_522_Suc__lessI,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.09       => ( ( ( suc @ M )
% 9.15/5.09           != N2 )
% 9.15/5.09         => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_523_less__SucE,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09       => ( ~ ( ord_less_nat @ M @ N2 )
% 9.15/5.09         => ( M = N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_524_less__SucI,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.09       => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_525_Ex__less__Suc,axiom,
% 9.15/5.09      ! [N2: nat,P: nat > $o] :
% 9.15/5.09        ( ( ? [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 9.15/5.09              & ( P @ I2 ) ) )
% 9.15/5.09        = ( ( P @ N2 )
% 9.15/5.09          | ? [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ N2 )
% 9.15/5.09              & ( P @ I2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_526_less__Suc__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09        = ( ( ord_less_nat @ M @ N2 )
% 9.15/5.09          | ( M = N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_527_not__less__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ~ ( ord_less_nat @ M @ N2 ) )
% 9.15/5.09        = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_528_All__less__Suc,axiom,
% 9.15/5.09      ! [N2: nat,P: nat > $o] :
% 9.15/5.09        ( ( ! [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 9.15/5.09             => ( P @ I2 ) ) )
% 9.15/5.09        = ( ( P @ N2 )
% 9.15/5.09          & ! [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ N2 )
% 9.15/5.09             => ( P @ I2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_529_Suc__less__eq2,axiom,
% 9.15/5.09      ! [N2: nat,M: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.15/5.09        = ( ? [M4: nat] :
% 9.15/5.09              ( ( M
% 9.15/5.09                = ( suc @ M4 ) )
% 9.15/5.09              & ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_530_less__antisym,axiom,
% 9.15/5.09      ! [N2: nat,M: nat] :
% 9.15/5.09        ( ~ ( ord_less_nat @ N2 @ M )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 9.15/5.09         => ( M = N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_531_Suc__less__SucD,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 9.15/5.09       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_532_less__trans__Suc,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ( ord_less_nat @ J @ K )
% 9.15/5.09         => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_533_less__Suc__induct,axiom,
% 9.15/5.09      ! [I: nat,J: nat,P: nat > nat > $o] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
% 9.15/5.09         => ( ! [I3: nat,J2: nat,K2: nat] :
% 9.15/5.09                ( ( ord_less_nat @ I3 @ J2 )
% 9.15/5.09               => ( ( ord_less_nat @ J2 @ K2 )
% 9.15/5.09                 => ( ( P @ I3 @ J2 )
% 9.15/5.09                   => ( ( P @ J2 @ K2 )
% 9.15/5.09                     => ( P @ I3 @ K2 ) ) ) ) )
% 9.15/5.09           => ( P @ I @ J ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_534_strict__inc__induct,axiom,
% 9.15/5.09      ! [I: nat,J: nat,P: nat > $o] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ! [I3: nat] :
% 9.15/5.09              ( ( J
% 9.15/5.09                = ( suc @ I3 ) )
% 9.15/5.09             => ( P @ I3 ) )
% 9.15/5.09         => ( ! [I3: nat] :
% 9.15/5.09                ( ( ord_less_nat @ I3 @ J )
% 9.15/5.09               => ( ( P @ ( suc @ I3 ) )
% 9.15/5.09                 => ( P @ I3 ) ) )
% 9.15/5.09           => ( P @ I ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_535_not__less__less__Suc__eq,axiom,
% 9.15/5.09      ! [N2: nat,M: nat] :
% 9.15/5.09        ( ~ ( ord_less_nat @ N2 @ M )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 9.15/5.09          = ( N2 = M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_536_add__lessD1,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.15/5.09       => ( ord_less_nat @ I @ K ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_537_add__less__mono,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ( ord_less_nat @ K @ L )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_538_not__add__less1,axiom,
% 9.15/5.09      ! [I: nat,J: nat] :
% 9.15/5.09        ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_539_not__add__less2,axiom,
% 9.15/5.09      ! [J: nat,I: nat] :
% 9.15/5.09        ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_540_add__less__mono1,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_541_trans__less__add1,axiom,
% 9.15/5.09      ! [I: nat,J: nat,M: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_542_trans__less__add2,axiom,
% 9.15/5.09      ! [I: nat,J: nat,M: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_543_less__add__eq__less,axiom,
% 9.15/5.09      ! [K: nat,L: nat,M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ K @ L )
% 9.15/5.09       => ( ( ( plus_plus_nat @ M @ L )
% 9.15/5.09            = ( plus_plus_nat @ K @ N2 ) )
% 9.15/5.09         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_544_Suc__leD,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 9.15/5.09       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_545_le__SucE,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09       => ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09         => ( M
% 9.15/5.09            = ( suc @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_546_le__SucI,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09       => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_547_Suc__le__D,axiom,
% 9.15/5.09      ! [N2: nat,M5: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
% 9.15/5.09       => ? [M3: nat] :
% 9.15/5.09            ( M5
% 9.15/5.09            = ( suc @ M3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_548_le__Suc__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09        = ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09          | ( M
% 9.15/5.09            = ( suc @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_549_Suc__n__not__le__n,axiom,
% 9.15/5.09      ! [N2: nat] :
% 9.15/5.09        ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_550_not__less__eq__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
% 9.15/5.09        = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_551_full__nat__induct,axiom,
% 9.15/5.09      ! [P: nat > $o,N2: nat] :
% 9.15/5.09        ( ! [N3: nat] :
% 9.15/5.09            ( ! [M6: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
% 9.15/5.09               => ( P @ M6 ) )
% 9.15/5.09           => ( P @ N3 ) )
% 9.15/5.09       => ( P @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_552_nat__induct__at__least,axiom,
% 9.15/5.09      ! [M: nat,N2: nat,P: nat > $o] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09       => ( ( P @ M )
% 9.15/5.09         => ( ! [N3: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ M @ N3 )
% 9.15/5.09               => ( ( P @ N3 )
% 9.15/5.09                 => ( P @ ( suc @ N3 ) ) ) )
% 9.15/5.09           => ( P @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_553_transitive__stepwise__le,axiom,
% 9.15/5.09      ! [M: nat,N2: nat,R: nat > nat > $o] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09       => ( ! [X4: nat] : ( R @ X4 @ X4 )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat,Z3: nat] :
% 9.15/5.09                ( ( R @ X4 @ Y5 )
% 9.15/5.09               => ( ( R @ Y5 @ Z3 )
% 9.15/5.09                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.09           => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
% 9.15/5.09             => ( R @ M @ N2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_554_add__leE,axiom,
% 9.15/5.09      ! [M: nat,K: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 9.15/5.09       => ~ ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09           => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_555_le__add1,axiom,
% 9.15/5.09      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_556_le__add2,axiom,
% 9.15/5.09      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_557_add__leD1,axiom,
% 9.15/5.09      ! [M: nat,K: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 9.15/5.09       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_558_add__leD2,axiom,
% 9.15/5.09      ! [M: nat,K: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
% 9.15/5.09       => ( ord_less_eq_nat @ K @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_559_le__Suc__ex,axiom,
% 9.15/5.09      ! [K: nat,L: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ K @ L )
% 9.15/5.09       => ? [N3: nat] :
% 9.15/5.09            ( L
% 9.15/5.09            = ( plus_plus_nat @ K @ N3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_560_add__le__mono,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ( ord_less_eq_nat @ K @ L )
% 9.15/5.09         => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_561_add__le__mono1,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_562_trans__le__add1,axiom,
% 9.15/5.09      ! [I: nat,J: nat,M: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_563_trans__le__add2,axiom,
% 9.15/5.09      ! [I: nat,J: nat,M: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_564_nat__le__iff__add,axiom,
% 9.15/5.09      ( ord_less_eq_nat
% 9.15/5.09      = ( ^ [M2: nat,N: nat] :
% 9.15/5.09          ? [K3: nat] :
% 9.15/5.09            ( N
% 9.15/5.09            = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_565_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.15/5.09        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: product_prod_nat_nat] :
% 9.15/5.09              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.09             => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_566_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_real,B2: set_real] :
% 9.15/5.09        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: real] :
% 9.15/5.09              ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.15/5.09             => ( member_real @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_567_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_set_nat,B2: set_set_nat] :
% 9.15/5.09        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: set_nat] :
% 9.15/5.09              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.15/5.09             => ( member_set_nat @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_568_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
% 9.15/5.09        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: vEBT_VEBT] :
% 9.15/5.09              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.09             => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_569_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_nat,B2: set_nat] :
% 9.15/5.09        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: nat] :
% 9.15/5.09              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.15/5.09             => ( member_nat @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_570_subset__code_I1_J,axiom,
% 9.15/5.09      ! [Xs: list_int,B2: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
% 9.15/5.09        = ( ! [X: int] :
% 9.15/5.09              ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.15/5.09             => ( member_int @ X @ B2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_571_add__decreasing,axiom,
% 9.15/5.09      ! [A: real,C: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_eq_real @ C @ B )
% 9.15/5.09         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_572_add__decreasing,axiom,
% 9.15/5.09      ! [A: rat,C: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_eq_rat @ C @ B )
% 9.15/5.09         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_573_add__decreasing,axiom,
% 9.15/5.09      ! [A: nat,C: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_eq_nat @ C @ B )
% 9.15/5.09         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_574_add__decreasing,axiom,
% 9.15/5.09      ! [A: int,C: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_eq_int @ C @ B )
% 9.15/5.09         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_575_add__increasing,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_eq_real @ B @ C )
% 9.15/5.09         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_576_add__increasing,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_577_add__increasing,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_578_add__increasing,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_579_add__decreasing2,axiom,
% 9.15/5.09      ! [C: real,A: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_eq_real @ A @ B )
% 9.15/5.09         => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_580_add__decreasing2,axiom,
% 9.15/5.09      ! [C: rat,A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_581_add__decreasing2,axiom,
% 9.15/5.09      ! [C: nat,A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ C @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_582_add__decreasing2,axiom,
% 9.15/5.09      ! [C: int,A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09         => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_583_add__increasing2,axiom,
% 9.15/5.09      ! [C: real,B: real,A: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.15/5.09       => ( ( ord_less_eq_real @ B @ A )
% 9.15/5.09         => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_584_add__increasing2,axiom,
% 9.15/5.09      ! [C: rat,B: rat,A: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.09         => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_585_add__increasing2,axiom,
% 9.15/5.09      ! [C: nat,B: nat,A: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.09         => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_586_add__increasing2,axiom,
% 9.15/5.09      ! [C: int,B: int,A: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ A )
% 9.15/5.09         => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_587_add__nonneg__nonneg,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.15/5.09         => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_588_add__nonneg__nonneg,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.15/5.09         => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_589_add__nonneg__nonneg,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.15/5.09         => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_590_add__nonneg__nonneg,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.15/5.09         => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_591_add__nonpos__nonpos,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.15/5.09         => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_592_add__nonpos__nonpos,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.15/5.09         => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_593_add__nonpos__nonpos,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.15/5.09         => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_594_add__nonpos__nonpos,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.15/5.09         => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_595_add__nonneg__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.15/5.09         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_real )
% 9.15/5.09            = ( ( X3 = zero_zero_real )
% 9.15/5.09              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_596_add__nonneg__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.15/5.09         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_rat )
% 9.15/5.09            = ( ( X3 = zero_zero_rat )
% 9.15/5.09              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_597_add__nonneg__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.15/5.09         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_nat )
% 9.15/5.09            = ( ( X3 = zero_zero_nat )
% 9.15/5.09              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_598_add__nonneg__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.15/5.09         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_int )
% 9.15/5.09            = ( ( X3 = zero_zero_int )
% 9.15/5.09              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_599_add__nonpos__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.15/5.09         => ( ( ( plus_plus_real @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_real )
% 9.15/5.09            = ( ( X3 = zero_zero_real )
% 9.15/5.09              & ( Y3 = zero_zero_real ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_600_add__nonpos__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.15/5.09         => ( ( ( plus_plus_rat @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_rat )
% 9.15/5.09            = ( ( X3 = zero_zero_rat )
% 9.15/5.09              & ( Y3 = zero_zero_rat ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_601_add__nonpos__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
% 9.15/5.09         => ( ( ( plus_plus_nat @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_nat )
% 9.15/5.09            = ( ( X3 = zero_zero_nat )
% 9.15/5.09              & ( Y3 = zero_zero_nat ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_602_add__nonpos__eq__0__iff,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.15/5.09         => ( ( ( plus_plus_int @ X3 @ Y3 )
% 9.15/5.09              = zero_zero_int )
% 9.15/5.09            = ( ( X3 = zero_zero_int )
% 9.15/5.09              & ( Y3 = zero_zero_int ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_603_add__less__le__mono,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real,D: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_real @ C @ D )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_604_add__less__le__mono,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ C @ D )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_605_add__less__le__mono,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ C @ D )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_606_add__less__le__mono,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int,D: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_int @ C @ D )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_607_add__le__less__mono,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real,D: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.09       => ( ( ord_less_real @ C @ D )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_608_add__le__less__mono,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_rat @ C @ D )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_609_add__le__less__mono,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_nat @ C @ D )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_610_add__le__less__mono,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int,D: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_int @ C @ D )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_611_add__mono__thms__linordered__field_I3_J,axiom,
% 9.15/5.09      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.09        ( ( ( ord_less_real @ I @ J )
% 9.15/5.09          & ( ord_less_eq_real @ K @ L ) )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_612_add__mono__thms__linordered__field_I3_J,axiom,
% 9.15/5.09      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.09        ( ( ( ord_less_rat @ I @ J )
% 9.15/5.09          & ( ord_less_eq_rat @ K @ L ) )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_613_add__mono__thms__linordered__field_I3_J,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ( ord_less_nat @ I @ J )
% 9.15/5.09          & ( ord_less_eq_nat @ K @ L ) )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_614_add__mono__thms__linordered__field_I3_J,axiom,
% 9.15/5.09      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.09        ( ( ( ord_less_int @ I @ J )
% 9.15/5.09          & ( ord_less_eq_int @ K @ L ) )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_615_add__mono__thms__linordered__field_I4_J,axiom,
% 9.15/5.09      ! [I: real,J: real,K: real,L: real] :
% 9.15/5.09        ( ( ( ord_less_eq_real @ I @ J )
% 9.15/5.09          & ( ord_less_real @ K @ L ) )
% 9.15/5.09       => ( ord_less_real @ ( plus_plus_real @ I @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_616_add__mono__thms__linordered__field_I4_J,axiom,
% 9.15/5.09      ! [I: rat,J: rat,K: rat,L: rat] :
% 9.15/5.09        ( ( ( ord_less_eq_rat @ I @ J )
% 9.15/5.09          & ( ord_less_rat @ K @ L ) )
% 9.15/5.09       => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_617_add__mono__thms__linordered__field_I4_J,axiom,
% 9.15/5.09      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.15/5.09        ( ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09          & ( ord_less_nat @ K @ L ) )
% 9.15/5.09       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_618_add__mono__thms__linordered__field_I4_J,axiom,
% 9.15/5.09      ! [I: int,J: int,K: int,L: int] :
% 9.15/5.09        ( ( ( ord_less_eq_int @ I @ J )
% 9.15/5.09          & ( ord_less_int @ K @ L ) )
% 9.15/5.09       => ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_619_pos__add__strict,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_620_pos__add__strict,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_rat @ B @ C )
% 9.15/5.09         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_621_pos__add__strict,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_nat @ B @ C )
% 9.15/5.09         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_622_pos__add__strict,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_int @ B @ C )
% 9.15/5.09         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_623_canonically__ordered__monoid__add__class_OlessE,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ~ ! [C2: nat] :
% 9.15/5.09              ( ( B
% 9.15/5.09                = ( plus_plus_nat @ A @ C2 ) )
% 9.15/5.09             => ( C2 = zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_624_add__pos__pos,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.15/5.09         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_625_add__pos__pos,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.15/5.09         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_626_add__pos__pos,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.15/5.09         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_627_add__pos__pos,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.15/5.09         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_628_add__neg__neg,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_629_add__neg__neg,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_630_add__neg__neg,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_631_add__neg__neg,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_632_lift__Suc__mono__less,axiom,
% 9.15/5.09      ! [F: nat > real,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_633_lift__Suc__mono__less,axiom,
% 9.15/5.09      ! [F: nat > rat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_634_lift__Suc__mono__less,axiom,
% 9.15/5.09      ! [F: nat > num,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_635_lift__Suc__mono__less,axiom,
% 9.15/5.09      ! [F: nat > nat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_636_lift__Suc__mono__less,axiom,
% 9.15/5.09      ! [F: nat > int,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_637_lift__Suc__mono__less__iff,axiom,
% 9.15/5.09      ! [F: nat > real,N2: nat,M: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
% 9.15/5.09          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_638_lift__Suc__mono__less__iff,axiom,
% 9.15/5.09      ! [F: nat > rat,N2: nat,M: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
% 9.15/5.09          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_639_lift__Suc__mono__less__iff,axiom,
% 9.15/5.09      ! [F: nat > num,N2: nat,M: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
% 9.15/5.09          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_640_lift__Suc__mono__less__iff,axiom,
% 9.15/5.09      ! [F: nat > nat,N2: nat,M: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
% 9.15/5.09          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_641_lift__Suc__mono__less__iff,axiom,
% 9.15/5.09      ! [F: nat > int,N2: nat,M: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
% 9.15/5.09          = ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_642_lift__Suc__mono__le,axiom,
% 9.15/5.09      ! [F: nat > set_int,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_643_lift__Suc__mono__le,axiom,
% 9.15/5.09      ! [F: nat > rat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_644_lift__Suc__mono__le,axiom,
% 9.15/5.09      ! [F: nat > num,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_645_lift__Suc__mono__le,axiom,
% 9.15/5.09      ! [F: nat > nat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_646_lift__Suc__mono__le,axiom,
% 9.15/5.09      ! [F: nat > int,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_647_lift__Suc__antimono__le,axiom,
% 9.15/5.09      ! [F: nat > set_int,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_set_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_648_lift__Suc__antimono__le,axiom,
% 9.15/5.09      ! [F: nat > rat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_649_lift__Suc__antimono__le,axiom,
% 9.15/5.09      ! [F: nat > num,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_650_lift__Suc__antimono__le,axiom,
% 9.15/5.09      ! [F: nat > nat,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_651_lift__Suc__antimono__le,axiom,
% 9.15/5.09      ! [F: nat > int,N2: nat,N4: nat] :
% 9.15/5.09        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ N2 @ N4 )
% 9.15/5.09         => ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_652_less__Suc__eq__0__disj,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09        = ( ( M = zero_zero_nat )
% 9.15/5.09          | ? [J3: nat] :
% 9.15/5.09              ( ( M
% 9.15/5.09                = ( suc @ J3 ) )
% 9.15/5.09              & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_653_gr0__implies__Suc,axiom,
% 9.15/5.09      ! [N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.09       => ? [M3: nat] :
% 9.15/5.09            ( N2
% 9.15/5.09            = ( suc @ M3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_654_All__less__Suc2,axiom,
% 9.15/5.09      ! [N2: nat,P: nat > $o] :
% 9.15/5.09        ( ( ! [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 9.15/5.09             => ( P @ I2 ) ) )
% 9.15/5.09        = ( ( P @ zero_zero_nat )
% 9.15/5.09          & ! [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ N2 )
% 9.15/5.09             => ( P @ ( suc @ I2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_655_gr0__conv__Suc,axiom,
% 9.15/5.09      ! [N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.09        = ( ? [M2: nat] :
% 9.15/5.09              ( N2
% 9.15/5.09              = ( suc @ M2 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_656_Ex__less__Suc2,axiom,
% 9.15/5.09      ! [N2: nat,P: nat > $o] :
% 9.15/5.09        ( ( ? [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
% 9.15/5.09              & ( P @ I2 ) ) )
% 9.15/5.09        = ( ( P @ zero_zero_nat )
% 9.15/5.09          | ? [I2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ I2 @ N2 )
% 9.15/5.09              & ( P @ ( suc @ I2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_657_less__imp__add__positive,axiom,
% 9.15/5.09      ! [I: nat,J: nat] :
% 9.15/5.09        ( ( ord_less_nat @ I @ J )
% 9.15/5.09       => ? [K2: nat] :
% 9.15/5.09            ( ( ord_less_nat @ zero_zero_nat @ K2 )
% 9.15/5.09            & ( ( plus_plus_nat @ I @ K2 )
% 9.15/5.09              = J ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_658_le__imp__less__Suc,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09       => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_659_less__eq__Suc__le,axiom,
% 9.15/5.09      ( ord_less_nat
% 9.15/5.09      = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_660_less__Suc__eq__le,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ ( suc @ N2 ) )
% 9.15/5.09        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_661_le__less__Suc__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.09       => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 9.15/5.09          = ( N2 = M ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_662_Suc__le__lessD,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 9.15/5.09       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_663_inc__induct,axiom,
% 9.15/5.09      ! [I: nat,J: nat,P: nat > $o] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ( P @ J )
% 9.15/5.09         => ( ! [N3: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ I @ N3 )
% 9.15/5.09               => ( ( ord_less_nat @ N3 @ J )
% 9.15/5.09                 => ( ( P @ ( suc @ N3 ) )
% 9.15/5.09                   => ( P @ N3 ) ) ) )
% 9.15/5.09           => ( P @ I ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_664_dec__induct,axiom,
% 9.15/5.09      ! [I: nat,J: nat,P: nat > $o] :
% 9.15/5.09        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.09       => ( ( P @ I )
% 9.15/5.09         => ( ! [N3: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ I @ N3 )
% 9.15/5.09               => ( ( ord_less_nat @ N3 @ J )
% 9.15/5.09                 => ( ( P @ N3 )
% 9.15/5.09                   => ( P @ ( suc @ N3 ) ) ) ) )
% 9.15/5.09           => ( P @ J ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_665_Suc__le__eq,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 9.15/5.09        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_666_Suc__leI,axiom,
% 9.15/5.09      ! [M: nat,N2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.09       => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_667_mono__nat__linear__lb,axiom,
% 9.15/5.09      ! [F: nat > nat,M: nat,K: nat] :
% 9.15/5.09        ( ! [M3: nat,N3: nat] :
% 9.15/5.09            ( ( ord_less_nat @ M3 @ N3 )
% 9.15/5.09           => ( ord_less_nat @ ( F @ M3 ) @ ( F @ N3 ) ) )
% 9.15/5.09       => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_668_add__strict__increasing2,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_669_add__strict__increasing2,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_rat @ B @ C )
% 9.15/5.09         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_670_add__strict__increasing2,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_nat @ B @ C )
% 9.15/5.09         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_671_add__strict__increasing2,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_int @ B @ C )
% 9.15/5.09         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_672_add__strict__increasing,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_eq_real @ B @ C )
% 9.15/5.09         => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_673_add__strict__increasing,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_674_add__strict__increasing,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_675_add__strict__increasing,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_676_add__pos__nonneg,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.15/5.09         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_677_add__pos__nonneg,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.15/5.09         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_678_add__pos__nonneg,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.15/5.09         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_679_add__pos__nonneg,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.15/5.09         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_680_add__nonpos__neg,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_681_add__nonpos__neg,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_682_add__nonpos__neg,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_683_add__nonpos__neg,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_684_add__nonneg__pos,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.09       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.15/5.09         => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_685_add__nonneg__pos,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.09       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.15/5.09         => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_686_add__nonneg__pos,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.09       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.15/5.09         => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_687_add__nonneg__pos,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.09       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.15/5.09         => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_688_add__neg__nonpos,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ zero_zero_real )
% 9.15/5.09       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.15/5.09         => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_689_add__neg__nonpos,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.15/5.09         => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_690_add__neg__nonpos,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.15/5.09         => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_691_add__neg__nonpos,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ zero_zero_int )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.15/5.09         => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_692_ex__least__nat__less,axiom,
% 9.15/5.09      ! [P: nat > $o,N2: nat] :
% 9.15/5.09        ( ( P @ N2 )
% 9.15/5.09       => ( ~ ( P @ zero_zero_nat )
% 9.15/5.09         => ? [K2: nat] :
% 9.15/5.09              ( ( ord_less_nat @ K2 @ N2 )
% 9.15/5.09              & ! [I4: nat] :
% 9.15/5.09                  ( ( ord_less_eq_nat @ I4 @ K2 )
% 9.15/5.09                 => ~ ( P @ I4 ) )
% 9.15/5.09              & ( P @ ( suc @ K2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_693_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,I: nat] :
% 9.15/5.09        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.15/5.09         => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_694_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_real,A2: set_real,X3: real,I: nat] :
% 9.15/5.09        ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member_real @ X3 @ A2 )
% 9.15/5.09         => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_695_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_set_nat,A2: set_set_nat,X3: set_nat,I: nat] :
% 9.15/5.09        ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member_set_nat @ X3 @ A2 )
% 9.15/5.09         => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_696_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
% 9.15/5.09        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member_VEBT_VEBT @ X3 @ A2 )
% 9.15/5.09         => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_697_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_nat,A2: set_nat,X3: nat,I: nat] :
% 9.15/5.09        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member_nat @ X3 @ A2 )
% 9.15/5.09         => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_698_set__update__subsetI,axiom,
% 9.15/5.09      ! [Xs: list_int,A2: set_int,X3: int,I: nat] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
% 9.15/5.09       => ( ( member_int @ X3 @ A2 )
% 9.15/5.09         => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_699_nle__le,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ~ ( ord_less_eq_rat @ A @ B ) )
% 9.15/5.09        = ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.09          & ( B != A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_700_nle__le,axiom,
% 9.15/5.09      ! [A: num,B: num] :
% 9.15/5.09        ( ( ~ ( ord_less_eq_num @ A @ B ) )
% 9.15/5.09        = ( ( ord_less_eq_num @ B @ A )
% 9.15/5.09          & ( B != A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_701_nle__le,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ~ ( ord_less_eq_nat @ A @ B ) )
% 9.15/5.09        = ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.09          & ( B != A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_702_nle__le,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ~ ( ord_less_eq_int @ A @ B ) )
% 9.15/5.09        = ( ( ord_less_eq_int @ B @ A )
% 9.15/5.09          & ( B != A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_703_le__cases3,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.09        ( ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09         => ~ ( ord_less_eq_rat @ Y3 @ Z2 ) )
% 9.15/5.09       => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.15/5.09           => ~ ( ord_less_eq_rat @ X3 @ Z2 ) )
% 9.15/5.09         => ( ( ( ord_less_eq_rat @ X3 @ Z2 )
% 9.15/5.09             => ~ ( ord_less_eq_rat @ Z2 @ Y3 ) )
% 9.15/5.09           => ( ( ( ord_less_eq_rat @ Z2 @ Y3 )
% 9.15/5.09               => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
% 9.15/5.09             => ( ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 9.15/5.09                 => ~ ( ord_less_eq_rat @ Z2 @ X3 ) )
% 9.15/5.09               => ~ ( ( ord_less_eq_rat @ Z2 @ X3 )
% 9.15/5.09                   => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_704_le__cases3,axiom,
% 9.15/5.09      ! [X3: num,Y3: num,Z2: num] :
% 9.15/5.09        ( ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09         => ~ ( ord_less_eq_num @ Y3 @ Z2 ) )
% 9.15/5.09       => ( ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.15/5.09           => ~ ( ord_less_eq_num @ X3 @ Z2 ) )
% 9.15/5.09         => ( ( ( ord_less_eq_num @ X3 @ Z2 )
% 9.15/5.09             => ~ ( ord_less_eq_num @ Z2 @ Y3 ) )
% 9.15/5.09           => ( ( ( ord_less_eq_num @ Z2 @ Y3 )
% 9.15/5.09               => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
% 9.15/5.09             => ( ( ( ord_less_eq_num @ Y3 @ Z2 )
% 9.15/5.09                 => ~ ( ord_less_eq_num @ Z2 @ X3 ) )
% 9.15/5.09               => ~ ( ( ord_less_eq_num @ Z2 @ X3 )
% 9.15/5.09                   => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_705_le__cases3,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.09        ( ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09         => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
% 9.15/5.09       => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.15/5.09           => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
% 9.15/5.09         => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
% 9.15/5.09             => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
% 9.15/5.09           => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
% 9.15/5.09               => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
% 9.15/5.09             => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 9.15/5.09                 => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
% 9.15/5.09               => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
% 9.15/5.09                   => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_706_le__cases3,axiom,
% 9.15/5.09      ! [X3: int,Y3: int,Z2: int] :
% 9.15/5.09        ( ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09         => ~ ( ord_less_eq_int @ Y3 @ Z2 ) )
% 9.15/5.09       => ( ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.15/5.09           => ~ ( ord_less_eq_int @ X3 @ Z2 ) )
% 9.15/5.09         => ( ( ( ord_less_eq_int @ X3 @ Z2 )
% 9.15/5.09             => ~ ( ord_less_eq_int @ Z2 @ Y3 ) )
% 9.15/5.09           => ( ( ( ord_less_eq_int @ Z2 @ Y3 )
% 9.15/5.09               => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
% 9.15/5.09             => ( ( ( ord_less_eq_int @ Y3 @ Z2 )
% 9.15/5.09                 => ~ ( ord_less_eq_int @ Z2 @ X3 ) )
% 9.15/5.09               => ~ ( ( ord_less_eq_int @ Z2 @ X3 )
% 9.15/5.09                   => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_707_order__class_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [X: set_int,Y: set_int] :
% 9.15/5.09            ( ( ord_less_eq_set_int @ X @ Y )
% 9.15/5.09            & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_708_order__class_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [X: rat,Y: rat] :
% 9.15/5.09            ( ( ord_less_eq_rat @ X @ Y )
% 9.15/5.09            & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_709_order__class_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [X: num,Y: num] :
% 9.15/5.09            ( ( ord_less_eq_num @ X @ Y )
% 9.15/5.09            & ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_710_order__class_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [X: nat,Y: nat] :
% 9.15/5.09            ( ( ord_less_eq_nat @ X @ Y )
% 9.15/5.09            & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_711_order__class_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [X: int,Y: int] :
% 9.15/5.09            ( ( ord_less_eq_int @ X @ Y )
% 9.15/5.09            & ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_712_ord__eq__le__trans,axiom,
% 9.15/5.09      ! [A: set_int,B: set_int,C: set_int] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ B @ C )
% 9.15/5.09         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_713_ord__eq__le__trans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_714_ord__eq__le__trans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_715_ord__eq__le__trans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_716_ord__eq__le__trans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_717_ord__le__eq__trans,axiom,
% 9.15/5.09      ! [A: set_int,B: set_int,C: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_718_ord__le__eq__trans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_719_ord__le__eq__trans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_720_ord__le__eq__trans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_721_ord__le__eq__trans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_722_order__antisym,axiom,
% 9.15/5.09      ! [X3: set_int,Y3: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.15/5.09         => ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_723_order__antisym,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.15/5.09         => ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_724_order__antisym,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.15/5.09         => ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_725_order__antisym,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.15/5.09         => ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_726_order__antisym,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.15/5.09         => ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_727_order_Otrans,axiom,
% 9.15/5.09      ! [A: set_int,B: set_int,C: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ B @ C )
% 9.15/5.09         => ( ord_less_eq_set_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_728_order_Otrans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_729_order_Otrans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ord_less_eq_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_730_order_Otrans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ord_less_eq_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_731_order_Otrans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ord_less_eq_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_732_order__trans,axiom,
% 9.15/5.09      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_eq_set_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_733_order__trans,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_734_order__trans,axiom,
% 9.15/5.09      ! [X3: num,Y3: num,Z2: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_num @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_735_order__trans,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_736_order__trans,axiom,
% 9.15/5.09      ! [X3: int,Y3: int,Z2: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_eq_int @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_737_linorder__wlog,axiom,
% 9.15/5.09      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.15/5.09        ( ! [A4: rat,B4: rat] :
% 9.15/5.09            ( ( ord_less_eq_rat @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: rat,B4: rat] :
% 9.15/5.09              ( ( P @ B4 @ A4 )
% 9.15/5.09             => ( P @ A4 @ B4 ) )
% 9.15/5.09         => ( P @ A @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_738_linorder__wlog,axiom,
% 9.15/5.09      ! [P: num > num > $o,A: num,B: num] :
% 9.15/5.09        ( ! [A4: num,B4: num] :
% 9.15/5.09            ( ( ord_less_eq_num @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: num,B4: num] :
% 9.15/5.09              ( ( P @ B4 @ A4 )
% 9.15/5.09             => ( P @ A4 @ B4 ) )
% 9.15/5.09         => ( P @ A @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_739_linorder__wlog,axiom,
% 9.15/5.09      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.15/5.09        ( ! [A4: nat,B4: nat] :
% 9.15/5.09            ( ( ord_less_eq_nat @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: nat,B4: nat] :
% 9.15/5.09              ( ( P @ B4 @ A4 )
% 9.15/5.09             => ( P @ A4 @ B4 ) )
% 9.15/5.09         => ( P @ A @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_740_linorder__wlog,axiom,
% 9.15/5.09      ! [P: int > int > $o,A: int,B: int] :
% 9.15/5.09        ( ! [A4: int,B4: int] :
% 9.15/5.09            ( ( ord_less_eq_int @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: int,B4: int] :
% 9.15/5.09              ( ( P @ B4 @ A4 )
% 9.15/5.09             => ( P @ A4 @ B4 ) )
% 9.15/5.09         => ( P @ A @ B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_741_dual__order_Oeq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: set_int,B3: set_int] :
% 9.15/5.09            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 9.15/5.09            & ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_742_dual__order_Oeq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: rat,B3: rat] :
% 9.15/5.09            ( ( ord_less_eq_rat @ B3 @ A3 )
% 9.15/5.09            & ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_743_dual__order_Oeq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: num,B3: num] :
% 9.15/5.09            ( ( ord_less_eq_num @ B3 @ A3 )
% 9.15/5.09            & ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_744_dual__order_Oeq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.09            ( ( ord_less_eq_nat @ B3 @ A3 )
% 9.15/5.09            & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_745_dual__order_Oeq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: int,B3: int] :
% 9.15/5.09            ( ( ord_less_eq_int @ B3 @ A3 )
% 9.15/5.09            & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_746_dual__order_Oantisym,axiom,
% 9.15/5.09      ! [B: set_int,A: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_747_dual__order_Oantisym,axiom,
% 9.15/5.09      ! [B: rat,A: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_748_dual__order_Oantisym,axiom,
% 9.15/5.09      ! [B: num,A: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_749_dual__order_Oantisym,axiom,
% 9.15/5.09      ! [B: nat,A: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_750_dual__order_Oantisym,axiom,
% 9.15/5.09      ! [B: int,A: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_751_dual__order_Otrans,axiom,
% 9.15/5.09      ! [B: set_int,A: set_int,C: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ C @ B )
% 9.15/5.09         => ( ord_less_eq_set_int @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_752_dual__order_Otrans,axiom,
% 9.15/5.09      ! [B: rat,A: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_rat @ C @ B )
% 9.15/5.09         => ( ord_less_eq_rat @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_753_dual__order_Otrans,axiom,
% 9.15/5.09      ! [B: num,A: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_num @ C @ B )
% 9.15/5.09         => ( ord_less_eq_num @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_754_dual__order_Otrans,axiom,
% 9.15/5.09      ! [B: nat,A: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_nat @ C @ B )
% 9.15/5.09         => ( ord_less_eq_nat @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_755_dual__order_Otrans,axiom,
% 9.15/5.09      ! [B: int,A: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ B @ A )
% 9.15/5.09       => ( ( ord_less_eq_int @ C @ B )
% 9.15/5.09         => ( ord_less_eq_int @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_756_antisym,axiom,
% 9.15/5.09      ! [A: set_int,B: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ B @ A )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_757_antisym,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_758_antisym,axiom,
% 9.15/5.09      ! [A: num,B: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ A )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_759_antisym,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_760_antisym,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ A )
% 9.15/5.09         => ( A = B ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_761_Orderings_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: set_int,B3: set_int] :
% 9.15/5.09            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 9.15/5.09            & ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_762_Orderings_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: rat,B3: rat] :
% 9.15/5.09            ( ( ord_less_eq_rat @ A3 @ B3 )
% 9.15/5.09            & ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_763_Orderings_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: num,Z: num] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: num,B3: num] :
% 9.15/5.09            ( ( ord_less_eq_num @ A3 @ B3 )
% 9.15/5.09            & ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_764_Orderings_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.09            ( ( ord_less_eq_nat @ A3 @ B3 )
% 9.15/5.09            & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_765_Orderings_Oorder__eq__iff,axiom,
% 9.15/5.09      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 9.15/5.09      = ( ^ [A3: int,B3: int] :
% 9.15/5.09            ( ( ord_less_eq_int @ A3 @ B3 )
% 9.15/5.09            & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_766_order__subst1,axiom,
% 9.15/5.09      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_767_order__subst1,axiom,
% 9.15/5.09      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_768_order__subst1,axiom,
% 9.15/5.09      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_769_order__subst1,axiom,
% 9.15/5.09      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ! [X4: int,Y5: int] :
% 9.15/5.09                ( ( ord_less_eq_int @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_770_order__subst1,axiom,
% 9.15/5.09      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_771_order__subst1,axiom,
% 9.15/5.09      ! [A: num,F: num > num,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_772_order__subst1,axiom,
% 9.15/5.09      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_773_order__subst1,axiom,
% 9.15/5.09      ! [A: num,F: int > num,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.09         => ( ! [X4: int,Y5: int] :
% 9.15/5.09                ( ( ord_less_eq_int @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_774_order__subst1,axiom,
% 9.15/5.09      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_775_order__subst1,axiom,
% 9.15/5.09      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_776_order__subst2,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_777_order__subst2,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_778_order__subst2,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_779_order__subst2,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_780_order__subst2,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_781_order__subst2,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_782_order__subst2,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_783_order__subst2,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_int @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_784_order__subst2,axiom,
% 9.15/5.09      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_785_order__subst2,axiom,
% 9.15/5.09      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_eq_num @ ( F @ B ) @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_786_order__eq__refl,axiom,
% 9.15/5.09      ! [X3: set_int,Y3: set_int] :
% 9.15/5.09        ( ( X3 = Y3 )
% 9.15/5.09       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_787_order__eq__refl,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( X3 = Y3 )
% 9.15/5.09       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_788_order__eq__refl,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( X3 = Y3 )
% 9.15/5.09       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_789_order__eq__refl,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( X3 = Y3 )
% 9.15/5.09       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_790_order__eq__refl,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( X3 = Y3 )
% 9.15/5.09       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_791_linorder__linear,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09        | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_792_linorder__linear,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09        | ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_793_linorder__linear,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09        | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_794_linorder__linear,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09        | ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_795_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_796_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_797_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_798_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_799_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_800_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: num,F: num > num,B: num,C: num] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_801_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_802_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: int,F: num > int,B: num,C: num] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_803_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_804_ord__eq__le__subst,axiom,
% 9.15/5.09      ! [A: num,F: nat > num,B: nat,C: nat] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_805_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_806_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_807_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_808_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.09                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_809_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_810_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_811_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_812_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: num,B: num,F: num > int,C: int] :
% 9.15/5.09        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: num,Y5: num] :
% 9.15/5.09                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_813_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_814_ord__le__eq__subst,axiom,
% 9.15/5.09      ! [A: nat,B: nat,F: nat > num,C: num] :
% 9.15/5.09        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.09       => ( ( ( F @ B )
% 9.15/5.09            = C )
% 9.15/5.09         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.09                ( ( ord_less_eq_nat @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_815_linorder__le__cases,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_816_linorder__le__cases,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ~ ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_817_linorder__le__cases,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_818_linorder__le__cases,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ~ ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_819_order__antisym__conv,axiom,
% 9.15/5.09      ! [Y3: set_int,X3: set_int] :
% 9.15/5.09        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_820_order__antisym__conv,axiom,
% 9.15/5.09      ! [Y3: rat,X3: rat] :
% 9.15/5.09        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_821_order__antisym__conv,axiom,
% 9.15/5.09      ! [Y3: num,X3: num] :
% 9.15/5.09        ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_822_order__antisym__conv,axiom,
% 9.15/5.09      ! [Y3: nat,X3: nat] :
% 9.15/5.09        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_823_order__antisym__conv,axiom,
% 9.15/5.09      ! [Y3: int,X3: int] :
% 9.15/5.09        ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.15/5.09       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_824_zero__reorient,axiom,
% 9.15/5.09      ! [X3: complex] :
% 9.15/5.09        ( ( zero_zero_complex = X3 )
% 9.15/5.09        = ( X3 = zero_zero_complex ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_825_zero__reorient,axiom,
% 9.15/5.09      ! [X3: real] :
% 9.15/5.09        ( ( zero_zero_real = X3 )
% 9.15/5.09        = ( X3 = zero_zero_real ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_826_zero__reorient,axiom,
% 9.15/5.09      ! [X3: rat] :
% 9.15/5.09        ( ( zero_zero_rat = X3 )
% 9.15/5.09        = ( X3 = zero_zero_rat ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_827_zero__reorient,axiom,
% 9.15/5.09      ! [X3: nat] :
% 9.15/5.09        ( ( zero_zero_nat = X3 )
% 9.15/5.09        = ( X3 = zero_zero_nat ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_828_zero__reorient,axiom,
% 9.15/5.09      ! [X3: int] :
% 9.15/5.09        ( ( zero_zero_int = X3 )
% 9.15/5.09        = ( X3 = zero_zero_int ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_829_lt__ex,axiom,
% 9.15/5.09      ! [X3: real] :
% 9.15/5.09      ? [Y5: real] : ( ord_less_real @ Y5 @ X3 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_830_lt__ex,axiom,
% 9.15/5.09      ! [X3: rat] :
% 9.15/5.09      ? [Y5: rat] : ( ord_less_rat @ Y5 @ X3 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_831_lt__ex,axiom,
% 9.15/5.09      ! [X3: int] :
% 9.15/5.09      ? [Y5: int] : ( ord_less_int @ Y5 @ X3 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_832_gt__ex,axiom,
% 9.15/5.09      ! [X3: real] :
% 9.15/5.09      ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_833_gt__ex,axiom,
% 9.15/5.09      ! [X3: rat] :
% 9.15/5.09      ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_834_gt__ex,axiom,
% 9.15/5.09      ! [X3: nat] :
% 9.15/5.09      ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_835_gt__ex,axiom,
% 9.15/5.09      ! [X3: int] :
% 9.15/5.09      ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_836_dense,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09       => ? [Z3: real] :
% 9.15/5.09            ( ( ord_less_real @ X3 @ Z3 )
% 9.15/5.09            & ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_837_dense,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09       => ? [Z3: rat] :
% 9.15/5.09            ( ( ord_less_rat @ X3 @ Z3 )
% 9.15/5.09            & ( ord_less_rat @ Z3 @ Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_838_less__imp__neq,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09       => ( X3 != Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_839_less__imp__neq,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09       => ( X3 != Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_840_less__imp__neq,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09       => ( X3 != Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_841_less__imp__neq,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09       => ( X3 != Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_842_less__imp__neq,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09       => ( X3 != Y3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_843_order_Oasym,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_844_order_Oasym,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_845_order_Oasym,axiom,
% 9.15/5.09      ! [A: num,B: num] :
% 9.15/5.09        ( ( ord_less_num @ A @ B )
% 9.15/5.09       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_846_order_Oasym,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_847_order_Oasym,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_848_ord__eq__less__trans,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ord_less_real @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_849_ord__eq__less__trans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_rat @ B @ C )
% 9.15/5.09         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_850_ord__eq__less__trans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_num @ B @ C )
% 9.15/5.09         => ( ord_less_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_851_ord__eq__less__trans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_nat @ B @ C )
% 9.15/5.09         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_852_ord__eq__less__trans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( A = B )
% 9.15/5.09       => ( ( ord_less_int @ B @ C )
% 9.15/5.09         => ( ord_less_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_853_ord__less__eq__trans,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_real @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_854_ord__less__eq__trans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_855_ord__less__eq__trans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_num @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_856_ord__less__eq__trans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_857_ord__less__eq__trans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ( B = C )
% 9.15/5.09         => ( ord_less_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_858_less__induct,axiom,
% 9.15/5.09      ! [P: nat > $o,A: nat] :
% 9.15/5.09        ( ! [X4: nat] :
% 9.15/5.09            ( ! [Y6: nat] :
% 9.15/5.09                ( ( ord_less_nat @ Y6 @ X4 )
% 9.15/5.09               => ( P @ Y6 ) )
% 9.15/5.09           => ( P @ X4 ) )
% 9.15/5.09       => ( P @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_859_antisym__conv3,axiom,
% 9.15/5.09      ! [Y3: real,X3: real] :
% 9.15/5.09        ( ~ ( ord_less_real @ Y3 @ X3 )
% 9.15/5.09       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_860_antisym__conv3,axiom,
% 9.15/5.09      ! [Y3: rat,X3: rat] :
% 9.15/5.09        ( ~ ( ord_less_rat @ Y3 @ X3 )
% 9.15/5.09       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_861_antisym__conv3,axiom,
% 9.15/5.09      ! [Y3: num,X3: num] :
% 9.15/5.09        ( ~ ( ord_less_num @ Y3 @ X3 )
% 9.15/5.09       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_862_antisym__conv3,axiom,
% 9.15/5.09      ! [Y3: nat,X3: nat] :
% 9.15/5.09        ( ~ ( ord_less_nat @ Y3 @ X3 )
% 9.15/5.09       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_863_antisym__conv3,axiom,
% 9.15/5.09      ! [Y3: int,X3: int] :
% 9.15/5.09        ( ~ ( ord_less_int @ Y3 @ X3 )
% 9.15/5.09       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.15/5.09          = ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_864_linorder__cases,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09       => ( ( X3 != Y3 )
% 9.15/5.09         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_865_linorder__cases,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09       => ( ( X3 != Y3 )
% 9.15/5.09         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_866_linorder__cases,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09       => ( ( X3 != Y3 )
% 9.15/5.09         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_867_linorder__cases,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09       => ( ( X3 != Y3 )
% 9.15/5.09         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_868_linorder__cases,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( X3 != Y3 )
% 9.15/5.09         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_869_dual__order_Oasym,axiom,
% 9.15/5.09      ! [B: real,A: real] :
% 9.15/5.09        ( ( ord_less_real @ B @ A )
% 9.15/5.09       => ~ ( ord_less_real @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_870_dual__order_Oasym,axiom,
% 9.15/5.09      ! [B: rat,A: rat] :
% 9.15/5.09        ( ( ord_less_rat @ B @ A )
% 9.15/5.09       => ~ ( ord_less_rat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_871_dual__order_Oasym,axiom,
% 9.15/5.09      ! [B: num,A: num] :
% 9.15/5.09        ( ( ord_less_num @ B @ A )
% 9.15/5.09       => ~ ( ord_less_num @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_872_dual__order_Oasym,axiom,
% 9.15/5.09      ! [B: nat,A: nat] :
% 9.15/5.09        ( ( ord_less_nat @ B @ A )
% 9.15/5.09       => ~ ( ord_less_nat @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_873_dual__order_Oasym,axiom,
% 9.15/5.09      ! [B: int,A: int] :
% 9.15/5.09        ( ( ord_less_int @ B @ A )
% 9.15/5.09       => ~ ( ord_less_int @ A @ B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_874_dual__order_Oirrefl,axiom,
% 9.15/5.09      ! [A: real] :
% 9.15/5.09        ~ ( ord_less_real @ A @ A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_875_dual__order_Oirrefl,axiom,
% 9.15/5.09      ! [A: rat] :
% 9.15/5.09        ~ ( ord_less_rat @ A @ A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_876_dual__order_Oirrefl,axiom,
% 9.15/5.09      ! [A: num] :
% 9.15/5.09        ~ ( ord_less_num @ A @ A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_877_dual__order_Oirrefl,axiom,
% 9.15/5.09      ! [A: nat] :
% 9.15/5.09        ~ ( ord_less_nat @ A @ A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_878_dual__order_Oirrefl,axiom,
% 9.15/5.09      ! [A: int] :
% 9.15/5.09        ~ ( ord_less_int @ A @ A ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_879_exists__least__iff,axiom,
% 9.15/5.09      ( ( ^ [P2: nat > $o] :
% 9.15/5.09          ? [X6: nat] : ( P2 @ X6 ) )
% 9.15/5.09      = ( ^ [P3: nat > $o] :
% 9.15/5.09          ? [N: nat] :
% 9.15/5.09            ( ( P3 @ N )
% 9.15/5.09            & ! [M2: nat] :
% 9.15/5.09                ( ( ord_less_nat @ M2 @ N )
% 9.15/5.09               => ~ ( P3 @ M2 ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_880_linorder__less__wlog,axiom,
% 9.15/5.09      ! [P: real > real > $o,A: real,B: real] :
% 9.15/5.09        ( ! [A4: real,B4: real] :
% 9.15/5.09            ( ( ord_less_real @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: real] : ( P @ A4 @ A4 )
% 9.15/5.09         => ( ! [A4: real,B4: real] :
% 9.15/5.09                ( ( P @ B4 @ A4 )
% 9.15/5.09               => ( P @ A4 @ B4 ) )
% 9.15/5.09           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_881_linorder__less__wlog,axiom,
% 9.15/5.09      ! [P: rat > rat > $o,A: rat,B: rat] :
% 9.15/5.09        ( ! [A4: rat,B4: rat] :
% 9.15/5.09            ( ( ord_less_rat @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: rat] : ( P @ A4 @ A4 )
% 9.15/5.09         => ( ! [A4: rat,B4: rat] :
% 9.15/5.09                ( ( P @ B4 @ A4 )
% 9.15/5.09               => ( P @ A4 @ B4 ) )
% 9.15/5.09           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_882_linorder__less__wlog,axiom,
% 9.15/5.09      ! [P: num > num > $o,A: num,B: num] :
% 9.15/5.09        ( ! [A4: num,B4: num] :
% 9.15/5.09            ( ( ord_less_num @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: num] : ( P @ A4 @ A4 )
% 9.15/5.09         => ( ! [A4: num,B4: num] :
% 9.15/5.09                ( ( P @ B4 @ A4 )
% 9.15/5.09               => ( P @ A4 @ B4 ) )
% 9.15/5.09           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_883_linorder__less__wlog,axiom,
% 9.15/5.09      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.15/5.09        ( ! [A4: nat,B4: nat] :
% 9.15/5.09            ( ( ord_less_nat @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: nat] : ( P @ A4 @ A4 )
% 9.15/5.09         => ( ! [A4: nat,B4: nat] :
% 9.15/5.09                ( ( P @ B4 @ A4 )
% 9.15/5.09               => ( P @ A4 @ B4 ) )
% 9.15/5.09           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_884_linorder__less__wlog,axiom,
% 9.15/5.09      ! [P: int > int > $o,A: int,B: int] :
% 9.15/5.09        ( ! [A4: int,B4: int] :
% 9.15/5.09            ( ( ord_less_int @ A4 @ B4 )
% 9.15/5.09           => ( P @ A4 @ B4 ) )
% 9.15/5.09       => ( ! [A4: int] : ( P @ A4 @ A4 )
% 9.15/5.09         => ( ! [A4: int,B4: int] :
% 9.15/5.09                ( ( P @ B4 @ A4 )
% 9.15/5.09               => ( P @ A4 @ B4 ) )
% 9.15/5.09           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_885_order_Ostrict__trans,axiom,
% 9.15/5.09      ! [A: real,B: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ord_less_real @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_886_order_Ostrict__trans,axiom,
% 9.15/5.09      ! [A: rat,B: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( ( ord_less_rat @ B @ C )
% 9.15/5.09         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_887_order_Ostrict__trans,axiom,
% 9.15/5.09      ! [A: num,B: num,C: num] :
% 9.15/5.09        ( ( ord_less_num @ A @ B )
% 9.15/5.09       => ( ( ord_less_num @ B @ C )
% 9.15/5.09         => ( ord_less_num @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_888_order_Ostrict__trans,axiom,
% 9.15/5.09      ! [A: nat,B: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( ( ord_less_nat @ B @ C )
% 9.15/5.09         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_889_order_Ostrict__trans,axiom,
% 9.15/5.09      ! [A: int,B: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( ( ord_less_int @ B @ C )
% 9.15/5.09         => ( ord_less_int @ A @ C ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_890_not__less__iff__gr__or__eq,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.15/5.09        = ( ( ord_less_real @ Y3 @ X3 )
% 9.15/5.09          | ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_891_not__less__iff__gr__or__eq,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.15/5.09        = ( ( ord_less_rat @ Y3 @ X3 )
% 9.15/5.09          | ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_892_not__less__iff__gr__or__eq,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.15/5.09        = ( ( ord_less_num @ Y3 @ X3 )
% 9.15/5.09          | ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_893_not__less__iff__gr__or__eq,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.15/5.09        = ( ( ord_less_nat @ Y3 @ X3 )
% 9.15/5.09          | ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_894_not__less__iff__gr__or__eq,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.15/5.09        = ( ( ord_less_int @ Y3 @ X3 )
% 9.15/5.09          | ( X3 = Y3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_895_dual__order_Ostrict__trans,axiom,
% 9.15/5.09      ! [B: real,A: real,C: real] :
% 9.15/5.09        ( ( ord_less_real @ B @ A )
% 9.15/5.09       => ( ( ord_less_real @ C @ B )
% 9.15/5.09         => ( ord_less_real @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_896_dual__order_Ostrict__trans,axiom,
% 9.15/5.09      ! [B: rat,A: rat,C: rat] :
% 9.15/5.09        ( ( ord_less_rat @ B @ A )
% 9.15/5.09       => ( ( ord_less_rat @ C @ B )
% 9.15/5.09         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_897_dual__order_Ostrict__trans,axiom,
% 9.15/5.09      ! [B: num,A: num,C: num] :
% 9.15/5.09        ( ( ord_less_num @ B @ A )
% 9.15/5.09       => ( ( ord_less_num @ C @ B )
% 9.15/5.09         => ( ord_less_num @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_898_dual__order_Ostrict__trans,axiom,
% 9.15/5.09      ! [B: nat,A: nat,C: nat] :
% 9.15/5.09        ( ( ord_less_nat @ B @ A )
% 9.15/5.09       => ( ( ord_less_nat @ C @ B )
% 9.15/5.09         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_899_dual__order_Ostrict__trans,axiom,
% 9.15/5.09      ! [B: int,A: int,C: int] :
% 9.15/5.09        ( ( ord_less_int @ B @ A )
% 9.15/5.09       => ( ( ord_less_int @ C @ B )
% 9.15/5.09         => ( ord_less_int @ C @ A ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_900_order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_901_order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_902_order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [A: num,B: num] :
% 9.15/5.09        ( ( ord_less_num @ A @ B )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_903_order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_904_order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_905_dual__order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [B: real,A: real] :
% 9.15/5.09        ( ( ord_less_real @ B @ A )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_906_dual__order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [B: rat,A: rat] :
% 9.15/5.09        ( ( ord_less_rat @ B @ A )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_907_dual__order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [B: num,A: num] :
% 9.15/5.09        ( ( ord_less_num @ B @ A )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_908_dual__order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [B: nat,A: nat] :
% 9.15/5.09        ( ( ord_less_nat @ B @ A )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_909_dual__order_Ostrict__implies__not__eq,axiom,
% 9.15/5.09      ! [B: int,A: int] :
% 9.15/5.09        ( ( ord_less_int @ B @ A )
% 9.15/5.09       => ( A != B ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_910_linorder__neqE,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_911_linorder__neqE,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_912_linorder__neqE,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09       => ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09         => ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_913_linorder__neqE,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_914_linorder__neqE,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_915_order__less__asym,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_916_order__less__asym,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_917_order__less__asym,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_918_order__less__asym,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_919_order__less__asym,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_920_linorder__neq__iff,axiom,
% 9.15/5.09      ! [X3: real,Y3: real] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09        = ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09          | ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_921_linorder__neq__iff,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09        = ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09          | ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_922_linorder__neq__iff,axiom,
% 9.15/5.09      ! [X3: num,Y3: num] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09        = ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09          | ( ord_less_num @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_923_linorder__neq__iff,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09        = ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09          | ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_924_linorder__neq__iff,axiom,
% 9.15/5.09      ! [X3: int,Y3: int] :
% 9.15/5.09        ( ( X3 != Y3 )
% 9.15/5.09        = ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09          | ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_925_order__less__asym_H,axiom,
% 9.15/5.09      ! [A: real,B: real] :
% 9.15/5.09        ( ( ord_less_real @ A @ B )
% 9.15/5.09       => ~ ( ord_less_real @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_926_order__less__asym_H,axiom,
% 9.15/5.09      ! [A: rat,B: rat] :
% 9.15/5.09        ( ( ord_less_rat @ A @ B )
% 9.15/5.09       => ~ ( ord_less_rat @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_927_order__less__asym_H,axiom,
% 9.15/5.09      ! [A: num,B: num] :
% 9.15/5.09        ( ( ord_less_num @ A @ B )
% 9.15/5.09       => ~ ( ord_less_num @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_928_order__less__asym_H,axiom,
% 9.15/5.09      ! [A: nat,B: nat] :
% 9.15/5.09        ( ( ord_less_nat @ A @ B )
% 9.15/5.09       => ~ ( ord_less_nat @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_929_order__less__asym_H,axiom,
% 9.15/5.09      ! [A: int,B: int] :
% 9.15/5.09        ( ( ord_less_int @ A @ B )
% 9.15/5.09       => ~ ( ord_less_int @ B @ A ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_930_order__less__trans,axiom,
% 9.15/5.09      ! [X3: real,Y3: real,Z2: real] :
% 9.15/5.09        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_real @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_931_order__less__trans,axiom,
% 9.15/5.09      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.09        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_rat @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_932_order__less__trans,axiom,
% 9.15/5.09      ! [X3: num,Y3: num,Z2: num] :
% 9.15/5.09        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_num @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_933_order__less__trans,axiom,
% 9.15/5.09      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.09        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_nat @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_934_order__less__trans,axiom,
% 9.15/5.09      ! [X3: int,Y3: int,Z2: int] :
% 9.15/5.09        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.09       => ( ( ord_less_int @ Y3 @ Z2 )
% 9.15/5.09         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_935_ord__eq__less__subst,axiom,
% 9.15/5.09      ! [A: real,F: real > real,B: real,C: real] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ! [X4: real,Y5: real] :
% 9.15/5.09                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_936_ord__eq__less__subst,axiom,
% 9.15/5.09      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ! [X4: real,Y5: real] :
% 9.15/5.09                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_937_ord__eq__less__subst,axiom,
% 9.15/5.09      ! [A: num,F: real > num,B: real,C: real] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ! [X4: real,Y5: real] :
% 9.15/5.09                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_938_ord__eq__less__subst,axiom,
% 9.15/5.09      ! [A: nat,F: real > nat,B: real,C: real] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.09         => ( ! [X4: real,Y5: real] :
% 9.15/5.09                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.09               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.09           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.09  
% 9.15/5.09  thf(fact_939_ord__eq__less__subst,axiom,
% 9.15/5.09      ! [A: int,F: real > int,B: real,C: real] :
% 9.15/5.09        ( ( A
% 9.15/5.09          = ( F @ B ) )
% 9.15/5.09       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_940_ord__eq__less__subst,axiom,
% 9.15/5.10      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.15/5.10        ( ( A
% 9.15/5.10          = ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_941_ord__eq__less__subst,axiom,
% 9.15/5.10      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.10        ( ( A
% 9.15/5.10          = ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_942_ord__eq__less__subst,axiom,
% 9.15/5.10      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.15/5.10        ( ( A
% 9.15/5.10          = ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_943_ord__eq__less__subst,axiom,
% 9.15/5.10      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.15/5.10        ( ( A
% 9.15/5.10          = ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_944_ord__eq__less__subst,axiom,
% 9.15/5.10      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.15/5.10        ( ( A
% 9.15/5.10          = ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_945_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_946_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_947_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > num,C: num] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_948_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_949_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > int,C: int] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_950_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_951_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_952_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_953_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_954_ord__less__eq__subst,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ( F @ B )
% 9.15/5.10            = C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_955_order__less__irrefl,axiom,
% 9.15/5.10      ! [X3: real] :
% 9.15/5.10        ~ ( ord_less_real @ X3 @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_956_order__less__irrefl,axiom,
% 9.15/5.10      ! [X3: rat] :
% 9.15/5.10        ~ ( ord_less_rat @ X3 @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_957_order__less__irrefl,axiom,
% 9.15/5.10      ! [X3: num] :
% 9.15/5.10        ~ ( ord_less_num @ X3 @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_958_order__less__irrefl,axiom,
% 9.15/5.10      ! [X3: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ X3 @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_959_order__less__irrefl,axiom,
% 9.15/5.10      ! [X3: int] :
% 9.15/5.10        ~ ( ord_less_int @ X3 @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_960_order__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: real > real,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_961_order__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_962_order__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: num > real,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_963_order__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_nat @ B @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_964_order__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: int > real,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_int @ B @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_965_order__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_966_order__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_967_order__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_968_order__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_nat @ B @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_969_order__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_int @ B @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_970_order__less__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_971_order__less__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_972_order__less__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > num,C: num] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_973_order__less__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_974_order__less__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > int,C: int] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_975_order__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_976_order__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_977_order__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_978_order__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_979_order__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_980_order__less__not__sym,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_981_order__less__not__sym,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_982_order__less__not__sym,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_983_order__less__not__sym,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_984_order__less__not__sym,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_985_order__less__imp__triv,axiom,
% 9.15/5.10      ! [X3: real,Y3: real,P: $o] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_real @ Y3 @ X3 )
% 9.15/5.10         => P ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_986_order__less__imp__triv,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat,P: $o] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_rat @ Y3 @ X3 )
% 9.15/5.10         => P ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_987_order__less__imp__triv,axiom,
% 9.15/5.10      ! [X3: num,Y3: num,P: $o] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_num @ Y3 @ X3 )
% 9.15/5.10         => P ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_988_order__less__imp__triv,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat,P: $o] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_nat @ Y3 @ X3 )
% 9.15/5.10         => P ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_989_order__less__imp__triv,axiom,
% 9.15/5.10      ! [X3: int,Y3: int,P: $o] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_int @ Y3 @ X3 )
% 9.15/5.10         => P ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_990_linorder__less__linear,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10        | ( X3 = Y3 )
% 9.15/5.10        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_991_linorder__less__linear,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10        | ( X3 = Y3 )
% 9.15/5.10        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_992_linorder__less__linear,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10        | ( X3 = Y3 )
% 9.15/5.10        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_993_linorder__less__linear,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10        | ( X3 = Y3 )
% 9.15/5.10        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_994_linorder__less__linear,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10        | ( X3 = Y3 )
% 9.15/5.10        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_995_order__less__imp__not__eq,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_996_order__less__imp__not__eq,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_997_order__less__imp__not__eq,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_998_order__less__imp__not__eq,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_999_order__less__imp__not__eq,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1000_order__less__imp__not__eq2,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( Y3 != X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1001_order__less__imp__not__eq2,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( Y3 != X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1002_order__less__imp__not__eq2,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( Y3 != X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1003_order__less__imp__not__eq2,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( Y3 != X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1004_order__less__imp__not__eq2,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( Y3 != X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1005_order__less__imp__not__less,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1006_order__less__imp__not__less,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1007_order__less__imp__not__less,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1008_order__less__imp__not__less,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1009_order__less__imp__not__less,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ~ ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1010_nat__neq__iff,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( M != N2 )
% 9.15/5.10        = ( ( ord_less_nat @ M @ N2 )
% 9.15/5.10          | ( ord_less_nat @ N2 @ M ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1011_less__not__refl,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ N2 @ N2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1012_less__not__refl2,axiom,
% 9.15/5.10      ! [N2: nat,M: nat] :
% 9.15/5.10        ( ( ord_less_nat @ N2 @ M )
% 9.15/5.10       => ( M != N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1013_less__not__refl3,axiom,
% 9.15/5.10      ! [S: nat,T: nat] :
% 9.15/5.10        ( ( ord_less_nat @ S @ T )
% 9.15/5.10       => ( S != T ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1014_less__irrefl__nat,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ N2 @ N2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1015_nat__less__induct,axiom,
% 9.15/5.10      ! [P: nat > $o,N2: nat] :
% 9.15/5.10        ( ! [N3: nat] :
% 9.15/5.10            ( ! [M6: nat] :
% 9.15/5.10                ( ( ord_less_nat @ M6 @ N3 )
% 9.15/5.10               => ( P @ M6 ) )
% 9.15/5.10           => ( P @ N3 ) )
% 9.15/5.10       => ( P @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1016_infinite__descent,axiom,
% 9.15/5.10      ! [P: nat > $o,N2: nat] :
% 9.15/5.10        ( ! [N3: nat] :
% 9.15/5.10            ( ~ ( P @ N3 )
% 9.15/5.10           => ? [M6: nat] :
% 9.15/5.10                ( ( ord_less_nat @ M6 @ N3 )
% 9.15/5.10                & ~ ( P @ M6 ) ) )
% 9.15/5.10       => ( P @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1017_linorder__neqE__nat,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( X3 != Y3 )
% 9.15/5.10       => ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10         => ( ord_less_nat @ Y3 @ X3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1018_le__refl,axiom,
% 9.15/5.10      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1019_le__trans,axiom,
% 9.15/5.10      ! [I: nat,J: nat,K: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.10       => ( ( ord_less_eq_nat @ J @ K )
% 9.15/5.10         => ( ord_less_eq_nat @ I @ K ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1020_eq__imp__le,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( M = N2 )
% 9.15/5.10       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1021_le__antisym,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.10       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.15/5.10         => ( M = N2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1022_nat__le__linear,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.10        | ( ord_less_eq_nat @ N2 @ M ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1023_Nat_Oex__has__greatest__nat,axiom,
% 9.15/5.10      ! [P: nat > $o,K: nat,B: nat] :
% 9.15/5.10        ( ( P @ K )
% 9.15/5.10       => ( ! [Y5: nat] :
% 9.15/5.10              ( ( P @ Y5 )
% 9.15/5.10             => ( ord_less_eq_nat @ Y5 @ B ) )
% 9.15/5.10         => ? [X4: nat] :
% 9.15/5.10              ( ( P @ X4 )
% 9.15/5.10              & ! [Y6: nat] :
% 9.15/5.10                  ( ( P @ Y6 )
% 9.15/5.10                 => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1024_size__neq__size__imp__neq,axiom,
% 9.15/5.10      ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
% 9.15/5.10        ( ( ( size_s6755466524823107622T_VEBT @ X3 )
% 9.15/5.10         != ( size_s6755466524823107622T_VEBT @ Y3 ) )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1025_size__neq__size__imp__neq,axiom,
% 9.15/5.10      ! [X3: list_o,Y3: list_o] :
% 9.15/5.10        ( ( ( size_size_list_o @ X3 )
% 9.15/5.10         != ( size_size_list_o @ Y3 ) )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1026_size__neq__size__imp__neq,axiom,
% 9.15/5.10      ! [X3: list_nat,Y3: list_nat] :
% 9.15/5.10        ( ( ( size_size_list_nat @ X3 )
% 9.15/5.10         != ( size_size_list_nat @ Y3 ) )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1027_size__neq__size__imp__neq,axiom,
% 9.15/5.10      ! [X3: list_int,Y3: list_int] :
% 9.15/5.10        ( ( ( size_size_list_int @ X3 )
% 9.15/5.10         != ( size_size_list_int @ Y3 ) )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1028_size__neq__size__imp__neq,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ( size_size_num @ X3 )
% 9.15/5.10         != ( size_size_num @ Y3 ) )
% 9.15/5.10       => ( X3 != Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1029_zero__le,axiom,
% 9.15/5.10      ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1030_leD,axiom,
% 9.15/5.10      ! [Y3: real,X3: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1031_leD,axiom,
% 9.15/5.10      ! [Y3: set_int,X3: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1032_leD,axiom,
% 9.15/5.10      ! [Y3: rat,X3: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1033_leD,axiom,
% 9.15/5.10      ! [Y3: num,X3: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_num @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1034_leD,axiom,
% 9.15/5.10      ! [Y3: nat,X3: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1035_leD,axiom,
% 9.15/5.10      ! [Y3: int,X3: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ Y3 @ X3 )
% 9.15/5.10       => ~ ( ord_less_int @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1036_leI,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1037_leI,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1038_leI,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1039_leI,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1040_leI,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1041_nless__le,axiom,
% 9.15/5.10      ! [A: real,B: real] :
% 9.15/5.10        ( ( ~ ( ord_less_real @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_real @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1042_nless__le,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int] :
% 9.15/5.10        ( ( ~ ( ord_less_set_int @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1043_nless__le,axiom,
% 9.15/5.10      ! [A: rat,B: rat] :
% 9.15/5.10        ( ( ~ ( ord_less_rat @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_rat @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1044_nless__le,axiom,
% 9.15/5.10      ! [A: num,B: num] :
% 9.15/5.10        ( ( ~ ( ord_less_num @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_num @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1045_nless__le,axiom,
% 9.15/5.10      ! [A: nat,B: nat] :
% 9.15/5.10        ( ( ~ ( ord_less_nat @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_nat @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1046_nless__le,axiom,
% 9.15/5.10      ! [A: int,B: int] :
% 9.15/5.10        ( ( ~ ( ord_less_int @ A @ B ) )
% 9.15/5.10        = ( ~ ( ord_less_eq_int @ A @ B )
% 9.15/5.10          | ( A = B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1047_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1048_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int] :
% 9.15/5.10        ( ~ ( ord_less_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1049_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1050_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ~ ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1051_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1052_antisym__conv1,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1053_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1054_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1055_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1056_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1057_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1058_antisym__conv2,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.15/5.10          = ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1059_dense__ge,axiom,
% 9.15/5.10      ! [Z2: real,Y3: real] :
% 9.15/5.10        ( ! [X4: real] :
% 9.15/5.10            ( ( ord_less_real @ Z2 @ X4 )
% 9.15/5.10           => ( ord_less_eq_real @ Y3 @ X4 ) )
% 9.15/5.10       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1060_dense__ge,axiom,
% 9.15/5.10      ! [Z2: rat,Y3: rat] :
% 9.15/5.10        ( ! [X4: rat] :
% 9.15/5.10            ( ( ord_less_rat @ Z2 @ X4 )
% 9.15/5.10           => ( ord_less_eq_rat @ Y3 @ X4 ) )
% 9.15/5.10       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1061_dense__le,axiom,
% 9.15/5.10      ! [Y3: real,Z2: real] :
% 9.15/5.10        ( ! [X4: real] :
% 9.15/5.10            ( ( ord_less_real @ X4 @ Y3 )
% 9.15/5.10           => ( ord_less_eq_real @ X4 @ Z2 ) )
% 9.15/5.10       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1062_dense__le,axiom,
% 9.15/5.10      ! [Y3: rat,Z2: rat] :
% 9.15/5.10        ( ! [X4: rat] :
% 9.15/5.10            ( ( ord_less_rat @ X4 @ Y3 )
% 9.15/5.10           => ( ord_less_eq_rat @ X4 @ Z2 ) )
% 9.15/5.10       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1063_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [X: real,Y: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1064_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [X: set_int,Y: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1065_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [X: rat,Y: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1066_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [X: num,Y: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1067_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [X: nat,Y: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1068_less__le__not__le,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [X: int,Y: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ X @ Y )
% 9.15/5.10            & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1069_not__le__imp__less,axiom,
% 9.15/5.10      ! [Y3: real,X3: real] :
% 9.15/5.10        ( ~ ( ord_less_eq_real @ Y3 @ X3 )
% 9.15/5.10       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1070_not__le__imp__less,axiom,
% 9.15/5.10      ! [Y3: rat,X3: rat] :
% 9.15/5.10        ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
% 9.15/5.10       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1071_not__le__imp__less,axiom,
% 9.15/5.10      ! [Y3: num,X3: num] :
% 9.15/5.10        ( ~ ( ord_less_eq_num @ Y3 @ X3 )
% 9.15/5.10       => ( ord_less_num @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1072_not__le__imp__less,axiom,
% 9.15/5.10      ! [Y3: nat,X3: nat] :
% 9.15/5.10        ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
% 9.15/5.10       => ( ord_less_nat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1073_not__le__imp__less,axiom,
% 9.15/5.10      ! [Y3: int,X3: int] :
% 9.15/5.10        ( ~ ( ord_less_eq_int @ Y3 @ X3 )
% 9.15/5.10       => ( ord_less_int @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1074_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_real
% 9.15/5.10      = ( ^ [A3: real,B3: real] :
% 9.15/5.10            ( ( ord_less_real @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1075_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [A3: set_int,B3: set_int] :
% 9.15/5.10            ( ( ord_less_set_int @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1076_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_rat
% 9.15/5.10      = ( ^ [A3: rat,B3: rat] :
% 9.15/5.10            ( ( ord_less_rat @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1077_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_num
% 9.15/5.10      = ( ^ [A3: num,B3: num] :
% 9.15/5.10            ( ( ord_less_num @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1078_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_nat
% 9.15/5.10      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.10            ( ( ord_less_nat @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1079_order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_int
% 9.15/5.10      = ( ^ [A3: int,B3: int] :
% 9.15/5.10            ( ( ord_less_int @ A3 @ B3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1080_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [A3: real,B3: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1081_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [A3: set_int,B3: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1082_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [A3: rat,B3: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1083_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [A3: num,B3: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1084_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1085_order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [A3: int,B3: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ A3 @ B3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1086_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: real,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ord_less_real @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1087_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int,C: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_set_int @ B @ C )
% 9.15/5.10         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1088_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: rat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1089_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: num,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_num @ B @ C )
% 9.15/5.10         => ( ord_less_num @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1090_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: nat,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.10       => ( ( ord_less_nat @ B @ C )
% 9.15/5.10         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1091_order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [A: int,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_int @ B @ C )
% 9.15/5.10         => ( ord_less_int @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1092_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: real,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ B @ C )
% 9.15/5.10         => ( ord_less_real @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1093_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int,C: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ B @ C )
% 9.15/5.10         => ( ord_less_set_int @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1094_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ord_less_rat @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1095_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: num,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ord_less_num @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1096_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: nat,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_nat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_nat @ B @ C )
% 9.15/5.10         => ( ord_less_nat @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1097_order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [A: int,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_int @ B @ C )
% 9.15/5.10         => ( ord_less_int @ A @ C ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1098_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [A3: real,B3: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1099_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [A3: set_int,B3: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_set_int @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1100_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [A3: rat,B3: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1101_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [A3: num,B3: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1102_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [A3: nat,B3: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1103_order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [A3: int,B3: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ A3 @ B3 )
% 9.15/5.10            & ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1104_dense__ge__bounded,axiom,
% 9.15/5.10      ! [Z2: real,X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ Z2 @ X3 )
% 9.15/5.10       => ( ! [W: real] :
% 9.15/5.10              ( ( ord_less_real @ Z2 @ W )
% 9.15/5.10             => ( ( ord_less_real @ W @ X3 )
% 9.15/5.10               => ( ord_less_eq_real @ Y3 @ W ) ) )
% 9.15/5.10         => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1105_dense__ge__bounded,axiom,
% 9.15/5.10      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ Z2 @ X3 )
% 9.15/5.10       => ( ! [W: rat] :
% 9.15/5.10              ( ( ord_less_rat @ Z2 @ W )
% 9.15/5.10             => ( ( ord_less_rat @ W @ X3 )
% 9.15/5.10               => ( ord_less_eq_rat @ Y3 @ W ) ) )
% 9.15/5.10         => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1106_dense__le__bounded,axiom,
% 9.15/5.10      ! [X3: real,Y3: real,Z2: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ! [W: real] :
% 9.15/5.10              ( ( ord_less_real @ X3 @ W )
% 9.15/5.10             => ( ( ord_less_real @ W @ Y3 )
% 9.15/5.10               => ( ord_less_eq_real @ W @ Z2 ) ) )
% 9.15/5.10         => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1107_dense__le__bounded,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ! [W: rat] :
% 9.15/5.10              ( ( ord_less_rat @ X3 @ W )
% 9.15/5.10             => ( ( ord_less_rat @ W @ Y3 )
% 9.15/5.10               => ( ord_less_eq_rat @ W @ Z2 ) ) )
% 9.15/5.10         => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1108_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_real
% 9.15/5.10      = ( ^ [B3: real,A3: real] :
% 9.15/5.10            ( ( ord_less_real @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1109_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [B3: set_int,A3: set_int] :
% 9.15/5.10            ( ( ord_less_set_int @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1110_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_rat
% 9.15/5.10      = ( ^ [B3: rat,A3: rat] :
% 9.15/5.10            ( ( ord_less_rat @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1111_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_num
% 9.15/5.10      = ( ^ [B3: num,A3: num] :
% 9.15/5.10            ( ( ord_less_num @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1112_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_nat
% 9.15/5.10      = ( ^ [B3: nat,A3: nat] :
% 9.15/5.10            ( ( ord_less_nat @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1113_dual__order_Oorder__iff__strict,axiom,
% 9.15/5.10      ( ord_less_eq_int
% 9.15/5.10      = ( ^ [B3: int,A3: int] :
% 9.15/5.10            ( ( ord_less_int @ B3 @ A3 )
% 9.15/5.10            | ( A3 = B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1114_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [B3: real,A3: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1115_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [B3: set_int,A3: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1116_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [B3: rat,A3: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1117_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [B3: num,A3: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1118_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [B3: nat,A3: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1119_dual__order_Ostrict__iff__order,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [B3: int,A3: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ B3 @ A3 )
% 9.15/5.10            & ( A3 != B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1120_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: real,A: real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ B @ A )
% 9.15/5.10       => ( ( ord_less_real @ C @ B )
% 9.15/5.10         => ( ord_less_real @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1121_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: set_int,A: set_int,C: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ B @ A )
% 9.15/5.10       => ( ( ord_less_set_int @ C @ B )
% 9.15/5.10         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1122_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: rat,A: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ B @ A )
% 9.15/5.10       => ( ( ord_less_rat @ C @ B )
% 9.15/5.10         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1123_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: num,A: num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ B @ A )
% 9.15/5.10       => ( ( ord_less_num @ C @ B )
% 9.15/5.10         => ( ord_less_num @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1124_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: nat,A: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ B @ A )
% 9.15/5.10       => ( ( ord_less_nat @ C @ B )
% 9.15/5.10         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1125_dual__order_Ostrict__trans1,axiom,
% 9.15/5.10      ! [B: int,A: int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ B @ A )
% 9.15/5.10       => ( ( ord_less_int @ C @ B )
% 9.15/5.10         => ( ord_less_int @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1126_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: real,A: real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_real @ C @ B )
% 9.15/5.10         => ( ord_less_real @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1127_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: set_int,A: set_int,C: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ C @ B )
% 9.15/5.10         => ( ord_less_set_int @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1128_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: rat,A: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_rat @ C @ B )
% 9.15/5.10         => ( ord_less_rat @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1129_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: num,A: num,C: num] :
% 9.15/5.10        ( ( ord_less_num @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_num @ C @ B )
% 9.15/5.10         => ( ord_less_num @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1130_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: nat,A: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_nat @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_nat @ C @ B )
% 9.15/5.10         => ( ord_less_nat @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1131_dual__order_Ostrict__trans2,axiom,
% 9.15/5.10      ! [B: int,A: int,C: int] :
% 9.15/5.10        ( ( ord_less_int @ B @ A )
% 9.15/5.10       => ( ( ord_less_eq_int @ C @ B )
% 9.15/5.10         => ( ord_less_int @ C @ A ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1132_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [B3: real,A3: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_real @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1133_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [B3: set_int,A3: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_set_int @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1134_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [B3: rat,A3: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1135_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [B3: num,A3: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1136_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [B3: nat,A3: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1137_dual__order_Ostrict__iff__not,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [B3: int,A3: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ B3 @ A3 )
% 9.15/5.10            & ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1138_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: real,B: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1139_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ A @ B )
% 9.15/5.10       => ( ord_less_eq_set_int @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1140_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: rat,B: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1141_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: num,B: num] :
% 9.15/5.10        ( ( ord_less_num @ A @ B )
% 9.15/5.10       => ( ord_less_eq_num @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1142_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: nat,B: nat] :
% 9.15/5.10        ( ( ord_less_nat @ A @ B )
% 9.15/5.10       => ( ord_less_eq_nat @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1143_order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [A: int,B: int] :
% 9.15/5.10        ( ( ord_less_int @ A @ B )
% 9.15/5.10       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1144_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: real,A: real] :
% 9.15/5.10        ( ( ord_less_real @ B @ A )
% 9.15/5.10       => ( ord_less_eq_real @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1145_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: set_int,A: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ B @ A )
% 9.15/5.10       => ( ord_less_eq_set_int @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1146_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: rat,A: rat] :
% 9.15/5.10        ( ( ord_less_rat @ B @ A )
% 9.15/5.10       => ( ord_less_eq_rat @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1147_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: num,A: num] :
% 9.15/5.10        ( ( ord_less_num @ B @ A )
% 9.15/5.10       => ( ord_less_eq_num @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1148_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: nat,A: nat] :
% 9.15/5.10        ( ( ord_less_nat @ B @ A )
% 9.15/5.10       => ( ord_less_eq_nat @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1149_dual__order_Ostrict__implies__order,axiom,
% 9.15/5.10      ! [B: int,A: int] :
% 9.15/5.10        ( ( ord_less_int @ B @ A )
% 9.15/5.10       => ( ord_less_eq_int @ B @ A ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1150_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_real
% 9.15/5.10      = ( ^ [X: real,Y: real] :
% 9.15/5.10            ( ( ord_less_real @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1151_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [X: set_int,Y: set_int] :
% 9.15/5.10            ( ( ord_less_set_int @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1152_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_rat
% 9.15/5.10      = ( ^ [X: rat,Y: rat] :
% 9.15/5.10            ( ( ord_less_rat @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1153_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_num
% 9.15/5.10      = ( ^ [X: num,Y: num] :
% 9.15/5.10            ( ( ord_less_num @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1154_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_nat
% 9.15/5.10      = ( ^ [X: nat,Y: nat] :
% 9.15/5.10            ( ( ord_less_nat @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1155_order__le__less,axiom,
% 9.15/5.10      ( ord_less_eq_int
% 9.15/5.10      = ( ^ [X: int,Y: int] :
% 9.15/5.10            ( ( ord_less_int @ X @ Y )
% 9.15/5.10            | ( X = Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1156_order__less__le,axiom,
% 9.15/5.10      ( ord_less_real
% 9.15/5.10      = ( ^ [X: real,Y: real] :
% 9.15/5.10            ( ( ord_less_eq_real @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1157_order__less__le,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [X: set_int,Y: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1158_order__less__le,axiom,
% 9.15/5.10      ( ord_less_rat
% 9.15/5.10      = ( ^ [X: rat,Y: rat] :
% 9.15/5.10            ( ( ord_less_eq_rat @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1159_order__less__le,axiom,
% 9.15/5.10      ( ord_less_num
% 9.15/5.10      = ( ^ [X: num,Y: num] :
% 9.15/5.10            ( ( ord_less_eq_num @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1160_order__less__le,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [X: nat,Y: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1161_order__less__le,axiom,
% 9.15/5.10      ( ord_less_int
% 9.15/5.10      = ( ^ [X: int,Y: int] :
% 9.15/5.10            ( ( ord_less_eq_int @ X @ Y )
% 9.15/5.10            & ( X != Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1162_linorder__not__le,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1163_linorder__not__le,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1164_linorder__not__le,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1165_linorder__not__le,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1166_linorder__not__le,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1167_linorder__not__less,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_eq_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1168_linorder__not__less,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1169_linorder__not__less,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_eq_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1170_linorder__not__less,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1171_linorder__not__less,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
% 9.15/5.10        = ( ord_less_eq_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1172_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1173_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1174_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1175_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1176_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1177_order__less__imp__le,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1178_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: real,B: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_real @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1179_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1180_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: rat,B: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1181_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: num,B: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_num @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1182_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: nat,B: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1183_order__le__neq__trans,axiom,
% 9.15/5.10      ! [A: int,B: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.10       => ( ( A != B )
% 9.15/5.10         => ( ord_less_int @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1184_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: real,B: real] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_real @ A @ B )
% 9.15/5.10         => ( ord_less_real @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1185_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: set_int,B: set_int] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ A @ B )
% 9.15/5.10         => ( ord_less_set_int @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1186_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: rat,B: rat] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1187_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: num,B: num] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10         => ( ord_less_num @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1188_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: nat,B: nat] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.10         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1189_order__neq__le__trans,axiom,
% 9.15/5.10      ! [A: int,B: int] :
% 9.15/5.10        ( ( A != B )
% 9.15/5.10       => ( ( ord_less_eq_int @ A @ B )
% 9.15/5.10         => ( ord_less_int @ A @ B ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1190_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: real,Y3: real,Z2: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_real @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1191_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_set_int @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1192_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_rat @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1193_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: num,Y3: num,Z2: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_num @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1194_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_nat @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1195_order__le__less__trans,axiom,
% 9.15/5.10      ! [X3: int,Y3: int,Z2: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_int @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1196_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: real,Y3: real,Z2: real] :
% 9.15/5.10        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_real @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_real @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1197_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int,Z2: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1198_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.15/5.10        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_rat @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_rat @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1199_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: num,Y3: num,Z2: num] :
% 9.15/5.10        ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_num @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_num @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1200_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.10        ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_nat @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_nat @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1201_order__less__le__trans,axiom,
% 9.15/5.10      ! [X3: int,Y3: int,Z2: int] :
% 9.15/5.10        ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_eq_int @ Y3 @ Z2 )
% 9.15/5.10         => ( ord_less_int @ X3 @ Z2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1202_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: real > real,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1203_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1204_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: num > real,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1205_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: nat > real,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_nat @ B @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1206_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: real,F: int > real,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_int @ B @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1207_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: real > rat,B: real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_real @ B @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1208_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1209_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1210_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: nat > rat,B: nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_nat @ B @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1211_order__le__less__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: int > rat,B: int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_int @ B @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1212_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1213_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1214_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1215_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1216_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1217_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1218_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1219_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > num,C: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_num @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1220_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_nat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1221_order__le__less__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_int @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1222_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: real,F: rat > real,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1223_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: rat > rat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1224_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: num,F: rat > num,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1225_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: nat,F: rat > nat,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1226_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: int,F: rat > int,B: rat,C: rat] :
% 9.15/5.10        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_rat @ B @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_eq_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1227_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: real,F: num > real,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_real @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1228_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: rat,F: num > rat,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_rat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1229_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: num,F: num > num,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_num @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1230_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: nat,F: num > nat,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_nat @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1231_order__less__le__subst1,axiom,
% 9.15/5.10      ! [A: int,F: num > int,B: num,C: num] :
% 9.15/5.10        ( ( ord_less_int @ A @ ( F @ B ) )
% 9.15/5.10       => ( ( ord_less_eq_num @ B @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_eq_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1232_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > real,C: real] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1233_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > real,C: real] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1234_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > real,C: real] :
% 9.15/5.10        ( ( ord_less_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1235_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: nat,B: nat,F: nat > real,C: real] :
% 9.15/5.10        ( ( ord_less_nat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1236_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: int,B: int,F: int > real,C: real] :
% 9.15/5.10        ( ( ord_less_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_real @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1237_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: real,B: real,F: real > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_real @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: real,Y5: real] :
% 9.15/5.10                ( ( ord_less_real @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1238_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: rat,B: rat,F: rat > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_rat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: rat,Y5: rat] :
% 9.15/5.10                ( ( ord_less_rat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1239_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: num,B: num,F: num > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_num @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: num,Y5: num] :
% 9.15/5.10                ( ( ord_less_num @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1240_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: nat,B: nat,F: nat > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_nat @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: nat,Y5: nat] :
% 9.15/5.10                ( ( ord_less_nat @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1241_order__less__le__subst2,axiom,
% 9.15/5.10      ! [A: int,B: int,F: int > rat,C: rat] :
% 9.15/5.10        ( ( ord_less_int @ A @ B )
% 9.15/5.10       => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
% 9.15/5.10         => ( ! [X4: int,Y5: int] :
% 9.15/5.10                ( ( ord_less_int @ X4 @ Y5 )
% 9.15/5.10               => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
% 9.15/5.10           => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1242_linorder__le__less__linear,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.10        | ( ord_less_real @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1243_linorder__le__less__linear,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.10        | ( ord_less_rat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1244_linorder__le__less__linear,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.10        | ( ord_less_num @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1245_linorder__le__less__linear,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.10        | ( ord_less_nat @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1246_linorder__le__less__linear,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.10        | ( ord_less_int @ Y3 @ X3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1247_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1248_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: set_int,Y3: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_set_int @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1249_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1250_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: num,Y3: num] :
% 9.15/5.10        ( ( ord_less_eq_num @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_num @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1251_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: nat,Y3: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_nat @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1252_order__le__imp__less__or__eq,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.15/5.10       => ( ( ord_less_int @ X3 @ Y3 )
% 9.15/5.10          | ( X3 = Y3 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1253_gr__zeroI,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ( ( N2 != zero_zero_nat )
% 9.15/5.10       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1254_not__less__zero,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1255_gr__implies__not__zero,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.10       => ( N2 != zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1256_zero__less__iff__neq__zero,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.10        = ( N2 != zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1257_bot__nat__0_Oextremum__strict,axiom,
% 9.15/5.10      ! [A: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1258_gr0I,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ( ( N2 != zero_zero_nat )
% 9.15/5.10       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1259_not__gr0,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
% 9.15/5.10        = ( N2 = zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1260_not__less0,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1261_less__zeroE,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1262_gr__implies__not0,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.10       => ( N2 != zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1263_infinite__descent0,axiom,
% 9.15/5.10      ! [P: nat > $o,N2: nat] :
% 9.15/5.10        ( ( P @ zero_zero_nat )
% 9.15/5.10       => ( ! [N3: nat] :
% 9.15/5.10              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.15/5.10             => ( ~ ( P @ N3 )
% 9.15/5.10               => ? [M6: nat] :
% 9.15/5.10                    ( ( ord_less_nat @ M6 @ N3 )
% 9.15/5.10                    & ~ ( P @ M6 ) ) ) )
% 9.15/5.10         => ( P @ N2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1264_less__eq__nat_Osimps_I1_J,axiom,
% 9.15/5.10      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1265_bot__nat__0_Oextremum__unique,axiom,
% 9.15/5.10      ! [A: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.15/5.10        = ( A = zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1266_bot__nat__0_Oextremum__uniqueI,axiom,
% 9.15/5.10      ! [A: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.15/5.10       => ( A = zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1267_le__0__eq,axiom,
% 9.15/5.10      ! [N2: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
% 9.15/5.10        = ( N2 = zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1268_nat__less__le,axiom,
% 9.15/5.10      ( ord_less_nat
% 9.15/5.10      = ( ^ [M2: nat,N: nat] :
% 9.15/5.10            ( ( ord_less_eq_nat @ M2 @ N )
% 9.15/5.10            & ( M2 != N ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1269_less__imp__le__nat,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_nat @ M @ N2 )
% 9.15/5.10       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1270_le__eq__less__or__eq,axiom,
% 9.15/5.10      ( ord_less_eq_nat
% 9.15/5.10      = ( ^ [M2: nat,N: nat] :
% 9.15/5.10            ( ( ord_less_nat @ M2 @ N )
% 9.15/5.10            | ( M2 = N ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1271_less__or__eq__imp__le,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ( ord_less_nat @ M @ N2 )
% 9.15/5.10          | ( M = N2 ) )
% 9.15/5.10       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1272_le__neq__implies__less,axiom,
% 9.15/5.10      ! [M: nat,N2: nat] :
% 9.15/5.10        ( ( ord_less_eq_nat @ M @ N2 )
% 9.15/5.10       => ( ( M != N2 )
% 9.15/5.10         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1273_less__mono__imp__le__mono,axiom,
% 9.15/5.10      ! [F: nat > nat,I: nat,J: nat] :
% 9.15/5.10        ( ! [I3: nat,J2: nat] :
% 9.15/5.10            ( ( ord_less_nat @ I3 @ J2 )
% 9.15/5.10           => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
% 9.15/5.10       => ( ( ord_less_eq_nat @ I @ J )
% 9.15/5.10         => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1274_double__eq__0__iff,axiom,
% 9.15/5.10      ! [A: real] :
% 9.15/5.10        ( ( ( plus_plus_real @ A @ A )
% 9.15/5.10          = zero_zero_real )
% 9.15/5.10        = ( A = zero_zero_real ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1275_double__eq__0__iff,axiom,
% 9.15/5.10      ! [A: rat] :
% 9.15/5.10        ( ( ( plus_plus_rat @ A @ A )
% 9.15/5.10          = zero_zero_rat )
% 9.15/5.10        = ( A = zero_zero_rat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1276_double__eq__0__iff,axiom,
% 9.15/5.10      ! [A: int] :
% 9.15/5.10        ( ( ( plus_plus_int @ A @ A )
% 9.15/5.10          = zero_zero_int )
% 9.15/5.10        = ( A = zero_zero_int ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1277_assms_I1_J,axiom,
% 9.15/5.10      vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1278_mint__corr__help__empty,axiom,
% 9.15/5.10      ! [T: vEBT_VEBT,N2: nat] :
% 9.15/5.10        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.10       => ( ( ( vEBT_vebt_mint @ T )
% 9.15/5.10            = none_nat )
% 9.15/5.10         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.15/5.10            = bot_bot_set_nat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1279_maxt__corr__help__empty,axiom,
% 9.15/5.10      ! [T: vEBT_VEBT,N2: nat] :
% 9.15/5.10        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.10       => ( ( ( vEBT_vebt_maxt @ T )
% 9.15/5.10            = none_nat )
% 9.15/5.10         => ( ( vEBT_VEBT_set_vebt @ T )
% 9.15/5.10            = bot_bot_set_nat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1280_field__le__epsilon,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ! [E: real] :
% 9.15/5.10            ( ( ord_less_real @ zero_zero_real @ E )
% 9.15/5.10           => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E ) ) )
% 9.15/5.10       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1281_field__le__epsilon,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ! [E: rat] :
% 9.15/5.10            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.15/5.10           => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E ) ) )
% 9.15/5.10       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1282_option_Osize__gen_I2_J,axiom,
% 9.15/5.10      ! [X3: nat > nat,X22: nat] :
% 9.15/5.10        ( ( size_option_nat @ X3 @ ( some_nat @ X22 ) )
% 9.15/5.10        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1283_option_Osize__gen_I2_J,axiom,
% 9.15/5.10      ! [X3: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
% 9.15/5.10        ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.15/5.10        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1284_option_Osize__gen_I2_J,axiom,
% 9.15/5.10      ! [X3: num > nat,X22: num] :
% 9.15/5.10        ( ( size_option_num @ X3 @ ( some_num @ X22 ) )
% 9.15/5.10        = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1285_add__less__zeroD,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 9.15/5.10       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.15/5.10          | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1286_add__less__zeroD,axiom,
% 9.15/5.10      ! [X3: rat,Y3: rat] :
% 9.15/5.10        ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
% 9.15/5.10       => ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.15/5.10          | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1287_add__less__zeroD,axiom,
% 9.15/5.10      ! [X3: int,Y3: int] :
% 9.15/5.10        ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
% 9.15/5.10       => ( ( ord_less_int @ X3 @ zero_zero_int )
% 9.15/5.10          | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1288_empty__subsetI,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1289_empty__subsetI,axiom,
% 9.15/5.10      ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1290_empty__subsetI,axiom,
% 9.15/5.10      ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1291_empty__subsetI,axiom,
% 9.15/5.10      ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1292_subset__empty,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.10        ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 9.15/5.10        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1293_subset__empty,axiom,
% 9.15/5.10      ! [A2: set_real] :
% 9.15/5.10        ( ( ord_less_eq_set_real @ A2 @ bot_bot_set_real )
% 9.15/5.10        = ( A2 = bot_bot_set_real ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1294_subset__empty,axiom,
% 9.15/5.10      ! [A2: set_nat] :
% 9.15/5.10        ( ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat )
% 9.15/5.10        = ( A2 = bot_bot_set_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1295_subset__empty,axiom,
% 9.15/5.10      ! [A2: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ bot_bot_set_int )
% 9.15/5.10        = ( A2 = bot_bot_set_int ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1296_empty__iff,axiom,
% 9.15/5.10      ! [C: set_nat] :
% 9.15/5.10        ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1297_empty__iff,axiom,
% 9.15/5.10      ! [C: product_prod_nat_nat] :
% 9.15/5.10        ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1298_empty__iff,axiom,
% 9.15/5.10      ! [C: real] :
% 9.15/5.10        ~ ( member_real @ C @ bot_bot_set_real ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1299_empty__iff,axiom,
% 9.15/5.10      ! [C: nat] :
% 9.15/5.10        ~ ( member_nat @ C @ bot_bot_set_nat ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1300_empty__iff,axiom,
% 9.15/5.10      ! [C: int] :
% 9.15/5.10        ~ ( member_int @ C @ bot_bot_set_int ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1301_all__not__in__conv,axiom,
% 9.15/5.10      ! [A2: set_set_nat] :
% 9.15/5.10        ( ( ! [X: set_nat] :
% 9.15/5.10              ~ ( member_set_nat @ X @ A2 ) )
% 9.15/5.10        = ( A2 = bot_bot_set_set_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1302_all__not__in__conv,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.10        ( ( ! [X: product_prod_nat_nat] :
% 9.15/5.10              ~ ( member8440522571783428010at_nat @ X @ A2 ) )
% 9.15/5.10        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1303_all__not__in__conv,axiom,
% 9.15/5.10      ! [A2: set_real] :
% 9.15/5.10        ( ( ! [X: real] :
% 9.15/5.10              ~ ( member_real @ X @ A2 ) )
% 9.15/5.10        = ( A2 = bot_bot_set_real ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1304_all__not__in__conv,axiom,
% 9.15/5.10      ! [A2: set_nat] :
% 9.15/5.10        ( ( ! [X: nat] :
% 9.15/5.10              ~ ( member_nat @ X @ A2 ) )
% 9.15/5.10        = ( A2 = bot_bot_set_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1305_all__not__in__conv,axiom,
% 9.15/5.10      ! [A2: set_int] :
% 9.15/5.10        ( ( ! [X: int] :
% 9.15/5.10              ~ ( member_int @ X @ A2 ) )
% 9.15/5.10        = ( A2 = bot_bot_set_int ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1306_deg__deg__n,axiom,
% 9.15/5.10      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 9.15/5.10        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 9.15/5.10       => ( Deg = N2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1307_deg__SUcn__Node,axiom,
% 9.15/5.10      ! [Tree: vEBT_VEBT,N2: nat] :
% 9.15/5.10        ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
% 9.15/5.10       => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 9.15/5.10            ( Tree
% 9.15/5.10            = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1308_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: list_nat > $o] :
% 9.15/5.10        ( ( bot_bot_set_list_nat
% 9.15/5.10          = ( collect_list_nat @ P ) )
% 9.15/5.10        = ( ! [X: list_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1309_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: set_nat > $o] :
% 9.15/5.10        ( ( bot_bot_set_set_nat
% 9.15/5.10          = ( collect_set_nat @ P ) )
% 9.15/5.10        = ( ! [X: set_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1310_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: product_prod_nat_nat > $o] :
% 9.15/5.10        ( ( bot_bo2099793752762293965at_nat
% 9.15/5.10          = ( collec3392354462482085612at_nat @ P ) )
% 9.15/5.10        = ( ! [X: product_prod_nat_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1311_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: real > $o] :
% 9.15/5.10        ( ( bot_bot_set_real
% 9.15/5.10          = ( collect_real @ P ) )
% 9.15/5.10        = ( ! [X: real] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1312_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: nat > $o] :
% 9.15/5.10        ( ( bot_bot_set_nat
% 9.15/5.10          = ( collect_nat @ P ) )
% 9.15/5.10        = ( ! [X: nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1313_empty__Collect__eq,axiom,
% 9.15/5.10      ! [P: int > $o] :
% 9.15/5.10        ( ( bot_bot_set_int
% 9.15/5.10          = ( collect_int @ P ) )
% 9.15/5.10        = ( ! [X: int] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1314_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: list_nat > $o] :
% 9.15/5.10        ( ( ( collect_list_nat @ P )
% 9.15/5.10          = bot_bot_set_list_nat )
% 9.15/5.10        = ( ! [X: list_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1315_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: set_nat > $o] :
% 9.15/5.10        ( ( ( collect_set_nat @ P )
% 9.15/5.10          = bot_bot_set_set_nat )
% 9.15/5.10        = ( ! [X: set_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1316_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: product_prod_nat_nat > $o] :
% 9.15/5.10        ( ( ( collec3392354462482085612at_nat @ P )
% 9.15/5.10          = bot_bo2099793752762293965at_nat )
% 9.15/5.10        = ( ! [X: product_prod_nat_nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1317_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: real > $o] :
% 9.15/5.10        ( ( ( collect_real @ P )
% 9.15/5.10          = bot_bot_set_real )
% 9.15/5.10        = ( ! [X: real] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1318_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: nat > $o] :
% 9.15/5.10        ( ( ( collect_nat @ P )
% 9.15/5.10          = bot_bot_set_nat )
% 9.15/5.10        = ( ! [X: nat] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1319_Collect__empty__eq,axiom,
% 9.15/5.10      ! [P: int > $o] :
% 9.15/5.10        ( ( ( collect_int @ P )
% 9.15/5.10          = bot_bot_set_int )
% 9.15/5.10        = ( ! [X: int] :
% 9.15/5.10              ~ ( P @ X ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1320_psubsetI,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( A2 != B2 )
% 9.15/5.10         => ( ord_less_set_int @ A2 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1321_subset__antisym,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.15/5.10         => ( A2 = B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1322_subsetI,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.15/5.10        ( ! [X4: product_prod_nat_nat] :
% 9.15/5.10            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.15/5.10           => ( member8440522571783428010at_nat @ X4 @ B2 ) )
% 9.15/5.10       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1323_subsetI,axiom,
% 9.15/5.10      ! [A2: set_real,B2: set_real] :
% 9.15/5.10        ( ! [X4: real] :
% 9.15/5.10            ( ( member_real @ X4 @ A2 )
% 9.15/5.10           => ( member_real @ X4 @ B2 ) )
% 9.15/5.10       => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1324_subsetI,axiom,
% 9.15/5.10      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.15/5.10        ( ! [X4: set_nat] :
% 9.15/5.10            ( ( member_set_nat @ X4 @ A2 )
% 9.15/5.10           => ( member_set_nat @ X4 @ B2 ) )
% 9.15/5.10       => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1325_subsetI,axiom,
% 9.15/5.10      ! [A2: set_nat,B2: set_nat] :
% 9.15/5.10        ( ! [X4: nat] :
% 9.15/5.10            ( ( member_nat @ X4 @ A2 )
% 9.15/5.10           => ( member_nat @ X4 @ B2 ) )
% 9.15/5.10       => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1326_subsetI,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ! [X4: int] :
% 9.15/5.10            ( ( member_int @ X4 @ A2 )
% 9.15/5.10           => ( member_int @ X4 @ B2 ) )
% 9.15/5.10       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1327_mi__eq__ma__no__ch,axiom,
% 9.15/5.10      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT] :
% 9.15/5.10        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg )
% 9.15/5.10       => ( ( Mi = Ma )
% 9.15/5.10         => ( ! [X2: vEBT_VEBT] :
% 9.15/5.10                ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.10               => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
% 9.15/5.10            & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1328_not__Some__eq,axiom,
% 9.15/5.10      ! [X3: option_nat] :
% 9.15/5.10        ( ( ! [Y: nat] :
% 9.15/5.10              ( X3
% 9.15/5.10             != ( some_nat @ Y ) ) )
% 9.15/5.10        = ( X3 = none_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1329_not__Some__eq,axiom,
% 9.15/5.10      ! [X3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( ! [Y: product_prod_nat_nat] :
% 9.15/5.10              ( X3
% 9.15/5.10             != ( some_P7363390416028606310at_nat @ Y ) ) )
% 9.15/5.10        = ( X3 = none_P5556105721700978146at_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1330_not__Some__eq,axiom,
% 9.15/5.10      ! [X3: option_num] :
% 9.15/5.10        ( ( ! [Y: num] :
% 9.15/5.10              ( X3
% 9.15/5.10             != ( some_num @ Y ) ) )
% 9.15/5.10        = ( X3 = none_num ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1331_not__None__eq,axiom,
% 9.15/5.10      ! [X3: option_nat] :
% 9.15/5.10        ( ( X3 != none_nat )
% 9.15/5.10        = ( ? [Y: nat] :
% 9.15/5.10              ( X3
% 9.15/5.10              = ( some_nat @ Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1332_not__None__eq,axiom,
% 9.15/5.10      ! [X3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( X3 != none_P5556105721700978146at_nat )
% 9.15/5.10        = ( ? [Y: product_prod_nat_nat] :
% 9.15/5.10              ( X3
% 9.15/5.10              = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1333_not__None__eq,axiom,
% 9.15/5.10      ! [X3: option_num] :
% 9.15/5.10        ( ( X3 != none_num )
% 9.15/5.10        = ( ? [Y: num] :
% 9.15/5.10              ( X3
% 9.15/5.10              = ( some_num @ Y ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1334_option_Ocollapse,axiom,
% 9.15/5.10      ! [Option: option_nat] :
% 9.15/5.10        ( ( Option != none_nat )
% 9.15/5.10       => ( ( some_nat @ ( the_nat @ Option ) )
% 9.15/5.10          = Option ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1335_option_Ocollapse,axiom,
% 9.15/5.10      ! [Option: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( Option != none_P5556105721700978146at_nat )
% 9.15/5.10       => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
% 9.15/5.10          = Option ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1336_option_Ocollapse,axiom,
% 9.15/5.10      ! [Option: option_num] :
% 9.15/5.10        ( ( Option != none_num )
% 9.15/5.10       => ( ( some_num @ ( the_num @ Option ) )
% 9.15/5.10          = Option ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1337_add__def,axiom,
% 9.15/5.10      ( vEBT_VEBT_add
% 9.15/5.10      = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1338_psubsetE,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ A2 @ B2 )
% 9.15/5.10       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10           => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1339_psubset__eq,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 9.15/5.10            & ( A5 != B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1340_psubset__imp__subset,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1341_psubset__subset__trans,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int,C4: set_int] :
% 9.15/5.10        ( ( ord_less_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 9.15/5.10         => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1342_subset__not__subset__eq,axiom,
% 9.15/5.10      ( ord_less_set_int
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 9.15/5.10            & ~ ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1343_subset__psubset__trans,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int,C4: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( ord_less_set_int @ B2 @ C4 )
% 9.15/5.10         => ( ord_less_set_int @ A2 @ C4 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1344_subset__iff__psubset__eq,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10            ( ( ord_less_set_int @ A5 @ B5 )
% 9.15/5.10            | ( A5 = B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1345_Collect__mono__iff,axiom,
% 9.15/5.10      ! [P: real > $o,Q: real > $o] :
% 9.15/5.10        ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
% 9.15/5.10        = ( ! [X: real] :
% 9.15/5.10              ( ( P @ X )
% 9.15/5.10             => ( Q @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1346_Collect__mono__iff,axiom,
% 9.15/5.10      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.15/5.10        ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
% 9.15/5.10        = ( ! [X: list_nat] :
% 9.15/5.10              ( ( P @ X )
% 9.15/5.10             => ( Q @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1347_Collect__mono__iff,axiom,
% 9.15/5.10      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.15/5.10        ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
% 9.15/5.10        = ( ! [X: set_nat] :
% 9.15/5.10              ( ( P @ X )
% 9.15/5.10             => ( Q @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1348_Collect__mono__iff,axiom,
% 9.15/5.10      ! [P: nat > $o,Q: nat > $o] :
% 9.15/5.10        ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
% 9.15/5.10        = ( ! [X: nat] :
% 9.15/5.10              ( ( P @ X )
% 9.15/5.10             => ( Q @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1349_Collect__mono__iff,axiom,
% 9.15/5.10      ! [P: int > $o,Q: int > $o] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
% 9.15/5.10        = ( ! [X: int] :
% 9.15/5.10              ( ( P @ X )
% 9.15/5.10             => ( Q @ X ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1350_set__eq__subset,axiom,
% 9.15/5.10      ( ( ^ [Y4: set_int,Z: set_int] : ( Y4 = Z ) )
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10            ( ( ord_less_eq_set_int @ A5 @ B5 )
% 9.15/5.10            & ( ord_less_eq_set_int @ B5 @ A5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1351_subset__trans,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int,C4: set_int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 9.15/5.10         => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1352_Collect__mono,axiom,
% 9.15/5.10      ! [P: real > $o,Q: real > $o] :
% 9.15/5.10        ( ! [X4: real] :
% 9.15/5.10            ( ( P @ X4 )
% 9.15/5.10           => ( Q @ X4 ) )
% 9.15/5.10       => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1353_Collect__mono,axiom,
% 9.15/5.10      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.15/5.10        ( ! [X4: list_nat] :
% 9.15/5.10            ( ( P @ X4 )
% 9.15/5.10           => ( Q @ X4 ) )
% 9.15/5.10       => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1354_Collect__mono,axiom,
% 9.15/5.10      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.15/5.10        ( ! [X4: set_nat] :
% 9.15/5.10            ( ( P @ X4 )
% 9.15/5.10           => ( Q @ X4 ) )
% 9.15/5.10       => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1355_Collect__mono,axiom,
% 9.15/5.10      ! [P: nat > $o,Q: nat > $o] :
% 9.15/5.10        ( ! [X4: nat] :
% 9.15/5.10            ( ( P @ X4 )
% 9.15/5.10           => ( Q @ X4 ) )
% 9.15/5.10       => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1356_Collect__mono,axiom,
% 9.15/5.10      ! [P: int > $o,Q: int > $o] :
% 9.15/5.10        ( ! [X4: int] :
% 9.15/5.10            ( ( P @ X4 )
% 9.15/5.10           => ( Q @ X4 ) )
% 9.15/5.10       => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1357_subset__refl,axiom,
% 9.15/5.10      ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1358_subset__iff,axiom,
% 9.15/5.10      ( ord_le3146513528884898305at_nat
% 9.15/5.10      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.15/5.10          ! [T2: product_prod_nat_nat] :
% 9.15/5.10            ( ( member8440522571783428010at_nat @ T2 @ A5 )
% 9.15/5.10           => ( member8440522571783428010at_nat @ T2 @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1359_subset__iff,axiom,
% 9.15/5.10      ( ord_less_eq_set_real
% 9.15/5.10      = ( ^ [A5: set_real,B5: set_real] :
% 9.15/5.10          ! [T2: real] :
% 9.15/5.10            ( ( member_real @ T2 @ A5 )
% 9.15/5.10           => ( member_real @ T2 @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1360_subset__iff,axiom,
% 9.15/5.10      ( ord_le6893508408891458716et_nat
% 9.15/5.10      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.15/5.10          ! [T2: set_nat] :
% 9.15/5.10            ( ( member_set_nat @ T2 @ A5 )
% 9.15/5.10           => ( member_set_nat @ T2 @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1361_subset__iff,axiom,
% 9.15/5.10      ( ord_less_eq_set_nat
% 9.15/5.10      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.15/5.10          ! [T2: nat] :
% 9.15/5.10            ( ( member_nat @ T2 @ A5 )
% 9.15/5.10           => ( member_nat @ T2 @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1362_subset__iff,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10          ! [T2: int] :
% 9.15/5.10            ( ( member_int @ T2 @ A5 )
% 9.15/5.10           => ( member_int @ T2 @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1363_equalityD2,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( A2 = B2 )
% 9.15/5.10       => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1364_equalityD1,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( A2 = B2 )
% 9.15/5.10       => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1365_subset__eq,axiom,
% 9.15/5.10      ( ord_le3146513528884898305at_nat
% 9.15/5.10      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.15/5.10          ! [X: product_prod_nat_nat] :
% 9.15/5.10            ( ( member8440522571783428010at_nat @ X @ A5 )
% 9.15/5.10           => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1366_subset__eq,axiom,
% 9.15/5.10      ( ord_less_eq_set_real
% 9.15/5.10      = ( ^ [A5: set_real,B5: set_real] :
% 9.15/5.10          ! [X: real] :
% 9.15/5.10            ( ( member_real @ X @ A5 )
% 9.15/5.10           => ( member_real @ X @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1367_subset__eq,axiom,
% 9.15/5.10      ( ord_le6893508408891458716et_nat
% 9.15/5.10      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.15/5.10          ! [X: set_nat] :
% 9.15/5.10            ( ( member_set_nat @ X @ A5 )
% 9.15/5.10           => ( member_set_nat @ X @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1368_subset__eq,axiom,
% 9.15/5.10      ( ord_less_eq_set_nat
% 9.15/5.10      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.15/5.10          ! [X: nat] :
% 9.15/5.10            ( ( member_nat @ X @ A5 )
% 9.15/5.10           => ( member_nat @ X @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1369_subset__eq,axiom,
% 9.15/5.10      ( ord_less_eq_set_int
% 9.15/5.10      = ( ^ [A5: set_int,B5: set_int] :
% 9.15/5.10          ! [X: int] :
% 9.15/5.10            ( ( member_int @ X @ A5 )
% 9.15/5.10           => ( member_int @ X @ B5 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1370_equalityE,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int] :
% 9.15/5.10        ( ( A2 = B2 )
% 9.15/5.10       => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10           => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1371_subsetD,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 9.15/5.10        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.15/5.10         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1372_subsetD,axiom,
% 9.15/5.10      ! [A2: set_real,B2: set_real,C: real] :
% 9.15/5.10        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.15/5.10       => ( ( member_real @ C @ A2 )
% 9.15/5.10         => ( member_real @ C @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1373_subsetD,axiom,
% 9.15/5.10      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.15/5.10        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member_set_nat @ C @ A2 )
% 9.15/5.10         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1374_subsetD,axiom,
% 9.15/5.10      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.15/5.10        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member_nat @ C @ A2 )
% 9.15/5.10         => ( member_nat @ C @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1375_subsetD,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int,C: int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( member_int @ C @ A2 )
% 9.15/5.10         => ( member_int @ C @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1376_in__mono,axiom,
% 9.15/5.10      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
% 9.15/5.10        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.15/5.10         => ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1377_in__mono,axiom,
% 9.15/5.10      ! [A2: set_real,B2: set_real,X3: real] :
% 9.15/5.10        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.15/5.10       => ( ( member_real @ X3 @ A2 )
% 9.15/5.10         => ( member_real @ X3 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1378_in__mono,axiom,
% 9.15/5.10      ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat] :
% 9.15/5.10        ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member_set_nat @ X3 @ A2 )
% 9.15/5.10         => ( member_set_nat @ X3 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1379_in__mono,axiom,
% 9.15/5.10      ! [A2: set_nat,B2: set_nat,X3: nat] :
% 9.15/5.10        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.15/5.10       => ( ( member_nat @ X3 @ A2 )
% 9.15/5.10         => ( member_nat @ X3 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1380_in__mono,axiom,
% 9.15/5.10      ! [A2: set_int,B2: set_int,X3: int] :
% 9.15/5.10        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.10       => ( ( member_int @ X3 @ A2 )
% 9.15/5.10         => ( member_int @ X3 @ B2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1381_bot__set__def,axiom,
% 9.15/5.10      ( bot_bot_set_list_nat
% 9.15/5.10      = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1382_bot__set__def,axiom,
% 9.15/5.10      ( bot_bot_set_set_nat
% 9.15/5.10      = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1383_bot__set__def,axiom,
% 9.15/5.10      ( bot_bo2099793752762293965at_nat
% 9.15/5.10      = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1384_bot__set__def,axiom,
% 9.15/5.10      ( bot_bot_set_real
% 9.15/5.10      = ( collect_real @ bot_bot_real_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1385_bot__set__def,axiom,
% 9.15/5.10      ( bot_bot_set_nat
% 9.15/5.10      = ( collect_nat @ bot_bot_nat_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1386_bot__set__def,axiom,
% 9.15/5.10      ( bot_bot_set_int
% 9.15/5.10      = ( collect_int @ bot_bot_int_o ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1387_bot__nat__def,axiom,
% 9.15/5.10      bot_bot_nat = zero_zero_nat ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1388_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
% 9.15/5.10        ( ( ( X3 = none_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: nat,B4: nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1389_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( ( X3 = none_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: nat,B4: product_prod_nat_nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1390_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
% 9.15/5.10        ( ( ( X3 = none_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_num )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: nat,B4: num] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_num @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1391_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
% 9.15/5.10        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: product_prod_nat_nat,B4: nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1392_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1393_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
% 9.15/5.10        ( ( ( X3 = none_P5556105721700978146at_nat )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_num )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: product_prod_nat_nat,B4: num] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_P7363390416028606310at_nat @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_num @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1394_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
% 9.15/5.10        ( ( ( X3 = none_num )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: num,B4: nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_num @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1395_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( ( X3 = none_num )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_P5556105721700978146at_nat )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: num,B4: product_prod_nat_nat] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_num @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_P7363390416028606310at_nat @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1396_combine__options__cases,axiom,
% 9.15/5.10      ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
% 9.15/5.10        ( ( ( X3 = none_num )
% 9.15/5.10         => ( P @ X3 @ Y3 ) )
% 9.15/5.10       => ( ( ( Y3 = none_num )
% 9.15/5.10           => ( P @ X3 @ Y3 ) )
% 9.15/5.10         => ( ! [A4: num,B4: num] :
% 9.15/5.10                ( ( X3
% 9.15/5.10                  = ( some_num @ A4 ) )
% 9.15/5.10               => ( ( Y3
% 9.15/5.10                    = ( some_num @ B4 ) )
% 9.15/5.10                 => ( P @ X3 @ Y3 ) ) )
% 9.15/5.10           => ( P @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1397_split__option__all,axiom,
% 9.15/5.10      ( ( ^ [P2: option_nat > $o] :
% 9.15/5.10          ! [X6: option_nat] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option_nat > $o] :
% 9.15/5.10            ( ( P3 @ none_nat )
% 9.15/5.10            & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1398_split__option__all,axiom,
% 9.15/5.10      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.15/5.10          ! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.15/5.10            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.15/5.10            & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1399_split__option__all,axiom,
% 9.15/5.10      ( ( ^ [P2: option_num > $o] :
% 9.15/5.10          ! [X6: option_num] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option_num > $o] :
% 9.15/5.10            ( ( P3 @ none_num )
% 9.15/5.10            & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1400_split__option__ex,axiom,
% 9.15/5.10      ( ( ^ [P2: option_nat > $o] :
% 9.15/5.10          ? [X6: option_nat] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option_nat > $o] :
% 9.15/5.10            ( ( P3 @ none_nat )
% 9.15/5.10            | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1401_split__option__ex,axiom,
% 9.15/5.10      ( ( ^ [P2: option4927543243414619207at_nat > $o] :
% 9.15/5.10          ? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option4927543243414619207at_nat > $o] :
% 9.15/5.10            ( ( P3 @ none_P5556105721700978146at_nat )
% 9.15/5.10            | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1402_split__option__ex,axiom,
% 9.15/5.10      ( ( ^ [P2: option_num > $o] :
% 9.15/5.10          ? [X6: option_num] : ( P2 @ X6 ) )
% 9.15/5.10      = ( ^ [P3: option_num > $o] :
% 9.15/5.10            ( ( P3 @ none_num )
% 9.15/5.10            | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1403_option_Oexhaust,axiom,
% 9.15/5.10      ! [Y3: option_nat] :
% 9.15/5.10        ( ( Y3 != none_nat )
% 9.15/5.10       => ~ ! [X23: nat] :
% 9.15/5.10              ( Y3
% 9.15/5.10             != ( some_nat @ X23 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1404_option_Oexhaust,axiom,
% 9.15/5.10      ! [Y3: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( Y3 != none_P5556105721700978146at_nat )
% 9.15/5.10       => ~ ! [X23: product_prod_nat_nat] :
% 9.15/5.10              ( Y3
% 9.15/5.10             != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1405_option_Oexhaust,axiom,
% 9.15/5.10      ! [Y3: option_num] :
% 9.15/5.10        ( ( Y3 != none_num )
% 9.15/5.10       => ~ ! [X23: num] :
% 9.15/5.10              ( Y3
% 9.15/5.10             != ( some_num @ X23 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1406_option_OdiscI,axiom,
% 9.15/5.10      ! [Option: option_nat,X22: nat] :
% 9.15/5.10        ( ( Option
% 9.15/5.10          = ( some_nat @ X22 ) )
% 9.15/5.10       => ( Option != none_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1407_option_OdiscI,axiom,
% 9.15/5.10      ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
% 9.15/5.10        ( ( Option
% 9.15/5.10          = ( some_P7363390416028606310at_nat @ X22 ) )
% 9.15/5.10       => ( Option != none_P5556105721700978146at_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1408_option_OdiscI,axiom,
% 9.15/5.10      ! [Option: option_num,X22: num] :
% 9.15/5.10        ( ( Option
% 9.15/5.10          = ( some_num @ X22 ) )
% 9.15/5.10       => ( Option != none_num ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1409_option_Odistinct_I1_J,axiom,
% 9.15/5.10      ! [X22: nat] :
% 9.15/5.10        ( none_nat
% 9.15/5.10       != ( some_nat @ X22 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1410_option_Odistinct_I1_J,axiom,
% 9.15/5.10      ! [X22: product_prod_nat_nat] :
% 9.15/5.10        ( none_P5556105721700978146at_nat
% 9.15/5.10       != ( some_P7363390416028606310at_nat @ X22 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1411_option_Odistinct_I1_J,axiom,
% 9.15/5.10      ! [X22: num] :
% 9.15/5.10        ( none_num
% 9.15/5.10       != ( some_num @ X22 ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1412_vebt__maxt_Osimps_I3_J,axiom,
% 9.15/5.10      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.15/5.10        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.15/5.10        = ( some_nat @ Ma ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1413_vebt__mint_Osimps_I3_J,axiom,
% 9.15/5.10      ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
% 9.15/5.10        ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
% 9.15/5.10        = ( some_nat @ Mi ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1414_option_Oexpand,axiom,
% 9.15/5.10      ! [Option: option_nat,Option2: option_nat] :
% 9.15/5.10        ( ( ( Option = none_nat )
% 9.15/5.10          = ( Option2 = none_nat ) )
% 9.15/5.10       => ( ( ( Option != none_nat )
% 9.15/5.10           => ( ( Option2 != none_nat )
% 9.15/5.10             => ( ( the_nat @ Option )
% 9.15/5.10                = ( the_nat @ Option2 ) ) ) )
% 9.15/5.10         => ( Option = Option2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1415_option_Oexpand,axiom,
% 9.15/5.10      ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( ( Option = none_P5556105721700978146at_nat )
% 9.15/5.10          = ( Option2 = none_P5556105721700978146at_nat ) )
% 9.15/5.10       => ( ( ( Option != none_P5556105721700978146at_nat )
% 9.15/5.10           => ( ( Option2 != none_P5556105721700978146at_nat )
% 9.15/5.10             => ( ( the_Pr8591224930841456533at_nat @ Option )
% 9.15/5.10                = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
% 9.15/5.10         => ( Option = Option2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1416_option_Oexpand,axiom,
% 9.15/5.10      ! [Option: option_num,Option2: option_num] :
% 9.15/5.10        ( ( ( Option = none_num )
% 9.15/5.10          = ( Option2 = none_num ) )
% 9.15/5.10       => ( ( ( Option != none_num )
% 9.15/5.10           => ( ( Option2 != none_num )
% 9.15/5.10             => ( ( the_num @ Option )
% 9.15/5.10                = ( the_num @ Option2 ) ) ) )
% 9.15/5.10         => ( Option = Option2 ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1417_option_Osize_I3_J,axiom,
% 9.15/5.10      ( ( size_size_option_nat @ none_nat )
% 9.15/5.10      = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1418_option_Osize_I3_J,axiom,
% 9.15/5.10      ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
% 9.15/5.10      = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1419_option_Osize_I3_J,axiom,
% 9.15/5.10      ( ( size_size_option_num @ none_num )
% 9.15/5.10      = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1420_option_Osize__gen_I1_J,axiom,
% 9.15/5.10      ! [X3: nat > nat] :
% 9.15/5.10        ( ( size_option_nat @ X3 @ none_nat )
% 9.15/5.10        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1421_option_Osize__gen_I1_J,axiom,
% 9.15/5.10      ! [X3: product_prod_nat_nat > nat] :
% 9.15/5.10        ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
% 9.15/5.10        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1422_option_Osize__gen_I1_J,axiom,
% 9.15/5.10      ! [X3: num > nat] :
% 9.15/5.10        ( ( size_option_num @ X3 @ none_num )
% 9.15/5.10        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1423_option_Oexhaust__sel,axiom,
% 9.15/5.10      ! [Option: option_nat] :
% 9.15/5.10        ( ( Option != none_nat )
% 9.15/5.10       => ( Option
% 9.15/5.10          = ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1424_option_Oexhaust__sel,axiom,
% 9.15/5.10      ! [Option: option4927543243414619207at_nat] :
% 9.15/5.10        ( ( Option != none_P5556105721700978146at_nat )
% 9.15/5.10       => ( Option
% 9.15/5.10          = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1425_option_Oexhaust__sel,axiom,
% 9.15/5.10      ! [Option: option_num] :
% 9.15/5.10        ( ( Option != none_num )
% 9.15/5.10       => ( Option
% 9.15/5.10          = ( some_num @ ( the_num @ Option ) ) ) ) ).
% 9.15/5.10  
% 9.15/5.10  thf(fact_1426_linorder__neqE__linordered__idom,axiom,
% 9.15/5.10      ! [X3: real,Y3: real] :
% 9.15/5.10        ( ( X3 != Y3 )
% 9.15/5.11       => ( ~ ( ord_less_real @ X3 @ Y3 )
% 9.15/5.11         => ( ord_less_real @ Y3 @ X3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1427_linorder__neqE__linordered__idom,axiom,
% 9.15/5.11      ! [X3: rat,Y3: rat] :
% 9.15/5.11        ( ( X3 != Y3 )
% 9.15/5.11       => ( ~ ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.11         => ( ord_less_rat @ Y3 @ X3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1428_linorder__neqE__linordered__idom,axiom,
% 9.15/5.11      ! [X3: int,Y3: int] :
% 9.15/5.11        ( ( X3 != Y3 )
% 9.15/5.11       => ( ~ ( ord_less_int @ X3 @ Y3 )
% 9.15/5.11         => ( ord_less_int @ Y3 @ X3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1429_linordered__field__no__ub,axiom,
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11      ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1430_linordered__field__no__ub,axiom,
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11      ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1431_linordered__field__no__lb,axiom,
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11      ? [Y5: real] : ( ord_less_real @ Y5 @ X2 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1432_linordered__field__no__lb,axiom,
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11      ? [Y5: rat] : ( ord_less_rat @ Y5 @ X2 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1433_not__psubset__empty,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1434_not__psubset__empty,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1435_not__psubset__empty,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1436_not__psubset__empty,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1437_ex__in__conv,axiom,
% 9.15/5.11      ! [A2: set_set_nat] :
% 9.15/5.11        ( ( ? [X: set_nat] : ( member_set_nat @ X @ A2 ) )
% 9.15/5.11        = ( A2 != bot_bot_set_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1438_ex__in__conv,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
% 9.15/5.11        = ( A2 != bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1439_ex__in__conv,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ( ( ? [X: real] : ( member_real @ X @ A2 ) )
% 9.15/5.11        = ( A2 != bot_bot_set_real ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1440_ex__in__conv,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
% 9.15/5.11        = ( A2 != bot_bot_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1441_ex__in__conv,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ( ? [X: int] : ( member_int @ X @ A2 ) )
% 9.15/5.11        = ( A2 != bot_bot_set_int ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1442_equals0I,axiom,
% 9.15/5.11      ! [A2: set_set_nat] :
% 9.15/5.11        ( ! [Y5: set_nat] :
% 9.15/5.11            ~ ( member_set_nat @ Y5 @ A2 )
% 9.15/5.11       => ( A2 = bot_bot_set_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1443_equals0I,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ! [Y5: product_prod_nat_nat] :
% 9.15/5.11            ~ ( member8440522571783428010at_nat @ Y5 @ A2 )
% 9.15/5.11       => ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1444_equals0I,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ( ! [Y5: real] :
% 9.15/5.11            ~ ( member_real @ Y5 @ A2 )
% 9.15/5.11       => ( A2 = bot_bot_set_real ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1445_equals0I,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ! [Y5: nat] :
% 9.15/5.11            ~ ( member_nat @ Y5 @ A2 )
% 9.15/5.11       => ( A2 = bot_bot_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1446_equals0I,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ! [Y5: int] :
% 9.15/5.11            ~ ( member_int @ Y5 @ A2 )
% 9.15/5.11       => ( A2 = bot_bot_set_int ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1447_equals0D,axiom,
% 9.15/5.11      ! [A2: set_set_nat,A: set_nat] :
% 9.15/5.11        ( ( A2 = bot_bot_set_set_nat )
% 9.15/5.11       => ~ ( member_set_nat @ A @ A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1448_equals0D,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.15/5.11        ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.15/5.11       => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1449_equals0D,axiom,
% 9.15/5.11      ! [A2: set_real,A: real] :
% 9.15/5.11        ( ( A2 = bot_bot_set_real )
% 9.15/5.11       => ~ ( member_real @ A @ A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1450_equals0D,axiom,
% 9.15/5.11      ! [A2: set_nat,A: nat] :
% 9.15/5.11        ( ( A2 = bot_bot_set_nat )
% 9.15/5.11       => ~ ( member_nat @ A @ A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1451_equals0D,axiom,
% 9.15/5.11      ! [A2: set_int,A: int] :
% 9.15/5.11        ( ( A2 = bot_bot_set_int )
% 9.15/5.11       => ~ ( member_int @ A @ A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1452_emptyE,axiom,
% 9.15/5.11      ! [A: set_nat] :
% 9.15/5.11        ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1453_emptyE,axiom,
% 9.15/5.11      ! [A: product_prod_nat_nat] :
% 9.15/5.11        ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1454_emptyE,axiom,
% 9.15/5.11      ! [A: real] :
% 9.15/5.11        ~ ( member_real @ A @ bot_bot_set_real ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1455_emptyE,axiom,
% 9.15/5.11      ! [A: nat] :
% 9.15/5.11        ~ ( member_nat @ A @ bot_bot_set_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1456_emptyE,axiom,
% 9.15/5.11      ! [A: int] :
% 9.15/5.11        ~ ( member_int @ A @ bot_bot_set_int ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1457_option_Osize_I4_J,axiom,
% 9.15/5.11      ! [X22: nat] :
% 9.15/5.11        ( ( size_size_option_nat @ ( some_nat @ X22 ) )
% 9.15/5.11        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1458_option_Osize_I4_J,axiom,
% 9.15/5.11      ! [X22: product_prod_nat_nat] :
% 9.15/5.11        ( ( size_s170228958280169651at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
% 9.15/5.11        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1459_option_Osize_I4_J,axiom,
% 9.15/5.11      ! [X22: num] :
% 9.15/5.11        ( ( size_size_option_num @ ( some_num @ X22 ) )
% 9.15/5.11        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1460_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
% 9.15/5.11      ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
% 9.15/5.11        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
% 9.15/5.11        = ( ( X3 = Mi )
% 9.15/5.11          | ( X3 = Ma ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1461_vebt__member_Osimps_I4_J,axiom,
% 9.15/5.11      ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 9.15/5.11        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1462_vebt__member_Osimps_I3_J,axiom,
% 9.15/5.11      ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
% 9.15/5.11        ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1463_VEBT__internal_OminNull_Osimps_I5_J,axiom,
% 9.15/5.11      ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
% 9.15/5.11        ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1464_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uu: option4927543243414619207at_nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
% 9.15/5.11        ~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ zero_zero_nat @ Uv @ Uw ) @ Ux ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1465_VEBT_Oinject_I1_J,axiom,
% 9.15/5.11      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
% 9.15/5.11        ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.15/5.11          = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
% 9.15/5.11        = ( ( X11 = Y11 )
% 9.15/5.11          & ( X12 = Y12 )
% 9.15/5.11          & ( X13 = Y13 )
% 9.15/5.11          & ( X14 = Y14 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1466_old_Oprod_Oinject,axiom,
% 9.15/5.11      ! [A: int,B: int,A6: int,B6: int] :
% 9.15/5.11        ( ( ( product_Pair_int_int @ A @ B )
% 9.15/5.11          = ( product_Pair_int_int @ A6 @ B6 ) )
% 9.15/5.11        = ( ( A = A6 )
% 9.15/5.11          & ( B = B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1467_old_Oprod_Oinject,axiom,
% 9.15/5.11      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc6137756002093451184nteger @ A @ B )
% 9.15/5.11          = ( produc6137756002093451184nteger @ A6 @ B6 ) )
% 9.15/5.11        = ( ( A = A6 )
% 9.15/5.11          & ( B = B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1468_old_Oprod_Oinject,axiom,
% 9.15/5.11      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc8603105652947943368nteger @ A @ B )
% 9.15/5.11          = ( produc8603105652947943368nteger @ A6 @ B6 ) )
% 9.15/5.11        = ( ( A = A6 )
% 9.15/5.11          & ( B = B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1469_old_Oprod_Oinject,axiom,
% 9.15/5.11      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 9.15/5.11          = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
% 9.15/5.11        = ( ( A = A6 )
% 9.15/5.11          & ( B = B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1470_old_Oprod_Oinject,axiom,
% 9.15/5.11      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 9.15/5.11          = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
% 9.15/5.11        = ( ( A = A6 )
% 9.15/5.11          & ( B = B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1471_prod_Oinject,axiom,
% 9.15/5.11      ! [X1: int,X22: int,Y1: int,Y2: int] :
% 9.15/5.11        ( ( ( product_Pair_int_int @ X1 @ X22 )
% 9.15/5.11          = ( product_Pair_int_int @ Y1 @ Y2 ) )
% 9.15/5.11        = ( ( X1 = Y1 )
% 9.15/5.11          & ( X22 = Y2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1472_prod_Oinject,axiom,
% 9.15/5.11      ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
% 9.15/5.11          = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
% 9.15/5.11        = ( ( X1 = Y1 )
% 9.15/5.11          & ( X22 = Y2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1473_prod_Oinject,axiom,
% 9.15/5.11      ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
% 9.15/5.11          = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
% 9.15/5.11        = ( ( X1 = Y1 )
% 9.15/5.11          & ( X22 = Y2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1474_prod_Oinject,axiom,
% 9.15/5.11      ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
% 9.15/5.11          = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
% 9.15/5.11        = ( ( X1 = Y1 )
% 9.15/5.11          & ( X22 = Y2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1475_prod_Oinject,axiom,
% 9.15/5.11      ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
% 9.15/5.11          = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
% 9.15/5.11        = ( ( X1 = Y1 )
% 9.15/5.11          & ( X22 = Y2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1476_nth__enumerate__eq,axiom,
% 9.15/5.11      ! [M: nat,Xs: list_VEBT_VEBT,N2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.11       => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) @ M )
% 9.15/5.11          = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1477_nth__enumerate__eq,axiom,
% 9.15/5.11      ! [M: nat,Xs: list_o,N2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
% 9.15/5.11       => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N2 @ Xs ) @ M )
% 9.15/5.11          = ( product_Pair_nat_o @ ( plus_plus_nat @ N2 @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1478_nth__enumerate__eq,axiom,
% 9.15/5.11      ! [M: nat,Xs: list_nat,N2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
% 9.15/5.11       => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M )
% 9.15/5.11          = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1479_nth__enumerate__eq,axiom,
% 9.15/5.11      ! [M: nat,Xs: list_int,N2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
% 9.15/5.11       => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M )
% 9.15/5.11          = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1480_set__vebt__finite,axiom,
% 9.15/5.11      ! [T: vEBT_VEBT,N2: nat] :
% 9.15/5.11        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.15/5.11       => ( finite_finite_nat @ ( vEBT_VEBT_set_vebt @ T ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1481_Euclid__induct,axiom,
% 9.15/5.11      ! [P: nat > nat > $o,A: nat,B: nat] :
% 9.15/5.11        ( ! [A4: nat,B4: nat] :
% 9.15/5.11            ( ( P @ A4 @ B4 )
% 9.15/5.11            = ( P @ B4 @ A4 ) )
% 9.15/5.11       => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
% 9.15/5.11         => ( ! [A4: nat,B4: nat] :
% 9.15/5.11                ( ( P @ A4 @ B4 )
% 9.15/5.11               => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
% 9.15/5.11           => ( P @ A @ B ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1482_List_Ofinite__set,axiom,
% 9.15/5.11      ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1483_List_Ofinite__set,axiom,
% 9.15/5.11      ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1484_List_Ofinite__set,axiom,
% 9.15/5.11      ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1485_List_Ofinite__set,axiom,
% 9.15/5.11      ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1486_List_Ofinite__set,axiom,
% 9.15/5.11      ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1487_length__enumerate,axiom,
% 9.15/5.11      ! [N2: nat,Xs: list_VEBT_VEBT] :
% 9.15/5.11        ( ( size_s4762443039079500285T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) )
% 9.15/5.11        = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1488_length__enumerate,axiom,
% 9.15/5.11      ! [N2: nat,Xs: list_o] :
% 9.15/5.11        ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N2 @ Xs ) )
% 9.15/5.11        = ( size_size_list_o @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1489_length__enumerate,axiom,
% 9.15/5.11      ! [N2: nat,Xs: list_nat] :
% 9.15/5.11        ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
% 9.15/5.11        = ( size_size_list_nat @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1490_length__enumerate,axiom,
% 9.15/5.11      ! [N2: nat,Xs: list_int] :
% 9.15/5.11        ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs ) )
% 9.15/5.11        = ( size_size_list_int @ Xs ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1491_psubsetD,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
% 9.15/5.11        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 9.15/5.11       => ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.15/5.11         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1492_psubsetD,axiom,
% 9.15/5.11      ! [A2: set_real,B2: set_real,C: real] :
% 9.15/5.11        ( ( ord_less_set_real @ A2 @ B2 )
% 9.15/5.11       => ( ( member_real @ C @ A2 )
% 9.15/5.11         => ( member_real @ C @ B2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1493_psubsetD,axiom,
% 9.15/5.11      ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
% 9.15/5.11        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.15/5.11       => ( ( member_set_nat @ C @ A2 )
% 9.15/5.11         => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1494_psubsetD,axiom,
% 9.15/5.11      ! [A2: set_nat,B2: set_nat,C: nat] :
% 9.15/5.11        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.15/5.11       => ( ( member_nat @ C @ A2 )
% 9.15/5.11         => ( member_nat @ C @ B2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1495_psubsetD,axiom,
% 9.15/5.11      ! [A2: set_int,B2: set_int,C: int] :
% 9.15/5.11        ( ( ord_less_set_int @ A2 @ B2 )
% 9.15/5.11       => ( ( member_int @ C @ A2 )
% 9.15/5.11         => ( member_int @ C @ B2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1496_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc8306885398267862888on_nat] :
% 9.15/5.11        ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: nat > nat > nat,V2: nat] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.15/5.11         => ~ ! [F2: nat > nat > nat,A4: nat,B4: nat] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1497_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc5542196010084753463at_nat] :
% 9.15/5.11        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.15/5.11         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1498_VEBT__internal_Ooption__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc1193250871479095198on_num] :
% 9.15/5.11        ( ! [Uu2: num > num > num,Uv2: option_num] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: num > num > num,V2: num] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.15/5.11         => ~ ! [F2: num > num > num,A4: num,B4: num] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc2233624965454879586on_nat] :
% 9.15/5.11        ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: nat > nat > $o,V2: nat] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
% 9.15/5.11         => ~ ! [F2: nat > nat > $o,X4: nat,Y5: nat] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc5491161045314408544at_nat] :
% 9.15/5.11        ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
% 9.15/5.11         => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
% 9.15/5.11      ! [X3: produc7036089656553540234on_num] :
% 9.15/5.11        ( ! [Uu2: num > num > $o,Uv2: option_num] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
% 9.15/5.11       => ( ! [Uw2: num > num > $o,V2: num] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
% 9.15/5.11         => ~ ! [F2: num > num > $o,X4: num,Y5: num] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y5 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1502_finite__list,axiom,
% 9.15/5.11      ! [A2: set_VEBT_VEBT] :
% 9.15/5.11        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.15/5.11       => ? [Xs3: list_VEBT_VEBT] :
% 9.15/5.11            ( ( set_VEBT_VEBT2 @ Xs3 )
% 9.15/5.11            = A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1503_finite__list,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ? [Xs3: list_nat] :
% 9.15/5.11            ( ( set_nat2 @ Xs3 )
% 9.15/5.11            = A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1504_finite__list,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ? [Xs3: list_int] :
% 9.15/5.11            ( ( set_int2 @ Xs3 )
% 9.15/5.11            = A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1505_finite__list,axiom,
% 9.15/5.11      ! [A2: set_complex] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ A2 )
% 9.15/5.11       => ? [Xs3: list_complex] :
% 9.15/5.11            ( ( set_complex2 @ Xs3 )
% 9.15/5.11            = A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1506_finite__list,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( finite6177210948735845034at_nat @ A2 )
% 9.15/5.11       => ? [Xs3: list_P6011104703257516679at_nat] :
% 9.15/5.11            ( ( set_Pr5648618587558075414at_nat @ Xs3 )
% 9.15/5.11            = A2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1507_vebt__member_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
% 9.15/5.11        ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1508_VEBT__internal_OminNull_Osimps_I4_J,axiom,
% 9.15/5.11      ! [Uw: nat,Ux: list_VEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw @ Ux @ Uy ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1509_ex__min__if__finite,axiom,
% 9.15/5.11      ! [S3: set_real] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ? [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ S3 )
% 9.15/5.11              & ~ ? [Xa: real] :
% 9.15/5.11                    ( ( member_real @ Xa @ S3 )
% 9.15/5.11                    & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1510_ex__min__if__finite,axiom,
% 9.15/5.11      ! [S3: set_rat] :
% 9.15/5.11        ( ( finite_finite_rat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_rat )
% 9.15/5.11         => ? [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ S3 )
% 9.15/5.11              & ~ ? [Xa: rat] :
% 9.15/5.11                    ( ( member_rat @ Xa @ S3 )
% 9.15/5.11                    & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1511_ex__min__if__finite,axiom,
% 9.15/5.11      ! [S3: set_num] :
% 9.15/5.11        ( ( finite_finite_num @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_num )
% 9.15/5.11         => ? [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ S3 )
% 9.15/5.11              & ~ ? [Xa: num] :
% 9.15/5.11                    ( ( member_num @ Xa @ S3 )
% 9.15/5.11                    & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1512_ex__min__if__finite,axiom,
% 9.15/5.11      ! [S3: set_nat] :
% 9.15/5.11        ( ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_nat )
% 9.15/5.11         => ? [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ S3 )
% 9.15/5.11              & ~ ? [Xa: nat] :
% 9.15/5.11                    ( ( member_nat @ Xa @ S3 )
% 9.15/5.11                    & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1513_ex__min__if__finite,axiom,
% 9.15/5.11      ! [S3: set_int] :
% 9.15/5.11        ( ( finite_finite_int @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_int )
% 9.15/5.11         => ? [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ S3 )
% 9.15/5.11              & ~ ? [Xa: int] :
% 9.15/5.11                    ( ( member_int @ Xa @ S3 )
% 9.15/5.11                    & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1514_infinite__growing,axiom,
% 9.15/5.11      ! [X7: set_real] :
% 9.15/5.11        ( ( X7 != bot_bot_set_real )
% 9.15/5.11       => ( ! [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ X7 )
% 9.15/5.11             => ? [Xa: real] :
% 9.15/5.11                  ( ( member_real @ Xa @ X7 )
% 9.15/5.11                  & ( ord_less_real @ X4 @ Xa ) ) )
% 9.15/5.11         => ~ ( finite_finite_real @ X7 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1515_infinite__growing,axiom,
% 9.15/5.11      ! [X7: set_rat] :
% 9.15/5.11        ( ( X7 != bot_bot_set_rat )
% 9.15/5.11       => ( ! [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ X7 )
% 9.15/5.11             => ? [Xa: rat] :
% 9.15/5.11                  ( ( member_rat @ Xa @ X7 )
% 9.15/5.11                  & ( ord_less_rat @ X4 @ Xa ) ) )
% 9.15/5.11         => ~ ( finite_finite_rat @ X7 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1516_infinite__growing,axiom,
% 9.15/5.11      ! [X7: set_num] :
% 9.15/5.11        ( ( X7 != bot_bot_set_num )
% 9.15/5.11       => ( ! [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ X7 )
% 9.15/5.11             => ? [Xa: num] :
% 9.15/5.11                  ( ( member_num @ Xa @ X7 )
% 9.15/5.11                  & ( ord_less_num @ X4 @ Xa ) ) )
% 9.15/5.11         => ~ ( finite_finite_num @ X7 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1517_infinite__growing,axiom,
% 9.15/5.11      ! [X7: set_nat] :
% 9.15/5.11        ( ( X7 != bot_bot_set_nat )
% 9.15/5.11       => ( ! [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ X7 )
% 9.15/5.11             => ? [Xa: nat] :
% 9.15/5.11                  ( ( member_nat @ Xa @ X7 )
% 9.15/5.11                  & ( ord_less_nat @ X4 @ Xa ) ) )
% 9.15/5.11         => ~ ( finite_finite_nat @ X7 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1518_infinite__growing,axiom,
% 9.15/5.11      ! [X7: set_int] :
% 9.15/5.11        ( ( X7 != bot_bot_set_int )
% 9.15/5.11       => ( ! [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ X7 )
% 9.15/5.11             => ? [Xa: int] :
% 9.15/5.11                  ( ( member_int @ Xa @ X7 )
% 9.15/5.11                  & ( ord_less_int @ X4 @ Xa ) ) )
% 9.15/5.11         => ~ ( finite_finite_int @ X7 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1519_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Ux: list_VEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
% 9.15/5.11        ~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux @ Uy ) @ Uz ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1520_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.15/5.11      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 9.15/5.11        ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
% 9.15/5.11        = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1521_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.15/5.11      ! [F: num > num > num,A: num,B: num] :
% 9.15/5.11        ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
% 9.15/5.11        = ( some_num @ ( F @ A @ B ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1522_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
% 9.15/5.11      ! [F: nat > nat > nat,A: nat,B: nat] :
% 9.15/5.11        ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
% 9.15/5.11        = ( some_nat @ ( F @ A @ B ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1523_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.15/5.11      ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
% 9.15/5.11        ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
% 9.15/5.11        = none_P5556105721700978146at_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1524_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.15/5.11      ! [Uu: num > num > num,Uv: option_num] :
% 9.15/5.11        ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
% 9.15/5.11        = none_num ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1525_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
% 9.15/5.11      ! [Uu: nat > nat > nat,Uv: option_nat] :
% 9.15/5.11        ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
% 9.15/5.11        = none_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1526_old_Oprod_Oexhaust,axiom,
% 9.15/5.11      ! [Y3: product_prod_int_int] :
% 9.15/5.11        ~ ! [A4: int,B4: int] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( product_Pair_int_int @ A4 @ B4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1527_old_Oprod_Oexhaust,axiom,
% 9.15/5.11      ! [Y3: produc8763457246119570046nteger] :
% 9.15/5.11        ~ ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc6137756002093451184nteger @ A4 @ B4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1528_old_Oprod_Oexhaust,axiom,
% 9.15/5.11      ! [Y3: produc1908205239877642774nteger] :
% 9.15/5.11        ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc8603105652947943368nteger @ A4 @ B4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1529_old_Oprod_Oexhaust,axiom,
% 9.15/5.11      ! [Y3: produc2285326912895808259nt_int] :
% 9.15/5.11        ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc5700946648718959541nt_int @ A4 @ B4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1530_old_Oprod_Oexhaust,axiom,
% 9.15/5.11      ! [Y3: produc7773217078559923341nt_int] :
% 9.15/5.11        ~ ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc4305682042979456191nt_int @ A4 @ B4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1531_surj__pair,axiom,
% 9.15/5.11      ! [P4: product_prod_int_int] :
% 9.15/5.11      ? [X4: int,Y5: int] :
% 9.15/5.11        ( P4
% 9.15/5.11        = ( product_Pair_int_int @ X4 @ Y5 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1532_surj__pair,axiom,
% 9.15/5.11      ! [P4: produc8763457246119570046nteger] :
% 9.15/5.11      ? [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.15/5.11        ( P4
% 9.15/5.11        = ( produc6137756002093451184nteger @ X4 @ Y5 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1533_surj__pair,axiom,
% 9.15/5.11      ! [P4: produc1908205239877642774nteger] :
% 9.15/5.11      ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.15/5.11        ( P4
% 9.15/5.11        = ( produc8603105652947943368nteger @ X4 @ Y5 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1534_surj__pair,axiom,
% 9.15/5.11      ! [P4: produc2285326912895808259nt_int] :
% 9.15/5.11      ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.15/5.11        ( P4
% 9.15/5.11        = ( produc5700946648718959541nt_int @ X4 @ Y5 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1535_surj__pair,axiom,
% 9.15/5.11      ! [P4: produc7773217078559923341nt_int] :
% 9.15/5.11      ? [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.15/5.11        ( P4
% 9.15/5.11        = ( produc4305682042979456191nt_int @ X4 @ Y5 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1536_prod__cases,axiom,
% 9.15/5.11      ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
% 9.15/5.11        ( ! [A4: int,B4: int] : ( P @ ( product_Pair_int_int @ A4 @ B4 ) )
% 9.15/5.11       => ( P @ P4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1537_prod__cases,axiom,
% 9.15/5.11      ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
% 9.15/5.11        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 9.15/5.11       => ( P @ P4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1538_prod__cases,axiom,
% 9.15/5.11      ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
% 9.15/5.11        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A4 @ B4 ) )
% 9.15/5.11       => ( P @ P4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1539_prod__cases,axiom,
% 9.15/5.11      ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
% 9.15/5.11        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ B4 ) )
% 9.15/5.11       => ( P @ P4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1540_prod__cases,axiom,
% 9.15/5.11      ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
% 9.15/5.11        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 9.15/5.11       => ( P @ P4 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1541_Pair__inject,axiom,
% 9.15/5.11      ! [A: int,B: int,A6: int,B6: int] :
% 9.15/5.11        ( ( ( product_Pair_int_int @ A @ B )
% 9.15/5.11          = ( product_Pair_int_int @ A6 @ B6 ) )
% 9.15/5.11       => ~ ( ( A = A6 )
% 9.15/5.11           => ( B != B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1542_Pair__inject,axiom,
% 9.15/5.11      ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc6137756002093451184nteger @ A @ B )
% 9.15/5.11          = ( produc6137756002093451184nteger @ A6 @ B6 ) )
% 9.15/5.11       => ~ ( ( A = A6 )
% 9.15/5.11           => ( B != B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1543_Pair__inject,axiom,
% 9.15/5.11      ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( ( produc8603105652947943368nteger @ A @ B )
% 9.15/5.11          = ( produc8603105652947943368nteger @ A6 @ B6 ) )
% 9.15/5.11       => ~ ( ( A = A6 )
% 9.15/5.11           => ( B != B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1544_Pair__inject,axiom,
% 9.15/5.11      ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc5700946648718959541nt_int @ A @ B )
% 9.15/5.11          = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
% 9.15/5.11       => ~ ( ( A = A6 )
% 9.15/5.11           => ( B != B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1545_Pair__inject,axiom,
% 9.15/5.11      ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
% 9.15/5.11        ( ( ( produc4305682042979456191nt_int @ A @ B )
% 9.15/5.11          = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
% 9.15/5.11       => ~ ( ( A = A6 )
% 9.15/5.11           => ( B != B6 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1546_prod__cases3,axiom,
% 9.15/5.11      ! [Y3: produc8763457246119570046nteger] :
% 9.15/5.11        ~ ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1547_prod__cases3,axiom,
% 9.15/5.11      ! [Y3: produc1908205239877642774nteger] :
% 9.15/5.11        ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1548_prod__cases3,axiom,
% 9.15/5.11      ! [Y3: produc2285326912895808259nt_int] :
% 9.15/5.11        ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1549_prod__cases3,axiom,
% 9.15/5.11      ! [Y3: produc7773217078559923341nt_int] :
% 9.15/5.11        ~ ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] :
% 9.15/5.11            ( Y3
% 9.15/5.11           != ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1550_prod__induct3,axiom,
% 9.15/5.11      ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
% 9.15/5.11        ( ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
% 9.15/5.11       => ( P @ X3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1551_prod__induct3,axiom,
% 9.15/5.11      ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
% 9.15/5.11        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
% 9.15/5.11       => ( P @ X3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1552_prod__induct3,axiom,
% 9.15/5.11      ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
% 9.15/5.11        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
% 9.15/5.11       => ( P @ X3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1553_prod__induct3,axiom,
% 9.15/5.11      ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
% 9.15/5.11        ( ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
% 9.15/5.11       => ( P @ X3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1554_vebt__maxt_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.15/5.11        ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.15/5.11        = none_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1555_vebt__mint_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
% 9.15/5.11        ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
% 9.15/5.11        = none_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1556_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
% 9.15/5.11        ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
% 9.15/5.11        = none_P5556105721700978146at_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1557_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uw: num > num > num,V: num] :
% 9.15/5.11        ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
% 9.15/5.11        = none_num ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1558_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
% 9.15/5.11      ! [Uw: nat > nat > nat,V: nat] :
% 9.15/5.11        ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
% 9.15/5.11        = none_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1559_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.15/5.11      ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
% 9.15/5.11        ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb )
% 9.15/5.11          = Y3 )
% 9.15/5.11       => ( ( ( Xa2 = none_P5556105721700978146at_nat )
% 9.15/5.11           => ( Y3 != none_P5556105721700978146at_nat ) )
% 9.15/5.11         => ( ( ? [V2: product_prod_nat_nat] :
% 9.15/5.11                  ( Xa2
% 9.15/5.11                  = ( some_P7363390416028606310at_nat @ V2 ) )
% 9.15/5.11             => ( ( Xb = none_P5556105721700978146at_nat )
% 9.15/5.11               => ( Y3 != none_P5556105721700978146at_nat ) ) )
% 9.15/5.11           => ~ ! [A4: product_prod_nat_nat] :
% 9.15/5.11                  ( ( Xa2
% 9.15/5.11                    = ( some_P7363390416028606310at_nat @ A4 ) )
% 9.15/5.11                 => ! [B4: product_prod_nat_nat] :
% 9.15/5.11                      ( ( Xb
% 9.15/5.11                        = ( some_P7363390416028606310at_nat @ B4 ) )
% 9.15/5.11                     => ( Y3
% 9.15/5.11                       != ( some_P7363390416028606310at_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1560_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.15/5.11      ! [X3: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
% 9.15/5.11        ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb )
% 9.15/5.11          = Y3 )
% 9.15/5.11       => ( ( ( Xa2 = none_num )
% 9.15/5.11           => ( Y3 != none_num ) )
% 9.15/5.11         => ( ( ? [V2: num] :
% 9.15/5.11                  ( Xa2
% 9.15/5.11                  = ( some_num @ V2 ) )
% 9.15/5.11             => ( ( Xb = none_num )
% 9.15/5.11               => ( Y3 != none_num ) ) )
% 9.15/5.11           => ~ ! [A4: num] :
% 9.15/5.11                  ( ( Xa2
% 9.15/5.11                    = ( some_num @ A4 ) )
% 9.15/5.11                 => ! [B4: num] :
% 9.15/5.11                      ( ( Xb
% 9.15/5.11                        = ( some_num @ B4 ) )
% 9.15/5.11                     => ( Y3
% 9.15/5.11                       != ( some_num @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1561_VEBT__internal_Ooption__shift_Oelims,axiom,
% 9.15/5.11      ! [X3: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
% 9.15/5.11        ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb )
% 9.15/5.11          = Y3 )
% 9.15/5.11       => ( ( ( Xa2 = none_nat )
% 9.15/5.11           => ( Y3 != none_nat ) )
% 9.15/5.11         => ( ( ? [V2: nat] :
% 9.15/5.11                  ( Xa2
% 9.15/5.11                  = ( some_nat @ V2 ) )
% 9.15/5.11             => ( ( Xb = none_nat )
% 9.15/5.11               => ( Y3 != none_nat ) ) )
% 9.15/5.11           => ~ ! [A4: nat] :
% 9.15/5.11                  ( ( Xa2
% 9.15/5.11                    = ( some_nat @ A4 ) )
% 9.15/5.11                 => ! [B4: nat] :
% 9.15/5.11                      ( ( Xb
% 9.15/5.11                        = ( some_nat @ B4 ) )
% 9.15/5.11                     => ( Y3
% 9.15/5.11                       != ( some_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1562_vebt__buildup_Ocases,axiom,
% 9.15/5.11      ! [X3: nat] :
% 9.15/5.11        ( ( X3 != zero_zero_nat )
% 9.15/5.11       => ( ( X3
% 9.15/5.11           != ( suc @ zero_zero_nat ) )
% 9.15/5.11         => ~ ! [Va2: nat] :
% 9.15/5.11                ( X3
% 9.15/5.11               != ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1563_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ( ( finite_finite_real @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_real )
% 9.15/5.11         => ? [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: real] :
% 9.15/5.11                  ( ( member_real @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1564_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_set_int] :
% 9.15/5.11        ( ( finite6197958912794628473et_int @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_set_int )
% 9.15/5.11         => ? [X4: set_int] :
% 9.15/5.11              ( ( member_set_int @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: set_int] :
% 9.15/5.11                  ( ( member_set_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1565_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_rat] :
% 9.15/5.11        ( ( finite_finite_rat @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_rat )
% 9.15/5.11         => ? [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: rat] :
% 9.15/5.11                  ( ( member_rat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1566_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_num] :
% 9.15/5.11        ( ( finite_finite_num @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_num )
% 9.15/5.11         => ? [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: num] :
% 9.15/5.11                  ( ( member_num @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1567_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_nat )
% 9.15/5.11         => ? [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: nat] :
% 9.15/5.11                  ( ( member_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1568_finite__has__minimal,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_int )
% 9.15/5.11         => ? [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: int] :
% 9.15/5.11                  ( ( member_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1569_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ( ( finite_finite_real @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_real )
% 9.15/5.11         => ? [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: real] :
% 9.15/5.11                  ( ( member_real @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1570_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_set_int] :
% 9.15/5.11        ( ( finite6197958912794628473et_int @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_set_int )
% 9.15/5.11         => ? [X4: set_int] :
% 9.15/5.11              ( ( member_set_int @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: set_int] :
% 9.15/5.11                  ( ( member_set_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1571_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_rat] :
% 9.15/5.11        ( ( finite_finite_rat @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_rat )
% 9.15/5.11         => ? [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: rat] :
% 9.15/5.11                  ( ( member_rat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1572_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_num] :
% 9.15/5.11        ( ( finite_finite_num @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_num )
% 9.15/5.11         => ? [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: num] :
% 9.15/5.11                  ( ( member_num @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1573_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_nat )
% 9.15/5.11         => ? [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: nat] :
% 9.15/5.11                  ( ( member_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1574_finite__has__maximal,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ( A2 != bot_bot_set_int )
% 9.15/5.11         => ? [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ A2 )
% 9.15/5.11              & ! [Xa: int] :
% 9.15/5.11                  ( ( member_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1575_bot__empty__eq,axiom,
% 9.15/5.11      ( bot_bot_set_nat_o
% 9.15/5.11      = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1576_bot__empty__eq,axiom,
% 9.15/5.11      ( bot_bo482883023278783056_nat_o
% 9.15/5.11      = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1577_bot__empty__eq,axiom,
% 9.15/5.11      ( bot_bot_real_o
% 9.15/5.11      = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1578_bot__empty__eq,axiom,
% 9.15/5.11      ( bot_bot_nat_o
% 9.15/5.11      = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1579_bot__empty__eq,axiom,
% 9.15/5.11      ( bot_bot_int_o
% 9.15/5.11      = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1580_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: list_nat > $o] :
% 9.15/5.11        ( ( ( collect_list_nat @ P )
% 9.15/5.11          = bot_bot_set_list_nat )
% 9.15/5.11        = ( P = bot_bot_list_nat_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1581_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: set_nat > $o] :
% 9.15/5.11        ( ( ( collect_set_nat @ P )
% 9.15/5.11          = bot_bot_set_set_nat )
% 9.15/5.11        = ( P = bot_bot_set_nat_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1582_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: product_prod_nat_nat > $o] :
% 9.15/5.11        ( ( ( collec3392354462482085612at_nat @ P )
% 9.15/5.11          = bot_bo2099793752762293965at_nat )
% 9.15/5.11        = ( P = bot_bo482883023278783056_nat_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1583_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: real > $o] :
% 9.15/5.11        ( ( ( collect_real @ P )
% 9.15/5.11          = bot_bot_set_real )
% 9.15/5.11        = ( P = bot_bot_real_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1584_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: nat > $o] :
% 9.15/5.11        ( ( ( collect_nat @ P )
% 9.15/5.11          = bot_bot_set_nat )
% 9.15/5.11        = ( P = bot_bot_nat_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1585_Collect__empty__eq__bot,axiom,
% 9.15/5.11      ! [P: int > $o] :
% 9.15/5.11        ( ( ( collect_int @ P )
% 9.15/5.11          = bot_bot_set_int )
% 9.15/5.11        = ( P = bot_bot_int_o ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1586_infinite__nat__iff__unbounded__le,axiom,
% 9.15/5.11      ! [S3: set_nat] :
% 9.15/5.11        ( ( ~ ( finite_finite_nat @ S3 ) )
% 9.15/5.11        = ( ! [M2: nat] :
% 9.15/5.11            ? [N: nat] :
% 9.15/5.11              ( ( ord_less_eq_nat @ M2 @ N )
% 9.15/5.11              & ( member_nat @ N @ S3 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1587_finite__nat__set__iff__bounded__le,axiom,
% 9.15/5.11      ( finite_finite_nat
% 9.15/5.11      = ( ^ [N5: set_nat] :
% 9.15/5.11          ? [M2: nat] :
% 9.15/5.11          ! [X: nat] :
% 9.15/5.11            ( ( member_nat @ X @ N5 )
% 9.15/5.11           => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1588_unbounded__k__infinite,axiom,
% 9.15/5.11      ! [K: nat,S3: set_nat] :
% 9.15/5.11        ( ! [M3: nat] :
% 9.15/5.11            ( ( ord_less_nat @ K @ M3 )
% 9.15/5.11           => ? [N6: nat] :
% 9.15/5.11                ( ( ord_less_nat @ M3 @ N6 )
% 9.15/5.11                & ( member_nat @ N6 @ S3 ) ) )
% 9.15/5.11       => ~ ( finite_finite_nat @ S3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1589_bounded__nat__set__is__finite,axiom,
% 9.15/5.11      ! [N7: set_nat,N2: nat] :
% 9.15/5.11        ( ! [X4: nat] :
% 9.15/5.11            ( ( member_nat @ X4 @ N7 )
% 9.15/5.11           => ( ord_less_nat @ X4 @ N2 ) )
% 9.15/5.11       => ( finite_finite_nat @ N7 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1590_finite__maxlen,axiom,
% 9.15/5.11      ! [M7: set_list_VEBT_VEBT] :
% 9.15/5.11        ( ( finite3004134309566078307T_VEBT @ M7 )
% 9.15/5.11       => ? [N3: nat] :
% 9.15/5.11          ! [X2: list_VEBT_VEBT] :
% 9.15/5.11            ( ( member2936631157270082147T_VEBT @ X2 @ M7 )
% 9.15/5.11           => ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ X2 ) @ N3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1591_finite__maxlen,axiom,
% 9.15/5.11      ! [M7: set_list_o] :
% 9.15/5.11        ( ( finite_finite_list_o @ M7 )
% 9.15/5.11       => ? [N3: nat] :
% 9.15/5.11          ! [X2: list_o] :
% 9.15/5.11            ( ( member_list_o @ X2 @ M7 )
% 9.15/5.11           => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1592_finite__maxlen,axiom,
% 9.15/5.11      ! [M7: set_list_nat] :
% 9.15/5.11        ( ( finite8100373058378681591st_nat @ M7 )
% 9.15/5.11       => ? [N3: nat] :
% 9.15/5.11          ! [X2: list_nat] :
% 9.15/5.11            ( ( member_list_nat @ X2 @ M7 )
% 9.15/5.11           => ( ord_less_nat @ ( size_size_list_nat @ X2 ) @ N3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1593_finite__maxlen,axiom,
% 9.15/5.11      ! [M7: set_list_int] :
% 9.15/5.11        ( ( finite3922522038869484883st_int @ M7 )
% 9.15/5.11       => ? [N3: nat] :
% 9.15/5.11          ! [X2: list_int] :
% 9.15/5.11            ( ( member_list_int @ X2 @ M7 )
% 9.15/5.11           => ( ord_less_nat @ ( size_size_list_int @ X2 ) @ N3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1594_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ! [X4: nat,Y5: nat] :
% 9.15/5.11            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1595_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
% 9.15/5.11        ( ! [X4: int,Y5: int] :
% 9.15/5.11            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1596_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
% 9.15/5.11        ( ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.15/5.11            ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le3216752416896350996nteger @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1597_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
% 9.15/5.11        ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.15/5.11            ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1598_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
% 9.15/5.11        ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.15/5.11            ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1599_subrelI,axiom,
% 9.15/5.11      ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
% 9.15/5.11        ( ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.15/5.11            ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ R2 )
% 9.15/5.11           => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ S ) )
% 9.15/5.11       => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1600_bounded__Max__nat,axiom,
% 9.15/5.11      ! [P: nat > $o,X3: nat,M7: nat] :
% 9.15/5.11        ( ( P @ X3 )
% 9.15/5.11       => ( ! [X4: nat] :
% 9.15/5.11              ( ( P @ X4 )
% 9.15/5.11             => ( ord_less_eq_nat @ X4 @ M7 ) )
% 9.15/5.11         => ~ ! [M3: nat] :
% 9.15/5.11                ( ( P @ M3 )
% 9.15/5.11               => ~ ! [X2: nat] :
% 9.15/5.11                      ( ( P @ X2 )
% 9.15/5.11                     => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1601_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_real,A: real] :
% 9.15/5.11        ( ( finite_finite_real @ A2 )
% 9.15/5.11       => ( ( member_real @ A @ A2 )
% 9.15/5.11         => ? [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_real @ X4 @ A )
% 9.15/5.11              & ! [Xa: real] :
% 9.15/5.11                  ( ( member_real @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_real @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1602_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_set_nat,A: set_nat] :
% 9.15/5.11        ( ( finite1152437895449049373et_nat @ A2 )
% 9.15/5.11       => ( ( member_set_nat @ A @ A2 )
% 9.15/5.11         => ? [X4: set_nat] :
% 9.15/5.11              ( ( member_set_nat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_set_nat @ X4 @ A )
% 9.15/5.11              & ! [Xa: set_nat] :
% 9.15/5.11                  ( ( member_set_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_nat @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1603_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_set_int,A: set_int] :
% 9.15/5.11        ( ( finite6197958912794628473et_int @ A2 )
% 9.15/5.11       => ( ( member_set_int @ A @ A2 )
% 9.15/5.11         => ? [X4: set_int] :
% 9.15/5.11              ( ( member_set_int @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_set_int @ X4 @ A )
% 9.15/5.11              & ! [Xa: set_int] :
% 9.15/5.11                  ( ( member_set_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_int @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1604_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_rat,A: rat] :
% 9.15/5.11        ( ( finite_finite_rat @ A2 )
% 9.15/5.11       => ( ( member_rat @ A @ A2 )
% 9.15/5.11         => ? [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_rat @ X4 @ A )
% 9.15/5.11              & ! [Xa: rat] :
% 9.15/5.11                  ( ( member_rat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_rat @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1605_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_num,A: num] :
% 9.15/5.11        ( ( finite_finite_num @ A2 )
% 9.15/5.11       => ( ( member_num @ A @ A2 )
% 9.15/5.11         => ? [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_num @ X4 @ A )
% 9.15/5.11              & ! [Xa: num] :
% 9.15/5.11                  ( ( member_num @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_num @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1606_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_nat,A: nat] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ( member_nat @ A @ A2 )
% 9.15/5.11         => ? [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_nat @ X4 @ A )
% 9.15/5.11              & ! [Xa: nat] :
% 9.15/5.11                  ( ( member_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_nat @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1607_finite__has__minimal2,axiom,
% 9.15/5.11      ! [A2: set_int,A: int] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ( member_int @ A @ A2 )
% 9.15/5.11         => ? [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_int @ X4 @ A )
% 9.15/5.11              & ! [Xa: int] :
% 9.15/5.11                  ( ( member_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_int @ Xa @ X4 )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1608_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_real,A: real] :
% 9.15/5.11        ( ( finite_finite_real @ A2 )
% 9.15/5.11       => ( ( member_real @ A @ A2 )
% 9.15/5.11         => ? [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_real @ A @ X4 )
% 9.15/5.11              & ! [Xa: real] :
% 9.15/5.11                  ( ( member_real @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_real @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1609_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_set_nat,A: set_nat] :
% 9.15/5.11        ( ( finite1152437895449049373et_nat @ A2 )
% 9.15/5.11       => ( ( member_set_nat @ A @ A2 )
% 9.15/5.11         => ? [X4: set_nat] :
% 9.15/5.11              ( ( member_set_nat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_set_nat @ A @ X4 )
% 9.15/5.11              & ! [Xa: set_nat] :
% 9.15/5.11                  ( ( member_set_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_nat @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1610_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_set_int,A: set_int] :
% 9.15/5.11        ( ( finite6197958912794628473et_int @ A2 )
% 9.15/5.11       => ( ( member_set_int @ A @ A2 )
% 9.15/5.11         => ? [X4: set_int] :
% 9.15/5.11              ( ( member_set_int @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_set_int @ A @ X4 )
% 9.15/5.11              & ! [Xa: set_int] :
% 9.15/5.11                  ( ( member_set_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_set_int @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1611_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_rat,A: rat] :
% 9.15/5.11        ( ( finite_finite_rat @ A2 )
% 9.15/5.11       => ( ( member_rat @ A @ A2 )
% 9.15/5.11         => ? [X4: rat] :
% 9.15/5.11              ( ( member_rat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_rat @ A @ X4 )
% 9.15/5.11              & ! [Xa: rat] :
% 9.15/5.11                  ( ( member_rat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_rat @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1612_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_num,A: num] :
% 9.15/5.11        ( ( finite_finite_num @ A2 )
% 9.15/5.11       => ( ( member_num @ A @ A2 )
% 9.15/5.11         => ? [X4: num] :
% 9.15/5.11              ( ( member_num @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_num @ A @ X4 )
% 9.15/5.11              & ! [Xa: num] :
% 9.15/5.11                  ( ( member_num @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_num @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1613_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_nat,A: nat] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ( member_nat @ A @ A2 )
% 9.15/5.11         => ? [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_nat @ A @ X4 )
% 9.15/5.11              & ! [Xa: nat] :
% 9.15/5.11                  ( ( member_nat @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_nat @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1614_finite__has__maximal2,axiom,
% 9.15/5.11      ! [A2: set_int,A: int] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ( member_int @ A @ A2 )
% 9.15/5.11         => ? [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ A2 )
% 9.15/5.11              & ( ord_less_eq_int @ A @ X4 )
% 9.15/5.11              & ! [Xa: int] :
% 9.15/5.11                  ( ( member_int @ Xa @ A2 )
% 9.15/5.11                 => ( ( ord_less_eq_int @ X4 @ Xa )
% 9.15/5.11                   => ( X4 = Xa ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1615_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
% 9.15/5.11        ( ( finite1152437895449049373et_nat @ A2 )
% 9.15/5.11       => ( ! [X4: set_nat] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: set_nat] :
% 9.15/5.11                  ( ( member_set_nat @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: set_nat] :
% 9.15/5.11                      ( ( member_set_nat @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1616_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_complex,R: complex > complex > $o] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ A2 )
% 9.15/5.11       => ( ! [X4: complex] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: complex,Y5: complex,Z3: complex] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: complex] :
% 9.15/5.11                  ( ( member_complex @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: complex] :
% 9.15/5.11                      ( ( member_complex @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bot_set_complex ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1617_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.15/5.11        ( ( finite6177210948735845034at_nat @ A2 )
% 9.15/5.11       => ( ! [X4: product_prod_nat_nat] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat,Z3: product_prod_nat_nat] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: product_prod_nat_nat] :
% 9.15/5.11                  ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: product_prod_nat_nat] :
% 9.15/5.11                      ( ( member8440522571783428010at_nat @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1618_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_real,R: real > real > $o] :
% 9.15/5.11        ( ( finite_finite_real @ A2 )
% 9.15/5.11       => ( ! [X4: real] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: real,Y5: real,Z3: real] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: real] :
% 9.15/5.11                  ( ( member_real @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: real] :
% 9.15/5.11                      ( ( member_real @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bot_set_real ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1619_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_nat,R: nat > nat > $o] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ! [X4: nat] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: nat,Y5: nat,Z3: nat] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: nat] :
% 9.15/5.11                  ( ( member_nat @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: nat] :
% 9.15/5.11                      ( ( member_nat @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bot_set_nat ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1620_finite__transitivity__chain,axiom,
% 9.15/5.11      ! [A2: set_int,R: int > int > $o] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ! [X4: int] :
% 9.15/5.11              ~ ( R @ X4 @ X4 )
% 9.15/5.11         => ( ! [X4: int,Y5: int,Z3: int] :
% 9.15/5.11                ( ( R @ X4 @ Y5 )
% 9.15/5.11               => ( ( R @ Y5 @ Z3 )
% 9.15/5.11                 => ( R @ X4 @ Z3 ) ) )
% 9.15/5.11           => ( ! [X4: int] :
% 9.15/5.11                  ( ( member_int @ X4 @ A2 )
% 9.15/5.11                 => ? [Y6: int] :
% 9.15/5.11                      ( ( member_int @ Y6 @ A2 )
% 9.15/5.11                      & ( R @ X4 @ Y6 ) ) )
% 9.15/5.11             => ( A2 = bot_bot_set_int ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1621_infinite__imp__nonempty,axiom,
% 9.15/5.11      ! [S3: set_complex] :
% 9.15/5.11        ( ~ ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( S3 != bot_bot_set_complex ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1622_infinite__imp__nonempty,axiom,
% 9.15/5.11      ! [S3: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ~ ( finite6177210948735845034at_nat @ S3 )
% 9.15/5.11       => ( S3 != bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1623_infinite__imp__nonempty,axiom,
% 9.15/5.11      ! [S3: set_real] :
% 9.15/5.11        ( ~ ( finite_finite_real @ S3 )
% 9.15/5.11       => ( S3 != bot_bot_set_real ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1624_infinite__imp__nonempty,axiom,
% 9.15/5.11      ! [S3: set_nat] :
% 9.15/5.11        ( ~ ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( S3 != bot_bot_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1625_infinite__imp__nonempty,axiom,
% 9.15/5.11      ! [S3: set_int] :
% 9.15/5.11        ( ~ ( finite_finite_int @ S3 )
% 9.15/5.11       => ( S3 != bot_bot_set_int ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1626_finite_OemptyI,axiom,
% 9.15/5.11      finite3207457112153483333omplex @ bot_bot_set_complex ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1627_finite_OemptyI,axiom,
% 9.15/5.11      finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1628_finite_OemptyI,axiom,
% 9.15/5.11      finite_finite_real @ bot_bot_set_real ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1629_finite_OemptyI,axiom,
% 9.15/5.11      finite_finite_nat @ bot_bot_set_nat ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1630_finite_OemptyI,axiom,
% 9.15/5.11      finite_finite_int @ bot_bot_set_int ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1631_rev__finite__subset,axiom,
% 9.15/5.11      ! [B2: set_nat,A2: set_nat] :
% 9.15/5.11        ( ( finite_finite_nat @ B2 )
% 9.15/5.11       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.15/5.11         => ( finite_finite_nat @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1632_rev__finite__subset,axiom,
% 9.15/5.11      ! [B2: set_complex,A2: set_complex] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ B2 )
% 9.15/5.11       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.15/5.11         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1633_rev__finite__subset,axiom,
% 9.15/5.11      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( finite6177210948735845034at_nat @ B2 )
% 9.15/5.11       => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.15/5.11         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1634_rev__finite__subset,axiom,
% 9.15/5.11      ! [B2: set_int,A2: set_int] :
% 9.15/5.11        ( ( finite_finite_int @ B2 )
% 9.15/5.11       => ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.11         => ( finite_finite_int @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1635_infinite__super,axiom,
% 9.15/5.11      ! [S3: set_nat,T3: set_nat] :
% 9.15/5.11        ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.15/5.11       => ( ~ ( finite_finite_nat @ S3 )
% 9.15/5.11         => ~ ( finite_finite_nat @ T3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1636_infinite__super,axiom,
% 9.15/5.11      ! [S3: set_complex,T3: set_complex] :
% 9.15/5.11        ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.15/5.11       => ( ~ ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11         => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1637_infinite__super,axiom,
% 9.15/5.11      ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( ord_le3146513528884898305at_nat @ S3 @ T3 )
% 9.15/5.11       => ( ~ ( finite6177210948735845034at_nat @ S3 )
% 9.15/5.11         => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1638_infinite__super,axiom,
% 9.15/5.11      ! [S3: set_int,T3: set_int] :
% 9.15/5.11        ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.15/5.11       => ( ~ ( finite_finite_int @ S3 )
% 9.15/5.11         => ~ ( finite_finite_int @ T3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1639_finite__subset,axiom,
% 9.15/5.11      ! [A2: set_nat,B2: set_nat] :
% 9.15/5.11        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.15/5.11       => ( ( finite_finite_nat @ B2 )
% 9.15/5.11         => ( finite_finite_nat @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1640_finite__subset,axiom,
% 9.15/5.11      ! [A2: set_complex,B2: set_complex] :
% 9.15/5.11        ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.15/5.11       => ( ( finite3207457112153483333omplex @ B2 )
% 9.15/5.11         => ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1641_finite__subset,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.15/5.11       => ( ( finite6177210948735845034at_nat @ B2 )
% 9.15/5.11         => ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1642_finite__subset,axiom,
% 9.15/5.11      ! [A2: set_int,B2: set_int] :
% 9.15/5.11        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.15/5.11       => ( ( finite_finite_int @ B2 )
% 9.15/5.11         => ( finite_finite_int @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1643_finite__psubset__induct,axiom,
% 9.15/5.11      ! [A2: set_nat,P: set_nat > $o] :
% 9.15/5.11        ( ( finite_finite_nat @ A2 )
% 9.15/5.11       => ( ! [A7: set_nat] :
% 9.15/5.11              ( ( finite_finite_nat @ A7 )
% 9.15/5.11             => ( ! [B7: set_nat] :
% 9.15/5.11                    ( ( ord_less_set_nat @ B7 @ A7 )
% 9.15/5.11                   => ( P @ B7 ) )
% 9.15/5.11               => ( P @ A7 ) ) )
% 9.15/5.11         => ( P @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1644_finite__psubset__induct,axiom,
% 9.15/5.11      ! [A2: set_int,P: set_int > $o] :
% 9.15/5.11        ( ( finite_finite_int @ A2 )
% 9.15/5.11       => ( ! [A7: set_int] :
% 9.15/5.11              ( ( finite_finite_int @ A7 )
% 9.15/5.11             => ( ! [B7: set_int] :
% 9.15/5.11                    ( ( ord_less_set_int @ B7 @ A7 )
% 9.15/5.11                   => ( P @ B7 ) )
% 9.15/5.11               => ( P @ A7 ) ) )
% 9.15/5.11         => ( P @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1645_finite__psubset__induct,axiom,
% 9.15/5.11      ! [A2: set_complex,P: set_complex > $o] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ A2 )
% 9.15/5.11       => ( ! [A7: set_complex] :
% 9.15/5.11              ( ( finite3207457112153483333omplex @ A7 )
% 9.15/5.11             => ( ! [B7: set_complex] :
% 9.15/5.11                    ( ( ord_less_set_complex @ B7 @ A7 )
% 9.15/5.11                   => ( P @ B7 ) )
% 9.15/5.11               => ( P @ A7 ) ) )
% 9.15/5.11         => ( P @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1646_finite__psubset__induct,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.15/5.11        ( ( finite6177210948735845034at_nat @ A2 )
% 9.15/5.11       => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 9.15/5.11              ( ( finite6177210948735845034at_nat @ A7 )
% 9.15/5.11             => ( ! [B7: set_Pr1261947904930325089at_nat] :
% 9.15/5.11                    ( ( ord_le7866589430770878221at_nat @ B7 @ A7 )
% 9.15/5.11                   => ( P @ B7 ) )
% 9.15/5.11               => ( P @ A7 ) ) )
% 9.15/5.11         => ( P @ A2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1647_finite__nat__set__iff__bounded,axiom,
% 9.15/5.11      ( finite_finite_nat
% 9.15/5.11      = ( ^ [N5: set_nat] :
% 9.15/5.11          ? [M2: nat] :
% 9.15/5.11          ! [X: nat] :
% 9.15/5.11            ( ( member_nat @ X @ N5 )
% 9.15/5.11           => ( ord_less_nat @ X @ M2 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1648_infinite__nat__iff__unbounded,axiom,
% 9.15/5.11      ! [S3: set_nat] :
% 9.15/5.11        ( ( ~ ( finite_finite_nat @ S3 ) )
% 9.15/5.11        = ( ! [M2: nat] :
% 9.15/5.11            ? [N: nat] :
% 9.15/5.11              ( ( ord_less_nat @ M2 @ N )
% 9.15/5.11              & ( member_nat @ N @ S3 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1649_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_complex,F: complex > real] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ~ ? [X2: complex] :
% 9.15/5.11                ( ( member_complex @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1650_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_real,F: real > real] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ~ ? [X2: real] :
% 9.15/5.11                ( ( member_real @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1651_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_nat,F: nat > real] :
% 9.15/5.11        ( ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_nat )
% 9.15/5.11         => ~ ? [X2: nat] :
% 9.15/5.11                ( ( member_nat @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1652_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_int,F: int > real] :
% 9.15/5.11        ( ( finite_finite_int @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_int )
% 9.15/5.11         => ~ ? [X2: int] :
% 9.15/5.11                ( ( member_int @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1653_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_complex,F: complex > rat] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ~ ? [X2: complex] :
% 9.15/5.11                ( ( member_complex @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1654_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_real,F: real > rat] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ~ ? [X2: real] :
% 9.15/5.11                ( ( member_real @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1655_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_nat,F: nat > rat] :
% 9.15/5.11        ( ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_nat )
% 9.15/5.11         => ~ ? [X2: nat] :
% 9.15/5.11                ( ( member_nat @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1656_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_int,F: int > rat] :
% 9.15/5.11        ( ( finite_finite_int @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_int )
% 9.15/5.11         => ~ ? [X2: int] :
% 9.15/5.11                ( ( member_int @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1657_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_complex,F: complex > num] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ~ ? [X2: complex] :
% 9.15/5.11                ( ( member_complex @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1658_arg__min__if__finite_I2_J,axiom,
% 9.15/5.11      ! [S3: set_real,F: real > num] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ~ ? [X2: real] :
% 9.15/5.11                ( ( member_real @ X2 @ S3 )
% 9.15/5.11                & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1659_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_complex,Y3: complex,F: complex > rat] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ( ( member_complex @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1660_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_real,Y3: real,F: real > rat] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ( ( member_real @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1661_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_nat,Y3: nat,F: nat > rat] :
% 9.15/5.11        ( ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_nat )
% 9.15/5.11         => ( ( member_nat @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1662_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_int,Y3: int,F: int > rat] :
% 9.15/5.11        ( ( finite_finite_int @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_int )
% 9.15/5.11         => ( ( member_int @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1663_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_complex,Y3: complex,F: complex > num] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ( ( member_complex @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1664_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_real,Y3: real,F: real > num] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ( ( member_real @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1665_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_nat,Y3: nat,F: nat > num] :
% 9.15/5.11        ( ( finite_finite_nat @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_nat )
% 9.15/5.11         => ( ( member_nat @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1666_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_int,Y3: int,F: int > num] :
% 9.15/5.11        ( ( finite_finite_int @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_int )
% 9.15/5.11         => ( ( member_int @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1667_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_complex,Y3: complex,F: complex > nat] :
% 9.15/5.11        ( ( finite3207457112153483333omplex @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_complex )
% 9.15/5.11         => ( ( member_complex @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1668_arg__min__least,axiom,
% 9.15/5.11      ! [S3: set_real,Y3: real,F: real > nat] :
% 9.15/5.11        ( ( finite_finite_real @ S3 )
% 9.15/5.11       => ( ( S3 != bot_bot_set_real )
% 9.15/5.11         => ( ( member_real @ Y3 @ S3 )
% 9.15/5.11           => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1669_nat__descend__induct,axiom,
% 9.15/5.11      ! [N2: nat,P: nat > $o,M: nat] :
% 9.15/5.11        ( ! [K2: nat] :
% 9.15/5.11            ( ( ord_less_nat @ N2 @ K2 )
% 9.15/5.11           => ( P @ K2 ) )
% 9.15/5.11       => ( ! [K2: nat] :
% 9.15/5.11              ( ( ord_less_eq_nat @ K2 @ N2 )
% 9.15/5.11             => ( ! [I4: nat] :
% 9.15/5.11                    ( ( ord_less_nat @ K2 @ I4 )
% 9.15/5.11                   => ( P @ I4 ) )
% 9.15/5.11               => ( P @ K2 ) ) )
% 9.15/5.11         => ( P @ M ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1670_subset__emptyI,axiom,
% 9.15/5.11      ! [A2: set_set_nat] :
% 9.15/5.11        ( ! [X4: set_nat] :
% 9.15/5.11            ~ ( member_set_nat @ X4 @ A2 )
% 9.15/5.11       => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1671_subset__emptyI,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ! [X4: product_prod_nat_nat] :
% 9.15/5.11            ~ ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.15/5.11       => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1672_subset__emptyI,axiom,
% 9.15/5.11      ! [A2: set_real] :
% 9.15/5.11        ( ! [X4: real] :
% 9.15/5.11            ~ ( member_real @ X4 @ A2 )
% 9.15/5.11       => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1673_subset__emptyI,axiom,
% 9.15/5.11      ! [A2: set_nat] :
% 9.15/5.11        ( ! [X4: nat] :
% 9.15/5.11            ~ ( member_nat @ X4 @ A2 )
% 9.15/5.11       => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1674_subset__emptyI,axiom,
% 9.15/5.11      ! [A2: set_int] :
% 9.15/5.11        ( ! [X4: int] :
% 9.15/5.11            ~ ( member_int @ X4 @ A2 )
% 9.15/5.11       => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1675_exists__least__lemma,axiom,
% 9.15/5.11      ! [P: nat > $o] :
% 9.15/5.11        ( ~ ( P @ zero_zero_nat )
% 9.15/5.11       => ( ? [X_12: nat] : ( P @ X_12 )
% 9.15/5.11         => ? [N3: nat] :
% 9.15/5.11              ( ~ ( P @ N3 )
% 9.15/5.11              & ( P @ ( suc @ N3 ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1676_list__decode_Ocases,axiom,
% 9.15/5.11      ! [X3: nat] :
% 9.15/5.11        ( ( X3 != zero_zero_nat )
% 9.15/5.11       => ~ ! [N3: nat] :
% 9.15/5.11              ( X3
% 9.15/5.11             != ( suc @ N3 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1677_verit__sum__simplify,axiom,
% 9.15/5.11      ! [A: complex] :
% 9.15/5.11        ( ( plus_plus_complex @ A @ zero_zero_complex )
% 9.15/5.11        = A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1678_verit__sum__simplify,axiom,
% 9.15/5.11      ! [A: real] :
% 9.15/5.11        ( ( plus_plus_real @ A @ zero_zero_real )
% 9.15/5.11        = A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1679_verit__sum__simplify,axiom,
% 9.15/5.11      ! [A: rat] :
% 9.15/5.11        ( ( plus_plus_rat @ A @ zero_zero_rat )
% 9.15/5.11        = A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1680_verit__sum__simplify,axiom,
% 9.15/5.11      ! [A: nat] :
% 9.15/5.11        ( ( plus_plus_nat @ A @ zero_zero_nat )
% 9.15/5.11        = A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1681_verit__sum__simplify,axiom,
% 9.15/5.11      ! [A: int] :
% 9.15/5.11        ( ( plus_plus_int @ A @ zero_zero_int )
% 9.15/5.11        = A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1682_add__0__iff,axiom,
% 9.15/5.11      ! [B: complex,A: complex] :
% 9.15/5.11        ( ( B
% 9.15/5.11          = ( plus_plus_complex @ B @ A ) )
% 9.15/5.11        = ( A = zero_zero_complex ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1683_add__0__iff,axiom,
% 9.15/5.11      ! [B: real,A: real] :
% 9.15/5.11        ( ( B
% 9.15/5.11          = ( plus_plus_real @ B @ A ) )
% 9.15/5.11        = ( A = zero_zero_real ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1684_add__0__iff,axiom,
% 9.15/5.11      ! [B: rat,A: rat] :
% 9.15/5.11        ( ( B
% 9.15/5.11          = ( plus_plus_rat @ B @ A ) )
% 9.15/5.11        = ( A = zero_zero_rat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1685_add__0__iff,axiom,
% 9.15/5.11      ! [B: nat,A: nat] :
% 9.15/5.11        ( ( B
% 9.15/5.11          = ( plus_plus_nat @ B @ A ) )
% 9.15/5.11        = ( A = zero_zero_nat ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1686_add__0__iff,axiom,
% 9.15/5.11      ! [B: int,A: int] :
% 9.15/5.11        ( ( B
% 9.15/5.11          = ( plus_plus_int @ B @ A ) )
% 9.15/5.11        = ( A = zero_zero_int ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1687_field__lbound__gt__zero,axiom,
% 9.15/5.11      ! [D1: real,D2: real] :
% 9.15/5.11        ( ( ord_less_real @ zero_zero_real @ D1 )
% 9.15/5.11       => ( ( ord_less_real @ zero_zero_real @ D2 )
% 9.15/5.11         => ? [E: real] :
% 9.15/5.11              ( ( ord_less_real @ zero_zero_real @ E )
% 9.15/5.11              & ( ord_less_real @ E @ D1 )
% 9.15/5.11              & ( ord_less_real @ E @ D2 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1688_field__lbound__gt__zero,axiom,
% 9.15/5.11      ! [D1: rat,D2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ zero_zero_rat @ D1 )
% 9.15/5.11       => ( ( ord_less_rat @ zero_zero_rat @ D2 )
% 9.15/5.11         => ? [E: rat] :
% 9.15/5.11              ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.15/5.11              & ( ord_less_rat @ E @ D1 )
% 9.15/5.11              & ( ord_less_rat @ E @ D2 ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1689_less__by__empty,axiom,
% 9.15/5.11      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.15/5.11        ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.15/5.11       => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1690_verit__comp__simplify1_I2_J,axiom,
% 9.15/5.11      ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1691_verit__comp__simplify1_I2_J,axiom,
% 9.15/5.11      ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1692_verit__comp__simplify1_I2_J,axiom,
% 9.15/5.11      ! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1693_verit__comp__simplify1_I2_J,axiom,
% 9.15/5.11      ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1694_verit__comp__simplify1_I2_J,axiom,
% 9.15/5.11      ! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1695_verit__la__disequality,axiom,
% 9.15/5.11      ! [A: rat,B: rat] :
% 9.15/5.11        ( ( A = B )
% 9.15/5.11        | ~ ( ord_less_eq_rat @ A @ B )
% 9.15/5.11        | ~ ( ord_less_eq_rat @ B @ A ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1696_verit__la__disequality,axiom,
% 9.15/5.11      ! [A: num,B: num] :
% 9.15/5.11        ( ( A = B )
% 9.15/5.11        | ~ ( ord_less_eq_num @ A @ B )
% 9.15/5.11        | ~ ( ord_less_eq_num @ B @ A ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1697_verit__la__disequality,axiom,
% 9.15/5.11      ! [A: nat,B: nat] :
% 9.15/5.11        ( ( A = B )
% 9.15/5.11        | ~ ( ord_less_eq_nat @ A @ B )
% 9.15/5.11        | ~ ( ord_less_eq_nat @ B @ A ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1698_verit__la__disequality,axiom,
% 9.15/5.11      ! [A: int,B: int] :
% 9.15/5.11        ( ( A = B )
% 9.15/5.11        | ~ ( ord_less_eq_int @ A @ B )
% 9.15/5.11        | ~ ( ord_less_eq_int @ B @ A ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1699_verit__comp__simplify1_I1_J,axiom,
% 9.15/5.11      ! [A: real] :
% 9.15/5.11        ~ ( ord_less_real @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1700_verit__comp__simplify1_I1_J,axiom,
% 9.15/5.11      ! [A: rat] :
% 9.15/5.11        ~ ( ord_less_rat @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1701_verit__comp__simplify1_I1_J,axiom,
% 9.15/5.11      ! [A: num] :
% 9.15/5.11        ~ ( ord_less_num @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1702_verit__comp__simplify1_I1_J,axiom,
% 9.15/5.11      ! [A: nat] :
% 9.15/5.11        ~ ( ord_less_nat @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1703_verit__comp__simplify1_I1_J,axiom,
% 9.15/5.11      ! [A: int] :
% 9.15/5.11        ~ ( ord_less_int @ A @ A ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1704_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
% 9.15/5.11        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1705_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
% 9.15/5.11        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1706_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S4: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1707_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S4: produc8923325533196201883nteger] :
% 9.15/5.11        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1708_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S4: product_prod_int_int] :
% 9.15/5.11        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1709_ssubst__Pair__rhs,axiom,
% 9.15/5.11      ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S4: product_prod_int_int] :
% 9.15/5.11        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R )
% 9.15/5.11       => ( ( S4 = S )
% 9.15/5.11         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S4 ) @ R ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1710_prod__decode__aux_Ocases,axiom,
% 9.15/5.11      ! [X3: product_prod_nat_nat] :
% 9.15/5.11        ~ ! [K2: nat,M3: nat] :
% 9.15/5.11            ( X3
% 9.15/5.11           != ( product_Pair_nat_nat @ K2 @ M3 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1711_verit__comp__simplify1_I3_J,axiom,
% 9.15/5.11      ! [B6: real,A6: real] :
% 9.15/5.11        ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
% 9.15/5.11        = ( ord_less_real @ A6 @ B6 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1712_verit__comp__simplify1_I3_J,axiom,
% 9.15/5.11      ! [B6: rat,A6: rat] :
% 9.15/5.11        ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
% 9.15/5.11        = ( ord_less_rat @ A6 @ B6 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1713_verit__comp__simplify1_I3_J,axiom,
% 9.15/5.11      ! [B6: num,A6: num] :
% 9.15/5.11        ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
% 9.15/5.11        = ( ord_less_num @ A6 @ B6 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1714_verit__comp__simplify1_I3_J,axiom,
% 9.15/5.11      ! [B6: nat,A6: nat] :
% 9.15/5.11        ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
% 9.15/5.11        = ( ord_less_nat @ A6 @ B6 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1715_verit__comp__simplify1_I3_J,axiom,
% 9.15/5.11      ! [B6: int,A6: int] :
% 9.15/5.11        ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
% 9.15/5.11        = ( ord_less_int @ A6 @ B6 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1716_triangle__Suc,axiom,
% 9.15/5.11      ! [N2: nat] :
% 9.15/5.11        ( ( nat_triangle @ ( suc @ N2 ) )
% 9.15/5.11        = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1717_less__numeral__extra_I3_J,axiom,
% 9.15/5.11      ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1718_less__numeral__extra_I3_J,axiom,
% 9.15/5.11      ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1719_less__numeral__extra_I3_J,axiom,
% 9.15/5.11      ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1720_less__numeral__extra_I3_J,axiom,
% 9.15/5.11      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1721_minf_I8_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1722_minf_I8_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1723_minf_I8_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1724_minf_I8_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1725_minf_I8_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1726_minf_I6_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1727_minf_I6_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1728_minf_I6_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1729_minf_I6_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1730_minf_I6_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1731_pinf_I8_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_eq_real @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1732_pinf_I8_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_eq_rat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1733_pinf_I8_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_eq_num @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1734_pinf_I8_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_eq_nat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1735_pinf_I8_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_eq_int @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1736_pinf_I6_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_eq_real @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1737_pinf_I6_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1738_pinf_I6_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_eq_num @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1739_pinf_I6_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1740_pinf_I6_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1741_complete__interval,axiom,
% 9.15/5.11      ! [A: real,B: real,P: real > $o] :
% 9.15/5.11        ( ( ord_less_real @ A @ B )
% 9.15/5.11       => ( ( P @ A )
% 9.15/5.11         => ( ~ ( P @ B )
% 9.15/5.11           => ? [C2: real] :
% 9.15/5.11                ( ( ord_less_eq_real @ A @ C2 )
% 9.15/5.11                & ( ord_less_eq_real @ C2 @ B )
% 9.15/5.11                & ! [X2: real] :
% 9.15/5.11                    ( ( ( ord_less_eq_real @ A @ X2 )
% 9.15/5.11                      & ( ord_less_real @ X2 @ C2 ) )
% 9.15/5.11                   => ( P @ X2 ) )
% 9.15/5.11                & ! [D3: real] :
% 9.15/5.11                    ( ! [X4: real] :
% 9.15/5.11                        ( ( ( ord_less_eq_real @ A @ X4 )
% 9.15/5.11                          & ( ord_less_real @ X4 @ D3 ) )
% 9.15/5.11                       => ( P @ X4 ) )
% 9.15/5.11                   => ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1742_complete__interval,axiom,
% 9.15/5.11      ! [A: nat,B: nat,P: nat > $o] :
% 9.15/5.11        ( ( ord_less_nat @ A @ B )
% 9.15/5.11       => ( ( P @ A )
% 9.15/5.11         => ( ~ ( P @ B )
% 9.15/5.11           => ? [C2: nat] :
% 9.15/5.11                ( ( ord_less_eq_nat @ A @ C2 )
% 9.15/5.11                & ( ord_less_eq_nat @ C2 @ B )
% 9.15/5.11                & ! [X2: nat] :
% 9.15/5.11                    ( ( ( ord_less_eq_nat @ A @ X2 )
% 9.15/5.11                      & ( ord_less_nat @ X2 @ C2 ) )
% 9.15/5.11                   => ( P @ X2 ) )
% 9.15/5.11                & ! [D3: nat] :
% 9.15/5.11                    ( ! [X4: nat] :
% 9.15/5.11                        ( ( ( ord_less_eq_nat @ A @ X4 )
% 9.15/5.11                          & ( ord_less_nat @ X4 @ D3 ) )
% 9.15/5.11                       => ( P @ X4 ) )
% 9.15/5.11                   => ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1743_complete__interval,axiom,
% 9.15/5.11      ! [A: int,B: int,P: int > $o] :
% 9.15/5.11        ( ( ord_less_int @ A @ B )
% 9.15/5.11       => ( ( P @ A )
% 9.15/5.11         => ( ~ ( P @ B )
% 9.15/5.11           => ? [C2: int] :
% 9.15/5.11                ( ( ord_less_eq_int @ A @ C2 )
% 9.15/5.11                & ( ord_less_eq_int @ C2 @ B )
% 9.15/5.11                & ! [X2: int] :
% 9.15/5.11                    ( ( ( ord_less_eq_int @ A @ X2 )
% 9.15/5.11                      & ( ord_less_int @ X2 @ C2 ) )
% 9.15/5.11                   => ( P @ X2 ) )
% 9.15/5.11                & ! [D3: int] :
% 9.15/5.11                    ( ! [X4: int] :
% 9.15/5.11                        ( ( ( ord_less_eq_int @ A @ X4 )
% 9.15/5.11                          & ( ord_less_int @ X4 @ D3 ) )
% 9.15/5.11                       => ( P @ X4 ) )
% 9.15/5.11                   => ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1744_le__numeral__extra_I3_J,axiom,
% 9.15/5.11      ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1745_le__numeral__extra_I3_J,axiom,
% 9.15/5.11      ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1746_le__numeral__extra_I3_J,axiom,
% 9.15/5.11      ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1747_le__numeral__extra_I3_J,axiom,
% 9.15/5.11      ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1748_find__Some__iff,axiom,
% 9.15/5.11      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
% 9.15/5.11        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.15/5.11          = ( some_P7363390416028606310at_nat @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1749_find__Some__iff,axiom,
% 9.15/5.11      ! [P: num > $o,Xs: list_num,X3: num] :
% 9.15/5.11        ( ( ( find_num @ P @ Xs )
% 9.15/5.11          = ( some_num @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_num @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_num @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1750_find__Some__iff,axiom,
% 9.15/5.11      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.11        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.15/5.11          = ( some_VEBT_VEBT @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1751_find__Some__iff,axiom,
% 9.15/5.11      ! [P: $o > $o,Xs: list_o,X3: $o] :
% 9.15/5.11        ( ( ( find_o @ P @ Xs )
% 9.15/5.11          = ( some_o @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_o @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_o @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1752_find__Some__iff,axiom,
% 9.15/5.11      ! [P: nat > $o,Xs: list_nat,X3: nat] :
% 9.15/5.11        ( ( ( find_nat @ P @ Xs )
% 9.15/5.11          = ( some_nat @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_nat @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_nat @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1753_find__Some__iff,axiom,
% 9.15/5.11      ! [P: int > $o,Xs: list_int,X3: int] :
% 9.15/5.11        ( ( ( find_int @ P @ Xs )
% 9.15/5.11          = ( some_int @ X3 ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_int @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_int @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1754_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.11        ( ( ( some_P7363390416028606310at_nat @ X3 )
% 9.15/5.11          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1755_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: num,P: num > $o,Xs: list_num] :
% 9.15/5.11        ( ( ( some_num @ X3 )
% 9.15/5.11          = ( find_num @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_num @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_num @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1756_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.15/5.11        ( ( ( some_VEBT_VEBT @ X3 )
% 9.15/5.11          = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_VEBT_VEBT @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1757_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: $o,P: $o > $o,Xs: list_o] :
% 9.15/5.11        ( ( ( some_o @ X3 )
% 9.15/5.11          = ( find_o @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_o @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_o @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1758_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: nat,P: nat > $o,Xs: list_nat] :
% 9.15/5.11        ( ( ( some_nat @ X3 )
% 9.15/5.11          = ( find_nat @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_nat @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_nat @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1759_find__Some__iff2,axiom,
% 9.15/5.11      ! [X3: int,P: int > $o,Xs: list_int] :
% 9.15/5.11        ( ( ( some_int @ X3 )
% 9.15/5.11          = ( find_int @ P @ Xs ) )
% 9.15/5.11        = ( ? [I2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
% 9.15/5.11              & ( P @ ( nth_int @ Xs @ I2 ) )
% 9.15/5.11              & ( X3
% 9.15/5.11                = ( nth_int @ Xs @ I2 ) )
% 9.15/5.11              & ! [J3: nat] :
% 9.15/5.11                  ( ( ord_less_nat @ J3 @ I2 )
% 9.15/5.11                 => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1760_triangle__0,axiom,
% 9.15/5.11      ( ( nat_triangle @ zero_zero_nat )
% 9.15/5.11      = zero_zero_nat ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1761_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: product_prod_nat_nat] :
% 9.15/5.11              ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.15/5.11            = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1762_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: real] :
% 9.15/5.11              ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_real @ P @ Xs )
% 9.15/5.11            = ( find_real @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1763_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: set_nat] :
% 9.15/5.11              ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_set_nat @ P @ Xs )
% 9.15/5.11            = ( find_set_nat @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1764_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: vEBT_VEBT] :
% 9.15/5.11              ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_VEBT_VEBT @ P @ Xs )
% 9.15/5.11            = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1765_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: int] :
% 9.15/5.11              ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_int @ P @ Xs )
% 9.15/5.11            = ( find_int @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1766_find__cong,axiom,
% 9.15/5.11      ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
% 9.15/5.11        ( ( Xs = Ys )
% 9.15/5.11       => ( ! [X4: nat] :
% 9.15/5.11              ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
% 9.15/5.11             => ( ( P @ X4 )
% 9.15/5.11                = ( Q @ X4 ) ) )
% 9.15/5.11         => ( ( find_nat @ P @ Xs )
% 9.15/5.11            = ( find_nat @ Q @ Ys ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1767_ex__gt__or__lt,axiom,
% 9.15/5.11      ! [A: real] :
% 9.15/5.11      ? [B4: real] :
% 9.15/5.11        ( ( ord_less_real @ A @ B4 )
% 9.15/5.11        | ( ord_less_real @ B4 @ A ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1768_pinf_I1_J,axiom,
% 9.15/5.11      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 9.15/5.11        ( ? [Z4: real] :
% 9.15/5.11          ! [X4: real] :
% 9.15/5.11            ( ( ord_less_real @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: real] :
% 9.15/5.11            ! [X4: real] :
% 9.15/5.11              ( ( ord_less_real @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: real] :
% 9.15/5.11            ! [X2: real] :
% 9.15/5.11              ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1769_pinf_I1_J,axiom,
% 9.15/5.11      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 9.15/5.11        ( ? [Z4: rat] :
% 9.15/5.11          ! [X4: rat] :
% 9.15/5.11            ( ( ord_less_rat @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: rat] :
% 9.15/5.11            ! [X4: rat] :
% 9.15/5.11              ( ( ord_less_rat @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: rat] :
% 9.15/5.11            ! [X2: rat] :
% 9.15/5.11              ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1770_pinf_I1_J,axiom,
% 9.15/5.11      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 9.15/5.11        ( ? [Z4: num] :
% 9.15/5.11          ! [X4: num] :
% 9.15/5.11            ( ( ord_less_num @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: num] :
% 9.15/5.11            ! [X4: num] :
% 9.15/5.11              ( ( ord_less_num @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: num] :
% 9.15/5.11            ! [X2: num] :
% 9.15/5.11              ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1771_pinf_I1_J,axiom,
% 9.15/5.11      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 9.15/5.11        ( ? [Z4: nat] :
% 9.15/5.11          ! [X4: nat] :
% 9.15/5.11            ( ( ord_less_nat @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: nat] :
% 9.15/5.11            ! [X4: nat] :
% 9.15/5.11              ( ( ord_less_nat @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: nat] :
% 9.15/5.11            ! [X2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1772_pinf_I1_J,axiom,
% 9.15/5.11      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 9.15/5.11        ( ? [Z4: int] :
% 9.15/5.11          ! [X4: int] :
% 9.15/5.11            ( ( ord_less_int @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: int] :
% 9.15/5.11            ! [X4: int] :
% 9.15/5.11              ( ( ord_less_int @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: int] :
% 9.15/5.11            ! [X2: int] :
% 9.15/5.11              ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1773_pinf_I2_J,axiom,
% 9.15/5.11      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 9.15/5.11        ( ? [Z4: real] :
% 9.15/5.11          ! [X4: real] :
% 9.15/5.11            ( ( ord_less_real @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: real] :
% 9.15/5.11            ! [X4: real] :
% 9.15/5.11              ( ( ord_less_real @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: real] :
% 9.15/5.11            ! [X2: real] :
% 9.15/5.11              ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1774_pinf_I2_J,axiom,
% 9.15/5.11      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 9.15/5.11        ( ? [Z4: rat] :
% 9.15/5.11          ! [X4: rat] :
% 9.15/5.11            ( ( ord_less_rat @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: rat] :
% 9.15/5.11            ! [X4: rat] :
% 9.15/5.11              ( ( ord_less_rat @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: rat] :
% 9.15/5.11            ! [X2: rat] :
% 9.15/5.11              ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1775_pinf_I2_J,axiom,
% 9.15/5.11      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 9.15/5.11        ( ? [Z4: num] :
% 9.15/5.11          ! [X4: num] :
% 9.15/5.11            ( ( ord_less_num @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: num] :
% 9.15/5.11            ! [X4: num] :
% 9.15/5.11              ( ( ord_less_num @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: num] :
% 9.15/5.11            ! [X2: num] :
% 9.15/5.11              ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1776_pinf_I2_J,axiom,
% 9.15/5.11      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 9.15/5.11        ( ? [Z4: nat] :
% 9.15/5.11          ! [X4: nat] :
% 9.15/5.11            ( ( ord_less_nat @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: nat] :
% 9.15/5.11            ! [X4: nat] :
% 9.15/5.11              ( ( ord_less_nat @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: nat] :
% 9.15/5.11            ! [X2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1777_pinf_I2_J,axiom,
% 9.15/5.11      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 9.15/5.11        ( ? [Z4: int] :
% 9.15/5.11          ! [X4: int] :
% 9.15/5.11            ( ( ord_less_int @ Z4 @ X4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: int] :
% 9.15/5.11            ! [X4: int] :
% 9.15/5.11              ( ( ord_less_int @ Z4 @ X4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: int] :
% 9.15/5.11            ! [X2: int] :
% 9.15/5.11              ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1778_pinf_I3_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1779_pinf_I3_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1780_pinf_I3_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1781_pinf_I3_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1782_pinf_I3_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1783_pinf_I4_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1784_pinf_I4_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1785_pinf_I4_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1786_pinf_I4_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1787_pinf_I4_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1788_pinf_I5_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_real @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1789_pinf_I5_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_rat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1790_pinf_I5_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_num @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1791_pinf_I5_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_nat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1792_pinf_I5_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ~ ( ord_less_int @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1793_pinf_I7_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_real @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1794_pinf_I7_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_rat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1795_pinf_I7_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_num @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1796_pinf_I7_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_nat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1797_pinf_I7_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ Z3 @ X2 )
% 9.15/5.11       => ( ord_less_int @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1798_minf_I1_J,axiom,
% 9.15/5.11      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 9.15/5.11        ( ? [Z4: real] :
% 9.15/5.11          ! [X4: real] :
% 9.15/5.11            ( ( ord_less_real @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: real] :
% 9.15/5.11            ! [X4: real] :
% 9.15/5.11              ( ( ord_less_real @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: real] :
% 9.15/5.11            ! [X2: real] :
% 9.15/5.11              ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1799_minf_I1_J,axiom,
% 9.15/5.11      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 9.15/5.11        ( ? [Z4: rat] :
% 9.15/5.11          ! [X4: rat] :
% 9.15/5.11            ( ( ord_less_rat @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: rat] :
% 9.15/5.11            ! [X4: rat] :
% 9.15/5.11              ( ( ord_less_rat @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: rat] :
% 9.15/5.11            ! [X2: rat] :
% 9.15/5.11              ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1800_minf_I1_J,axiom,
% 9.15/5.11      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 9.15/5.11        ( ? [Z4: num] :
% 9.15/5.11          ! [X4: num] :
% 9.15/5.11            ( ( ord_less_num @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: num] :
% 9.15/5.11            ! [X4: num] :
% 9.15/5.11              ( ( ord_less_num @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: num] :
% 9.15/5.11            ! [X2: num] :
% 9.15/5.11              ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1801_minf_I1_J,axiom,
% 9.15/5.11      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 9.15/5.11        ( ? [Z4: nat] :
% 9.15/5.11          ! [X4: nat] :
% 9.15/5.11            ( ( ord_less_nat @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: nat] :
% 9.15/5.11            ! [X4: nat] :
% 9.15/5.11              ( ( ord_less_nat @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: nat] :
% 9.15/5.11            ! [X2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1802_minf_I1_J,axiom,
% 9.15/5.11      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 9.15/5.11        ( ? [Z4: int] :
% 9.15/5.11          ! [X4: int] :
% 9.15/5.11            ( ( ord_less_int @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: int] :
% 9.15/5.11            ! [X4: int] :
% 9.15/5.11              ( ( ord_less_int @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: int] :
% 9.15/5.11            ! [X2: int] :
% 9.15/5.11              ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  & ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  & ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1803_minf_I2_J,axiom,
% 9.15/5.11      ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
% 9.15/5.11        ( ? [Z4: real] :
% 9.15/5.11          ! [X4: real] :
% 9.15/5.11            ( ( ord_less_real @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: real] :
% 9.15/5.11            ! [X4: real] :
% 9.15/5.11              ( ( ord_less_real @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: real] :
% 9.15/5.11            ! [X2: real] :
% 9.15/5.11              ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1804_minf_I2_J,axiom,
% 9.15/5.11      ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
% 9.15/5.11        ( ? [Z4: rat] :
% 9.15/5.11          ! [X4: rat] :
% 9.15/5.11            ( ( ord_less_rat @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: rat] :
% 9.15/5.11            ! [X4: rat] :
% 9.15/5.11              ( ( ord_less_rat @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: rat] :
% 9.15/5.11            ! [X2: rat] :
% 9.15/5.11              ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1805_minf_I2_J,axiom,
% 9.15/5.11      ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
% 9.15/5.11        ( ? [Z4: num] :
% 9.15/5.11          ! [X4: num] :
% 9.15/5.11            ( ( ord_less_num @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: num] :
% 9.15/5.11            ! [X4: num] :
% 9.15/5.11              ( ( ord_less_num @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: num] :
% 9.15/5.11            ! [X2: num] :
% 9.15/5.11              ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1806_minf_I2_J,axiom,
% 9.15/5.11      ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
% 9.15/5.11        ( ? [Z4: nat] :
% 9.15/5.11          ! [X4: nat] :
% 9.15/5.11            ( ( ord_less_nat @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: nat] :
% 9.15/5.11            ! [X4: nat] :
% 9.15/5.11              ( ( ord_less_nat @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: nat] :
% 9.15/5.11            ! [X2: nat] :
% 9.15/5.11              ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1807_minf_I2_J,axiom,
% 9.15/5.11      ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
% 9.15/5.11        ( ? [Z4: int] :
% 9.15/5.11          ! [X4: int] :
% 9.15/5.11            ( ( ord_less_int @ X4 @ Z4 )
% 9.15/5.11           => ( ( P @ X4 )
% 9.15/5.11              = ( P5 @ X4 ) ) )
% 9.15/5.11       => ( ? [Z4: int] :
% 9.15/5.11            ! [X4: int] :
% 9.15/5.11              ( ( ord_less_int @ X4 @ Z4 )
% 9.15/5.11             => ( ( Q @ X4 )
% 9.15/5.11                = ( Q3 @ X4 ) ) )
% 9.15/5.11         => ? [Z3: int] :
% 9.15/5.11            ! [X2: int] :
% 9.15/5.11              ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11             => ( ( ( P @ X2 )
% 9.15/5.11                  | ( Q @ X2 ) )
% 9.15/5.11                = ( ( P5 @ X2 )
% 9.15/5.11                  | ( Q3 @ X2 ) ) ) ) ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1808_minf_I3_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1809_minf_I3_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1810_minf_I3_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1811_minf_I3_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1812_minf_I3_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1813_minf_I4_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1814_minf_I4_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1815_minf_I4_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1816_minf_I4_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1817_minf_I4_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11       => ( X2 != T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1818_minf_I5_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_real @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1819_minf_I5_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_rat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1820_minf_I5_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_num @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1821_minf_I5_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.11      ? [Z3: nat] :
% 9.15/5.11      ! [X2: nat] :
% 9.15/5.11        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_nat @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1822_minf_I5_J,axiom,
% 9.15/5.11      ! [T: int] :
% 9.15/5.11      ? [Z3: int] :
% 9.15/5.11      ! [X2: int] :
% 9.15/5.11        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.11       => ( ord_less_int @ X2 @ T ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1823_minf_I7_J,axiom,
% 9.15/5.11      ! [T: real] :
% 9.15/5.11      ? [Z3: real] :
% 9.15/5.11      ! [X2: real] :
% 9.15/5.11        ( ( ord_less_real @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_real @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1824_minf_I7_J,axiom,
% 9.15/5.11      ! [T: rat] :
% 9.15/5.11      ? [Z3: rat] :
% 9.15/5.11      ! [X2: rat] :
% 9.15/5.11        ( ( ord_less_rat @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_rat @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1825_minf_I7_J,axiom,
% 9.15/5.11      ! [T: num] :
% 9.15/5.11      ? [Z3: num] :
% 9.15/5.11      ! [X2: num] :
% 9.15/5.11        ( ( ord_less_num @ X2 @ Z3 )
% 9.15/5.11       => ~ ( ord_less_num @ T @ X2 ) ) ).
% 9.15/5.11  
% 9.15/5.11  thf(fact_1826_minf_I7_J,axiom,
% 9.15/5.11      ! [T: nat] :
% 9.15/5.12      ? [Z3: nat] :
% 9.15/5.12      ! [X2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ X2 @ Z3 )
% 9.15/5.12       => ~ ( ord_less_nat @ T @ X2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1827_minf_I7_J,axiom,
% 9.15/5.12      ! [T: int] :
% 9.15/5.12      ? [Z3: int] :
% 9.15/5.12      ! [X2: int] :
% 9.15/5.12        ( ( ord_less_int @ X2 @ Z3 )
% 9.15/5.12       => ~ ( ord_less_int @ T @ X2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1828_is__num__normalize_I1_J,axiom,
% 9.15/5.12      ! [A: real,B: real,C: real] :
% 9.15/5.12        ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.15/5.12        = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1829_is__num__normalize_I1_J,axiom,
% 9.15/5.12      ! [A: rat,B: rat,C: rat] :
% 9.15/5.12        ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.15/5.12        = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1830_is__num__normalize_I1_J,axiom,
% 9.15/5.12      ! [A: int,B: int,C: int] :
% 9.15/5.12        ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.15/5.12        = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1831_find__None__iff2,axiom,
% 9.15/5.12      ! [P: real > $o,Xs: list_real] :
% 9.15/5.12        ( ( none_real
% 9.15/5.12          = ( find_real @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: real] :
% 9.15/5.12                ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1832_find__None__iff2,axiom,
% 9.15/5.12      ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.15/5.12        ( ( none_set_nat
% 9.15/5.12          = ( find_set_nat @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: set_nat] :
% 9.15/5.12                ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1833_find__None__iff2,axiom,
% 9.15/5.12      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.15/5.12        ( ( none_VEBT_VEBT
% 9.15/5.12          = ( find_VEBT_VEBT @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: vEBT_VEBT] :
% 9.15/5.12                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1834_find__None__iff2,axiom,
% 9.15/5.12      ! [P: int > $o,Xs: list_int] :
% 9.15/5.12        ( ( none_int
% 9.15/5.12          = ( find_int @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: int] :
% 9.15/5.12                ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1835_find__None__iff2,axiom,
% 9.15/5.12      ! [P: nat > $o,Xs: list_nat] :
% 9.15/5.12        ( ( none_nat
% 9.15/5.12          = ( find_nat @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: nat] :
% 9.15/5.12                ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1836_find__None__iff2,axiom,
% 9.15/5.12      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.12        ( ( none_P5556105721700978146at_nat
% 9.15/5.12          = ( find_P8199882355184865565at_nat @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: product_prod_nat_nat] :
% 9.15/5.12                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1837_find__None__iff2,axiom,
% 9.15/5.12      ! [P: num > $o,Xs: list_num] :
% 9.15/5.12        ( ( none_num
% 9.15/5.12          = ( find_num @ P @ Xs ) )
% 9.15/5.12        = ( ~ ? [X: num] :
% 9.15/5.12                ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1838_find__None__iff,axiom,
% 9.15/5.12      ! [P: real > $o,Xs: list_real] :
% 9.15/5.12        ( ( ( find_real @ P @ Xs )
% 9.15/5.12          = none_real )
% 9.15/5.12        = ( ~ ? [X: real] :
% 9.15/5.12                ( ( member_real @ X @ ( set_real2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1839_find__None__iff,axiom,
% 9.15/5.12      ! [P: set_nat > $o,Xs: list_set_nat] :
% 9.15/5.12        ( ( ( find_set_nat @ P @ Xs )
% 9.15/5.12          = none_set_nat )
% 9.15/5.12        = ( ~ ? [X: set_nat] :
% 9.15/5.12                ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1840_find__None__iff,axiom,
% 9.15/5.12      ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
% 9.15/5.12        ( ( ( find_VEBT_VEBT @ P @ Xs )
% 9.15/5.12          = none_VEBT_VEBT )
% 9.15/5.12        = ( ~ ? [X: vEBT_VEBT] :
% 9.15/5.12                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1841_find__None__iff,axiom,
% 9.15/5.12      ! [P: int > $o,Xs: list_int] :
% 9.15/5.12        ( ( ( find_int @ P @ Xs )
% 9.15/5.12          = none_int )
% 9.15/5.12        = ( ~ ? [X: int] :
% 9.15/5.12                ( ( member_int @ X @ ( set_int2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1842_find__None__iff,axiom,
% 9.15/5.12      ! [P: nat > $o,Xs: list_nat] :
% 9.15/5.12        ( ( ( find_nat @ P @ Xs )
% 9.15/5.12          = none_nat )
% 9.15/5.12        = ( ~ ? [X: nat] :
% 9.15/5.12                ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1843_find__None__iff,axiom,
% 9.15/5.12      ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.12        ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
% 9.15/5.12          = none_P5556105721700978146at_nat )
% 9.15/5.12        = ( ~ ? [X: product_prod_nat_nat] :
% 9.15/5.12                ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1844_find__None__iff,axiom,
% 9.15/5.12      ! [P: num > $o,Xs: list_num] :
% 9.15/5.12        ( ( ( find_num @ P @ Xs )
% 9.15/5.12          = none_num )
% 9.15/5.12        = ( ~ ? [X: num] :
% 9.15/5.12                ( ( member_num @ X @ ( set_num2 @ Xs ) )
% 9.15/5.12                & ( P @ X ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1845_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1846_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1847_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1848_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1849_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1850_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_o,Ys: list_o] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.15/5.12         => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( product_Pair_o_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1851_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_o,Ys: list_nat] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( product_Pair_o_nat @ ( nth_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1852_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_o,Ys: list_int] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1853_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1854_nth__zip,axiom,
% 9.15/5.12      ! [I: nat,Xs: list_nat,Ys: list_o] :
% 9.15/5.12        ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
% 9.15/5.12       => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
% 9.15/5.12         => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys ) @ I )
% 9.15/5.12            = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1855__C5_C_I10_J,axiom,
% 9.15/5.12      ( ( mi != ma )
% 9.15/5.12     => ! [I4: nat] :
% 9.15/5.12          ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.15/5.12         => ( ( ( ( vEBT_VEBT_high @ ma @ n )
% 9.15/5.12                = I4 )
% 9.15/5.12             => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ n ) ) )
% 9.15/5.12            & ! [X2: nat] :
% 9.15/5.12                ( ( ( ( vEBT_VEBT_high @ X2 @ n )
% 9.15/5.12                    = I4 )
% 9.15/5.12                  & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) )
% 9.15/5.12               => ( ( ord_less_nat @ mi @ X2 )
% 9.15/5.12                  & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1856_vebt__member_Ocases,axiom,
% 9.15/5.12      ! [X3: produc9072475918466114483BT_nat] :
% 9.15/5.12        ( ! [A4: $o,B4: $o,X4: nat] :
% 9.15/5.12            ( X3
% 9.15/5.12           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 9.15/5.12       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
% 9.15/5.12         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
% 9.15/5.12           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                  ( X3
% 9.15/5.12                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
% 9.15/5.12             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                    ( X3
% 9.15/5.12                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1857_VEBT__internal_Omembermima_Ocases,axiom,
% 9.15/5.12      ! [X3: produc9072475918466114483BT_nat] :
% 9.15/5.12        ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
% 9.15/5.12            ( X3
% 9.15/5.12           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
% 9.15/5.12       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
% 9.15/5.12         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
% 9.15/5.12           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                  ( X3
% 9.15/5.12                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
% 9.15/5.12             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
% 9.15/5.12                    ( X3
% 9.15/5.12                   != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X4 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1858_count__notin,axiom,
% 9.15/5.12      ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.12        ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.12       => ( ( count_4203492906077236349at_nat @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1859_count__notin,axiom,
% 9.15/5.12      ! [X3: real,Xs: list_real] :
% 9.15/5.12        ( ~ ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12       => ( ( count_list_real @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1860_count__notin,axiom,
% 9.15/5.12      ! [X3: set_nat,Xs: list_set_nat] :
% 9.15/5.12        ( ~ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
% 9.15/5.12       => ( ( count_list_set_nat @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1861_count__notin,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.15/5.12        ( ~ ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12       => ( ( count_list_VEBT_VEBT @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1862_count__notin,axiom,
% 9.15/5.12      ! [X3: int,Xs: list_int] :
% 9.15/5.12        ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.12       => ( ( count_list_int @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1863_count__notin,axiom,
% 9.15/5.12      ! [X3: nat,Xs: list_nat] :
% 9.15/5.12        ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
% 9.15/5.12       => ( ( count_list_nat @ Xs @ X3 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1864_listrel1__iff__update,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 9.15/5.12        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 9.15/5.12        = ( ? [Y: vEBT_VEBT,N: nat] :
% 9.15/5.12              ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ Y ) @ R2 )
% 9.15/5.12              & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
% 9.15/5.12              & ( Ys
% 9.15/5.12                = ( list_u1324408373059187874T_VEBT @ Xs @ N @ Y ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1865_listrel1__iff__update,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 9.15/5.12        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 9.15/5.12        = ( ? [Y: $o,N: nat] :
% 9.15/5.12              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N ) @ Y ) @ R2 )
% 9.15/5.12              & ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
% 9.15/5.12              & ( Ys
% 9.15/5.12                = ( list_update_o @ Xs @ N @ Y ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1866_listrel1__iff__update,axiom,
% 9.15/5.12      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 9.15/5.12        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 9.15/5.12        = ( ? [Y: nat,N: nat] :
% 9.15/5.12              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ Y ) @ R2 )
% 9.15/5.12              & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
% 9.15/5.12              & ( Ys
% 9.15/5.12                = ( list_update_nat @ Xs @ N @ Y ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1867_listrel1__iff__update,axiom,
% 9.15/5.12      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 9.15/5.12        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 9.15/5.12        = ( ? [Y: int,N: nat] :
% 9.15/5.12              ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ Y ) @ R2 )
% 9.15/5.12              & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
% 9.15/5.12              & ( Ys
% 9.15/5.12                = ( list_update_int @ Xs @ N @ Y ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1868_pair__lessI2,axiom,
% 9.15/5.12      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.15/5.12        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.12       => ( ( ord_less_nat @ S @ T )
% 9.15/5.12         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1869_pair__less__iff1,axiom,
% 9.15/5.12      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.12        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ X3 @ Z2 ) ) @ fun_pair_less )
% 9.15/5.12        = ( ord_less_nat @ Y3 @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1870_gen__length__def,axiom,
% 9.15/5.12      ( gen_length_VEBT_VEBT
% 9.15/5.12      = ( ^ [N: nat,Xs2: list_VEBT_VEBT] : ( plus_plus_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1871_gen__length__def,axiom,
% 9.15/5.12      ( gen_length_o
% 9.15/5.12      = ( ^ [N: nat,Xs2: list_o] : ( plus_plus_nat @ N @ ( size_size_list_o @ Xs2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1872_gen__length__def,axiom,
% 9.15/5.12      ( gen_length_nat
% 9.15/5.12      = ( ^ [N: nat,Xs2: list_nat] : ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1873_gen__length__def,axiom,
% 9.15/5.12      ( gen_length_int
% 9.15/5.12      = ( ^ [N: nat,Xs2: list_int] : ( plus_plus_nat @ N @ ( size_size_list_int @ Xs2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1874_set__encode__empty,axiom,
% 9.15/5.12      ( ( nat_set_encode @ bot_bot_set_nat )
% 9.15/5.12      = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1875_Leaf__0__not,axiom,
% 9.15/5.12      ! [A: $o,B: $o] :
% 9.15/5.12        ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1876_power__shift,axiom,
% 9.15/5.12      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.15/5.12        ( ( ( power_power_nat @ X3 @ Y3 )
% 9.15/5.12          = Z2 )
% 9.15/5.12        = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.15/5.12          = ( some_nat @ Z2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1877__C5_C_I3_J,axiom,
% 9.15/5.12      ( ( size_s6755466524823107622T_VEBT @ treeList )
% 9.15/5.12      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1878__C5_C_I9_J,axiom,
% 9.15/5.12      ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1879_VEBT_Oinject_I2_J,axiom,
% 9.15/5.12      ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
% 9.15/5.12        ( ( ( vEBT_Leaf @ X21 @ X222 )
% 9.15/5.12          = ( vEBT_Leaf @ Y21 @ Y22 ) )
% 9.15/5.12        = ( ( X21 = Y21 )
% 9.15/5.12          & ( X222 = Y22 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1880_local_Opower__def,axiom,
% 9.15/5.12      ( vEBT_VEBT_power
% 9.15/5.12      = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1881_high__bound__aux,axiom,
% 9.15/5.12      ! [Ma: nat,N2: nat,M: nat] :
% 9.15/5.12        ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 9.15/5.12       => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1882_member__bound,axiom,
% 9.15/5.12      ! [Tree: vEBT_VEBT,X3: nat,N2: nat] :
% 9.15/5.12        ( ( vEBT_vebt_member @ Tree @ X3 )
% 9.15/5.12       => ( ( vEBT_invar_vebt @ Tree @ N2 )
% 9.15/5.12         => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1883__C5_C_I6_J,axiom,
% 9.15/5.12      ! [I4: nat] :
% 9.15/5.12        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
% 9.15/5.12       => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X5 ) )
% 9.15/5.12          = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1884_numeral__le__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.15/5.12        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1885_numeral__le__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 9.15/5.12        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1886_numeral__le__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 9.15/5.12        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1887_numeral__le__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.15/5.12        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1888_numeral__le__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.15/5.12        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1889_numeral__less__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.15/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1890_numeral__less__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 9.15/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1891_numeral__less__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 9.15/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1892_numeral__less__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.15/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1893_numeral__less__iff,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.15/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1894_numeral__plus__numeral,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.15/5.12        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1895_numeral__plus__numeral,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
% 9.15/5.12        = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1896_numeral__plus__numeral,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
% 9.15/5.12        = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1897_numeral__plus__numeral,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.15/5.12        = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1898_numeral__plus__numeral,axiom,
% 9.15/5.12      ! [M: num,N2: num] :
% 9.15/5.12        ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.15/5.12        = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1899_add__numeral__left,axiom,
% 9.15/5.12      ! [V: num,W2: num,Z2: extended_enat] :
% 9.15/5.12        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
% 9.15/5.12        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1900_add__numeral__left,axiom,
% 9.15/5.12      ! [V: num,W2: num,Z2: real] :
% 9.15/5.12        ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
% 9.15/5.12        = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1901_add__numeral__left,axiom,
% 9.15/5.12      ! [V: num,W2: num,Z2: rat] :
% 9.15/5.12        ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
% 9.15/5.12        = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1902_add__numeral__left,axiom,
% 9.15/5.12      ! [V: num,W2: num,Z2: nat] :
% 9.15/5.12        ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
% 9.15/5.12        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1903_add__numeral__left,axiom,
% 9.15/5.12      ! [V: num,W2: num,Z2: int] :
% 9.15/5.12        ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
% 9.15/5.12        = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1904_mi__ma__2__deg,axiom,
% 9.15/5.12      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat] :
% 9.15/5.12        ( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ N2 )
% 9.15/5.12       => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.15/5.12          & ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1905_Suc__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( suc @ ( numeral_numeral_nat @ N2 ) )
% 9.15/5.12        = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1906_add__2__eq__Suc_H,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( plus_plus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12        = ( suc @ ( suc @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1907_add__2__eq__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.15/5.12        = ( suc @ ( suc @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1908_numeral__Bit0,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
% 9.15/5.12        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1909_numeral__Bit0,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 9.15/5.12        = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1910_numeral__Bit0,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 9.15/5.12        = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1911_numeral__Bit0,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 9.15/5.12        = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1912_numeral__Bit0,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 9.15/5.12        = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1913_numeral__2__eq__2,axiom,
% 9.15/5.12      ( ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.15/5.12      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1914_less__2__cases,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12       => ( ( N2 = zero_zero_nat )
% 9.15/5.12          | ( N2
% 9.15/5.12            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1915_less__2__cases__iff,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12        = ( ( N2 = zero_zero_nat )
% 9.15/5.12          | ( N2
% 9.15/5.12            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1916_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
% 9.15/5.12      ! [X3: nat,N2: nat,M: nat] :
% 9.15/5.12        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 9.15/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.15/5.12           => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1917_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
% 9.15/5.12      ! [X3: nat,N2: nat,M: nat] :
% 9.15/5.12        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
% 9.15/5.12       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.15/5.12           => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1918_numeral__1__eq__Suc__0,axiom,
% 9.15/5.12      ( ( numeral_numeral_nat @ one )
% 9.15/5.12      = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1919_Suc__nat__number__of__add,axiom,
% 9.15/5.12      ! [V: num,N2: nat] :
% 9.15/5.12        ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
% 9.15/5.12        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1920_VEBT_Osize_I4_J,axiom,
% 9.15/5.12      ! [X21: $o,X222: $o] :
% 9.15/5.12        ( ( size_size_VEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.15/5.12        = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1921_VEBT__internal_Ovalid_H_Ocases,axiom,
% 9.15/5.12      ! [X3: produc9072475918466114483BT_nat] :
% 9.15/5.12        ( ! [Uu2: $o,Uv2: $o,D4: nat] :
% 9.15/5.12            ( X3
% 9.15/5.12           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
% 9.15/5.12       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1922_VEBT_Odistinct_I1_J,axiom,
% 9.15/5.12      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
% 9.15/5.12        ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
% 9.15/5.12       != ( vEBT_Leaf @ X21 @ X222 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1923_VEBT_Oexhaust,axiom,
% 9.15/5.12      ! [Y3: vEBT_VEBT] :
% 9.15/5.12        ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
% 9.15/5.12            ( Y3
% 9.15/5.12           != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
% 9.15/5.12       => ~ ! [X212: $o,X223: $o] :
% 9.15/5.12              ( Y3
% 9.15/5.12             != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1924_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_zero_complex
% 9.15/5.12       != ( numera6690914467698888265omplex @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1925_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_z5237406670263579293d_enat
% 9.15/5.12       != ( numera1916890842035813515d_enat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1926_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_zero_real
% 9.15/5.12       != ( numeral_numeral_real @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1927_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_zero_rat
% 9.15/5.12       != ( numeral_numeral_rat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1928_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_zero_nat
% 9.15/5.12       != ( numeral_numeral_nat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1929_zero__neq__numeral,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ( zero_zero_int
% 9.15/5.12       != ( numeral_numeral_int @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1930_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
% 9.15/5.12      ! [Uu: $o,Uv: $o,Uw: nat] :
% 9.15/5.12        ~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1931_num_Osize_I4_J,axiom,
% 9.15/5.12      ( ( size_size_num @ one )
% 9.15/5.12      = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1932_listrel1__eq__len,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,R2: set_Pr6192946355708809607T_VEBT] :
% 9.15/5.12        ( ( member4439316823752958928T_VEBT @ ( produc3897820843166775703T_VEBT @ Xs @ Ys ) @ ( listrel1_VEBT_VEBT @ R2 ) )
% 9.15/5.12       => ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1933_listrel1__eq__len,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
% 9.15/5.12        ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
% 9.15/5.12       => ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1934_listrel1__eq__len,axiom,
% 9.15/5.12      ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
% 9.15/5.12        ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
% 9.15/5.12       => ( ( size_size_list_nat @ Xs )
% 9.15/5.12          = ( size_size_list_nat @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1935_listrel1__eq__len,axiom,
% 9.15/5.12      ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
% 9.15/5.12        ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
% 9.15/5.12       => ( ( size_size_list_int @ Xs )
% 9.15/5.12          = ( size_size_list_int @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1936_VEBT__internal_OminNull_Osimps_I3_J,axiom,
% 9.15/5.12      ! [Uu: $o] :
% 9.15/5.12        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1937_VEBT__internal_OminNull_Osimps_I2_J,axiom,
% 9.15/5.12      ! [Uv: $o] :
% 9.15/5.12        ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1938_VEBT__internal_OminNull_Osimps_I1_J,axiom,
% 9.15/5.12      vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1939_invar__vebt_Ointros_I2_J,axiom,
% 9.15/5.12      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.15/5.12        ( ! [X4: vEBT_VEBT] :
% 9.15/5.12            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 9.15/5.12       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.15/5.12         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.15/5.12              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12           => ( ( M = N2 )
% 9.15/5.12             => ( ( Deg
% 9.15/5.12                  = ( plus_plus_nat @ N2 @ M ) )
% 9.15/5.12               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.15/5.12                 => ( ! [X4: vEBT_VEBT] :
% 9.15/5.12                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1940_invar__vebt_Ointros_I3_J,axiom,
% 9.15/5.12      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
% 9.15/5.12        ( ! [X4: vEBT_VEBT] :
% 9.15/5.12            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 9.15/5.12       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.15/5.12         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.15/5.12              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12           => ( ( M
% 9.15/5.12                = ( suc @ N2 ) )
% 9.15/5.12             => ( ( Deg
% 9.15/5.12                  = ( plus_plus_nat @ N2 @ M ) )
% 9.15/5.12               => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
% 9.15/5.12                 => ( ! [X4: vEBT_VEBT] :
% 9.15/5.12                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1941_zip__same,axiom,
% 9.15/5.12      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
% 9.15/5.12        ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1942_zip__same,axiom,
% 9.15/5.12      ! [A: real,B: real,Xs: list_real] :
% 9.15/5.12        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member_real @ A @ ( set_real2 @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1943_zip__same,axiom,
% 9.15/5.12      ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
% 9.15/5.12        ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1944_zip__same,axiom,
% 9.15/5.12      ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
% 9.15/5.12        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1945_zip__same,axiom,
% 9.15/5.12      ! [A: nat,B: nat,Xs: list_nat] :
% 9.15/5.12        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1946_zip__same,axiom,
% 9.15/5.12      ! [A: int,B: int,Xs: list_int] :
% 9.15/5.12        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
% 9.15/5.12        = ( ( member_int @ A @ ( set_int2 @ Xs ) )
% 9.15/5.12          & ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1947_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: real,Y3: real,Xs: list_real,Ys: list_real] :
% 9.15/5.12        ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y3 ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1948_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: real,Y3: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1949_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: real,Y3: int,Xs: list_real,Ys: list_int] :
% 9.15/5.12        ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12           => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1950_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: real,Y3: nat,Xs: list_real,Ys: list_nat] :
% 9.15/5.12        ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12           => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1951_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Y3: real,Xs: list_VEBT_VEBT,Ys: list_real] :
% 9.15/5.12        ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X3 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1952_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1953_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Y3: int,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.15/5.12        ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12           => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1954_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Y3: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.15/5.12        ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12           => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1955_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: int,Y3: real,Xs: list_int,Ys: list_real] :
% 9.15/5.12        ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X3 @ Y3 ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.12           => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1956_in__set__zipE,axiom,
% 9.15/5.12      ! [X3: int,Y3: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
% 9.15/5.12        ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X3 @ Y3 ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
% 9.15/5.12       => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
% 9.15/5.12           => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1957_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
% 9.15/5.12        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1958_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
% 9.15/5.12        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member_int @ X3 @ ( set_int2 @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1959_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 9.15/5.12        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member1535805642427569193e_term @ X3 @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1960_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 9.15/5.12        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member4242434998011752849e_term @ X3 @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1961_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 9.15/5.12        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member3222579708246209666e_term @ X3 @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1962_set__zip__leftD,axiom,
% 9.15/5.12      ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 9.15/5.12        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member8845023287901829240e_term @ X3 @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1963_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
% 9.15/5.12        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1964_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
% 9.15/5.12        ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1965_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
% 9.15/5.12        ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1966_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
% 9.15/5.12        ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1967_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
% 9.15/5.12        ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1968_set__zip__rightD,axiom,
% 9.15/5.12      ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
% 9.15/5.12        ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
% 9.15/5.12       => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1969_zip__update,axiom,
% 9.15/5.12      ! [Xs: list_int,I: nat,X3: int,Ys: list_int,Y3: int] :
% 9.15/5.12        ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X3 ) @ ( list_update_int @ Ys @ I @ Y3 ) )
% 9.15/5.12        = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X3 @ Y3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1970_zip__update,axiom,
% 9.15/5.12      ! [Xs: list_C878401137130745250e_term,I: nat,X3: code_integer > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
% 9.15/5.12        ( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
% 9.15/5.12        = ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) @ I @ ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1971_zip__update,axiom,
% 9.15/5.12      ! [Xs: list_P1316552470764441098e_term,I: nat,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
% 9.15/5.12        ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
% 9.15/5.12        = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1972_zip__update,axiom,
% 9.15/5.12      ! [Xs: list_P1743416141875011707e_term,I: nat,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
% 9.15/5.12        ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
% 9.15/5.12        = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1973_zip__update,axiom,
% 9.15/5.12      ! [Xs: list_i8448526496819171953e_term,I: nat,X3: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
% 9.15/5.12        ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
% 9.15/5.12        = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1974_zero__le__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1975_zero__le__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1976_zero__le__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1977_zero__le__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1978_zero__le__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1979_not__numeral__le__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1980_not__numeral__le__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1981_not__numeral__le__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1982_not__numeral__le__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1983_not__numeral__le__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1984_not__numeral__less__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1985_not__numeral__less__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1986_not__numeral__less__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1987_not__numeral__less__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1988_not__numeral__less__zero,axiom,
% 9.15/5.12      ! [N2: num] :
% 9.15/5.12        ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1989_zero__less__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1990_zero__less__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1991_zero__less__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1992_zero__less__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1993_zero__less__numeral,axiom,
% 9.15/5.12      ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1994_set__encode__eq,axiom,
% 9.15/5.12      ! [A2: set_nat,B2: set_nat] :
% 9.15/5.12        ( ( finite_finite_nat @ A2 )
% 9.15/5.12       => ( ( finite_finite_nat @ B2 )
% 9.15/5.12         => ( ( ( nat_set_encode @ A2 )
% 9.15/5.12              = ( nat_set_encode @ B2 ) )
% 9.15/5.12            = ( A2 = B2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1995_vebt__buildup_Osimps_I1_J,axiom,
% 9.15/5.12      ( ( vEBT_vebt_buildup @ zero_zero_nat )
% 9.15/5.12      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1996_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_real,Y3: real] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_real @ Ys ) )
% 9.15/5.12       => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: vEBT_VEBT] :
% 9.15/5.12                ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1997_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: vEBT_VEBT] :
% 9.15/5.12                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1998_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_o,Y3: $o] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) )
% 9.15/5.12       => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: vEBT_VEBT] :
% 9.15/5.12                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_1999_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y3: nat] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_nat @ Ys ) )
% 9.15/5.12       => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: vEBT_VEBT] :
% 9.15/5.12                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2000_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_int,Y3: int] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_int @ Ys ) )
% 9.15/5.12       => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: vEBT_VEBT] :
% 9.15/5.12                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2001_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_real,Y3: real] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_real @ Ys ) )
% 9.15/5.12       => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: $o] :
% 9.15/5.12                ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y3 ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2002_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: $o] :
% 9.15/5.12                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X4 @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2003_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_o,Y3: $o] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) )
% 9.15/5.12       => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: $o] :
% 9.15/5.12                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2004_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_nat,Y3: nat] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_nat @ Ys ) )
% 9.15/5.12       => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: $o] :
% 9.15/5.12                ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X4 @ Y3 ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2005_in__set__impl__in__set__zip2,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_int,Y3: int] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_int @ Ys ) )
% 9.15/5.12       => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
% 9.15/5.12         => ~ ! [X4: $o] :
% 9.15/5.12                ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X4 @ Y3 ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2006_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_real,Ys: list_VEBT_VEBT,X3: real] :
% 9.15/5.12        ( ( ( size_size_list_real @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: vEBT_VEBT] :
% 9.15/5.12                ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y5 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2007_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_real,Ys: list_o,X3: real] :
% 9.15/5.12        ( ( ( size_size_list_real @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) )
% 9.15/5.12       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: $o] :
% 9.15/5.12                ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X3 @ Y5 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2008_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_real,Ys: list_nat,X3: real] :
% 9.15/5.12        ( ( ( size_size_list_real @ Xs )
% 9.15/5.12          = ( size_size_list_nat @ Ys ) )
% 9.15/5.12       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: nat] :
% 9.15/5.12                ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y5 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2009_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_real,Ys: list_int,X3: real] :
% 9.15/5.12        ( ( ( size_size_list_real @ Xs )
% 9.15/5.12          = ( size_size_list_int @ Ys ) )
% 9.15/5.12       => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: int] :
% 9.15/5.12                ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y5 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2010_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: vEBT_VEBT] :
% 9.15/5.12                ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y5 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2011_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_o,X3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: $o] :
% 9.15/5.12                ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X3 @ Y5 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2012_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_nat,X3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_nat @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: nat] :
% 9.15/5.12                ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y5 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2013_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,Ys: list_int,X3: vEBT_VEBT] :
% 9.15/5.12        ( ( ( size_s6755466524823107622T_VEBT @ Xs )
% 9.15/5.12          = ( size_size_list_int @ Ys ) )
% 9.15/5.12       => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: int] :
% 9.15/5.12                ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y5 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2014_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_VEBT_VEBT,X3: $o] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_s6755466524823107622T_VEBT @ Ys ) )
% 9.15/5.12       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: vEBT_VEBT] :
% 9.15/5.12                ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X3 @ Y5 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2015_in__set__impl__in__set__zip1,axiom,
% 9.15/5.12      ! [Xs: list_o,Ys: list_o,X3: $o] :
% 9.15/5.12        ( ( ( size_size_list_o @ Xs )
% 9.15/5.12          = ( size_size_list_o @ Ys ) )
% 9.15/5.12       => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
% 9.15/5.12         => ~ ! [Y5: $o] :
% 9.15/5.12                ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y5 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2016_VEBT__internal_Onaive__member_Ocases,axiom,
% 9.15/5.12      ! [X3: produc9072475918466114483BT_nat] :
% 9.15/5.12        ( ! [A4: $o,B4: $o,X4: nat] :
% 9.15/5.12            ( X3
% 9.15/5.12           != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
% 9.15/5.12       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
% 9.15/5.12         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X4 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2017_invar__vebt_Ointros_I1_J,axiom,
% 9.15/5.12      ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2018_vebt__buildup_Osimps_I2_J,axiom,
% 9.15/5.12      ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
% 9.15/5.12      = ( vEBT_Leaf @ $false @ $false ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2019_invar__vebt_Osimps,axiom,
% 9.15/5.12      ( vEBT_invar_vebt
% 9.15/5.12      = ( ^ [A1: vEBT_VEBT,A22: nat] :
% 9.15/5.12            ( ( ? [A3: $o,B3: $o] :
% 9.15/5.12                  ( A1
% 9.15/5.12                  = ( vEBT_Leaf @ A3 @ B3 ) )
% 9.15/5.12              & ( A22
% 9.15/5.12                = ( suc @ zero_zero_nat ) ) )
% 9.15/5.12            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 9.15/5.12                ( ( A1
% 9.15/5.12                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ X @ N ) )
% 9.15/5.12                & ( vEBT_invar_vebt @ Summary3 @ N )
% 9.15/5.12                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.15/5.12                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.15/5.12                & ( A22
% 9.15/5.12                  = ( plus_plus_nat @ N @ N ) )
% 9.15/5.12                & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.15/5.12            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
% 9.15/5.12                ( ( A1
% 9.15/5.12                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ X @ N ) )
% 9.15/5.12                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 9.15/5.12                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.15/5.12                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 9.15/5.12                & ( A22
% 9.15/5.12                  = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 9.15/5.12                & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.15/5.12            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.15/5.12                ( ( A1
% 9.15/5.12                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ X @ N ) )
% 9.15/5.12                & ( vEBT_invar_vebt @ Summary3 @ N )
% 9.15/5.12                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.15/5.12                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.15/5.12                & ( A22
% 9.15/5.12                  = ( plus_plus_nat @ N @ N ) )
% 9.15/5.12                & ! [I2: nat] :
% 9.15/5.12                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.15/5.12                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
% 9.15/5.12                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 9.15/5.12                & ( ( Mi3 = Ma3 )
% 9.15/5.12                 => ! [X: vEBT_VEBT] :
% 9.15/5.12                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.15/5.12                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.15/5.12                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.15/5.12                & ( ( Mi3 != Ma3 )
% 9.15/5.12                 => ! [I2: nat] :
% 9.15/5.12                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
% 9.15/5.12                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 9.15/5.12                            = I2 )
% 9.15/5.12                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 9.15/5.12                        & ! [X: nat] :
% 9.15/5.12                            ( ( ( ( vEBT_VEBT_high @ X @ N )
% 9.15/5.12                                = I2 )
% 9.15/5.12                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
% 9.15/5.12                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.15/5.12                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
% 9.15/5.12            | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
% 9.15/5.12                ( ( A1
% 9.15/5.12                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
% 9.15/5.12                & ! [X: vEBT_VEBT] :
% 9.15/5.12                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                   => ( vEBT_invar_vebt @ X @ N ) )
% 9.15/5.12                & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
% 9.15/5.12                & ( ( size_s6755466524823107622T_VEBT @ TreeList )
% 9.15/5.12                  = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 9.15/5.12                & ( A22
% 9.15/5.12                  = ( plus_plus_nat @ N @ ( suc @ N ) ) )
% 9.15/5.12                & ! [I2: nat] :
% 9.15/5.12                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 9.15/5.12                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
% 9.15/5.12                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
% 9.15/5.12                & ( ( Mi3 = Ma3 )
% 9.15/5.12                 => ! [X: vEBT_VEBT] :
% 9.15/5.12                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
% 9.15/5.12                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.15/5.12                & ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.15/5.12                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
% 9.15/5.12                & ( ( Mi3 != Ma3 )
% 9.15/5.12                 => ! [I2: nat] :
% 9.15/5.12                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
% 9.15/5.12                     => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
% 9.15/5.12                            = I2 )
% 9.15/5.12                         => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
% 9.15/5.12                        & ! [X: nat] :
% 9.15/5.12                            ( ( ( ( vEBT_VEBT_high @ X @ N )
% 9.15/5.12                                = I2 )
% 9.15/5.12                              & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
% 9.15/5.12                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.15/5.12                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2020_invar__vebt_Ocases,axiom,
% 9.15/5.12      ! [A12: vEBT_VEBT,A23: nat] :
% 9.15/5.12        ( ( vEBT_invar_vebt @ A12 @ A23 )
% 9.15/5.12       => ( ( ? [A4: $o,B4: $o] :
% 9.15/5.12                ( A12
% 9.15/5.12                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.15/5.12           => ( A23
% 9.15/5.12             != ( suc @ zero_zero_nat ) ) )
% 9.15/5.12         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
% 9.15/5.12                ( ( A12
% 9.15/5.12                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.15/5.12               => ( ( A23 = Deg2 )
% 9.15/5.12                 => ( ! [X2: vEBT_VEBT] :
% 9.15/5.12                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.15/5.12                   => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 9.15/5.12                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.15/5.12                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                       => ( ( M3 = N3 )
% 9.15/5.12                         => ( ( Deg2
% 9.15/5.12                              = ( plus_plus_nat @ N3 @ M3 ) )
% 9.15/5.12                           => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.15/5.12                             => ~ ! [X2: vEBT_VEBT] :
% 9.15/5.12                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.15/5.12           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
% 9.15/5.12                  ( ( A12
% 9.15/5.12                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.15/5.12                 => ( ( A23 = Deg2 )
% 9.15/5.12                   => ( ! [X2: vEBT_VEBT] :
% 9.15/5.12                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.15/5.12                     => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 9.15/5.12                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.15/5.12                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                         => ( ( M3
% 9.15/5.12                              = ( suc @ N3 ) )
% 9.15/5.12                           => ( ( Deg2
% 9.15/5.12                                = ( plus_plus_nat @ N3 @ M3 ) )
% 9.15/5.12                             => ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
% 9.15/5.12                               => ~ ! [X2: vEBT_VEBT] :
% 9.15/5.12                                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
% 9.15/5.12             => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.15/5.12                    ( ( A12
% 9.15/5.12                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.15/5.12                   => ( ( A23 = Deg2 )
% 9.15/5.12                     => ( ! [X2: vEBT_VEBT] :
% 9.15/5.12                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.15/5.12                       => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 9.15/5.12                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.15/5.12                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                           => ( ( M3 = N3 )
% 9.15/5.12                             => ( ( Deg2
% 9.15/5.12                                  = ( plus_plus_nat @ N3 @ M3 ) )
% 9.15/5.12                               => ( ! [I4: nat] :
% 9.15/5.12                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
% 9.15/5.12                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.15/5.12                                 => ( ( ( Mi2 = Ma2 )
% 9.15/5.12                                     => ! [X2: vEBT_VEBT] :
% 9.15/5.12                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 9.15/5.12                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.15/5.12                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.15/5.12                                       => ~ ( ( Mi2 != Ma2 )
% 9.15/5.12                                           => ! [I4: nat] :
% 9.15/5.12                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.15/5.12                                                      = I4 )
% 9.15/5.12                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.15/5.12                                                  & ! [X2: nat] :
% 9.15/5.12                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 9.15/5.12                                                          = I4 )
% 9.15/5.12                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 9.15/5.12                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
% 9.15/5.12                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
% 9.15/5.12               => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
% 9.15/5.12                      ( ( A12
% 9.15/5.12                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.15/5.12                     => ( ( A23 = Deg2 )
% 9.15/5.12                       => ( ! [X2: vEBT_VEBT] :
% 9.15/5.12                              ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                             => ( vEBT_invar_vebt @ X2 @ N3 ) )
% 9.15/5.12                         => ( ( vEBT_invar_vebt @ Summary2 @ M3 )
% 9.15/5.12                           => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.15/5.12                                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                             => ( ( M3
% 9.15/5.12                                  = ( suc @ N3 ) )
% 9.15/5.12                               => ( ( Deg2
% 9.15/5.12                                    = ( plus_plus_nat @ N3 @ M3 ) )
% 9.15/5.12                                 => ( ! [I4: nat] :
% 9.15/5.12                                        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                                       => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
% 9.15/5.12                                          = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
% 9.15/5.12                                   => ( ( ( Mi2 = Ma2 )
% 9.15/5.12                                       => ! [X2: vEBT_VEBT] :
% 9.15/5.12                                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.15/5.12                                           => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
% 9.15/5.12                                     => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
% 9.15/5.12                                       => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.15/5.12                                         => ~ ( ( Mi2 != Ma2 )
% 9.15/5.12                                             => ! [I4: nat] :
% 9.15/5.12                                                  ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
% 9.15/5.12                                                 => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
% 9.15/5.12                                                        = I4 )
% 9.15/5.12                                                     => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
% 9.15/5.12                                                    & ! [X2: nat] :
% 9.15/5.12                                                        ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
% 9.15/5.12                                                            = I4 )
% 9.15/5.12                                                          & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
% 9.15/5.12                                                       => ( ( ord_less_nat @ Mi2 @ X2 )
% 9.15/5.12                                                          & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2021_VEBT__internal_OminNull_Ocases,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT] :
% 9.15/5.12        ( ( X3
% 9.15/5.12         != ( vEBT_Leaf @ $false @ $false ) )
% 9.15/5.12       => ( ! [Uv2: $o] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.15/5.12         => ( ! [Uu2: $o] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.15/5.12           => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.15/5.12                  ( X3
% 9.15/5.12                 != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.15/5.12             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.15/5.12                    ( X3
% 9.15/5.12                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2022_num_Osize_I5_J,axiom,
% 9.15/5.12      ! [X22: num] :
% 9.15/5.12        ( ( size_size_num @ ( bit0 @ X22 ) )
% 9.15/5.12        = ( plus_plus_nat @ ( size_size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2023_invar__vebt_Ointros_I4_J,axiom,
% 9.15/5.12      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.15/5.12        ( ! [X4: vEBT_VEBT] :
% 9.15/5.12            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 9.15/5.12       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.15/5.12         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.15/5.12              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12           => ( ( M = N2 )
% 9.15/5.12             => ( ( Deg
% 9.15/5.12                  = ( plus_plus_nat @ N2 @ M ) )
% 9.15/5.12               => ( ! [I3: nat] :
% 9.15/5.12                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
% 9.15/5.12                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 9.15/5.12                 => ( ( ( Mi = Ma )
% 9.15/5.12                     => ! [X4: vEBT_VEBT] :
% 9.15/5.12                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 9.15/5.12                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.15/5.12                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.15/5.12                       => ( ( ( Mi != Ma )
% 9.15/5.12                           => ! [I3: nat] :
% 9.15/5.12                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 9.15/5.12                                      = I3 )
% 9.15/5.12                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 9.15/5.12                                  & ! [X4: nat] :
% 9.15/5.12                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 9.15/5.12                                          = I3 )
% 9.15/5.12                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 9.15/5.12                                     => ( ( ord_less_nat @ Mi @ X4 )
% 9.15/5.12                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 9.15/5.12                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2024_VEBT__internal_OminNull_Oelims_I3_J,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT] :
% 9.15/5.12        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 9.15/5.12       => ( ! [Uv2: $o] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.15/5.12         => ( ! [Uu2: $o] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.15/5.12           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.15/5.12                  ( X3
% 9.15/5.12                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2025_VEBT__internal_OminNull_Oelims_I2_J,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT] :
% 9.15/5.12        ( ( vEBT_VEBT_minNull @ X3 )
% 9.15/5.12       => ( ( X3
% 9.15/5.12           != ( vEBT_Leaf @ $false @ $false ) )
% 9.15/5.12         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2026_invar__vebt_Ointros_I5_J,axiom,
% 9.15/5.12      ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
% 9.15/5.12        ( ! [X4: vEBT_VEBT] :
% 9.15/5.12            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 9.15/5.12       => ( ( vEBT_invar_vebt @ Summary @ M )
% 9.15/5.12         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.15/5.12              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12           => ( ( M
% 9.15/5.12                = ( suc @ N2 ) )
% 9.15/5.12             => ( ( Deg
% 9.15/5.12                  = ( plus_plus_nat @ N2 @ M ) )
% 9.15/5.12               => ( ! [I3: nat] :
% 9.15/5.12                      ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
% 9.15/5.12                        = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
% 9.15/5.12                 => ( ( ( Mi = Ma )
% 9.15/5.12                     => ! [X4: vEBT_VEBT] :
% 9.15/5.12                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.15/5.12                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
% 9.15/5.12                   => ( ( ord_less_eq_nat @ Mi @ Ma )
% 9.15/5.12                     => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.15/5.12                       => ( ( ( Mi != Ma )
% 9.15/5.12                           => ! [I3: nat] :
% 9.15/5.12                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.15/5.12                               => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
% 9.15/5.12                                      = I3 )
% 9.15/5.12                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
% 9.15/5.12                                  & ! [X4: nat] :
% 9.15/5.12                                      ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
% 9.15/5.12                                          = I3 )
% 9.15/5.12                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
% 9.15/5.12                                     => ( ( ord_less_nat @ Mi @ X4 )
% 9.15/5.12                                        & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
% 9.15/5.12                         => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2027_set__encode__inf,axiom,
% 9.15/5.12      ! [A2: set_nat] :
% 9.15/5.12        ( ~ ( finite_finite_nat @ A2 )
% 9.15/5.12       => ( ( nat_set_encode @ A2 )
% 9.15/5.12          = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2028_pair__lessI1,axiom,
% 9.15/5.12      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.15/5.12        ( ( ord_less_nat @ A @ B )
% 9.15/5.12       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2029_count__le__length,axiom,
% 9.15/5.12      ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X3 ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2030_count__le__length,axiom,
% 9.15/5.12      ! [Xs: list_o,X3: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X3 ) @ ( size_size_list_o @ Xs ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2031_count__le__length,axiom,
% 9.15/5.12      ! [Xs: list_nat,X3: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X3 ) @ ( size_size_list_nat @ Xs ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2032_count__le__length,axiom,
% 9.15/5.12      ! [Xs: list_int,X3: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X3 ) @ ( size_size_list_int @ Xs ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2033_vebt__mint_Ocases,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT] :
% 9.15/5.12        ( ! [A4: $o,B4: $o] :
% 9.15/5.12            ( X3
% 9.15/5.12           != ( vEBT_Leaf @ A4 @ B4 ) )
% 9.15/5.12       => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.15/5.12              ( X3
% 9.15/5.12             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.15/5.12         => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.15/5.12                ( X3
% 9.15/5.12               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2034_VEBT__internal_OminNull_Oelims_I1_J,axiom,
% 9.15/5.12      ! [X3: vEBT_VEBT,Y3: $o] :
% 9.15/5.12        ( ( ( vEBT_VEBT_minNull @ X3 )
% 9.15/5.12          = Y3 )
% 9.15/5.12       => ( ( ( X3
% 9.15/5.12              = ( vEBT_Leaf @ $false @ $false ) )
% 9.15/5.12           => ~ Y3 )
% 9.15/5.12         => ( ( ? [Uv2: $o] :
% 9.15/5.12                  ( X3
% 9.15/5.12                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.15/5.12             => Y3 )
% 9.15/5.12           => ( ( ? [Uu2: $o] :
% 9.15/5.12                    ( X3
% 9.15/5.12                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.15/5.12               => Y3 )
% 9.15/5.12             => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.15/5.12                      ( X3
% 9.15/5.12                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.15/5.12                 => ~ Y3 )
% 9.15/5.12               => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.15/5.12                        ( X3
% 9.15/5.12                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.15/5.12                   => Y3 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2035_length__code,axiom,
% 9.15/5.12      ( size_s6755466524823107622T_VEBT
% 9.15/5.12      = ( gen_length_VEBT_VEBT @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2036_length__code,axiom,
% 9.15/5.12      ( size_size_list_o
% 9.15/5.12      = ( gen_length_o @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2037_length__code,axiom,
% 9.15/5.12      ( size_size_list_nat
% 9.15/5.12      = ( gen_length_nat @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2038_length__code,axiom,
% 9.15/5.12      ( size_size_list_int
% 9.15/5.12      = ( gen_length_int @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2039_sum__power2__eq__zero__iff,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12          = zero_zero_rat )
% 9.15/5.12        = ( ( X3 = zero_zero_rat )
% 9.15/5.12          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2040_sum__power2__eq__zero__iff,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12          = zero_zero_real )
% 9.15/5.12        = ( ( X3 = zero_zero_real )
% 9.15/5.12          & ( Y3 = zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2041_sum__power2__eq__zero__iff,axiom,
% 9.15/5.12      ! [X3: int,Y3: int] :
% 9.15/5.12        ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12          = zero_zero_int )
% 9.15/5.12        = ( ( X3 = zero_zero_int )
% 9.15/5.12          & ( Y3 = zero_zero_int ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2042_zero__less__power2,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = ( A != zero_zero_real ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2043_zero__less__power2,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = ( A != zero_zero_rat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2044_zero__less__power2,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = ( A != zero_zero_int ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2045_power2__less__eq__zero__iff,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real )
% 9.15/5.12        = ( A = zero_zero_real ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2046_power2__less__eq__zero__iff,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
% 9.15/5.12        = ( A = zero_zero_rat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2047_power2__less__eq__zero__iff,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.15/5.12        = ( A = zero_zero_int ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2048_power2__eq__iff__nonneg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12              = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12            = ( X3 = Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2049_power2__eq__iff__nonneg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12              = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12            = ( X3 = Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2050_power2__eq__iff__nonneg,axiom,
% 9.15/5.12      ! [X3: nat,Y3: nat] :
% 9.15/5.12        ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.15/5.12         => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12              = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12            = ( X3 = Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2051_power2__eq__iff__nonneg,axiom,
% 9.15/5.12      ! [X3: int,Y3: int] :
% 9.15/5.12        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.15/5.12         => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12              = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.15/5.12            = ( X3 = Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2052_zero__eq__power2,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12          = zero_zero_rat )
% 9.15/5.12        = ( A = zero_zero_rat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2053_zero__eq__power2,axiom,
% 9.15/5.12      ! [A: nat] :
% 9.15/5.12        ( ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12          = zero_zero_nat )
% 9.15/5.12        = ( A = zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2054_zero__eq__power2,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12          = zero_zero_real )
% 9.15/5.12        = ( A = zero_zero_real ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2055_zero__eq__power2,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12          = zero_zero_int )
% 9.15/5.12        = ( A = zero_zero_int ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2056_zero__eq__power2,axiom,
% 9.15/5.12      ! [A: complex] :
% 9.15/5.12        ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.15/5.12          = zero_zero_complex )
% 9.15/5.12        = ( A = zero_zero_complex ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2057_power__mono__iff,axiom,
% 9.15/5.12      ! [A: real,B: real,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12           => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 9.15/5.12              = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2058_power__mono__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12           => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 9.15/5.12              = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2059_power__mono__iff,axiom,
% 9.15/5.12      ! [A: nat,B: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12           => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 9.15/5.12              = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2060_power__mono__iff,axiom,
% 9.15/5.12      ! [A: int,B: int,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.15/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.15/5.12           => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 9.15/5.12              = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2061_power__eq__0__iff,axiom,
% 9.15/5.12      ! [A: rat,N2: nat] :
% 9.15/5.12        ( ( ( power_power_rat @ A @ N2 )
% 9.15/5.12          = zero_zero_rat )
% 9.15/5.12        = ( ( A = zero_zero_rat )
% 9.15/5.12          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2062_power__eq__0__iff,axiom,
% 9.15/5.12      ! [A: nat,N2: nat] :
% 9.15/5.12        ( ( ( power_power_nat @ A @ N2 )
% 9.15/5.12          = zero_zero_nat )
% 9.15/5.12        = ( ( A = zero_zero_nat )
% 9.15/5.12          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2063_power__eq__0__iff,axiom,
% 9.15/5.12      ! [A: real,N2: nat] :
% 9.15/5.12        ( ( ( power_power_real @ A @ N2 )
% 9.15/5.12          = zero_zero_real )
% 9.15/5.12        = ( ( A = zero_zero_real )
% 9.15/5.12          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2064_power__eq__0__iff,axiom,
% 9.15/5.12      ! [A: int,N2: nat] :
% 9.15/5.12        ( ( ( power_power_int @ A @ N2 )
% 9.15/5.12          = zero_zero_int )
% 9.15/5.12        = ( ( A = zero_zero_int )
% 9.15/5.12          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2065_power__eq__0__iff,axiom,
% 9.15/5.12      ! [A: complex,N2: nat] :
% 9.15/5.12        ( ( ( power_power_complex @ A @ N2 )
% 9.15/5.12          = zero_zero_complex )
% 9.15/5.12        = ( ( A = zero_zero_complex )
% 9.15/5.12          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2066_member__inv,axiom,
% 9.15/5.12      ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.15/5.12        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
% 9.15/5.12          & ( ( X3 = Mi )
% 9.15/5.12            | ( X3 = Ma )
% 9.15/5.12            | ( ( ord_less_nat @ X3 @ Ma )
% 9.15/5.12              & ( ord_less_nat @ Mi @ X3 )
% 9.15/5.12              & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.15/5.12              & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2067_both__member__options__ding,axiom,
% 9.15/5.12      ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] :
% 9.15/5.12        ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
% 9.15/5.12       => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.15/5.12         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.15/5.12           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2068_nat__zero__less__power__iff,axiom,
% 9.15/5.12      ! [X3: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N2 ) )
% 9.15/5.12        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.15/5.12          | ( N2 = zero_zero_nat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2069_pow__sum,axiom,
% 9.15/5.12      ! [A: nat,B: nat] :
% 9.15/5.12        ( ( 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 ) )
% 9.15/5.12        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2070_high__def,axiom,
% 9.15/5.12      ( vEBT_VEBT_high
% 9.15/5.12      = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2071_division__ring__divide__zero,axiom,
% 9.15/5.12      ! [A: complex] :
% 9.15/5.12        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.15/5.12        = zero_zero_complex ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2072_division__ring__divide__zero,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.15/5.12        = zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2073_division__ring__divide__zero,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.15/5.12        = zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2074_divide__cancel__right,axiom,
% 9.15/5.12      ! [A: complex,C: complex,B: complex] :
% 9.15/5.12        ( ( ( divide1717551699836669952omplex @ A @ C )
% 9.15/5.12          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.15/5.12        = ( ( C = zero_zero_complex )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2075_divide__cancel__right,axiom,
% 9.15/5.12      ! [A: real,C: real,B: real] :
% 9.15/5.12        ( ( ( divide_divide_real @ A @ C )
% 9.15/5.12          = ( divide_divide_real @ B @ C ) )
% 9.15/5.12        = ( ( C = zero_zero_real )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2076_divide__cancel__right,axiom,
% 9.15/5.12      ! [A: rat,C: rat,B: rat] :
% 9.15/5.12        ( ( ( divide_divide_rat @ A @ C )
% 9.15/5.12          = ( divide_divide_rat @ B @ C ) )
% 9.15/5.12        = ( ( C = zero_zero_rat )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2077_divide__cancel__left,axiom,
% 9.15/5.12      ! [C: complex,A: complex,B: complex] :
% 9.15/5.12        ( ( ( divide1717551699836669952omplex @ C @ A )
% 9.15/5.12          = ( divide1717551699836669952omplex @ C @ B ) )
% 9.15/5.12        = ( ( C = zero_zero_complex )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2078_divide__cancel__left,axiom,
% 9.15/5.12      ! [C: real,A: real,B: real] :
% 9.15/5.12        ( ( ( divide_divide_real @ C @ A )
% 9.15/5.12          = ( divide_divide_real @ C @ B ) )
% 9.15/5.12        = ( ( C = zero_zero_real )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2079_divide__cancel__left,axiom,
% 9.15/5.12      ! [C: rat,A: rat,B: rat] :
% 9.15/5.12        ( ( ( divide_divide_rat @ C @ A )
% 9.15/5.12          = ( divide_divide_rat @ C @ B ) )
% 9.15/5.12        = ( ( C = zero_zero_rat )
% 9.15/5.12          | ( A = B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2080_div__by__0,axiom,
% 9.15/5.12      ! [A: complex] :
% 9.15/5.12        ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
% 9.15/5.12        = zero_zero_complex ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2081_div__by__0,axiom,
% 9.15/5.12      ! [A: nat] :
% 9.15/5.12        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.15/5.12        = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2082_div__by__0,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.15/5.12        = zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2083_div__by__0,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( divide_divide_real @ A @ zero_zero_real )
% 9.15/5.12        = zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2084_div__by__0,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( divide_divide_rat @ A @ zero_zero_rat )
% 9.15/5.12        = zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2085_div__by__0,axiom,
% 9.15/5.12      ! [A: code_integer] :
% 9.15/5.12        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 9.15/5.12        = zero_z3403309356797280102nteger ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2086_divide__eq__0__iff,axiom,
% 9.15/5.12      ! [A: complex,B: complex] :
% 9.15/5.12        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.15/5.12          = zero_zero_complex )
% 9.15/5.12        = ( ( A = zero_zero_complex )
% 9.15/5.12          | ( B = zero_zero_complex ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2087_divide__eq__0__iff,axiom,
% 9.15/5.12      ! [A: real,B: real] :
% 9.15/5.12        ( ( ( divide_divide_real @ A @ B )
% 9.15/5.12          = zero_zero_real )
% 9.15/5.12        = ( ( A = zero_zero_real )
% 9.15/5.12          | ( B = zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2088_divide__eq__0__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat] :
% 9.15/5.12        ( ( ( divide_divide_rat @ A @ B )
% 9.15/5.12          = zero_zero_rat )
% 9.15/5.12        = ( ( A = zero_zero_rat )
% 9.15/5.12          | ( B = zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2089_div__0,axiom,
% 9.15/5.12      ! [A: complex] :
% 9.15/5.12        ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
% 9.15/5.12        = zero_zero_complex ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2090_div__0,axiom,
% 9.15/5.12      ! [A: nat] :
% 9.15/5.12        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.15/5.12        = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2091_div__0,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.15/5.12        = zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2092_div__0,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( divide_divide_real @ zero_zero_real @ A )
% 9.15/5.12        = zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2093_div__0,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( divide_divide_rat @ zero_zero_rat @ A )
% 9.15/5.12        = zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2094_div__0,axiom,
% 9.15/5.12      ! [A: code_integer] :
% 9.15/5.12        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 9.15/5.12        = zero_z3403309356797280102nteger ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2095_power__0__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
% 9.15/5.12        = zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2096_power__0__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 9.15/5.12        = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2097_power__0__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
% 9.15/5.12        = zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2098_power__0__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
% 9.15/5.12        = zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2099_power__0__Suc,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
% 9.15/5.12        = zero_zero_complex ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2100_power__zero__numeral,axiom,
% 9.15/5.12      ! [K: num] :
% 9.15/5.12        ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
% 9.15/5.12        = zero_zero_rat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2101_power__zero__numeral,axiom,
% 9.15/5.12      ! [K: num] :
% 9.15/5.12        ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
% 9.15/5.12        = zero_zero_nat ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2102_power__zero__numeral,axiom,
% 9.15/5.12      ! [K: num] :
% 9.15/5.12        ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
% 9.15/5.12        = zero_zero_real ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2103_power__zero__numeral,axiom,
% 9.15/5.12      ! [K: num] :
% 9.15/5.12        ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
% 9.15/5.12        = zero_zero_int ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2104_power__zero__numeral,axiom,
% 9.15/5.12      ! [K: num] :
% 9.15/5.12        ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
% 9.15/5.12        = zero_zero_complex ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2105_power__Suc0__right,axiom,
% 9.15/5.12      ! [A: nat] :
% 9.15/5.12        ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
% 9.15/5.12        = A ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2106_power__Suc0__right,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
% 9.15/5.12        = A ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2107_power__Suc0__right,axiom,
% 9.15/5.12      ! [A: int] :
% 9.15/5.12        ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
% 9.15/5.12        = A ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2108_power__Suc0__right,axiom,
% 9.15/5.12      ! [A: complex] :
% 9.15/5.12        ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
% 9.15/5.12        = A ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2109_power__Suc__0,axiom,
% 9.15/5.12      ! [N2: nat] :
% 9.15/5.12        ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.15/5.12        = ( suc @ zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2110_nat__power__eq__Suc__0__iff,axiom,
% 9.15/5.12      ! [X3: nat,M: nat] :
% 9.15/5.12        ( ( ( power_power_nat @ X3 @ M )
% 9.15/5.12          = ( suc @ zero_zero_nat ) )
% 9.15/5.12        = ( ( M = zero_zero_nat )
% 9.15/5.12          | ( X3
% 9.15/5.12            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2111_add__divide__distrib,axiom,
% 9.15/5.12      ! [A: real,B: real,C: real] :
% 9.15/5.12        ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.15/5.12        = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2112_add__divide__distrib,axiom,
% 9.15/5.12      ! [A: rat,B: rat,C: rat] :
% 9.15/5.12        ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.15/5.12        = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2113_divide__right__mono__neg,axiom,
% 9.15/5.12      ! [A: real,B: real,C: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2114_divide__right__mono__neg,axiom,
% 9.15/5.12      ! [A: rat,B: rat,C: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2115_divide__nonpos__nonpos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2116_divide__nonpos__nonpos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2117_divide__nonpos__nonneg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2118_divide__nonpos__nonneg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2119_divide__nonneg__nonpos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2120_divide__nonneg__nonpos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2121_divide__nonneg__nonneg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2122_divide__nonneg__nonneg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2123_zero__le__divide__iff,axiom,
% 9.15/5.12      ! [A: real,B: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.15/5.12        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.12            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.15/5.12          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.15/5.12            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2124_zero__le__divide__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.15/5.12        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.12            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.15/5.12          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.15/5.12            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2125_divide__right__mono,axiom,
% 9.15/5.12      ! [A: real,B: real,C: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2126_divide__right__mono,axiom,
% 9.15/5.12      ! [A: rat,B: rat,C: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2127_divide__le__0__iff,axiom,
% 9.15/5.12      ! [A: real,B: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.15/5.12        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.12            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.15/5.12          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.15/5.12            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2128_divide__le__0__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.15/5.12        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.12            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.15/5.12          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.15/5.12            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2129_divide__neg__neg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2130_divide__neg__neg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2131_divide__neg__pos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2132_divide__neg__pos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2133_divide__pos__neg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2134_divide__pos__neg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2135_divide__pos__pos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2136_divide__pos__pos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2137_divide__less__0__iff,axiom,
% 9.15/5.12      ! [A: real,B: real] :
% 9.15/5.12        ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
% 9.15/5.12        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.12            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.15/5.12          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.15/5.12            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2138_divide__less__0__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
% 9.15/5.12        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.12            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.15/5.12          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.15/5.12            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2139_divide__less__cancel,axiom,
% 9.15/5.12      ! [A: real,C: real,B: real] :
% 9.15/5.12        ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.15/5.12        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.15/5.12           => ( ord_less_real @ A @ B ) )
% 9.15/5.12          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.15/5.12           => ( ord_less_real @ B @ A ) )
% 9.15/5.12          & ( C != zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2140_divide__less__cancel,axiom,
% 9.15/5.12      ! [A: rat,C: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.15/5.12        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.15/5.12           => ( ord_less_rat @ A @ B ) )
% 9.15/5.12          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.15/5.12           => ( ord_less_rat @ B @ A ) )
% 9.15/5.12          & ( C != zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2141_zero__less__divide__iff,axiom,
% 9.15/5.12      ! [A: real,B: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
% 9.15/5.12        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.12            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.15/5.12          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.15/5.12            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2142_zero__less__divide__iff,axiom,
% 9.15/5.12      ! [A: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
% 9.15/5.12        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.12            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.15/5.12          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.15/5.12            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2143_divide__strict__right__mono,axiom,
% 9.15/5.12      ! [A: real,B: real,C: real] :
% 9.15/5.12        ( ( ord_less_real @ A @ B )
% 9.15/5.12       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.15/5.12         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2144_divide__strict__right__mono,axiom,
% 9.15/5.12      ! [A: rat,B: rat,C: rat] :
% 9.15/5.12        ( ( ord_less_rat @ A @ B )
% 9.15/5.12       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.15/5.12         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2145_divide__strict__right__mono__neg,axiom,
% 9.15/5.12      ! [B: real,A: real,C: real] :
% 9.15/5.12        ( ( ord_less_real @ B @ A )
% 9.15/5.12       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.15/5.12         => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2146_divide__strict__right__mono__neg,axiom,
% 9.15/5.12      ! [B: rat,A: rat,C: rat] :
% 9.15/5.12        ( ( ord_less_rat @ B @ A )
% 9.15/5.12       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2147_frac__le,axiom,
% 9.15/5.12      ! [Y3: real,X3: real,W2: real,Z2: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.15/5.12       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.15/5.12           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2148_frac__le,axiom,
% 9.15/5.12      ! [Y3: rat,X3: rat,W2: rat,Z2: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.15/5.12           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2149_frac__less,axiom,
% 9.15/5.12      ! [X3: real,Y3: real,W2: real,Z2: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.15/5.12           => ( ( ord_less_eq_real @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2150_frac__less,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.15/5.12           => ( ( ord_less_eq_rat @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2151_frac__less2,axiom,
% 9.15/5.12      ! [X3: real,Y3: real,W2: real,Z2: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_real @ zero_zero_real @ W2 )
% 9.15/5.12           => ( ( ord_less_real @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2152_frac__less2,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.15/5.12         => ( ( ord_less_rat @ zero_zero_rat @ W2 )
% 9.15/5.12           => ( ( ord_less_rat @ W2 @ Z2 )
% 9.15/5.12             => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2153_divide__le__cancel,axiom,
% 9.15/5.12      ! [A: real,C: real,B: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
% 9.15/5.12        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.15/5.12           => ( ord_less_eq_real @ A @ B ) )
% 9.15/5.12          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.15/5.12           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2154_divide__le__cancel,axiom,
% 9.15/5.12      ! [A: rat,C: rat,B: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
% 9.15/5.12        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.15/5.12           => ( ord_less_eq_rat @ A @ B ) )
% 9.15/5.12          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.15/5.12           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2155_divide__nonneg__neg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2156_divide__nonneg__neg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2157_divide__nonneg__pos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.15/5.12       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2158_divide__nonneg__pos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.15/5.12       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2159_divide__nonpos__neg,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_real @ Y3 @ zero_zero_real )
% 9.15/5.12         => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2160_divide__nonpos__neg,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
% 9.15/5.12         => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2161_divide__nonpos__pos,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.15/5.12       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.15/5.12         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2162_divide__nonpos__pos,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
% 9.15/5.12       => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.15/5.12         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2163_field__sum__of__halves,axiom,
% 9.15/5.12      ! [X3: real] :
% 9.15/5.12        ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = X3 ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2164_field__sum__of__halves,axiom,
% 9.15/5.12      ! [X3: rat] :
% 9.15/5.12        ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = X3 ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2165_half__gt__zero,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.12       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2166_half__gt__zero,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.12       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2167_half__gt__zero__iff,axiom,
% 9.15/5.12      ! [A: real] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2168_half__gt__zero__iff,axiom,
% 9.15/5.12      ! [A: rat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.15/5.12        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2169_field__less__half__sum,axiom,
% 9.15/5.12      ! [X3: real,Y3: real] :
% 9.15/5.12        ( ( ord_less_real @ X3 @ Y3 )
% 9.15/5.12       => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2170_field__less__half__sum,axiom,
% 9.15/5.12      ! [X3: rat,Y3: rat] :
% 9.15/5.12        ( ( ord_less_rat @ X3 @ Y3 )
% 9.15/5.12       => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2171_power__not__zero,axiom,
% 9.15/5.12      ! [A: rat,N2: nat] :
% 9.15/5.12        ( ( A != zero_zero_rat )
% 9.15/5.12       => ( ( power_power_rat @ A @ N2 )
% 9.15/5.12         != zero_zero_rat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2172_power__not__zero,axiom,
% 9.15/5.12      ! [A: nat,N2: nat] :
% 9.15/5.12        ( ( A != zero_zero_nat )
% 9.15/5.12       => ( ( power_power_nat @ A @ N2 )
% 9.15/5.12         != zero_zero_nat ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2173_power__not__zero,axiom,
% 9.15/5.12      ! [A: real,N2: nat] :
% 9.15/5.12        ( ( A != zero_zero_real )
% 9.15/5.12       => ( ( power_power_real @ A @ N2 )
% 9.15/5.12         != zero_zero_real ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2174_power__not__zero,axiom,
% 9.15/5.12      ! [A: int,N2: nat] :
% 9.15/5.12        ( ( A != zero_zero_int )
% 9.15/5.12       => ( ( power_power_int @ A @ N2 )
% 9.15/5.12         != zero_zero_int ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2175_power__not__zero,axiom,
% 9.15/5.12      ! [A: complex,N2: nat] :
% 9.15/5.12        ( ( A != zero_zero_complex )
% 9.15/5.12       => ( ( power_power_complex @ A @ N2 )
% 9.15/5.12         != zero_zero_complex ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2176_zero__le__power,axiom,
% 9.15/5.12      ! [A: real,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.12       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2177_zero__le__power,axiom,
% 9.15/5.12      ! [A: rat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.12       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2178_zero__le__power,axiom,
% 9.15/5.12      ! [A: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.12       => ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2179_zero__le__power,axiom,
% 9.15/5.12      ! [A: int,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.12       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2180_power__mono,axiom,
% 9.15/5.12      ! [A: real,B: real,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_real @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.15/5.12         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2181_power__mono,axiom,
% 9.15/5.12      ! [A: rat,B: rat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_rat @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.15/5.12         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2182_power__mono,axiom,
% 9.15/5.12      ! [A: nat,B: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_nat @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.15/5.12         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2183_power__mono,axiom,
% 9.15/5.12      ! [A: int,B: int,N2: nat] :
% 9.15/5.12        ( ( ord_less_eq_int @ A @ B )
% 9.15/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.15/5.12         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2184_zero__less__power,axiom,
% 9.15/5.12      ! [A: real,N2: nat] :
% 9.15/5.12        ( ( ord_less_real @ zero_zero_real @ A )
% 9.15/5.12       => ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2185_zero__less__power,axiom,
% 9.15/5.12      ! [A: rat,N2: nat] :
% 9.15/5.12        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.15/5.12       => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2186_zero__less__power,axiom,
% 9.15/5.12      ! [A: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.15/5.12       => ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2187_zero__less__power,axiom,
% 9.15/5.12      ! [A: int,N2: nat] :
% 9.15/5.12        ( ( ord_less_int @ zero_zero_int @ A )
% 9.15/5.12       => ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2188_nat__power__less__imp__less,axiom,
% 9.15/5.12      ! [I: nat,M: nat,N2: nat] :
% 9.15/5.12        ( ( ord_less_nat @ zero_zero_nat @ I )
% 9.15/5.12       => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
% 9.15/5.12         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2189_power__less__imp__less__base,axiom,
% 9.15/5.12      ! [A: real,N2: nat,B: real] :
% 9.15/5.12        ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
% 9.15/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.15/5.12         => ( ord_less_real @ A @ B ) ) ) ).
% 9.15/5.12  
% 9.15/5.12  thf(fact_2190_power__less__imp__less__base,axiom,
% 9.15/5.12      ! [A: rat,N2: nat,B: rat] :
% 9.15/5.12        ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
% 9.15/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.15/5.12         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2191_power__less__imp__less__base,axiom,
% 9.20/5.12      ! [A: nat,N2: nat,B: nat] :
% 9.20/5.12        ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.12         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2192_power__less__imp__less__base,axiom,
% 9.20/5.12      ! [A: int,N2: nat,B: int] :
% 9.20/5.12        ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.12         => ( ord_less_int @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2193_power__le__imp__le__base,axiom,
% 9.20/5.12      ! [A: real,N2: nat,B: real] :
% 9.20/5.12        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.12         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2194_power__le__imp__le__base,axiom,
% 9.20/5.12      ! [A: rat,N2: nat,B: rat] :
% 9.20/5.12        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.12         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2195_power__le__imp__le__base,axiom,
% 9.20/5.12      ! [A: nat,N2: nat,B: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.12         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2196_power__le__imp__le__base,axiom,
% 9.20/5.12      ! [A: int,N2: nat,B: int] :
% 9.20/5.12        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.12         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2197_power__inject__base,axiom,
% 9.20/5.12      ! [A: real,N2: nat,B: real] :
% 9.20/5.12        ( ( ( power_power_real @ A @ ( suc @ N2 ) )
% 9.20/5.12          = ( power_power_real @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.12         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.12           => ( A = B ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2198_power__inject__base,axiom,
% 9.20/5.12      ! [A: rat,N2: nat,B: rat] :
% 9.20/5.12        ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 9.20/5.12          = ( power_power_rat @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.12         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.12           => ( A = B ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2199_power__inject__base,axiom,
% 9.20/5.12      ! [A: nat,N2: nat,B: nat] :
% 9.20/5.12        ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 9.20/5.12          = ( power_power_nat @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.12         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.12           => ( A = B ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2200_power__inject__base,axiom,
% 9.20/5.12      ! [A: int,N2: nat,B: int] :
% 9.20/5.12        ( ( ( power_power_int @ A @ ( suc @ N2 ) )
% 9.20/5.12          = ( power_power_int @ B @ ( suc @ N2 ) ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.12         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.12           => ( A = B ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2201_zero__power,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 9.20/5.12          = zero_zero_rat ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2202_zero__power,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 9.20/5.12          = zero_zero_nat ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2203_zero__power,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( power_power_real @ zero_zero_real @ N2 )
% 9.20/5.12          = zero_zero_real ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2204_zero__power,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( power_power_int @ zero_zero_int @ N2 )
% 9.20/5.12          = zero_zero_int ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2205_zero__power,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 9.20/5.12          = zero_zero_complex ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2206_power__gt__expt,axiom,
% 9.20/5.12      ! [N2: nat,K: nat] :
% 9.20/5.12        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.12       => ( ord_less_nat @ K @ ( power_power_nat @ N2 @ K ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2207_nat__one__le__power,axiom,
% 9.20/5.12      ! [I: nat,N2: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
% 9.20/5.12       => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2208_power__eq__iff__eq__base,axiom,
% 9.20/5.12      ! [N2: nat,A: real,B: real] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.12         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.12           => ( ( ( power_power_real @ A @ N2 )
% 9.20/5.12                = ( power_power_real @ B @ N2 ) )
% 9.20/5.12              = ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2209_power__eq__iff__eq__base,axiom,
% 9.20/5.12      ! [N2: nat,A: rat,B: rat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.12         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.12           => ( ( ( power_power_rat @ A @ N2 )
% 9.20/5.12                = ( power_power_rat @ B @ N2 ) )
% 9.20/5.12              = ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2210_power__eq__iff__eq__base,axiom,
% 9.20/5.12      ! [N2: nat,A: nat,B: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.12         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.12           => ( ( ( power_power_nat @ A @ N2 )
% 9.20/5.12                = ( power_power_nat @ B @ N2 ) )
% 9.20/5.12              = ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2211_power__eq__iff__eq__base,axiom,
% 9.20/5.12      ! [N2: nat,A: int,B: int] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.12         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.12           => ( ( ( power_power_int @ A @ N2 )
% 9.20/5.12                = ( power_power_int @ B @ N2 ) )
% 9.20/5.12              = ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2212_power__eq__imp__eq__base,axiom,
% 9.20/5.12      ! [A: real,N2: nat,B: real] :
% 9.20/5.12        ( ( ( power_power_real @ A @ N2 )
% 9.20/5.12          = ( power_power_real @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.12         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.12           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12             => ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2213_power__eq__imp__eq__base,axiom,
% 9.20/5.12      ! [A: rat,N2: nat,B: rat] :
% 9.20/5.12        ( ( ( power_power_rat @ A @ N2 )
% 9.20/5.12          = ( power_power_rat @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.12         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.12           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12             => ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2214_power__eq__imp__eq__base,axiom,
% 9.20/5.12      ! [A: nat,N2: nat,B: nat] :
% 9.20/5.12        ( ( ( power_power_nat @ A @ N2 )
% 9.20/5.12          = ( power_power_nat @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.12         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.12           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12             => ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2215_power__eq__imp__eq__base,axiom,
% 9.20/5.12      ! [A: int,N2: nat,B: int] :
% 9.20/5.12        ( ( ( power_power_int @ A @ N2 )
% 9.20/5.12          = ( power_power_int @ B @ N2 ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.12         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.12           => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12             => ( A = B ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2216_zero__power2,axiom,
% 9.20/5.12      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12      = zero_zero_rat ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2217_zero__power2,axiom,
% 9.20/5.12      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12      = zero_zero_nat ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2218_zero__power2,axiom,
% 9.20/5.12      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12      = zero_zero_real ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2219_zero__power2,axiom,
% 9.20/5.12      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12      = zero_zero_int ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2220_zero__power2,axiom,
% 9.20/5.12      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12      = zero_zero_complex ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2221_less__exp,axiom,
% 9.20/5.12      ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2222_power2__nat__le__imp__le,axiom,
% 9.20/5.12      ! [M: nat,N2: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 9.20/5.12       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2223_power2__nat__le__eq__le,axiom,
% 9.20/5.12      ! [M: nat,N2: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2224_self__le__ge2__pow,axiom,
% 9.20/5.12      ! [K: nat,M: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.20/5.12       => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2225_power2__le__imp__le,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.12         => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2226_power2__le__imp__le,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.12         => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2227_power2__le__imp__le,axiom,
% 9.20/5.12      ! [X3: nat,Y3: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.20/5.12         => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2228_power2__le__imp__le,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.12         => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2229_power2__eq__imp__eq,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12          = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.12         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.12           => ( X3 = Y3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2230_power2__eq__imp__eq,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12          = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.12         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.12           => ( X3 = Y3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2231_power2__eq__imp__eq,axiom,
% 9.20/5.12      ! [X3: nat,Y3: nat] :
% 9.20/5.12        ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12          = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.20/5.12         => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.20/5.12           => ( X3 = Y3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2232_power2__eq__imp__eq,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12          = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.12         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.12           => ( X3 = Y3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2233_zero__le__power2,axiom,
% 9.20/5.12      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2234_zero__le__power2,axiom,
% 9.20/5.12      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2235_zero__le__power2,axiom,
% 9.20/5.12      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2236_power__strict__mono,axiom,
% 9.20/5.12      ! [A: real,B: real,N2: nat] :
% 9.20/5.12        ( ( ord_less_real @ A @ B )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12           => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2237_power__strict__mono,axiom,
% 9.20/5.12      ! [A: rat,B: rat,N2: nat] :
% 9.20/5.12        ( ( ord_less_rat @ A @ B )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12           => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2238_power__strict__mono,axiom,
% 9.20/5.12      ! [A: nat,B: nat,N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ A @ B )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12           => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2239_power__strict__mono,axiom,
% 9.20/5.12      ! [A: int,B: int,N2: nat] :
% 9.20/5.12        ( ( ord_less_int @ A @ B )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.12         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12           => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2240_power2__less__0,axiom,
% 9.20/5.12      ! [A: real] :
% 9.20/5.12        ~ ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_real ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2241_power2__less__0,axiom,
% 9.20/5.12      ! [A: rat] :
% 9.20/5.12        ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2242_power2__less__0,axiom,
% 9.20/5.12      ! [A: int] :
% 9.20/5.12        ~ ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_int ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2243_power2__less__imp__less,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.12         => ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2244_power2__less__imp__less,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.12         => ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2245_power2__less__imp__less,axiom,
% 9.20/5.12      ! [X3: nat,Y3: nat] :
% 9.20/5.12        ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
% 9.20/5.12         => ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2246_power2__less__imp__less,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.12       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.12         => ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2247_sum__power2__le__zero__iff,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
% 9.20/5.12        = ( ( X3 = zero_zero_real )
% 9.20/5.12          & ( Y3 = zero_zero_real ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2248_sum__power2__le__zero__iff,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
% 9.20/5.12        = ( ( X3 = zero_zero_rat )
% 9.20/5.12          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2249_sum__power2__le__zero__iff,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
% 9.20/5.12        = ( ( X3 = zero_zero_int )
% 9.20/5.12          & ( Y3 = zero_zero_int ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2250_sum__power2__ge__zero,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2251_sum__power2__ge__zero,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2252_sum__power2__ge__zero,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2253_not__sum__power2__lt__zero,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2254_not__sum__power2__lt__zero,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2255_not__sum__power2__lt__zero,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2256_sum__power2__gt__zero__iff,axiom,
% 9.20/5.12      ! [X3: real,Y3: real] :
% 9.20/5.12        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.12        = ( ( X3 != zero_zero_real )
% 9.20/5.12          | ( Y3 != zero_zero_real ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2257_sum__power2__gt__zero__iff,axiom,
% 9.20/5.12      ! [X3: rat,Y3: rat] :
% 9.20/5.12        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.12        = ( ( X3 != zero_zero_rat )
% 9.20/5.12          | ( Y3 != zero_zero_rat ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2258_sum__power2__gt__zero__iff,axiom,
% 9.20/5.12      ! [X3: int,Y3: int] :
% 9.20/5.12        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.12        = ( ( X3 != zero_zero_int )
% 9.20/5.12          | ( Y3 != zero_zero_int ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2259_both__member__options__from__chilf__to__complete__tree,axiom,
% 9.20/5.12      ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
% 9.20/5.12        ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.20/5.12       => ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.20/5.12         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.12           => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2260_add__self__div__2,axiom,
% 9.20/5.12      ! [M: nat] :
% 9.20/5.12        ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12        = M ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2261_div2__Suc__Suc,axiom,
% 9.20/5.12      ! [M: nat] :
% 9.20/5.12        ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.12        = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2262_both__member__options__from__complete__tree__to__child,axiom,
% 9.20/5.12      ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.20/5.12        ( ( ord_less_eq_nat @ one_one_nat @ Deg )
% 9.20/5.12       => ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
% 9.20/5.12         => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.12            | ( X3 = Mi )
% 9.20/5.12            | ( X3 = Ma ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2263_semiring__norm_I76_J,axiom,
% 9.20/5.12      ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2264_div__less,axiom,
% 9.20/5.12      ! [M: nat,N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.12       => ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.12          = zero_zero_nat ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2265_div__by__Suc__0,axiom,
% 9.20/5.12      ! [M: nat] :
% 9.20/5.12        ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.20/5.12        = M ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2266_set__n__deg__not__0,axiom,
% 9.20/5.12      ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
% 9.20/5.12        ( ! [X4: vEBT_VEBT] :
% 9.20/5.12            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.20/5.12           => ( vEBT_invar_vebt @ X4 @ N2 ) )
% 9.20/5.12       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.20/5.12            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.12         => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2267_Suc__n__div__2__gt__zero,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.12       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2268_div__2__gt__zero,axiom,
% 9.20/5.12      ! [N2: nat] :
% 9.20/5.12        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.12       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2269_deg1Leaf,axiom,
% 9.20/5.12      ! [T: vEBT_VEBT] :
% 9.20/5.12        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.20/5.12        = ( ? [A3: $o,B3: $o] :
% 9.20/5.12              ( T
% 9.20/5.12              = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2270_deg__1__Leaf,axiom,
% 9.20/5.12      ! [T: vEBT_VEBT] :
% 9.20/5.12        ( ( vEBT_invar_vebt @ T @ one_one_nat )
% 9.20/5.12       => ? [A4: $o,B4: $o] :
% 9.20/5.12            ( T
% 9.20/5.12            = ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2271_deg__1__Leafy,axiom,
% 9.20/5.12      ! [T: vEBT_VEBT,N2: nat] :
% 9.20/5.12        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.20/5.12       => ( ( N2 = one_one_nat )
% 9.20/5.12         => ? [A4: $o,B4: $o] :
% 9.20/5.12              ( T
% 9.20/5.12              = ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2272_div__by__1,axiom,
% 9.20/5.12      ! [A: complex] :
% 9.20/5.12        ( ( divide1717551699836669952omplex @ A @ one_one_complex )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2273_div__by__1,axiom,
% 9.20/5.12      ! [A: nat] :
% 9.20/5.12        ( ( divide_divide_nat @ A @ one_one_nat )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2274_div__by__1,axiom,
% 9.20/5.12      ! [A: int] :
% 9.20/5.12        ( ( divide_divide_int @ A @ one_one_int )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2275_div__by__1,axiom,
% 9.20/5.12      ! [A: real] :
% 9.20/5.12        ( ( divide_divide_real @ A @ one_one_real )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2276_div__by__1,axiom,
% 9.20/5.12      ! [A: rat] :
% 9.20/5.12        ( ( divide_divide_rat @ A @ one_one_rat )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2277_div__by__1,axiom,
% 9.20/5.12      ! [A: code_integer] :
% 9.20/5.12        ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
% 9.20/5.12        = A ) ).
% 9.20/5.12  
% 9.20/5.12  thf(fact_2278_semiring__norm_I78_J,axiom,
% 9.20/5.12      ! [M: num,N2: num] :
% 9.20/5.12        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.12        = ( ord_less_num @ M @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2279_semiring__norm_I75_J,axiom,
% 9.20/5.13      ! [M: num] :
% 9.20/5.13        ~ ( ord_less_num @ M @ one ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2280_divide__eq__1__iff,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.20/5.13          = one_one_complex )
% 9.20/5.13        = ( ( B != zero_zero_complex )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2281_divide__eq__1__iff,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ( divide_divide_real @ A @ B )
% 9.20/5.13          = one_one_real )
% 9.20/5.13        = ( ( B != zero_zero_real )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2282_divide__eq__1__iff,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ( divide_divide_rat @ A @ B )
% 9.20/5.13          = one_one_rat )
% 9.20/5.13        = ( ( B != zero_zero_rat )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2283_div__self,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( A != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.20/5.13          = one_one_complex ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2284_div__self,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( A != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ A @ A )
% 9.20/5.13          = one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2285_div__self,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( A != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ A @ A )
% 9.20/5.13          = one_one_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2286_div__self,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( A != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ A @ A )
% 9.20/5.13          = one_one_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2287_div__self,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( A != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ A @ A )
% 9.20/5.13          = one_one_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2288_div__self,axiom,
% 9.20/5.13      ! [A: code_integer] :
% 9.20/5.13        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ A @ A )
% 9.20/5.13          = one_one_Code_integer ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2289_one__eq__divide__iff,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( one_one_complex
% 9.20/5.13          = ( divide1717551699836669952omplex @ A @ B ) )
% 9.20/5.13        = ( ( B != zero_zero_complex )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2290_one__eq__divide__iff,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( one_one_real
% 9.20/5.13          = ( divide_divide_real @ A @ B ) )
% 9.20/5.13        = ( ( B != zero_zero_real )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2291_one__eq__divide__iff,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( one_one_rat
% 9.20/5.13          = ( divide_divide_rat @ A @ B ) )
% 9.20/5.13        = ( ( B != zero_zero_rat )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2292_divide__self,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( A != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ A @ A )
% 9.20/5.13          = one_one_complex ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2293_divide__self,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( A != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ A @ A )
% 9.20/5.13          = one_one_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2294_divide__self,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( A != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ A @ A )
% 9.20/5.13          = one_one_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2295_divide__self__if,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( ( A = zero_zero_complex )
% 9.20/5.13         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.20/5.13            = zero_zero_complex ) )
% 9.20/5.13        & ( ( A != zero_zero_complex )
% 9.20/5.13         => ( ( divide1717551699836669952omplex @ A @ A )
% 9.20/5.13            = one_one_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2296_divide__self__if,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ( A = zero_zero_real )
% 9.20/5.13         => ( ( divide_divide_real @ A @ A )
% 9.20/5.13            = zero_zero_real ) )
% 9.20/5.13        & ( ( A != zero_zero_real )
% 9.20/5.13         => ( ( divide_divide_real @ A @ A )
% 9.20/5.13            = one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2297_divide__self__if,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ( A = zero_zero_rat )
% 9.20/5.13         => ( ( divide_divide_rat @ A @ A )
% 9.20/5.13            = zero_zero_rat ) )
% 9.20/5.13        & ( ( A != zero_zero_rat )
% 9.20/5.13         => ( ( divide_divide_rat @ A @ A )
% 9.20/5.13            = one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2298_divide__eq__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( ( divide_divide_real @ B @ A )
% 9.20/5.13          = one_one_real )
% 9.20/5.13        = ( ( A != zero_zero_real )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2299_divide__eq__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( ( divide_divide_rat @ B @ A )
% 9.20/5.13          = one_one_rat )
% 9.20/5.13        = ( ( A != zero_zero_rat )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2300_eq__divide__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( one_one_real
% 9.20/5.13          = ( divide_divide_real @ B @ A ) )
% 9.20/5.13        = ( ( A != zero_zero_real )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2301_eq__divide__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( one_one_rat
% 9.20/5.13          = ( divide_divide_rat @ B @ A ) )
% 9.20/5.13        = ( ( A != zero_zero_rat )
% 9.20/5.13          & ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2302_one__divide__eq__0__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ( divide_divide_real @ one_one_real @ A )
% 9.20/5.13          = zero_zero_real )
% 9.20/5.13        = ( A = zero_zero_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2303_one__divide__eq__0__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ( divide_divide_rat @ one_one_rat @ A )
% 9.20/5.13          = zero_zero_rat )
% 9.20/5.13        = ( A = zero_zero_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2304_zero__eq__1__divide__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( zero_zero_real
% 9.20/5.13          = ( divide_divide_real @ one_one_real @ A ) )
% 9.20/5.13        = ( A = zero_zero_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2305_zero__eq__1__divide__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( zero_zero_rat
% 9.20/5.13          = ( divide_divide_rat @ one_one_rat @ A ) )
% 9.20/5.13        = ( A = zero_zero_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2306_power__inject__exp,axiom,
% 9.20/5.13      ! [A: real,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13       => ( ( ( power_power_real @ A @ M )
% 9.20/5.13            = ( power_power_real @ A @ N2 ) )
% 9.20/5.13          = ( M = N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2307_power__inject__exp,axiom,
% 9.20/5.13      ! [A: rat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ( ( power_power_rat @ A @ M )
% 9.20/5.13            = ( power_power_rat @ A @ N2 ) )
% 9.20/5.13          = ( M = N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2308_power__inject__exp,axiom,
% 9.20/5.13      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ( ( power_power_nat @ A @ M )
% 9.20/5.13            = ( power_power_nat @ A @ N2 ) )
% 9.20/5.13          = ( M = N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2309_power__inject__exp,axiom,
% 9.20/5.13      ! [A: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13       => ( ( ( power_power_int @ A @ M )
% 9.20/5.13            = ( power_power_int @ A @ N2 ) )
% 9.20/5.13          = ( M = N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2310_less__one,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ one_one_nat )
% 9.20/5.13        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2311_divide__le__0__1__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.20/5.13        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2312_divide__le__0__1__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.20/5.13        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2313_zero__le__divide__1__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.20/5.13        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2314_zero__le__divide__1__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.20/5.13        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2315_zero__less__divide__1__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ one_one_real @ A ) )
% 9.20/5.13        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2316_zero__less__divide__1__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
% 9.20/5.13        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2317_less__divide__eq__1__pos,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13          = ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2318_less__divide__eq__1__pos,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2319_less__divide__eq__1__neg,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13       => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13          = ( ord_less_real @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2320_less__divide__eq__1__neg,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13       => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2321_divide__less__eq__1__pos,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13          = ( ord_less_real @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2322_divide__less__eq__1__pos,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2323_divide__less__eq__1__neg,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13       => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13          = ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2324_divide__less__eq__1__neg,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2325_divide__less__0__1__iff,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( ord_less_real @ ( divide_divide_real @ one_one_real @ A ) @ zero_zero_real )
% 9.20/5.13        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2326_divide__less__0__1__iff,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
% 9.20/5.13        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2327_power__strict__increasing__iff,axiom,
% 9.20/5.13      ! [B: real,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.13       => ( ( ord_less_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2328_power__strict__increasing__iff,axiom,
% 9.20/5.13      ! [B: rat,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ B )
% 9.20/5.13       => ( ( ord_less_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2329_power__strict__increasing__iff,axiom,
% 9.20/5.13      ! [B: nat,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ B )
% 9.20/5.13       => ( ( ord_less_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2330_power__strict__increasing__iff,axiom,
% 9.20/5.13      ! [B: int,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ B )
% 9.20/5.13       => ( ( ord_less_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2331_le__divide__eq__1__pos,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2332_le__divide__eq__1__pos,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2333_le__divide__eq__1__neg,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13       => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2334_le__divide__eq__1__neg,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13       => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2335_divide__le__eq__1__pos,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2336_divide__le__eq__1__pos,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2337_divide__le__eq__1__neg,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2338_divide__le__eq__1__neg,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2339_one__add__one,axiom,
% 9.20/5.13      ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
% 9.20/5.13      = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2340_one__add__one,axiom,
% 9.20/5.13      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
% 9.20/5.13      = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2341_one__add__one,axiom,
% 9.20/5.13      ( ( plus_plus_real @ one_one_real @ one_one_real )
% 9.20/5.13      = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2342_one__add__one,axiom,
% 9.20/5.13      ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
% 9.20/5.13      = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2343_one__add__one,axiom,
% 9.20/5.13      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.20/5.13      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2344_one__add__one,axiom,
% 9.20/5.13      ( ( plus_plus_int @ one_one_int @ one_one_int )
% 9.20/5.13      = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2345_power__strict__decreasing__iff,axiom,
% 9.20/5.13      ! [B: real,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.13       => ( ( ord_less_real @ B @ one_one_real )
% 9.20/5.13         => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2346_power__strict__decreasing__iff,axiom,
% 9.20/5.13      ! [B: rat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.20/5.13       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.20/5.13         => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2347_power__strict__decreasing__iff,axiom,
% 9.20/5.13      ! [B: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.13       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.20/5.13         => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2348_power__strict__decreasing__iff,axiom,
% 9.20/5.13      ! [B: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.13       => ( ( ord_less_int @ B @ one_one_int )
% 9.20/5.13         => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2349_power__increasing__iff,axiom,
% 9.20/5.13      ! [B: real,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.13       => ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2350_power__increasing__iff,axiom,
% 9.20/5.13      ! [B: rat,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ B )
% 9.20/5.13       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2351_power__increasing__iff,axiom,
% 9.20/5.13      ! [B: nat,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ B )
% 9.20/5.13       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2352_power__increasing__iff,axiom,
% 9.20/5.13      ! [B: int,X3: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ B )
% 9.20/5.13       => ( ( ord_less_eq_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
% 9.20/5.13          = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2353_Suc__1,axiom,
% 9.20/5.13      ( ( suc @ one_one_nat )
% 9.20/5.13      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2354_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ N2 ) )
% 9.20/5.13        = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2355_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.20/5.13        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2356_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 9.20/5.13        = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2357_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 9.20/5.13        = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2358_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.13        = ( numeral_numeral_nat @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2359_one__plus__numeral,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.13        = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2360_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ one_one_complex )
% 9.20/5.13        = ( numera6690914467698888265omplex @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2361_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
% 9.20/5.13        = ( numera1916890842035813515d_enat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2362_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 9.20/5.13        = ( numeral_numeral_real @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2363_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 9.20/5.13        = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2364_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 9.20/5.13        = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2365_numeral__plus__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 9.20/5.13        = ( numeral_numeral_int @ ( plus_plus_num @ N2 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2366_numeral__le__one__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat )
% 9.20/5.13        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2367_numeral__le__one__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ one_one_real )
% 9.20/5.13        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2368_numeral__le__one__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
% 9.20/5.13        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2369_numeral__le__one__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat )
% 9.20/5.13        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2370_numeral__le__one__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ one_one_int )
% 9.20/5.13        = ( ord_less_eq_num @ N2 @ one ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2371_one__less__numeral__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_le72135733267957522d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.20/5.13        = ( ord_less_num @ one @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2372_one__less__numeral__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ ( numeral_numeral_real @ N2 ) )
% 9.20/5.13        = ( ord_less_num @ one @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2373_one__less__numeral__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
% 9.20/5.13        = ( ord_less_num @ one @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2374_one__less__numeral__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.13        = ( ord_less_num @ one @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2375_one__less__numeral__iff,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.13        = ( ord_less_num @ one @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2376_one__div__two__eq__zero,axiom,
% 9.20/5.13      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2377_one__div__two__eq__zero,axiom,
% 9.20/5.13      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2378_one__div__two__eq__zero,axiom,
% 9.20/5.13      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_z3403309356797280102nteger ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2379_power__decreasing__iff,axiom,
% 9.20/5.13      ! [B: real,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.13       => ( ( ord_less_real @ B @ one_one_real )
% 9.20/5.13         => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2380_power__decreasing__iff,axiom,
% 9.20/5.13      ! [B: rat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.20/5.13       => ( ( ord_less_rat @ B @ one_one_rat )
% 9.20/5.13         => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2381_power__decreasing__iff,axiom,
% 9.20/5.13      ! [B: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.13       => ( ( ord_less_nat @ B @ one_one_nat )
% 9.20/5.13         => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2382_power__decreasing__iff,axiom,
% 9.20/5.13      ! [B: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.13       => ( ( ord_less_int @ B @ one_one_int )
% 9.20/5.13         => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
% 9.20/5.13            = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2383_one__reorient,axiom,
% 9.20/5.13      ! [X3: complex] :
% 9.20/5.13        ( ( one_one_complex = X3 )
% 9.20/5.13        = ( X3 = one_one_complex ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2384_one__reorient,axiom,
% 9.20/5.13      ! [X3: real] :
% 9.20/5.13        ( ( one_one_real = X3 )
% 9.20/5.13        = ( X3 = one_one_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2385_one__reorient,axiom,
% 9.20/5.13      ! [X3: rat] :
% 9.20/5.13        ( ( one_one_rat = X3 )
% 9.20/5.13        = ( X3 = one_one_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2386_one__reorient,axiom,
% 9.20/5.13      ! [X3: nat] :
% 9.20/5.13        ( ( one_one_nat = X3 )
% 9.20/5.13        = ( X3 = one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2387_one__reorient,axiom,
% 9.20/5.13      ! [X3: int] :
% 9.20/5.13        ( ( one_one_int = X3 )
% 9.20/5.13        = ( X3 = one_one_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2388_le__numeral__extra_I4_J,axiom,
% 9.20/5.13      ord_less_eq_real @ one_one_real @ one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2389_le__numeral__extra_I4_J,axiom,
% 9.20/5.13      ord_less_eq_rat @ one_one_rat @ one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2390_le__numeral__extra_I4_J,axiom,
% 9.20/5.13      ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2391_le__numeral__extra_I4_J,axiom,
% 9.20/5.13      ord_less_eq_int @ one_one_int @ one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2392_zero__neq__one,axiom,
% 9.20/5.13      zero_zero_complex != one_one_complex ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2393_zero__neq__one,axiom,
% 9.20/5.13      zero_zero_real != one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2394_zero__neq__one,axiom,
% 9.20/5.13      zero_zero_rat != one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2395_zero__neq__one,axiom,
% 9.20/5.13      zero_zero_nat != one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2396_zero__neq__one,axiom,
% 9.20/5.13      zero_zero_int != one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2397_less__numeral__extra_I4_J,axiom,
% 9.20/5.13      ~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2398_less__numeral__extra_I4_J,axiom,
% 9.20/5.13      ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2399_less__numeral__extra_I4_J,axiom,
% 9.20/5.13      ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2400_less__numeral__extra_I4_J,axiom,
% 9.20/5.13      ~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2401_div__add__self2,axiom,
% 9.20/5.13      ! [B: nat,A: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2402_div__add__self2,axiom,
% 9.20/5.13      ! [B: int,A: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.20/5.13          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2403_div__add__self2,axiom,
% 9.20/5.13      ! [B: code_integer,A: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2404_div__add__self1,axiom,
% 9.20/5.13      ! [B: nat,A: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2405_div__add__self1,axiom,
% 9.20/5.13      ! [B: int,A: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.20/5.13          = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2406_div__add__self1,axiom,
% 9.20/5.13      ! [B: code_integer,A: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2407_zero__less__one__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2408_zero__less__one__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2409_zero__less__one__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2410_zero__less__one__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2411_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_real @ zero_zero_real @ one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2412_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2413_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2414_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
% 9.20/5.13      ord_less_eq_int @ zero_zero_int @ one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2415_not__one__le__zero,axiom,
% 9.20/5.13      ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2416_not__one__le__zero,axiom,
% 9.20/5.13      ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2417_not__one__le__zero,axiom,
% 9.20/5.13      ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2418_not__one__le__zero,axiom,
% 9.20/5.13      ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2419_not__one__less__zero,axiom,
% 9.20/5.13      ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2420_not__one__less__zero,axiom,
% 9.20/5.13      ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2421_not__one__less__zero,axiom,
% 9.20/5.13      ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2422_not__one__less__zero,axiom,
% 9.20/5.13      ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2423_zero__less__one,axiom,
% 9.20/5.13      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2424_zero__less__one,axiom,
% 9.20/5.13      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2425_zero__less__one,axiom,
% 9.20/5.13      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2426_zero__less__one,axiom,
% 9.20/5.13      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2427_less__numeral__extra_I1_J,axiom,
% 9.20/5.13      ord_less_real @ zero_zero_real @ one_one_real ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2428_less__numeral__extra_I1_J,axiom,
% 9.20/5.13      ord_less_rat @ zero_zero_rat @ one_one_rat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2429_less__numeral__extra_I1_J,axiom,
% 9.20/5.13      ord_less_nat @ zero_zero_nat @ one_one_nat ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2430_less__numeral__extra_I1_J,axiom,
% 9.20/5.13      ord_less_int @ zero_zero_int @ one_one_int ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2431_one__le__numeral,axiom,
% 9.20/5.13      ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2432_one__le__numeral,axiom,
% 9.20/5.13      ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2433_one__le__numeral,axiom,
% 9.20/5.13      ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2434_one__le__numeral,axiom,
% 9.20/5.13      ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2435_one__le__numeral,axiom,
% 9.20/5.13      ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2436_not__numeral__less__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2437_not__numeral__less__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2438_not__numeral__less__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2439_not__numeral__less__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2440_not__numeral__less__one,axiom,
% 9.20/5.13      ! [N2: num] :
% 9.20/5.13        ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2441_add__mono1,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ B )
% 9.20/5.13       => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2442_add__mono1,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ B )
% 9.20/5.13       => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2443_add__mono1,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( ord_less_nat @ A @ B )
% 9.20/5.13       => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2444_add__mono1,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( ord_less_int @ A @ B )
% 9.20/5.13       => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2445_less__add__one,axiom,
% 9.20/5.13      ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2446_less__add__one,axiom,
% 9.20/5.13      ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2447_less__add__one,axiom,
% 9.20/5.13      ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2448_less__add__one,axiom,
% 9.20/5.13      ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2449_right__inverse__eq,axiom,
% 9.20/5.13      ! [B: complex,A: complex] :
% 9.20/5.13        ( ( B != zero_zero_complex )
% 9.20/5.13       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.20/5.13            = one_one_complex )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2450_right__inverse__eq,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( B != zero_zero_real )
% 9.20/5.13       => ( ( ( divide_divide_real @ A @ B )
% 9.20/5.13            = one_one_real )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2451_right__inverse__eq,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( B != zero_zero_rat )
% 9.20/5.13       => ( ( ( divide_divide_rat @ A @ B )
% 9.20/5.13            = one_one_rat )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2452_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
% 9.20/5.13        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2453_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
% 9.20/5.13        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2454_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
% 9.20/5.13        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2455_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
% 9.20/5.13        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2456_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
% 9.20/5.13        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2457_one__plus__numeral__commute,axiom,
% 9.20/5.13      ! [X3: num] :
% 9.20/5.13        ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
% 9.20/5.13        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2458_one__le__power,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.20/5.13       => ( ord_less_eq_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2459_one__le__power,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2460_one__le__power,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2461_one__le__power,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.20/5.13       => ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2462_power__0,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( power_power_rat @ A @ zero_zero_nat )
% 9.20/5.13        = one_one_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2463_power__0,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( power_power_nat @ A @ zero_zero_nat )
% 9.20/5.13        = one_one_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2464_power__0,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( power_power_real @ A @ zero_zero_nat )
% 9.20/5.13        = one_one_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2465_power__0,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( power_power_int @ A @ zero_zero_nat )
% 9.20/5.13        = one_one_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2466_power__0,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( power_power_complex @ A @ zero_zero_nat )
% 9.20/5.13        = one_one_complex ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2467_div__eq__dividend__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.13       => ( ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.13            = M )
% 9.20/5.13          = ( N2 = one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2468_div__less__dividend,axiom,
% 9.20/5.13      ! [N2: nat,M: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ N2 )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.13         => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2469_One__nat__def,axiom,
% 9.20/5.13      ( one_one_nat
% 9.20/5.13      = ( suc @ zero_zero_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2470_Suc__eq__plus1,axiom,
% 9.20/5.13      ( suc
% 9.20/5.13      = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2471_plus__1__eq__Suc,axiom,
% 9.20/5.13      ( ( plus_plus_nat @ one_one_nat )
% 9.20/5.13      = suc ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2472_Suc__eq__plus1__left,axiom,
% 9.20/5.13      ( suc
% 9.20/5.13      = ( plus_plus_nat @ one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2473_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
% 9.20/5.13      ! [Uu: $o,Uv: $o,D: nat] :
% 9.20/5.13        ( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D )
% 9.20/5.13        = ( D = one_one_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2474_zero__less__two,axiom,
% 9.20/5.13      ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2475_zero__less__two,axiom,
% 9.20/5.13      ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2476_zero__less__two,axiom,
% 9.20/5.13      ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2477_zero__less__two,axiom,
% 9.20/5.13      ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2478_power__le__one,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.20/5.13         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2479_power__le__one,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.20/5.13         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2480_power__le__one,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.13       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.20/5.13         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2481_power__le__one,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.13       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.20/5.13         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2482_less__divide__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13            & ( ord_less_real @ A @ B ) )
% 9.20/5.13          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2483_less__divide__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13            & ( ord_less_rat @ A @ B ) )
% 9.20/5.13          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2484_divide__less__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13            & ( ord_less_real @ B @ A ) )
% 9.20/5.13          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13            & ( ord_less_real @ A @ B ) )
% 9.20/5.13          | ( A = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2485_divide__less__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13            & ( ord_less_rat @ B @ A ) )
% 9.20/5.13          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13            & ( ord_less_rat @ A @ B ) )
% 9.20/5.13          | ( A = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2486_less__half__sum,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ B )
% 9.20/5.13       => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2487_less__half__sum,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ B )
% 9.20/5.13       => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2488_gt__half__sum,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ord_less_real @ A @ B )
% 9.20/5.13       => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2489_gt__half__sum,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ord_less_rat @ A @ B )
% 9.20/5.13       => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2490_power__0__left,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ( N2 = zero_zero_nat )
% 9.20/5.13         => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 9.20/5.13            = one_one_rat ) )
% 9.20/5.13        & ( ( N2 != zero_zero_nat )
% 9.20/5.13         => ( ( power_power_rat @ zero_zero_rat @ N2 )
% 9.20/5.13            = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2491_power__0__left,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ( N2 = zero_zero_nat )
% 9.20/5.13         => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 9.20/5.13            = one_one_nat ) )
% 9.20/5.13        & ( ( N2 != zero_zero_nat )
% 9.20/5.13         => ( ( power_power_nat @ zero_zero_nat @ N2 )
% 9.20/5.13            = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2492_power__0__left,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ( N2 = zero_zero_nat )
% 9.20/5.13         => ( ( power_power_real @ zero_zero_real @ N2 )
% 9.20/5.13            = one_one_real ) )
% 9.20/5.13        & ( ( N2 != zero_zero_nat )
% 9.20/5.13         => ( ( power_power_real @ zero_zero_real @ N2 )
% 9.20/5.13            = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2493_power__0__left,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ( N2 = zero_zero_nat )
% 9.20/5.13         => ( ( power_power_int @ zero_zero_int @ N2 )
% 9.20/5.13            = one_one_int ) )
% 9.20/5.13        & ( ( N2 != zero_zero_nat )
% 9.20/5.13         => ( ( power_power_int @ zero_zero_int @ N2 )
% 9.20/5.13            = zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2494_power__0__left,axiom,
% 9.20/5.13      ! [N2: nat] :
% 9.20/5.13        ( ( ( N2 = zero_zero_nat )
% 9.20/5.13         => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 9.20/5.13            = one_one_complex ) )
% 9.20/5.13        & ( ( N2 != zero_zero_nat )
% 9.20/5.13         => ( ( power_power_complex @ zero_zero_complex @ N2 )
% 9.20/5.13            = zero_zero_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2495_power__gt1,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13       => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2496_power__gt1,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2497_power__gt1,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2498_power__gt1,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13       => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2499_power__less__imp__less__exp,axiom,
% 9.20/5.13      ! [A: real,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13       => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2500_power__less__imp__less__exp,axiom,
% 9.20/5.13      ! [A: rat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2501_power__less__imp__less__exp,axiom,
% 9.20/5.13      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2502_power__less__imp__less__exp,axiom,
% 9.20/5.13      ! [A: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13       => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2503_power__strict__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: real] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13         => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2504_power__strict__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: rat] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2505_power__strict__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: nat] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13         => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2506_power__strict__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: int] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13         => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2507_power__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: real] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_real @ one_one_real @ A )
% 9.20/5.13         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2508_power__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: rat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.20/5.13         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2509_power__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.20/5.13         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2510_power__increasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: int] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_int @ one_one_int @ A )
% 9.20/5.13         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2511_nat__induct__non__zero,axiom,
% 9.20/5.13      ! [N2: nat,P: nat > $o] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13       => ( ( P @ one_one_nat )
% 9.20/5.13         => ( ! [N3: nat] :
% 9.20/5.13                ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.13               => ( ( P @ N3 )
% 9.20/5.13                 => ( P @ ( suc @ N3 ) ) ) )
% 9.20/5.13           => ( P @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2512_vebt__member_Osimps_I1_J,axiom,
% 9.20/5.13      ! [A: $o,B: $o,X3: nat] :
% 9.20/5.13        ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X3 )
% 9.20/5.13        = ( ( ( X3 = zero_zero_nat )
% 9.20/5.13           => A )
% 9.20/5.13          & ( ( X3 != zero_zero_nat )
% 9.20/5.13           => ( ( ( X3 = one_one_nat )
% 9.20/5.13               => B )
% 9.20/5.13              & ( X3 = one_one_nat ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2513_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
% 9.20/5.13      ! [A: $o,B: $o,X3: nat] :
% 9.20/5.13        ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X3 )
% 9.20/5.13        = ( ( ( X3 = zero_zero_nat )
% 9.20/5.13           => A )
% 9.20/5.13          & ( ( X3 != zero_zero_nat )
% 9.20/5.13           => ( ( ( X3 = one_one_nat )
% 9.20/5.13               => B )
% 9.20/5.13              & ( X3 = one_one_nat ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2514_le__divide__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
% 9.20/5.13        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13            & ( ord_less_eq_real @ A @ B ) )
% 9.20/5.13          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13            & ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2515_le__divide__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
% 9.20/5.13        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13            & ( ord_less_eq_rat @ A @ B ) )
% 9.20/5.13          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13            & ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2516_divide__le__eq__1,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
% 9.20/5.13        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13            & ( ord_less_eq_real @ B @ A ) )
% 9.20/5.13          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.13            & ( ord_less_eq_real @ A @ B ) )
% 9.20/5.13          | ( A = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2517_divide__le__eq__1,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
% 9.20/5.13        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13            & ( ord_less_eq_rat @ B @ A ) )
% 9.20/5.13          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.13            & ( ord_less_eq_rat @ A @ B ) )
% 9.20/5.13          | ( A = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2518_power__Suc__le__self,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.20/5.13         => ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2519_power__Suc__le__self,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.20/5.13         => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2520_power__Suc__le__self,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.13       => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.20/5.13         => ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2521_power__Suc__le__self,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.13       => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.20/5.13         => ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2522_power__Suc__less__one,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13       => ( ( ord_less_real @ A @ one_one_real )
% 9.20/5.13         => ( ord_less_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2523_power__Suc__less__one,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.20/5.13         => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2524_power__Suc__less__one,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.20/5.13         => ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2525_power__Suc__less__one,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.13       => ( ( ord_less_int @ A @ one_one_int )
% 9.20/5.13         => ( ord_less_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2526_power__strict__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: real] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.13         => ( ( ord_less_real @ A @ one_one_real )
% 9.20/5.13           => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2527_power__strict__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: rat] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.13         => ( ( ord_less_rat @ A @ one_one_rat )
% 9.20/5.13           => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2528_power__strict__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: nat] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.13         => ( ( ord_less_nat @ A @ one_one_nat )
% 9.20/5.13           => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2529_power__strict__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: int] :
% 9.20/5.13        ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.13         => ( ( ord_less_int @ A @ one_one_int )
% 9.20/5.13           => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2530_power__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: real] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.13         => ( ( ord_less_eq_real @ A @ one_one_real )
% 9.20/5.13           => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2531_power__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: rat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.13         => ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.20/5.13           => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2532_power__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.13         => ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.20/5.13           => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2533_power__decreasing,axiom,
% 9.20/5.13      ! [N2: nat,N7: nat,A: int] :
% 9.20/5.13        ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.13       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.13         => ( ( ord_less_eq_int @ A @ one_one_int )
% 9.20/5.13           => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2534_power__le__imp__le__exp,axiom,
% 9.20/5.13      ! [A: real,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13       => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2535_power__le__imp__le__exp,axiom,
% 9.20/5.13      ! [A: rat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2536_power__le__imp__le__exp,axiom,
% 9.20/5.13      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2537_power__le__imp__le__exp,axiom,
% 9.20/5.13      ! [A: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13       => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2538_self__le__power,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_real @ one_one_real @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_eq_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2539_self__le__power,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2540_self__le__power,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2541_self__le__power,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_int @ one_one_int @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_eq_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2542_one__less__power,axiom,
% 9.20/5.13      ! [A: real,N2: nat] :
% 9.20/5.13        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2543_one__less__power,axiom,
% 9.20/5.13      ! [A: rat,N2: nat] :
% 9.20/5.13        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2544_one__less__power,axiom,
% 9.20/5.13      ! [A: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2545_one__less__power,axiom,
% 9.20/5.13      ! [A: int,N2: nat] :
% 9.20/5.13        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.13       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13         => ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2546_nat__1__add__1,axiom,
% 9.20/5.13      ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
% 9.20/5.13      = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2547_vebt__maxt_Osimps_I1_J,axiom,
% 9.20/5.13      ! [B: $o,A: $o] :
% 9.20/5.13        ( ( B
% 9.20/5.13         => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13            = ( some_nat @ one_one_nat ) ) )
% 9.20/5.13        & ( ~ B
% 9.20/5.13         => ( ( A
% 9.20/5.13             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13                = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.13            & ( ~ A
% 9.20/5.13             => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13                = none_nat ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2548_vebt__mint_Osimps_I1_J,axiom,
% 9.20/5.13      ! [A: $o,B: $o] :
% 9.20/5.13        ( ( A
% 9.20/5.13         => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13            = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.13        & ( ~ A
% 9.20/5.13         => ( ( B
% 9.20/5.13             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13                = ( some_nat @ one_one_nat ) ) )
% 9.20/5.13            & ( ~ B
% 9.20/5.13             => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
% 9.20/5.13                = none_nat ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2549_div__le__mono,axiom,
% 9.20/5.13      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.13       => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2550_div__le__dividend,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2551_ex__power__ivl2,axiom,
% 9.20/5.13      ! [B: nat,K: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.13       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.20/5.13         => ? [N3: nat] :
% 9.20/5.13              ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 9.20/5.13              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2552_ex__power__ivl1,axiom,
% 9.20/5.13      ! [B: nat,K: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.13       => ( ( ord_less_eq_nat @ one_one_nat @ K )
% 9.20/5.13         => ? [N3: nat] :
% 9.20/5.13              ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
% 9.20/5.13              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2553_Euclidean__Division_Odiv__eq__0__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.13          = zero_zero_nat )
% 9.20/5.13        = ( ( ord_less_nat @ M @ N2 )
% 9.20/5.13          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2554_Suc__div__le__mono,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2555_vebt__maxt_Oelims,axiom,
% 9.20/5.13      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.20/5.13        ( ( ( vEBT_vebt_maxt @ X3 )
% 9.20/5.13          = Y3 )
% 9.20/5.13       => ( ! [A4: $o,B4: $o] :
% 9.20/5.13              ( ( X3
% 9.20/5.13                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.13             => ~ ( ( B4
% 9.20/5.13                   => ( Y3
% 9.20/5.13                      = ( some_nat @ one_one_nat ) ) )
% 9.20/5.13                  & ( ~ B4
% 9.20/5.13                   => ( ( A4
% 9.20/5.13                       => ( Y3
% 9.20/5.13                          = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.13                      & ( ~ A4
% 9.20/5.13                       => ( Y3 = none_nat ) ) ) ) ) )
% 9.20/5.13         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.13                  ( X3
% 9.20/5.13                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.13             => ( Y3 != none_nat ) )
% 9.20/5.13           => ~ ! [Mi2: nat,Ma2: nat] :
% 9.20/5.13                  ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.13                      ( X3
% 9.20/5.13                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.20/5.13                 => ( Y3
% 9.20/5.13                   != ( some_nat @ Ma2 ) ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2556_vebt__mint_Oelims,axiom,
% 9.20/5.13      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.20/5.13        ( ( ( vEBT_vebt_mint @ X3 )
% 9.20/5.13          = Y3 )
% 9.20/5.13       => ( ! [A4: $o,B4: $o] :
% 9.20/5.13              ( ( X3
% 9.20/5.13                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.13             => ~ ( ( A4
% 9.20/5.13                   => ( Y3
% 9.20/5.13                      = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.13                  & ( ~ A4
% 9.20/5.13                   => ( ( B4
% 9.20/5.13                       => ( Y3
% 9.20/5.13                          = ( some_nat @ one_one_nat ) ) )
% 9.20/5.13                      & ( ~ B4
% 9.20/5.13                       => ( Y3 = none_nat ) ) ) ) ) )
% 9.20/5.13         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.13                  ( X3
% 9.20/5.13                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.13             => ( Y3 != none_nat ) )
% 9.20/5.13           => ~ ! [Mi2: nat] :
% 9.20/5.13                  ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.13                      ( X3
% 9.20/5.13                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.20/5.13                 => ( Y3
% 9.20/5.13                   != ( some_nat @ Mi2 ) ) ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2557_div__greater__zero__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
% 9.20/5.13        = ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.13          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2558_div__le__mono2,axiom,
% 9.20/5.13      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.13       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.13         => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2559_bits__1__div__2,axiom,
% 9.20/5.13      ( ( divide_divide_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2560_bits__1__div__2,axiom,
% 9.20/5.13      ( ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2561_bits__1__div__2,axiom,
% 9.20/5.13      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.13      = zero_z3403309356797280102nteger ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2562_nat__induct2,axiom,
% 9.20/5.13      ! [P: nat > $o,N2: nat] :
% 9.20/5.13        ( ( P @ zero_zero_nat )
% 9.20/5.13       => ( ( P @ one_one_nat )
% 9.20/5.13         => ( ! [N3: nat] :
% 9.20/5.13                ( ( P @ N3 )
% 9.20/5.13               => ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.13           => ( P @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2563_div__exp__eq,axiom,
% 9.20/5.13      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.13        = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2564_div__exp__eq,axiom,
% 9.20/5.13      ! [A: int,M: nat,N2: nat] :
% 9.20/5.13        ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.13        = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2565_div__exp__eq,axiom,
% 9.20/5.13      ! [A: code_integer,M: nat,N2: nat] :
% 9.20/5.13        ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.13        = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2566_exp__add__not__zero__imp__right,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.13         != zero_zero_nat )
% 9.20/5.13       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.13         != zero_zero_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2567_exp__add__not__zero__imp__right,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.13         != zero_zero_int )
% 9.20/5.13       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.13         != zero_zero_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2568_exp__add__not__zero__imp__left,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.13         != zero_zero_nat )
% 9.20/5.13       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.20/5.13         != zero_zero_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2569_exp__add__not__zero__imp__left,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.13         != zero_zero_int )
% 9.20/5.13       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.20/5.13         != zero_zero_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2570_bit__concat__def,axiom,
% 9.20/5.13      ( vEBT_VEBT_bit_concat
% 9.20/5.13      = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2571_low__inv,axiom,
% 9.20/5.13      ! [X3: nat,N2: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.13       => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
% 9.20/5.13          = X3 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2572_high__inv,axiom,
% 9.20/5.13      ! [X3: nat,N2: nat,Y3: nat] :
% 9.20/5.13        ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.13       => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
% 9.20/5.13          = Y3 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2573_enat__ord__number_I1_J,axiom,
% 9.20/5.13      ! [M: num,N2: num] :
% 9.20/5.13        ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.20/5.13        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2574_enat__ord__number_I2_J,axiom,
% 9.20/5.13      ! [M: num,N2: num] :
% 9.20/5.13        ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
% 9.20/5.13        = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2575_mul__shift,axiom,
% 9.20/5.13      ! [X3: nat,Y3: nat,Z2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ X3 @ Y3 )
% 9.20/5.13          = Z2 )
% 9.20/5.13        = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
% 9.20/5.13          = ( some_nat @ Z2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2576_mul__def,axiom,
% 9.20/5.13      ( vEBT_VEBT_mul
% 9.20/5.13      = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2577_i0__less,axiom,
% 9.20/5.13      ! [N2: extended_enat] :
% 9.20/5.13        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
% 9.20/5.13        = ( N2 != zero_z5237406670263579293d_enat ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2578_mult__cancel__right,axiom,
% 9.20/5.13      ! [A: complex,C: complex,B: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ A @ C )
% 9.20/5.13          = ( times_times_complex @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2579_mult__cancel__right,axiom,
% 9.20/5.13      ! [A: real,C: real,B: real] :
% 9.20/5.13        ( ( ( times_times_real @ A @ C )
% 9.20/5.13          = ( times_times_real @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2580_mult__cancel__right,axiom,
% 9.20/5.13      ! [A: rat,C: rat,B: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ A @ C )
% 9.20/5.13          = ( times_times_rat @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2581_mult__cancel__right,axiom,
% 9.20/5.13      ! [A: nat,C: nat,B: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ A @ C )
% 9.20/5.13          = ( times_times_nat @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_nat )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2582_mult__cancel__right,axiom,
% 9.20/5.13      ! [A: int,C: int,B: int] :
% 9.20/5.13        ( ( ( times_times_int @ A @ C )
% 9.20/5.13          = ( times_times_int @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2583_mult__cancel__left,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ C @ A )
% 9.20/5.13          = ( times_times_complex @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2584_mult__cancel__left,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( ( times_times_real @ C @ A )
% 9.20/5.13          = ( times_times_real @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2585_mult__cancel__left,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ C @ A )
% 9.20/5.13          = ( times_times_rat @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2586_mult__cancel__left,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ C @ A )
% 9.20/5.13          = ( times_times_nat @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_nat )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2587_mult__cancel__left,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( ( times_times_int @ C @ A )
% 9.20/5.13          = ( times_times_int @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2588_mult__eq__0__iff,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ A @ B )
% 9.20/5.13          = zero_zero_complex )
% 9.20/5.13        = ( ( A = zero_zero_complex )
% 9.20/5.13          | ( B = zero_zero_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2589_mult__eq__0__iff,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ( times_times_real @ A @ B )
% 9.20/5.13          = zero_zero_real )
% 9.20/5.13        = ( ( A = zero_zero_real )
% 9.20/5.13          | ( B = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2590_mult__eq__0__iff,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ A @ B )
% 9.20/5.13          = zero_zero_rat )
% 9.20/5.13        = ( ( A = zero_zero_rat )
% 9.20/5.13          | ( B = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2591_mult__eq__0__iff,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ A @ B )
% 9.20/5.13          = zero_zero_nat )
% 9.20/5.13        = ( ( A = zero_zero_nat )
% 9.20/5.13          | ( B = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2592_mult__eq__0__iff,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( ( times_times_int @ A @ B )
% 9.20/5.13          = zero_zero_int )
% 9.20/5.13        = ( ( A = zero_zero_int )
% 9.20/5.13          | ( B = zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2593_mult__zero__right,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ A @ zero_zero_complex )
% 9.20/5.13        = zero_zero_complex ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2594_mult__zero__right,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ A @ zero_zero_real )
% 9.20/5.13        = zero_zero_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2595_mult__zero__right,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ A @ zero_zero_rat )
% 9.20/5.13        = zero_zero_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2596_mult__zero__right,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ A @ zero_zero_nat )
% 9.20/5.13        = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2597_mult__zero__right,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ A @ zero_zero_int )
% 9.20/5.13        = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2598_mult__zero__left,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ zero_zero_complex @ A )
% 9.20/5.13        = zero_zero_complex ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2599_mult__zero__left,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ zero_zero_real @ A )
% 9.20/5.13        = zero_zero_real ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2600_mult__zero__left,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ zero_zero_rat @ A )
% 9.20/5.13        = zero_zero_rat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2601_mult__zero__left,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ zero_zero_nat @ A )
% 9.20/5.13        = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2602_mult__zero__left,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ zero_zero_int @ A )
% 9.20/5.13        = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2603_bits__div__by__0,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( divide_divide_nat @ A @ zero_zero_nat )
% 9.20/5.13        = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2604_bits__div__by__0,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( divide_divide_int @ A @ zero_zero_int )
% 9.20/5.13        = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2605_bits__div__by__0,axiom,
% 9.20/5.13      ! [A: code_integer] :
% 9.20/5.13        ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
% 9.20/5.13        = zero_z3403309356797280102nteger ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2606_bits__div__0,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( divide_divide_nat @ zero_zero_nat @ A )
% 9.20/5.13        = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2607_bits__div__0,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( divide_divide_int @ zero_zero_int @ A )
% 9.20/5.13        = zero_zero_int ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2608_bits__div__0,axiom,
% 9.20/5.13      ! [A: code_integer] :
% 9.20/5.13        ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.13        = zero_z3403309356797280102nteger ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2609_mult_Oright__neutral,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ A @ one_one_complex )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2610_mult_Oright__neutral,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ A @ one_one_real )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2611_mult_Oright__neutral,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ A @ one_one_rat )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2612_mult_Oright__neutral,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ A @ one_one_nat )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2613_mult_Oright__neutral,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ A @ one_one_int )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2614_mult__1,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ one_one_complex @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2615_mult__1,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ one_one_real @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2616_mult__1,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ one_one_rat @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2617_mult__1,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ one_one_nat @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2618_mult__1,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ one_one_int @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2619_half__negative__int__iff,axiom,
% 9.20/5.13      ! [K: int] :
% 9.20/5.13        ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
% 9.20/5.13        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2620_times__divide__eq__left,axiom,
% 9.20/5.13      ! [B: complex,C: complex,A: complex] :
% 9.20/5.13        ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
% 9.20/5.13        = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2621_times__divide__eq__left,axiom,
% 9.20/5.13      ! [B: real,C: real,A: real] :
% 9.20/5.13        ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.13        = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2622_times__divide__eq__left,axiom,
% 9.20/5.13      ! [B: rat,C: rat,A: rat] :
% 9.20/5.13        ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.13        = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2623_divide__divide__eq__left,axiom,
% 9.20/5.13      ! [A: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 9.20/5.13        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2624_divide__divide__eq__left,axiom,
% 9.20/5.13      ! [A: real,B: real,C: real] :
% 9.20/5.13        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 9.20/5.13        = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2625_divide__divide__eq__left,axiom,
% 9.20/5.13      ! [A: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 9.20/5.13        = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2626_divide__divide__eq__right,axiom,
% 9.20/5.13      ! [A: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.13        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2627_divide__divide__eq__right,axiom,
% 9.20/5.13      ! [A: real,B: real,C: real] :
% 9.20/5.13        ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.13        = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2628_divide__divide__eq__right,axiom,
% 9.20/5.13      ! [A: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.13        = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2629_times__divide__eq__right,axiom,
% 9.20/5.13      ! [A: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.13        = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2630_times__divide__eq__right,axiom,
% 9.20/5.13      ! [A: real,B: real,C: real] :
% 9.20/5.13        ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.13        = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2631_times__divide__eq__right,axiom,
% 9.20/5.13      ! [A: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.13        = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2632_mult__cancel2,axiom,
% 9.20/5.13      ! [M: nat,K: nat,N2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ M @ K )
% 9.20/5.13          = ( times_times_nat @ N2 @ K ) )
% 9.20/5.13        = ( ( M = N2 )
% 9.20/5.13          | ( K = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2633_mult__cancel1,axiom,
% 9.20/5.13      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ K @ M )
% 9.20/5.13          = ( times_times_nat @ K @ N2 ) )
% 9.20/5.13        = ( ( M = N2 )
% 9.20/5.13          | ( K = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2634_mult__0__right,axiom,
% 9.20/5.13      ! [M: nat] :
% 9.20/5.13        ( ( times_times_nat @ M @ zero_zero_nat )
% 9.20/5.13        = zero_zero_nat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2635_mult__is__0,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ M @ N2 )
% 9.20/5.13          = zero_zero_nat )
% 9.20/5.13        = ( ( M = zero_zero_nat )
% 9.20/5.13          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2636_nat__1__eq__mult__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( one_one_nat
% 9.20/5.13          = ( times_times_nat @ M @ N2 ) )
% 9.20/5.13        = ( ( M = one_one_nat )
% 9.20/5.13          & ( N2 = one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2637_nat__mult__eq__1__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ M @ N2 )
% 9.20/5.13          = one_one_nat )
% 9.20/5.13        = ( ( M = one_one_nat )
% 9.20/5.13          & ( N2 = one_one_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2638_mult__cancel__right2,axiom,
% 9.20/5.13      ! [A: complex,C: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ A @ C )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( A = one_one_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2639_mult__cancel__right2,axiom,
% 9.20/5.13      ! [A: real,C: real] :
% 9.20/5.13        ( ( ( times_times_real @ A @ C )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( A = one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2640_mult__cancel__right2,axiom,
% 9.20/5.13      ! [A: rat,C: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ A @ C )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( A = one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2641_mult__cancel__right2,axiom,
% 9.20/5.13      ! [A: int,C: int] :
% 9.20/5.13        ( ( ( times_times_int @ A @ C )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( A = one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2642_mult__cancel__right1,axiom,
% 9.20/5.13      ! [C: complex,B: complex] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_complex @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( B = one_one_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2643_mult__cancel__right1,axiom,
% 9.20/5.13      ! [C: real,B: real] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_real @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( B = one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2644_mult__cancel__right1,axiom,
% 9.20/5.13      ! [C: rat,B: rat] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_rat @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( B = one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2645_mult__cancel__right1,axiom,
% 9.20/5.13      ! [C: int,B: int] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_int @ B @ C ) )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( B = one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2646_mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: complex,A: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ C @ A )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( A = one_one_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2647_mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: real,A: real] :
% 9.20/5.13        ( ( ( times_times_real @ C @ A )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( A = one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2648_mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: rat,A: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ C @ A )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( A = one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2649_mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: int,A: int] :
% 9.20/5.13        ( ( ( times_times_int @ C @ A )
% 9.20/5.13          = C )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( A = one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2650_mult__cancel__left1,axiom,
% 9.20/5.13      ! [C: complex,B: complex] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_complex @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_complex )
% 9.20/5.13          | ( B = one_one_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2651_mult__cancel__left1,axiom,
% 9.20/5.13      ! [C: real,B: real] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_real @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_real )
% 9.20/5.13          | ( B = one_one_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2652_mult__cancel__left1,axiom,
% 9.20/5.13      ! [C: rat,B: rat] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_rat @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_rat )
% 9.20/5.13          | ( B = one_one_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2653_mult__cancel__left1,axiom,
% 9.20/5.13      ! [C: int,B: int] :
% 9.20/5.13        ( ( C
% 9.20/5.13          = ( times_times_int @ C @ B ) )
% 9.20/5.13        = ( ( C = zero_zero_int )
% 9.20/5.13          | ( B = one_one_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2654_sum__squares__eq__zero__iff,axiom,
% 9.20/5.13      ! [X3: real,Y3: real] :
% 9.20/5.13        ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 9.20/5.13          = zero_zero_real )
% 9.20/5.13        = ( ( X3 = zero_zero_real )
% 9.20/5.13          & ( Y3 = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2655_sum__squares__eq__zero__iff,axiom,
% 9.20/5.13      ! [X3: rat,Y3: rat] :
% 9.20/5.13        ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 9.20/5.13          = zero_zero_rat )
% 9.20/5.13        = ( ( X3 = zero_zero_rat )
% 9.20/5.13          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2656_sum__squares__eq__zero__iff,axiom,
% 9.20/5.13      ! [X3: int,Y3: int] :
% 9.20/5.13        ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 9.20/5.13          = zero_zero_int )
% 9.20/5.13        = ( ( X3 = zero_zero_int )
% 9.20/5.13          & ( Y3 = zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2657_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2658_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
% 9.20/5.13          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2659_nonzero__mult__divide__mult__cancel__right2,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.13          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2660_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: complex,A: complex] :
% 9.20/5.13        ( ( B != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2661_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: nat,A: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2662_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: int,A: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2663_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( B != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2664_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( B != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2665_nonzero__mult__div__cancel__right,axiom,
% 9.20/5.13      ! [B: code_integer,A: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 9.20/5.13          = A ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2666_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2667_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.13          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2668_nonzero__mult__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.13          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2669_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2670_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
% 9.20/5.13          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2671_nonzero__mult__divide__mult__cancel__left2,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.13          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2672_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( A != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2673_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( A != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2674_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( A != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2675_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( A != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2676_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( A != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2677_nonzero__mult__div__cancel__left,axiom,
% 9.20/5.13      ! [A: code_integer,B: code_integer] :
% 9.20/5.13        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
% 9.20/5.13          = B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2678_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2679_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.13          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2680_nonzero__mult__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.13          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2681_mult__divide__mult__cancel__left__if,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( ( C = zero_zero_complex )
% 9.20/5.13         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.20/5.13            = zero_zero_complex ) )
% 9.20/5.13        & ( ( C != zero_zero_complex )
% 9.20/5.13         => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.20/5.13            = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2682_mult__divide__mult__cancel__left__if,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( ( C = zero_zero_real )
% 9.20/5.13         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.13            = zero_zero_real ) )
% 9.20/5.13        & ( ( C != zero_zero_real )
% 9.20/5.13         => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.13            = ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2683_mult__divide__mult__cancel__left__if,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( ( C = zero_zero_rat )
% 9.20/5.13         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.13            = zero_zero_rat ) )
% 9.20/5.13        & ( ( C != zero_zero_rat )
% 9.20/5.13         => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.13            = ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2684_div__mult__mult1__if,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( ( C = zero_zero_nat )
% 9.20/5.13         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.20/5.13            = zero_zero_nat ) )
% 9.20/5.13        & ( ( C != zero_zero_nat )
% 9.20/5.13         => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.20/5.13            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2685_div__mult__mult1__if,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( ( C = zero_zero_int )
% 9.20/5.13         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.13            = zero_zero_int ) )
% 9.20/5.13        & ( ( C != zero_zero_int )
% 9.20/5.13         => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.13            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2686_div__mult__mult1__if,axiom,
% 9.20/5.13      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.13        ( ( ( C = zero_z3403309356797280102nteger )
% 9.20/5.13         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.20/5.13            = zero_z3403309356797280102nteger ) )
% 9.20/5.13        & ( ( C != zero_z3403309356797280102nteger )
% 9.20/5.13         => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.20/5.13            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2687_div__mult__mult2,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( C != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.20/5.13          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2688_div__mult__mult2,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( C != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.13          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2689_div__mult__mult2,axiom,
% 9.20/5.13      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.13        ( ( C != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.13          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2690_div__mult__mult1,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( C != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.20/5.13          = ( divide_divide_nat @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2691_div__mult__mult1,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( C != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.13          = ( divide_divide_int @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2692_div__mult__mult1,axiom,
% 9.20/5.13      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.13        ( ( C != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.20/5.13          = ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2693_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: complex,B: complex,V: num] :
% 9.20/5.13        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
% 9.20/5.13        = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2694_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: extended_enat,B: extended_enat,V: num] :
% 9.20/5.13        ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
% 9.20/5.13        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2695_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: real,B: real,V: num] :
% 9.20/5.13        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
% 9.20/5.13        = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2696_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: rat,B: rat,V: num] :
% 9.20/5.13        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
% 9.20/5.13        = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2697_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: nat,B: nat,V: num] :
% 9.20/5.13        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
% 9.20/5.13        = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2698_distrib__right__numeral,axiom,
% 9.20/5.13      ! [A: int,B: int,V: num] :
% 9.20/5.13        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.13        = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2699_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: complex,C: complex] :
% 9.20/5.13        ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
% 9.20/5.13        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2700_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: extended_enat,C: extended_enat] :
% 9.20/5.13        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
% 9.20/5.13        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2701_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: real,C: real] :
% 9.20/5.13        ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
% 9.20/5.13        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2702_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: rat,C: rat] :
% 9.20/5.13        ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.13        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2703_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: nat,C: nat] :
% 9.20/5.13        ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
% 9.20/5.13        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2704_distrib__left__numeral,axiom,
% 9.20/5.13      ! [V: num,B: int,C: int] :
% 9.20/5.13        ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
% 9.20/5.13        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2705_mult__eq__1__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ M @ N2 )
% 9.20/5.13          = ( suc @ zero_zero_nat ) )
% 9.20/5.13        = ( ( M
% 9.20/5.13            = ( suc @ zero_zero_nat ) )
% 9.20/5.13          & ( N2
% 9.20/5.13            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2706_one__eq__mult__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ( suc @ zero_zero_nat )
% 9.20/5.13          = ( times_times_nat @ M @ N2 ) )
% 9.20/5.13        = ( ( M
% 9.20/5.13            = ( suc @ zero_zero_nat ) )
% 9.20/5.13          & ( N2
% 9.20/5.13            = ( suc @ zero_zero_nat ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2707_mult__less__cancel2,axiom,
% 9.20/5.13      ! [M: nat,K: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 9.20/5.13        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.13          & ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2708_nat__0__less__mult__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.13        = ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.13          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2709_nat__mult__less__cancel__disj,axiom,
% 9.20/5.13      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.13        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.13          & ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2710_mult__Suc__right,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( times_times_nat @ M @ ( suc @ N2 ) )
% 9.20/5.13        = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2711_nat__mult__div__cancel__disj,axiom,
% 9.20/5.13      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ( K = zero_zero_nat )
% 9.20/5.13         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.13            = zero_zero_nat ) )
% 9.20/5.13        & ( ( K != zero_zero_nat )
% 9.20/5.13         => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.13            = ( divide_divide_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2712_divide__le__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: real,W2: num,A: real] :
% 9.20/5.13        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 9.20/5.13        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2713_divide__le__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: rat,W2: num,A: rat] :
% 9.20/5.13        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 9.20/5.13        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2714_le__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: real,B: real,W2: num] :
% 9.20/5.13        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.13        = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2715_le__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: rat,B: rat,W2: num] :
% 9.20/5.13        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.13        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2716_divide__eq__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: complex,W2: num,A: complex] :
% 9.20/5.13        ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.13          = A )
% 9.20/5.13        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.13             != zero_zero_complex )
% 9.20/5.13           => ( B
% 9.20/5.13              = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.20/5.13          & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.13              = zero_zero_complex )
% 9.20/5.13           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2717_divide__eq__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: real,W2: num,A: real] :
% 9.20/5.13        ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.13          = A )
% 9.20/5.13        = ( ( ( ( numeral_numeral_real @ W2 )
% 9.20/5.13             != zero_zero_real )
% 9.20/5.13           => ( B
% 9.20/5.13              = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.20/5.13          & ( ( ( numeral_numeral_real @ W2 )
% 9.20/5.13              = zero_zero_real )
% 9.20/5.13           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2718_divide__eq__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: rat,W2: num,A: rat] :
% 9.20/5.13        ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.13          = A )
% 9.20/5.13        = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.20/5.13             != zero_zero_rat )
% 9.20/5.13           => ( B
% 9.20/5.13              = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.20/5.13          & ( ( ( numeral_numeral_rat @ W2 )
% 9.20/5.13              = zero_zero_rat )
% 9.20/5.13           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2719_eq__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: complex,B: complex,W2: num] :
% 9.20/5.13        ( ( A
% 9.20/5.13          = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.20/5.13        = ( ( ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.13             != zero_zero_complex )
% 9.20/5.13           => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.13              = B ) )
% 9.20/5.13          & ( ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.13              = zero_zero_complex )
% 9.20/5.13           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2720_eq__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: real,B: real,W2: num] :
% 9.20/5.13        ( ( A
% 9.20/5.13          = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.13        = ( ( ( ( numeral_numeral_real @ W2 )
% 9.20/5.13             != zero_zero_real )
% 9.20/5.13           => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.13              = B ) )
% 9.20/5.13          & ( ( ( numeral_numeral_real @ W2 )
% 9.20/5.13              = zero_zero_real )
% 9.20/5.13           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2721_eq__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: rat,B: rat,W2: num] :
% 9.20/5.13        ( ( A
% 9.20/5.13          = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.13        = ( ( ( ( numeral_numeral_rat @ W2 )
% 9.20/5.13             != zero_zero_rat )
% 9.20/5.13           => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.13              = B ) )
% 9.20/5.13          & ( ( ( numeral_numeral_rat @ W2 )
% 9.20/5.13              = zero_zero_rat )
% 9.20/5.13           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2722_less__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: real,B: real,W2: num] :
% 9.20/5.13        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.13        = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2723_less__divide__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [A: rat,B: rat,W2: num] :
% 9.20/5.13        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.13        = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2724_divide__less__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: real,W2: num,A: real] :
% 9.20/5.13        ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
% 9.20/5.13        = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2725_divide__less__eq__numeral1_I1_J,axiom,
% 9.20/5.13      ! [B: rat,W2: num,A: rat] :
% 9.20/5.13        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
% 9.20/5.13        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2726_nonzero__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( A != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2727_nonzero__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( A != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
% 9.20/5.13          = ( divide_divide_real @ one_one_real @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2728_nonzero__divide__mult__cancel__left,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( A != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
% 9.20/5.13          = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2729_nonzero__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [B: complex,A: complex] :
% 9.20/5.13        ( ( B != zero_zero_complex )
% 9.20/5.13       => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
% 9.20/5.13          = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2730_nonzero__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [B: real,A: real] :
% 9.20/5.13        ( ( B != zero_zero_real )
% 9.20/5.13       => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
% 9.20/5.13          = ( divide_divide_real @ one_one_real @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2731_nonzero__divide__mult__cancel__right,axiom,
% 9.20/5.13      ! [B: rat,A: rat] :
% 9.20/5.13        ( ( B != zero_zero_rat )
% 9.20/5.13       => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
% 9.20/5.13          = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2732_div__mult__self1,axiom,
% 9.20/5.13      ! [B: nat,A: nat,C: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2733_div__mult__self1,axiom,
% 9.20/5.13      ! [B: int,A: int,C: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 9.20/5.13          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2734_div__mult__self1,axiom,
% 9.20/5.13      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2735_div__mult__self2,axiom,
% 9.20/5.13      ! [B: nat,A: nat,C: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2736_div__mult__self2,axiom,
% 9.20/5.13      ! [B: int,A: int,C: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 9.20/5.13          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2737_div__mult__self2,axiom,
% 9.20/5.13      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2738_div__mult__self3,axiom,
% 9.20/5.13      ! [B: nat,C: nat,A: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2739_div__mult__self3,axiom,
% 9.20/5.13      ! [B: int,C: int,A: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 9.20/5.13          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2740_div__mult__self3,axiom,
% 9.20/5.13      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2741_div__mult__self4,axiom,
% 9.20/5.13      ! [B: nat,C: nat,A: nat] :
% 9.20/5.13        ( ( B != zero_zero_nat )
% 9.20/5.13       => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 9.20/5.13          = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2742_div__mult__self4,axiom,
% 9.20/5.13      ! [B: int,C: int,A: int] :
% 9.20/5.13        ( ( B != zero_zero_int )
% 9.20/5.13       => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 9.20/5.13          = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2743_div__mult__self4,axiom,
% 9.20/5.13      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.20/5.13        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.13       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 9.20/5.13          = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2744_one__le__mult__iff,axiom,
% 9.20/5.13      ! [M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.13        = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.20/5.13          & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2745_mult__le__cancel2,axiom,
% 9.20/5.13      ! [M: nat,K: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
% 9.20/5.13        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2746_nat__mult__le__cancel__disj,axiom,
% 9.20/5.13      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.13        ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.13        = ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.13         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2747_div__mult__self__is__m,axiom,
% 9.20/5.13      ! [N2: nat,M: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
% 9.20/5.13          = M ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2748_div__mult__self1__is__m,axiom,
% 9.20/5.13      ! [N2: nat,M: nat] :
% 9.20/5.13        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.13       => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
% 9.20/5.13          = M ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2749_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.20/5.13      ! [A: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2750_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.20/5.13      ! [A: real,B: real,C: real] :
% 9.20/5.13        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2751_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.20/5.13      ! [A: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2752_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.20/5.13      ! [A: nat,B: nat,C: nat] :
% 9.20/5.13        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2753_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
% 9.20/5.13      ! [A: int,B: int,C: int] :
% 9.20/5.13        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2754_mult_Oassoc,axiom,
% 9.20/5.13      ! [A: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2755_mult_Oassoc,axiom,
% 9.20/5.13      ! [A: real,B: real,C: real] :
% 9.20/5.13        ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2756_mult_Oassoc,axiom,
% 9.20/5.13      ! [A: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2757_mult_Oassoc,axiom,
% 9.20/5.13      ! [A: nat,B: nat,C: nat] :
% 9.20/5.13        ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2758_mult_Oassoc,axiom,
% 9.20/5.13      ! [A: int,B: int,C: int] :
% 9.20/5.13        ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.13        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2759_mult_Ocommute,axiom,
% 9.20/5.13      ( times_times_complex
% 9.20/5.13      = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2760_mult_Ocommute,axiom,
% 9.20/5.13      ( times_times_real
% 9.20/5.13      = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2761_mult_Ocommute,axiom,
% 9.20/5.13      ( times_times_rat
% 9.20/5.13      = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2762_mult_Ocommute,axiom,
% 9.20/5.13      ( times_times_nat
% 9.20/5.13      = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2763_mult_Ocommute,axiom,
% 9.20/5.13      ( times_times_int
% 9.20/5.13      = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2764_mult_Oleft__commute,axiom,
% 9.20/5.13      ! [B: complex,A: complex,C: complex] :
% 9.20/5.13        ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
% 9.20/5.13        = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2765_mult_Oleft__commute,axiom,
% 9.20/5.13      ! [B: real,A: real,C: real] :
% 9.20/5.13        ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
% 9.20/5.13        = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2766_mult_Oleft__commute,axiom,
% 9.20/5.13      ! [B: rat,A: rat,C: rat] :
% 9.20/5.13        ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
% 9.20/5.13        = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2767_mult_Oleft__commute,axiom,
% 9.20/5.13      ! [B: nat,A: nat,C: nat] :
% 9.20/5.13        ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
% 9.20/5.13        = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2768_mult_Oleft__commute,axiom,
% 9.20/5.13      ! [B: int,A: int,C: int] :
% 9.20/5.13        ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
% 9.20/5.13        = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2769_not__iless0,axiom,
% 9.20/5.13      ! [N2: extended_enat] :
% 9.20/5.13        ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2770_enat__less__induct,axiom,
% 9.20/5.13      ! [P: extended_enat > $o,N2: extended_enat] :
% 9.20/5.13        ( ! [N3: extended_enat] :
% 9.20/5.13            ( ! [M6: extended_enat] :
% 9.20/5.13                ( ( ord_le72135733267957522d_enat @ M6 @ N3 )
% 9.20/5.13               => ( P @ M6 ) )
% 9.20/5.13           => ( P @ N3 ) )
% 9.20/5.13       => ( P @ N2 ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2771_mult__right__cancel,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( ( times_times_complex @ A @ C )
% 9.20/5.13            = ( times_times_complex @ B @ C ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2772_mult__right__cancel,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( ( times_times_real @ A @ C )
% 9.20/5.13            = ( times_times_real @ B @ C ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2773_mult__right__cancel,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( ( times_times_rat @ A @ C )
% 9.20/5.13            = ( times_times_rat @ B @ C ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2774_mult__right__cancel,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( C != zero_zero_nat )
% 9.20/5.13       => ( ( ( times_times_nat @ A @ C )
% 9.20/5.13            = ( times_times_nat @ B @ C ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2775_mult__right__cancel,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( C != zero_zero_int )
% 9.20/5.13       => ( ( ( times_times_int @ A @ C )
% 9.20/5.13            = ( times_times_int @ B @ C ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2776_mult__left__cancel,axiom,
% 9.20/5.13      ! [C: complex,A: complex,B: complex] :
% 9.20/5.13        ( ( C != zero_zero_complex )
% 9.20/5.13       => ( ( ( times_times_complex @ C @ A )
% 9.20/5.13            = ( times_times_complex @ C @ B ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2777_mult__left__cancel,axiom,
% 9.20/5.13      ! [C: real,A: real,B: real] :
% 9.20/5.13        ( ( C != zero_zero_real )
% 9.20/5.13       => ( ( ( times_times_real @ C @ A )
% 9.20/5.13            = ( times_times_real @ C @ B ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2778_mult__left__cancel,axiom,
% 9.20/5.13      ! [C: rat,A: rat,B: rat] :
% 9.20/5.13        ( ( C != zero_zero_rat )
% 9.20/5.13       => ( ( ( times_times_rat @ C @ A )
% 9.20/5.13            = ( times_times_rat @ C @ B ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2779_mult__left__cancel,axiom,
% 9.20/5.13      ! [C: nat,A: nat,B: nat] :
% 9.20/5.13        ( ( C != zero_zero_nat )
% 9.20/5.13       => ( ( ( times_times_nat @ C @ A )
% 9.20/5.13            = ( times_times_nat @ C @ B ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2780_mult__left__cancel,axiom,
% 9.20/5.13      ! [C: int,A: int,B: int] :
% 9.20/5.13        ( ( C != zero_zero_int )
% 9.20/5.13       => ( ( ( times_times_int @ C @ A )
% 9.20/5.13            = ( times_times_int @ C @ B ) )
% 9.20/5.13          = ( A = B ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2781_no__zero__divisors,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( A != zero_zero_complex )
% 9.20/5.13       => ( ( B != zero_zero_complex )
% 9.20/5.13         => ( ( times_times_complex @ A @ B )
% 9.20/5.13           != zero_zero_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2782_no__zero__divisors,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( A != zero_zero_real )
% 9.20/5.13       => ( ( B != zero_zero_real )
% 9.20/5.13         => ( ( times_times_real @ A @ B )
% 9.20/5.13           != zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2783_no__zero__divisors,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( A != zero_zero_rat )
% 9.20/5.13       => ( ( B != zero_zero_rat )
% 9.20/5.13         => ( ( times_times_rat @ A @ B )
% 9.20/5.13           != zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2784_no__zero__divisors,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( A != zero_zero_nat )
% 9.20/5.13       => ( ( B != zero_zero_nat )
% 9.20/5.13         => ( ( times_times_nat @ A @ B )
% 9.20/5.13           != zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2785_no__zero__divisors,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( A != zero_zero_int )
% 9.20/5.13       => ( ( B != zero_zero_int )
% 9.20/5.13         => ( ( times_times_int @ A @ B )
% 9.20/5.13           != zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2786_divisors__zero,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ A @ B )
% 9.20/5.13          = zero_zero_complex )
% 9.20/5.13       => ( ( A = zero_zero_complex )
% 9.20/5.13          | ( B = zero_zero_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2787_divisors__zero,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ( times_times_real @ A @ B )
% 9.20/5.13          = zero_zero_real )
% 9.20/5.13       => ( ( A = zero_zero_real )
% 9.20/5.13          | ( B = zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2788_divisors__zero,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ A @ B )
% 9.20/5.13          = zero_zero_rat )
% 9.20/5.13       => ( ( A = zero_zero_rat )
% 9.20/5.13          | ( B = zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2789_divisors__zero,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ A @ B )
% 9.20/5.13          = zero_zero_nat )
% 9.20/5.13       => ( ( A = zero_zero_nat )
% 9.20/5.13          | ( B = zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2790_divisors__zero,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( ( times_times_int @ A @ B )
% 9.20/5.13          = zero_zero_int )
% 9.20/5.13       => ( ( A = zero_zero_int )
% 9.20/5.13          | ( B = zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2791_mult__not__zero,axiom,
% 9.20/5.13      ! [A: complex,B: complex] :
% 9.20/5.13        ( ( ( times_times_complex @ A @ B )
% 9.20/5.13         != zero_zero_complex )
% 9.20/5.13       => ( ( A != zero_zero_complex )
% 9.20/5.13          & ( B != zero_zero_complex ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2792_mult__not__zero,axiom,
% 9.20/5.13      ! [A: real,B: real] :
% 9.20/5.13        ( ( ( times_times_real @ A @ B )
% 9.20/5.13         != zero_zero_real )
% 9.20/5.13       => ( ( A != zero_zero_real )
% 9.20/5.13          & ( B != zero_zero_real ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2793_mult__not__zero,axiom,
% 9.20/5.13      ! [A: rat,B: rat] :
% 9.20/5.13        ( ( ( times_times_rat @ A @ B )
% 9.20/5.13         != zero_zero_rat )
% 9.20/5.13       => ( ( A != zero_zero_rat )
% 9.20/5.13          & ( B != zero_zero_rat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2794_mult__not__zero,axiom,
% 9.20/5.13      ! [A: nat,B: nat] :
% 9.20/5.13        ( ( ( times_times_nat @ A @ B )
% 9.20/5.13         != zero_zero_nat )
% 9.20/5.13       => ( ( A != zero_zero_nat )
% 9.20/5.13          & ( B != zero_zero_nat ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2795_mult__not__zero,axiom,
% 9.20/5.13      ! [A: int,B: int] :
% 9.20/5.13        ( ( ( times_times_int @ A @ B )
% 9.20/5.13         != zero_zero_int )
% 9.20/5.13       => ( ( A != zero_zero_int )
% 9.20/5.13          & ( B != zero_zero_int ) ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2796_comm__monoid__mult__class_Omult__1,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ one_one_complex @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2797_comm__monoid__mult__class_Omult__1,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ one_one_real @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2798_comm__monoid__mult__class_Omult__1,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ one_one_rat @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2799_comm__monoid__mult__class_Omult__1,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ one_one_nat @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2800_comm__monoid__mult__class_Omult__1,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ one_one_int @ A )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2801_mult_Ocomm__neutral,axiom,
% 9.20/5.13      ! [A: complex] :
% 9.20/5.13        ( ( times_times_complex @ A @ one_one_complex )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2802_mult_Ocomm__neutral,axiom,
% 9.20/5.13      ! [A: real] :
% 9.20/5.13        ( ( times_times_real @ A @ one_one_real )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2803_mult_Ocomm__neutral,axiom,
% 9.20/5.13      ! [A: rat] :
% 9.20/5.13        ( ( times_times_rat @ A @ one_one_rat )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2804_mult_Ocomm__neutral,axiom,
% 9.20/5.13      ! [A: nat] :
% 9.20/5.13        ( ( times_times_nat @ A @ one_one_nat )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2805_mult_Ocomm__neutral,axiom,
% 9.20/5.13      ! [A: int] :
% 9.20/5.13        ( ( times_times_int @ A @ one_one_int )
% 9.20/5.13        = A ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2806_combine__common__factor,axiom,
% 9.20/5.13      ! [A: complex,E2: complex,B: complex,C: complex] :
% 9.20/5.13        ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
% 9.20/5.13        = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2807_combine__common__factor,axiom,
% 9.20/5.13      ! [A: real,E2: real,B: real,C: real] :
% 9.20/5.13        ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
% 9.20/5.13        = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).
% 9.20/5.13  
% 9.20/5.13  thf(fact_2808_combine__common__factor,axiom,
% 9.20/5.13      ! [A: rat,E2: rat,B: rat,C: rat] :
% 9.20/5.13        ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2809_combine__common__factor,axiom,
% 9.20/5.14      ! [A: nat,E2: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2810_combine__common__factor,axiom,
% 9.20/5.14      ! [A: int,E2: int,B: int,C: int] :
% 9.20/5.14        ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2811_distrib__right,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2812_distrib__right,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2813_distrib__right,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2814_distrib__right,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2815_distrib__right,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2816_distrib__left,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2817_distrib__left,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2818_distrib__left,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2819_distrib__left,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2820_distrib__left,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2821_comm__semiring__class_Odistrib,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2822_comm__semiring__class_Odistrib,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2823_comm__semiring__class_Odistrib,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2824_comm__semiring__class_Odistrib,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2825_comm__semiring__class_Odistrib,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2826_ring__class_Oring__distribs_I1_J,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2827_ring__class_Oring__distribs_I1_J,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2828_ring__class_Oring__distribs_I1_J,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2829_ring__class_Oring__distribs_I1_J,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2830_ring__class_Oring__distribs_I2_J,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2832_ring__class_Oring__distribs_I2_J,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2833_ring__class_Oring__distribs_I2_J,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2834_crossproduct__noteq,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex,D: complex] :
% 9.20/5.14        ( ( ( A != B )
% 9.20/5.14          & ( C != D ) )
% 9.20/5.14        = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
% 9.20/5.14         != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2835_crossproduct__noteq,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ( A != B )
% 9.20/5.14          & ( C != D ) )
% 9.20/5.14        = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
% 9.20/5.14         != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2836_crossproduct__noteq,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ( A != B )
% 9.20/5.14          & ( C != D ) )
% 9.20/5.14        = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
% 9.20/5.14         != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2837_crossproduct__noteq,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ( A != B )
% 9.20/5.14          & ( C != D ) )
% 9.20/5.14        = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
% 9.20/5.14         != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2838_crossproduct__noteq,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ( A != B )
% 9.20/5.14          & ( C != D ) )
% 9.20/5.14        = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
% 9.20/5.14         != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2839_crossproduct__eq,axiom,
% 9.20/5.14      ! [W2: complex,Y3: complex,X3: complex,Z2: complex] :
% 9.20/5.14        ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y3 ) @ ( times_times_complex @ X3 @ Z2 ) )
% 9.20/5.14          = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z2 ) @ ( times_times_complex @ X3 @ Y3 ) ) )
% 9.20/5.14        = ( ( W2 = X3 )
% 9.20/5.14          | ( Y3 = Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2840_crossproduct__eq,axiom,
% 9.20/5.14      ! [W2: real,Y3: real,X3: real,Z2: real] :
% 9.20/5.14        ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y3 ) @ ( times_times_real @ X3 @ Z2 ) )
% 9.20/5.14          = ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X3 @ Y3 ) ) )
% 9.20/5.14        = ( ( W2 = X3 )
% 9.20/5.14          | ( Y3 = Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2841_crossproduct__eq,axiom,
% 9.20/5.14      ! [W2: rat,Y3: rat,X3: rat,Z2: rat] :
% 9.20/5.14        ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y3 ) @ ( times_times_rat @ X3 @ Z2 ) )
% 9.20/5.14          = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X3 @ Y3 ) ) )
% 9.20/5.14        = ( ( W2 = X3 )
% 9.20/5.14          | ( Y3 = Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2842_crossproduct__eq,axiom,
% 9.20/5.14      ! [W2: nat,Y3: nat,X3: nat,Z2: nat] :
% 9.20/5.14        ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y3 ) @ ( times_times_nat @ X3 @ Z2 ) )
% 9.20/5.14          = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X3 @ Y3 ) ) )
% 9.20/5.14        = ( ( W2 = X3 )
% 9.20/5.14          | ( Y3 = Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2843_crossproduct__eq,axiom,
% 9.20/5.14      ! [W2: int,Y3: int,X3: int,Z2: int] :
% 9.20/5.14        ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y3 ) @ ( times_times_int @ X3 @ Z2 ) )
% 9.20/5.14          = ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X3 @ Y3 ) ) )
% 9.20/5.14        = ( ( W2 = X3 )
% 9.20/5.14          | ( Y3 = Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2844_times__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
% 9.20/5.14        ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ Y3 @ W2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2845_times__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: real,Y3: real,Z2: real,W2: real] :
% 9.20/5.14        ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide_divide_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ W2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2846_times__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
% 9.20/5.14        ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ W2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2847_divide__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
% 9.20/5.14        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W2 ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2848_divide__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: real,Y3: real,Z2: real,W2: real] :
% 9.20/5.14        ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide_divide_real @ ( times_times_real @ X3 @ W2 ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2849_divide__divide__times__eq,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
% 9.20/5.14        ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
% 9.20/5.14        = ( divide_divide_rat @ ( times_times_rat @ X3 @ W2 ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2850_divide__divide__eq__left_H,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
% 9.20/5.14        = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2851_divide__divide__eq__left_H,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
% 9.20/5.14        = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2852_divide__divide__eq__left_H,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
% 9.20/5.14        = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2853_Suc__mult__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ( times_times_nat @ ( suc @ K ) @ M )
% 9.20/5.14          = ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 9.20/5.14        = ( M = N2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2854_mult__0,axiom,
% 9.20/5.14      ! [N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ zero_zero_nat @ N2 )
% 9.20/5.14        = zero_zero_nat ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2855_nat__mult__eq__cancel__disj,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ( times_times_nat @ K @ M )
% 9.20/5.14          = ( times_times_nat @ K @ N2 ) )
% 9.20/5.14        = ( ( K = zero_zero_nat )
% 9.20/5.14          | ( M = N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2856_mult__le__mono2,axiom,
% 9.20/5.14      ! [I: nat,J: nat,K: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.14       => ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2857_mult__le__mono1,axiom,
% 9.20/5.14      ! [I: nat,J: nat,K: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.14       => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2858_mult__le__mono,axiom,
% 9.20/5.14      ! [I: nat,J: nat,K: nat,L: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.14       => ( ( ord_less_eq_nat @ K @ L )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2859_le__square,axiom,
% 9.20/5.14      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2860_le__cube,axiom,
% 9.20/5.14      ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2861_add__mult__distrib2,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2862_add__mult__distrib,axiom,
% 9.20/5.14      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2863_left__add__mult__distrib,axiom,
% 9.20/5.14      ! [I: nat,U: nat,J: nat,K: nat] :
% 9.20/5.14        ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2864_nat__mult__1,axiom,
% 9.20/5.14      ! [N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ one_one_nat @ N2 )
% 9.20/5.14        = N2 ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2865_nat__mult__1__right,axiom,
% 9.20/5.14      ! [N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ N2 @ one_one_nat )
% 9.20/5.14        = N2 ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2866_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2867_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2868_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2869_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2870_zero__le__mult__iff,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2871_zero__le__mult__iff,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2872_zero__le__mult__iff,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2873_mult__nonneg__nonpos2,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2874_mult__nonneg__nonpos2,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2875_mult__nonneg__nonpos2,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2876_mult__nonneg__nonpos2,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2877_mult__nonpos__nonneg,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2878_mult__nonpos__nonneg,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2879_mult__nonpos__nonneg,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2880_mult__nonpos__nonneg,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2881_mult__nonneg__nonpos,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2882_mult__nonneg__nonpos,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2883_mult__nonneg__nonpos,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2884_mult__nonneg__nonpos,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2885_mult__nonneg__nonneg,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.14         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2886_mult__nonneg__nonneg,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.14         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2887_mult__nonneg__nonneg,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.14         => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2888_mult__nonneg__nonneg,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.14         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2889_split__mult__neg__le,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.20/5.14          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
% 9.20/5.14       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2890_split__mult__neg__le,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.20/5.14          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
% 9.20/5.14       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2891_split__mult__neg__le,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14            & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
% 9.20/5.14          | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
% 9.20/5.14            & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
% 9.20/5.14       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2892_split__mult__neg__le,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.20/5.14          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
% 9.20/5.14       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2893_mult__le__0__iff,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_eq_real @ B @ zero_zero_real ) )
% 9.20/5.14          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2894_mult__le__0__iff,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
% 9.20/5.14          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2895_mult__le__0__iff,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_eq_int @ B @ zero_zero_int ) )
% 9.20/5.14          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2896_mult__right__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2897_mult__right__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2898_mult__right__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2899_mult__right__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2900_mult__right__mono__neg,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2901_mult__right__mono__neg,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2902_mult__right__mono__neg,axiom,
% 9.20/5.14      ! [B: int,A: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2903_mult__left__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2904_mult__left__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2905_mult__left__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2906_mult__left__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2907_mult__nonpos__nonpos,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2908_mult__nonpos__nonpos,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2909_mult__nonpos__nonpos,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2910_mult__left__mono__neg,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2911_mult__left__mono__neg,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2912_mult__left__mono__neg,axiom,
% 9.20/5.14      ! [B: int,A: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2913_split__mult__pos__le,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_eq_real @ zero_zero_real @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.20/5.14       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2914_split__mult__pos__le,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.20/5.14       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2915_split__mult__pos__le,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_eq_int @ zero_zero_int @ B ) )
% 9.20/5.14          | ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.20/5.14       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2916_zero__le__square,axiom,
% 9.20/5.14      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2917_zero__le__square,axiom,
% 9.20/5.14      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2918_zero__le__square,axiom,
% 9.20/5.14      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2919_mult__mono_H,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2920_mult__mono_H,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2921_mult__mono_H,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2922_mult__mono_H,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2923_mult__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2924_mult__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2925_mult__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.14           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2926_mult__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2927_mult__neg__neg,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2928_mult__neg__neg,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2929_mult__neg__neg,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2930_not__square__less__zero,axiom,
% 9.20/5.14      ! [A: real] :
% 9.20/5.14        ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2931_not__square__less__zero,axiom,
% 9.20/5.14      ! [A: rat] :
% 9.20/5.14        ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2932_not__square__less__zero,axiom,
% 9.20/5.14      ! [A: int] :
% 9.20/5.14        ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2933_mult__less__0__iff,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_real @ B @ zero_zero_real ) )
% 9.20/5.14          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2934_mult__less__0__iff,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_rat @ B @ zero_zero_rat ) )
% 9.20/5.14          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2935_mult__less__0__iff,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_int @ B @ zero_zero_int ) )
% 9.20/5.14          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2936_mult__neg__pos,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2937_mult__neg__pos,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2938_mult__neg__pos,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ zero_zero_nat )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2939_mult__neg__pos,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2940_mult__pos__neg,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2941_mult__pos__neg,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2942_mult__pos__neg,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2943_mult__pos__neg,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2944_mult__pos__pos,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.14         => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2945_mult__pos__pos,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.20/5.14         => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2946_mult__pos__pos,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.14         => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2947_mult__pos__pos,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.14         => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2948_mult__pos__neg2,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_real @ B @ zero_zero_real )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2949_mult__pos__neg2,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_rat @ B @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2950_mult__pos__neg2,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_nat @ B @ zero_zero_nat )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2951_mult__pos__neg2,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2952_zero__less__mult__iff,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14            & ( ord_less_real @ zero_zero_real @ B ) )
% 9.20/5.14          | ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.14            & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2953_zero__less__mult__iff,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14            & ( ord_less_rat @ zero_zero_rat @ B ) )
% 9.20/5.14          | ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2954_zero__less__mult__iff,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14            & ( ord_less_int @ zero_zero_int @ B ) )
% 9.20/5.14          | ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.14            & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2955_zero__less__mult__pos,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2956_zero__less__mult__pos,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2957_zero__less__mult__pos,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2958_zero__less__mult__pos,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2959_zero__less__mult__pos2,axiom,
% 9.20/5.14      ! [B: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14         => ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2960_zero__less__mult__pos2,axiom,
% 9.20/5.14      ! [B: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14         => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2961_zero__less__mult__pos2,axiom,
% 9.20/5.14      ! [B: nat,A: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14         => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2962_zero__less__mult__pos2,axiom,
% 9.20/5.14      ! [B: int,A: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14         => ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2963_mult__less__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14          = ( ord_less_real @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2964_mult__less__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14          = ( ord_less_rat @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2965_mult__less__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14          = ( ord_less_int @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2966_mult__less__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14          = ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2967_mult__less__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14          = ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2968_mult__less__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14       => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14          = ( ord_less_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2969_mult__strict__left__mono__neg,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2970_mult__strict__left__mono__neg,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2971_mult__strict__left__mono__neg,axiom,
% 9.20/5.14      ! [B: int,A: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ B @ A )
% 9.20/5.14       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2972_mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2973_mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2974_mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2975_mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2976_mult__less__cancel__left__disj,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14            & ( ord_less_real @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2977_mult__less__cancel__left__disj,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14            & ( ord_less_rat @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2978_mult__less__cancel__left__disj,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14            & ( ord_less_int @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2979_mult__strict__right__mono__neg,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2980_mult__strict__right__mono__neg,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2981_mult__strict__right__mono__neg,axiom,
% 9.20/5.14      ! [B: int,A: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ B @ A )
% 9.20/5.14       => ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2982_mult__strict__right__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2983_mult__strict__right__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2984_mult__strict__right__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2985_mult__strict__right__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2986_mult__less__cancel__right__disj,axiom,
% 9.20/5.14      ! [A: real,C: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14            & ( ord_less_real @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14            & ( ord_less_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2987_mult__less__cancel__right__disj,axiom,
% 9.20/5.14      ! [A: rat,C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14            & ( ord_less_rat @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14            & ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2988_mult__less__cancel__right__disj,axiom,
% 9.20/5.14      ! [A: int,C: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14            & ( ord_less_int @ A @ B ) )
% 9.20/5.14          | ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14            & ( ord_less_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2989_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2990_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2991_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2992_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2993_add__scale__eq__noteq,axiom,
% 9.20/5.14      ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
% 9.20/5.14        ( ( R2 != zero_zero_complex )
% 9.20/5.14       => ( ( ( A = B )
% 9.20/5.14            & ( C != D ) )
% 9.20/5.14         => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
% 9.20/5.14           != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2994_add__scale__eq__noteq,axiom,
% 9.20/5.14      ! [R2: real,A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( R2 != zero_zero_real )
% 9.20/5.14       => ( ( ( A = B )
% 9.20/5.14            & ( C != D ) )
% 9.20/5.14         => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
% 9.20/5.14           != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2995_add__scale__eq__noteq,axiom,
% 9.20/5.14      ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( R2 != zero_zero_rat )
% 9.20/5.14       => ( ( ( A = B )
% 9.20/5.14            & ( C != D ) )
% 9.20/5.14         => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
% 9.20/5.14           != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2996_add__scale__eq__noteq,axiom,
% 9.20/5.14      ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( R2 != zero_zero_nat )
% 9.20/5.14       => ( ( ( A = B )
% 9.20/5.14            & ( C != D ) )
% 9.20/5.14         => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
% 9.20/5.14           != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2997_add__scale__eq__noteq,axiom,
% 9.20/5.14      ! [R2: int,A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( R2 != zero_zero_int )
% 9.20/5.14       => ( ( ( A = B )
% 9.20/5.14            & ( C != D ) )
% 9.20/5.14         => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
% 9.20/5.14           != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2998_less__1__mult,axiom,
% 9.20/5.14      ! [M: real,N2: real] :
% 9.20/5.14        ( ( ord_less_real @ one_one_real @ M )
% 9.20/5.14       => ( ( ord_less_real @ one_one_real @ N2 )
% 9.20/5.14         => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_2999_less__1__mult,axiom,
% 9.20/5.14      ! [M: rat,N2: rat] :
% 9.20/5.14        ( ( ord_less_rat @ one_one_rat @ M )
% 9.20/5.14       => ( ( ord_less_rat @ one_one_rat @ N2 )
% 9.20/5.14         => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3000_less__1__mult,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ one_one_nat @ M )
% 9.20/5.14       => ( ( ord_less_nat @ one_one_nat @ N2 )
% 9.20/5.14         => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3001_less__1__mult,axiom,
% 9.20/5.14      ! [M: int,N2: int] :
% 9.20/5.14        ( ( ord_less_int @ one_one_int @ M )
% 9.20/5.14       => ( ( ord_less_int @ one_one_int @ N2 )
% 9.20/5.14         => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3002_frac__eq__eq,axiom,
% 9.20/5.14      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 9.20/5.14        ( ( Y3 != zero_zero_complex )
% 9.20/5.14       => ( ( Z2 != zero_zero_complex )
% 9.20/5.14         => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
% 9.20/5.14              = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.20/5.14            = ( ( times_times_complex @ X3 @ Z2 )
% 9.20/5.14              = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3003_frac__eq__eq,axiom,
% 9.20/5.14      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 9.20/5.14        ( ( Y3 != zero_zero_real )
% 9.20/5.14       => ( ( Z2 != zero_zero_real )
% 9.20/5.14         => ( ( ( divide_divide_real @ X3 @ Y3 )
% 9.20/5.14              = ( divide_divide_real @ W2 @ Z2 ) )
% 9.20/5.14            = ( ( times_times_real @ X3 @ Z2 )
% 9.20/5.14              = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3004_frac__eq__eq,axiom,
% 9.20/5.14      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 9.20/5.14        ( ( Y3 != zero_zero_rat )
% 9.20/5.14       => ( ( Z2 != zero_zero_rat )
% 9.20/5.14         => ( ( ( divide_divide_rat @ X3 @ Y3 )
% 9.20/5.14              = ( divide_divide_rat @ W2 @ Z2 ) )
% 9.20/5.14            = ( ( times_times_rat @ X3 @ Z2 )
% 9.20/5.14              = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3005_divide__eq__eq,axiom,
% 9.20/5.14      ! [B: complex,C: complex,A: complex] :
% 9.20/5.14        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.20/5.14          = A )
% 9.20/5.14        = ( ( ( C != zero_zero_complex )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_complex @ A @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_complex )
% 9.20/5.14           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3006_divide__eq__eq,axiom,
% 9.20/5.14      ! [B: real,C: real,A: real] :
% 9.20/5.14        ( ( ( divide_divide_real @ B @ C )
% 9.20/5.14          = A )
% 9.20/5.14        = ( ( ( C != zero_zero_real )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_real @ A @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_real )
% 9.20/5.14           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3007_divide__eq__eq,axiom,
% 9.20/5.14      ! [B: rat,C: rat,A: rat] :
% 9.20/5.14        ( ( ( divide_divide_rat @ B @ C )
% 9.20/5.14          = A )
% 9.20/5.14        = ( ( ( C != zero_zero_rat )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_rat @ A @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_rat )
% 9.20/5.14           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3008_eq__divide__eq,axiom,
% 9.20/5.14      ! [A: complex,B: complex,C: complex] :
% 9.20/5.14        ( ( A
% 9.20/5.14          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_complex )
% 9.20/5.14           => ( ( times_times_complex @ A @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_complex )
% 9.20/5.14           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3009_eq__divide__eq,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( A
% 9.20/5.14          = ( divide_divide_real @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_real )
% 9.20/5.14           => ( ( times_times_real @ A @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_real )
% 9.20/5.14           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3010_eq__divide__eq,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( A
% 9.20/5.14          = ( divide_divide_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_rat )
% 9.20/5.14           => ( ( times_times_rat @ A @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_rat )
% 9.20/5.14           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3011_divide__eq__imp,axiom,
% 9.20/5.14      ! [C: complex,B: complex,A: complex] :
% 9.20/5.14        ( ( C != zero_zero_complex )
% 9.20/5.14       => ( ( B
% 9.20/5.14            = ( times_times_complex @ A @ C ) )
% 9.20/5.14         => ( ( divide1717551699836669952omplex @ B @ C )
% 9.20/5.14            = A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3012_divide__eq__imp,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( C != zero_zero_real )
% 9.20/5.14       => ( ( B
% 9.20/5.14            = ( times_times_real @ A @ C ) )
% 9.20/5.14         => ( ( divide_divide_real @ B @ C )
% 9.20/5.14            = A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3013_divide__eq__imp,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( C != zero_zero_rat )
% 9.20/5.14       => ( ( B
% 9.20/5.14            = ( times_times_rat @ A @ C ) )
% 9.20/5.14         => ( ( divide_divide_rat @ B @ C )
% 9.20/5.14            = A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3014_eq__divide__imp,axiom,
% 9.20/5.14      ! [C: complex,A: complex,B: complex] :
% 9.20/5.14        ( ( C != zero_zero_complex )
% 9.20/5.14       => ( ( ( times_times_complex @ A @ C )
% 9.20/5.14            = B )
% 9.20/5.14         => ( A
% 9.20/5.14            = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3015_eq__divide__imp,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( C != zero_zero_real )
% 9.20/5.14       => ( ( ( times_times_real @ A @ C )
% 9.20/5.14            = B )
% 9.20/5.14         => ( A
% 9.20/5.14            = ( divide_divide_real @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3016_eq__divide__imp,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( C != zero_zero_rat )
% 9.20/5.14       => ( ( ( times_times_rat @ A @ C )
% 9.20/5.14            = B )
% 9.20/5.14         => ( A
% 9.20/5.14            = ( divide_divide_rat @ B @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3017_nonzero__divide__eq__eq,axiom,
% 9.20/5.14      ! [C: complex,B: complex,A: complex] :
% 9.20/5.14        ( ( C != zero_zero_complex )
% 9.20/5.14       => ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.20/5.14            = A )
% 9.20/5.14          = ( B
% 9.20/5.14            = ( times_times_complex @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3018_nonzero__divide__eq__eq,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( C != zero_zero_real )
% 9.20/5.14       => ( ( ( divide_divide_real @ B @ C )
% 9.20/5.14            = A )
% 9.20/5.14          = ( B
% 9.20/5.14            = ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3019_nonzero__divide__eq__eq,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( C != zero_zero_rat )
% 9.20/5.14       => ( ( ( divide_divide_rat @ B @ C )
% 9.20/5.14            = A )
% 9.20/5.14          = ( B
% 9.20/5.14            = ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3020_nonzero__eq__divide__eq,axiom,
% 9.20/5.14      ! [C: complex,A: complex,B: complex] :
% 9.20/5.14        ( ( C != zero_zero_complex )
% 9.20/5.14       => ( ( A
% 9.20/5.14            = ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.14          = ( ( times_times_complex @ A @ C )
% 9.20/5.14            = B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3021_nonzero__eq__divide__eq,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( C != zero_zero_real )
% 9.20/5.14       => ( ( A
% 9.20/5.14            = ( divide_divide_real @ B @ C ) )
% 9.20/5.14          = ( ( times_times_real @ A @ C )
% 9.20/5.14            = B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3022_nonzero__eq__divide__eq,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( C != zero_zero_rat )
% 9.20/5.14       => ( ( A
% 9.20/5.14            = ( divide_divide_rat @ B @ C ) )
% 9.20/5.14          = ( ( times_times_rat @ A @ C )
% 9.20/5.14            = B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3023_power__Suc,axiom,
% 9.20/5.14      ! [A: complex,N2: nat] :
% 9.20/5.14        ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3024_power__Suc,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( power_power_real @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3025_power__Suc,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3026_power__Suc,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3027_power__Suc,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( power_power_int @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3028_power__Suc2,axiom,
% 9.20/5.14      ! [A: complex,N2: nat] :
% 9.20/5.14        ( ( power_power_complex @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3029_power__Suc2,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( power_power_real @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_real @ ( power_power_real @ A @ N2 ) @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3030_power__Suc2,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( power_power_rat @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3031_power__Suc2,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_nat @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3032_power__Suc2,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( power_power_int @ A @ ( suc @ N2 ) )
% 9.20/5.14        = ( times_times_int @ ( power_power_int @ A @ N2 ) @ A ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3033_Suc__mult__less__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 9.20/5.14        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3034_power__add,axiom,
% 9.20/5.14      ! [A: complex,M: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3035_power__add,axiom,
% 9.20/5.14      ! [A: real,M: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3036_power__add,axiom,
% 9.20/5.14      ! [A: rat,M: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3037_power__add,axiom,
% 9.20/5.14      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3038_power__add,axiom,
% 9.20/5.14      ! [A: int,M: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.14        = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3039_mult__less__mono1,axiom,
% 9.20/5.14      ! [I: nat,J: nat,K: nat] :
% 9.20/5.14        ( ( ord_less_nat @ I @ J )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3040_mult__less__mono2,axiom,
% 9.20/5.14      ! [I: nat,J: nat,K: nat] :
% 9.20/5.14        ( ( ord_less_nat @ I @ J )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3041_nat__mult__eq__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14       => ( ( ( times_times_nat @ K @ M )
% 9.20/5.14            = ( times_times_nat @ K @ N2 ) )
% 9.20/5.14          = ( M = N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3042_nat__mult__less__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14       => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.14          = ( ord_less_nat @ M @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3043_Suc__mult__le__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
% 9.20/5.14        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3044_mult__Suc,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( suc @ M ) @ N2 )
% 9.20/5.14        = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3045_mult__eq__self__implies__10,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] :
% 9.20/5.14        ( ( M
% 9.20/5.14          = ( times_times_nat @ M @ N2 ) )
% 9.20/5.14       => ( ( N2 = one_one_nat )
% 9.20/5.14          | ( M = zero_zero_nat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3046_less__mult__imp__div__less,axiom,
% 9.20/5.14      ! [M: nat,I: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
% 9.20/5.14       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3047_times__div__less__eq__dividend,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3048_div__times__less__eq__dividend,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3049_power__odd__eq,axiom,
% 9.20/5.14      ! [A: complex,N2: nat] :
% 9.20/5.14        ( ( power_power_complex @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.14        = ( times_times_complex @ A @ ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3050_power__odd__eq,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.14        = ( times_times_real @ A @ ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3051_power__odd__eq,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.14        = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3052_power__odd__eq,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( power_power_nat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.14        = ( times_times_nat @ A @ ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3053_power__odd__eq,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.14        = ( times_times_int @ A @ ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3054_Suc__double__not__eq__double,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] :
% 9.20/5.14        ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.14       != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3055_double__not__eq__Suc__double,axiom,
% 9.20/5.14      ! [M: nat,N2: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.20/5.14       != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3056_mult__le__cancel__left,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3057_mult__le__cancel__left,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3058_mult__le__cancel__left,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3059_mult__le__cancel__right,axiom,
% 9.20/5.14      ! [A: real,C: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3060_mult__le__cancel__right,axiom,
% 9.20/5.14      ! [A: rat,C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3061_mult__le__cancel__right,axiom,
% 9.20/5.14      ! [A: int,C: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3062_mult__left__less__imp__less,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3063_mult__left__less__imp__less,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3064_mult__left__less__imp__less,axiom,
% 9.20/5.14      ! [C: nat,A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3065_mult__left__less__imp__less,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3066_mult__strict__mono,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3067_mult__strict__mono,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ B )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3068_mult__strict__mono,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.14           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3069_mult__strict__mono,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3070_mult__less__cancel__left,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3071_mult__less__cancel__left,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3072_mult__less__cancel__left,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3073_mult__right__less__imp__less,axiom,
% 9.20/5.14      ! [A: real,C: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3074_mult__right__less__imp__less,axiom,
% 9.20/5.14      ! [A: rat,C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3075_mult__right__less__imp__less,axiom,
% 9.20/5.14      ! [A: nat,C: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_nat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3076_mult__right__less__imp__less,axiom,
% 9.20/5.14      ! [A: int,C: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3077_mult__strict__mono_H,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3078_mult__strict__mono_H,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3079_mult__strict__mono_H,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3080_mult__strict__mono_H,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3081_mult__less__cancel__right,axiom,
% 9.20/5.14      ! [A: real,C: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3082_mult__less__cancel__right,axiom,
% 9.20/5.14      ! [A: rat,C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3083_mult__less__cancel__right,axiom,
% 9.20/5.14      ! [A: int,C: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ A @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ B @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3084_mult__le__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_real @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3085_mult__le__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_rat @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3086_mult__le__cancel__left__neg,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3087_mult__le__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3088_mult__le__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3089_mult__le__cancel__left__pos,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14       => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3090_mult__left__le__imp__le,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3091_mult__left__le__imp__le,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3092_mult__left__le__imp__le,axiom,
% 9.20/5.14      ! [C: nat,A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3093_mult__left__le__imp__le,axiom,
% 9.20/5.14      ! [C: int,A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3094_mult__right__le__imp__le,axiom,
% 9.20/5.14      ! [A: real,C: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3095_mult__right__le__imp__le,axiom,
% 9.20/5.14      ! [A: rat,C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3096_mult__right__le__imp__le,axiom,
% 9.20/5.14      ! [A: nat,C: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14         => ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3097_mult__right__le__imp__le,axiom,
% 9.20/5.14      ! [A: int,C: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.14       => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14         => ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3098_mult__le__less__imp__less,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3099_mult__le__less__imp__less,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3100_mult__le__less__imp__less,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14           => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3101_mult__le__less__imp__less,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3102_mult__less__le__imp__less,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14           => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3103_mult__less__le__imp__less,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14           => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3104_mult__less__le__imp__less,axiom,
% 9.20/5.14      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.14        ( ( ord_less_nat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_nat @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14           => ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.14             => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3105_mult__less__le__imp__less,axiom,
% 9.20/5.14      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.14        ( ( ord_less_int @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_int @ C @ D )
% 9.20/5.14         => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14           => ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14             => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3106_mult__left__le,axiom,
% 9.20/5.14      ! [C: real,A: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ C @ one_one_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.14         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3107_mult__left__le,axiom,
% 9.20/5.14      ! [C: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ C @ one_one_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.14         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3108_mult__left__le,axiom,
% 9.20/5.14      ! [C: nat,A: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ C @ one_one_nat )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.14         => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3109_mult__left__le,axiom,
% 9.20/5.14      ! [C: int,A: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ C @ one_one_int )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.14         => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3110_mult__le__one,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ one_one_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.14         => ( ( ord_less_eq_real @ B @ one_one_real )
% 9.20/5.14           => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3111_mult__le__one,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ one_one_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.14         => ( ( ord_less_eq_rat @ B @ one_one_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3112_mult__le__one,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ A @ one_one_nat )
% 9.20/5.14       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
% 9.20/5.14         => ( ( ord_less_eq_nat @ B @ one_one_nat )
% 9.20/5.14           => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3113_mult__le__one,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ A @ one_one_int )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.14         => ( ( ord_less_eq_int @ B @ one_one_int )
% 9.20/5.14           => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3114_mult__right__le__one__le,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.20/5.14           => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3115_mult__right__le__one__le,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3116_mult__right__le__one__le,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.20/5.14           => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3117_mult__left__le__one__le,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.20/5.14           => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3118_mult__left__le__one__le,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3119_mult__left__le__one__le,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.14       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.14         => ( ( ord_less_eq_int @ Y3 @ one_one_int )
% 9.20/5.14           => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3120_sum__squares__le__zero__iff,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
% 9.20/5.14        = ( ( X3 = zero_zero_real )
% 9.20/5.14          & ( Y3 = zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3121_sum__squares__le__zero__iff,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
% 9.20/5.14        = ( ( X3 = zero_zero_rat )
% 9.20/5.14          & ( Y3 = zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3122_sum__squares__le__zero__iff,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
% 9.20/5.14        = ( ( X3 = zero_zero_int )
% 9.20/5.14          & ( Y3 = zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3123_sum__squares__ge__zero,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3124_sum__squares__ge__zero,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3125_sum__squares__ge__zero,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3126_sum__squares__gt__zero__iff,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
% 9.20/5.14        = ( ( X3 != zero_zero_real )
% 9.20/5.14          | ( Y3 != zero_zero_real ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3127_sum__squares__gt__zero__iff,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
% 9.20/5.14        = ( ( X3 != zero_zero_rat )
% 9.20/5.14          | ( Y3 != zero_zero_rat ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3128_sum__squares__gt__zero__iff,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
% 9.20/5.14        = ( ( X3 != zero_zero_int )
% 9.20/5.14          | ( Y3 != zero_zero_int ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3129_not__sum__squares__lt__zero,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3130_not__sum__squares__lt__zero,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3131_not__sum__squares__lt__zero,axiom,
% 9.20/5.14      ! [X3: int,Y3: int] :
% 9.20/5.14        ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3132_divide__less__eq,axiom,
% 9.20/5.14      ! [B: real,C: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3133_divide__less__eq,axiom,
% 9.20/5.14      ! [B: rat,C: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3134_less__divide__eq,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3135_less__divide__eq,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3136_neg__divide__less__eq,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3137_neg__divide__less__eq,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3138_neg__less__divide__eq,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3139_neg__less__divide__eq,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3140_pos__divide__less__eq,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3141_pos__divide__less__eq,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3142_pos__less__divide__eq,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14          = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3143_pos__less__divide__eq,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3144_mult__imp__div__pos__less,axiom,
% 9.20/5.14      ! [Y3: real,X3: real,Z2: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.14       => ( ( ord_less_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
% 9.20/5.14         => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3145_mult__imp__div__pos__less,axiom,
% 9.20/5.14      ! [Y3: rat,X3: rat,Z2: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14       => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
% 9.20/5.14         => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3146_mult__imp__less__div__pos,axiom,
% 9.20/5.14      ! [Y3: real,Z2: real,X3: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.14       => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
% 9.20/5.14         => ( ord_less_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3147_mult__imp__less__div__pos,axiom,
% 9.20/5.14      ! [Y3: rat,Z2: rat,X3: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14       => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
% 9.20/5.14         => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3148_divide__strict__left__mono,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3149_divide__strict__left__mono,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3150_divide__strict__left__mono__neg,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14           => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3151_divide__strict__left__mono__neg,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14           => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3152_divide__eq__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [B: complex,C: complex,W2: num] :
% 9.20/5.14        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.20/5.14          = ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.14        = ( ( ( C != zero_zero_complex )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_complex )
% 9.20/5.14           => ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.14              = zero_zero_complex ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3153_divide__eq__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [B: real,C: real,W2: num] :
% 9.20/5.14        ( ( ( divide_divide_real @ B @ C )
% 9.20/5.14          = ( numeral_numeral_real @ W2 ) )
% 9.20/5.14        = ( ( ( C != zero_zero_real )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_real )
% 9.20/5.14           => ( ( numeral_numeral_real @ W2 )
% 9.20/5.14              = zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3154_divide__eq__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [B: rat,C: rat,W2: num] :
% 9.20/5.14        ( ( ( divide_divide_rat @ B @ C )
% 9.20/5.14          = ( numeral_numeral_rat @ W2 ) )
% 9.20/5.14        = ( ( ( C != zero_zero_rat )
% 9.20/5.14           => ( B
% 9.20/5.14              = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.20/5.14          & ( ( C = zero_zero_rat )
% 9.20/5.14           => ( ( numeral_numeral_rat @ W2 )
% 9.20/5.14              = zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3155_eq__divide__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [W2: num,B: complex,C: complex] :
% 9.20/5.14        ( ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.14          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_complex )
% 9.20/5.14           => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_complex )
% 9.20/5.14           => ( ( numera6690914467698888265omplex @ W2 )
% 9.20/5.14              = zero_zero_complex ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3156_eq__divide__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [W2: num,B: real,C: real] :
% 9.20/5.14        ( ( ( numeral_numeral_real @ W2 )
% 9.20/5.14          = ( divide_divide_real @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_real )
% 9.20/5.14           => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_real )
% 9.20/5.14           => ( ( numeral_numeral_real @ W2 )
% 9.20/5.14              = zero_zero_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3157_eq__divide__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [W2: num,B: rat,C: rat] :
% 9.20/5.14        ( ( ( numeral_numeral_rat @ W2 )
% 9.20/5.14          = ( divide_divide_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( C != zero_zero_rat )
% 9.20/5.14           => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
% 9.20/5.14              = B ) )
% 9.20/5.14          & ( ( C = zero_zero_rat )
% 9.20/5.14           => ( ( numeral_numeral_rat @ W2 )
% 9.20/5.14              = zero_zero_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3158_add__divide__eq__if__simps_I2_J,axiom,
% 9.20/5.14      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.14         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.20/5.14            = B ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_complex )
% 9.20/5.14         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.20/5.14            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3159_add__divide__eq__if__simps_I2_J,axiom,
% 9.20/5.14      ! [Z2: real,A: real,B: real] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_real )
% 9.20/5.14         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.20/5.14            = B ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_real )
% 9.20/5.14         => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.20/5.14            = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3160_add__divide__eq__if__simps_I2_J,axiom,
% 9.20/5.14      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.14         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.20/5.14            = B ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_rat )
% 9.20/5.14         => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.20/5.14            = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3161_add__divide__eq__if__simps_I1_J,axiom,
% 9.20/5.14      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.14         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.20/5.14            = A ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_complex )
% 9.20/5.14         => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.20/5.14            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3162_add__divide__eq__if__simps_I1_J,axiom,
% 9.20/5.14      ! [Z2: real,A: real,B: real] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_real )
% 9.20/5.14         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.20/5.14            = A ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_real )
% 9.20/5.14         => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.20/5.14            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3163_add__divide__eq__if__simps_I1_J,axiom,
% 9.20/5.14      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.14         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.20/5.14            = A ) )
% 9.20/5.14        & ( ( Z2 != zero_zero_rat )
% 9.20/5.14         => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.20/5.14            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3164_add__frac__eq,axiom,
% 9.20/5.14      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 9.20/5.14        ( ( Y3 != zero_zero_complex )
% 9.20/5.14       => ( ( Z2 != zero_zero_complex )
% 9.20/5.14         => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.20/5.14            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3165_add__frac__eq,axiom,
% 9.20/5.14      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 9.20/5.14        ( ( Y3 != zero_zero_real )
% 9.20/5.14       => ( ( Z2 != zero_zero_real )
% 9.20/5.14         => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.20/5.14            = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3166_add__frac__eq,axiom,
% 9.20/5.14      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 9.20/5.14        ( ( Y3 != zero_zero_rat )
% 9.20/5.14       => ( ( Z2 != zero_zero_rat )
% 9.20/5.14         => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.20/5.14            = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3167_add__frac__num,axiom,
% 9.20/5.14      ! [Y3: complex,X3: complex,Z2: complex] :
% 9.20/5.14        ( ( Y3 != zero_zero_complex )
% 9.20/5.14       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z2 )
% 9.20/5.14          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3168_add__frac__num,axiom,
% 9.20/5.14      ! [Y3: real,X3: real,Z2: real] :
% 9.20/5.14        ( ( Y3 != zero_zero_real )
% 9.20/5.14       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 )
% 9.20/5.14          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3169_add__frac__num,axiom,
% 9.20/5.14      ! [Y3: rat,X3: rat,Z2: rat] :
% 9.20/5.14        ( ( Y3 != zero_zero_rat )
% 9.20/5.14       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 )
% 9.20/5.14          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3170_add__num__frac,axiom,
% 9.20/5.14      ! [Y3: complex,Z2: complex,X3: complex] :
% 9.20/5.14        ( ( Y3 != zero_zero_complex )
% 9.20/5.14       => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
% 9.20/5.14          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3171_add__num__frac,axiom,
% 9.20/5.14      ! [Y3: real,Z2: real,X3: real] :
% 9.20/5.14        ( ( Y3 != zero_zero_real )
% 9.20/5.14       => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.20/5.14          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3172_add__num__frac,axiom,
% 9.20/5.14      ! [Y3: rat,Z2: rat,X3: rat] :
% 9.20/5.14        ( ( Y3 != zero_zero_rat )
% 9.20/5.14       => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) )
% 9.20/5.14          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3173_add__divide__eq__iff,axiom,
% 9.20/5.14      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.14        ( ( Z2 != zero_zero_complex )
% 9.20/5.14       => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
% 9.20/5.14          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3174_add__divide__eq__iff,axiom,
% 9.20/5.14      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.14        ( ( Z2 != zero_zero_real )
% 9.20/5.14       => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
% 9.20/5.14          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3175_add__divide__eq__iff,axiom,
% 9.20/5.14      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.14        ( ( Z2 != zero_zero_rat )
% 9.20/5.14       => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
% 9.20/5.14          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3176_divide__add__eq__iff,axiom,
% 9.20/5.14      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.14        ( ( Z2 != zero_zero_complex )
% 9.20/5.14       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
% 9.20/5.14          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3177_divide__add__eq__iff,axiom,
% 9.20/5.14      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.14        ( ( Z2 != zero_zero_real )
% 9.20/5.14       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
% 9.20/5.14          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3178_divide__add__eq__iff,axiom,
% 9.20/5.14      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.14        ( ( Z2 != zero_zero_rat )
% 9.20/5.14       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
% 9.20/5.14          = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3179_power__gt1__lemma,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.14       => ( ord_less_real @ one_one_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3180_power__gt1__lemma,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.14       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3181_power__gt1__lemma,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.14       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3182_power__gt1__lemma,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.14       => ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3183_power__less__power__Suc,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.14       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3184_power__less__power__Suc,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( ord_less_rat @ one_one_rat @ A )
% 9.20/5.14       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3185_power__less__power__Suc,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ one_one_nat @ A )
% 9.20/5.14       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3186_power__less__power__Suc,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( ord_less_int @ one_one_int @ A )
% 9.20/5.14       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3187_one__less__mult,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.14       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.20/5.14         => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3188_n__less__m__mult__n,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.14       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.20/5.14         => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3189_n__less__n__mult__m,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.14       => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.20/5.14         => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3190_nat__mult__le__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14       => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.14          = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3191_nat__mult__div__cancel1,axiom,
% 9.20/5.14      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.14       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.14          = ( divide_divide_nat @ M @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3192_div__less__iff__less__mult,axiom,
% 9.20/5.14      ! [Q4: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 9.20/5.14       => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N2 )
% 9.20/5.14          = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3193_mult__le__cancel__left1,axiom,
% 9.20/5.14      ! [C: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3194_mult__le__cancel__left1,axiom,
% 9.20/5.14      ! [C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3195_mult__le__cancel__left1,axiom,
% 9.20/5.14      ! [C: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3196_mult__le__cancel__left2,axiom,
% 9.20/5.14      ! [C: real,A: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3197_mult__le__cancel__left2,axiom,
% 9.20/5.14      ! [C: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3198_mult__le__cancel__left2,axiom,
% 9.20/5.14      ! [C: int,A: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3199_mult__le__cancel__right1,axiom,
% 9.20/5.14      ! [C: real,B: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ one_one_real @ B ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3200_mult__le__cancel__right1,axiom,
% 9.20/5.14      ! [C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ one_one_rat @ B ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3201_mult__le__cancel__right1,axiom,
% 9.20/5.14      ! [C: int,B: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ one_one_int @ B ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3202_mult__le__cancel__right2,axiom,
% 9.20/5.14      ! [A: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ A @ one_one_real ) )
% 9.20/5.14          & ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_eq_real @ one_one_real @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3203_mult__le__cancel__right2,axiom,
% 9.20/5.14      ! [A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ A @ one_one_rat ) )
% 9.20/5.14          & ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3204_mult__le__cancel__right2,axiom,
% 9.20/5.14      ! [A: int,C: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_eq_int @ A @ one_one_int ) )
% 9.20/5.14          & ( ( ord_less_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3205_mult__less__cancel__left1,axiom,
% 9.20/5.14      ! [C: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ one_one_real @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3206_mult__less__cancel__left1,axiom,
% 9.20/5.14      ! [C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3207_mult__less__cancel__left1,axiom,
% 9.20/5.14      ! [C: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ one_one_int @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3208_mult__less__cancel__left2,axiom,
% 9.20/5.14      ! [C: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ A @ one_one_real ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3209_mult__less__cancel__left2,axiom,
% 9.20/5.14      ! [C: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3210_mult__less__cancel__left2,axiom,
% 9.20/5.14      ! [C: int,A: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ A @ one_one_int ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3211_mult__less__cancel__right1,axiom,
% 9.20/5.14      ! [C: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ one_one_real @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ B @ one_one_real ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3212_mult__less__cancel__right1,axiom,
% 9.20/5.14      ! [C: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ one_one_rat @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3213_mult__less__cancel__right1,axiom,
% 9.20/5.14      ! [C: int,B: int] :
% 9.20/5.14        ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ one_one_int @ B ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3214_mult__less__cancel__right2,axiom,
% 9.20/5.14      ! [A: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ ( times_times_real @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ A @ one_one_real ) )
% 9.20/5.14          & ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14           => ( ord_less_real @ one_one_real @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3215_mult__less__cancel__right2,axiom,
% 9.20/5.14      ! [A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ A @ one_one_rat ) )
% 9.20/5.14          & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14           => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3216_mult__less__cancel__right2,axiom,
% 9.20/5.14      ! [A: int,C: int] :
% 9.20/5.14        ( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
% 9.20/5.14        = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.14           => ( ord_less_int @ A @ one_one_int ) )
% 9.20/5.14          & ( ( ord_less_eq_int @ C @ zero_zero_int )
% 9.20/5.14           => ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3217_field__le__mult__one__interval,axiom,
% 9.20/5.14      ! [X3: real,Y3: real] :
% 9.20/5.14        ( ! [Z3: real] :
% 9.20/5.14            ( ( ord_less_real @ zero_zero_real @ Z3 )
% 9.20/5.14           => ( ( ord_less_real @ Z3 @ one_one_real )
% 9.20/5.14             => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y3 ) ) )
% 9.20/5.14       => ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3218_field__le__mult__one__interval,axiom,
% 9.20/5.14      ! [X3: rat,Y3: rat] :
% 9.20/5.14        ( ! [Z3: rat] :
% 9.20/5.14            ( ( ord_less_rat @ zero_zero_rat @ Z3 )
% 9.20/5.14           => ( ( ord_less_rat @ Z3 @ one_one_rat )
% 9.20/5.14             => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X3 ) @ Y3 ) ) )
% 9.20/5.14       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3219_divide__le__eq,axiom,
% 9.20/5.14      ! [B: real,C: real,A: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3220_divide__le__eq,axiom,
% 9.20/5.14      ! [B: rat,C: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3221_le__divide__eq,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3222_le__divide__eq,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3223_divide__left__mono,axiom,
% 9.20/5.14      ! [B: real,A: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3224_divide__left__mono,axiom,
% 9.20/5.14      ! [B: rat,A: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3225_neg__divide__le__eq,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3226_neg__divide__le__eq,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3227_neg__le__divide__eq,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3228_neg__le__divide__eq,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3229_pos__divide__le__eq,axiom,
% 9.20/5.14      ! [C: real,B: real,A: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3230_pos__divide__le__eq,axiom,
% 9.20/5.14      ! [C: rat,B: rat,A: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
% 9.20/5.14          = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3231_pos__le__divide__eq,axiom,
% 9.20/5.14      ! [C: real,A: real,B: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14       => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3232_pos__le__divide__eq,axiom,
% 9.20/5.14      ! [C: rat,A: rat,B: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14       => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3233_mult__imp__div__pos__le,axiom,
% 9.20/5.14      ! [Y3: real,X3: real,Z2: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.14       => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
% 9.20/5.14         => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3234_mult__imp__div__pos__le,axiom,
% 9.20/5.14      ! [Y3: rat,X3: rat,Z2: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14       => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
% 9.20/5.14         => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3235_mult__imp__le__div__pos,axiom,
% 9.20/5.14      ! [Y3: real,Z2: real,X3: real] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.14       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
% 9.20/5.14         => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3236_mult__imp__le__div__pos,axiom,
% 9.20/5.14      ! [Y3: rat,Z2: rat,X3: rat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.20/5.14       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
% 9.20/5.14         => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3237_divide__left__mono__neg,axiom,
% 9.20/5.14      ! [A: real,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_real @ C @ zero_zero_real )
% 9.20/5.14         => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
% 9.20/5.14           => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3238_divide__left__mono__neg,axiom,
% 9.20/5.14      ! [A: rat,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.14       => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
% 9.20/5.14         => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.14           => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3239_convex__bound__le,axiom,
% 9.20/5.14      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 9.20/5.14        ( ( ord_less_eq_real @ X3 @ A )
% 9.20/5.14       => ( ( ord_less_eq_real @ Y3 @ A )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.20/5.14             => ( ( ( plus_plus_real @ U @ V )
% 9.20/5.14                  = one_one_real )
% 9.20/5.14               => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3240_convex__bound__le,axiom,
% 9.20/5.14      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 9.20/5.14        ( ( ord_less_eq_rat @ X3 @ A )
% 9.20/5.14       => ( ( ord_less_eq_rat @ Y3 @ A )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.20/5.14             => ( ( ( plus_plus_rat @ U @ V )
% 9.20/5.14                  = one_one_rat )
% 9.20/5.14               => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3241_convex__bound__le,axiom,
% 9.20/5.14      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 9.20/5.14        ( ( ord_less_eq_int @ X3 @ A )
% 9.20/5.14       => ( ( ord_less_eq_int @ Y3 @ A )
% 9.20/5.14         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.20/5.14           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.20/5.14             => ( ( ( plus_plus_int @ U @ V )
% 9.20/5.14                  = one_one_int )
% 9.20/5.14               => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3242_divide__less__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [B: real,C: real,W2: num] :
% 9.20/5.14        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3243_divide__less__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [B: rat,C: rat,W2: num] :
% 9.20/5.14        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3244_less__divide__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [W2: num,B: real,C: real] :
% 9.20/5.14        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.14           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.14               => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3245_less__divide__eq__numeral_I1_J,axiom,
% 9.20/5.14      ! [W2: num,B: rat,C: rat] :
% 9.20/5.14        ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.14        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.20/5.14          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.14           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.20/5.14              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.14               => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3246_power__Suc__less,axiom,
% 9.20/5.14      ! [A: real,N2: nat] :
% 9.20/5.14        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.14       => ( ( ord_less_real @ A @ one_one_real )
% 9.20/5.14         => ( ord_less_real @ ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) @ ( power_power_real @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3247_power__Suc__less,axiom,
% 9.20/5.14      ! [A: rat,N2: nat] :
% 9.20/5.14        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.14       => ( ( ord_less_rat @ A @ one_one_rat )
% 9.20/5.14         => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3248_power__Suc__less,axiom,
% 9.20/5.14      ! [A: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ A )
% 9.20/5.14       => ( ( ord_less_nat @ A @ one_one_nat )
% 9.20/5.14         => ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) @ ( power_power_nat @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3249_power__Suc__less,axiom,
% 9.20/5.14      ! [A: int,N2: nat] :
% 9.20/5.14        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.14       => ( ( ord_less_int @ A @ one_one_int )
% 9.20/5.14         => ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) @ ( power_power_int @ A @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3250_mult__2,axiom,
% 9.20/5.14      ! [Z2: complex] :
% 9.20/5.14        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3251_mult__2,axiom,
% 9.20/5.14      ! [Z2: extended_enat] :
% 9.20/5.14        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3252_mult__2,axiom,
% 9.20/5.14      ! [Z2: real] :
% 9.20/5.14        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3253_mult__2,axiom,
% 9.20/5.14      ! [Z2: rat] :
% 9.20/5.14        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3254_mult__2,axiom,
% 9.20/5.14      ! [Z2: nat] :
% 9.20/5.14        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3255_mult__2,axiom,
% 9.20/5.14      ! [Z2: int] :
% 9.20/5.14        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
% 9.20/5.14        = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3256_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: complex] :
% 9.20/5.14        ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_plus_complex @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3257_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: extended_enat] :
% 9.20/5.14        ( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3258_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: real] :
% 9.20/5.14        ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_plus_real @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3259_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: rat] :
% 9.20/5.14        ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_plus_rat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3260_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: nat] :
% 9.20/5.14        ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_plus_nat @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3261_mult__2__right,axiom,
% 9.20/5.14      ! [Z2: int] :
% 9.20/5.14        ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.14        = ( plus_plus_int @ Z2 @ Z2 ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3262_left__add__twice,axiom,
% 9.20/5.14      ! [A: complex,B: complex] :
% 9.20/5.14        ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
% 9.20/5.14        = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3263_left__add__twice,axiom,
% 9.20/5.14      ! [A: extended_enat,B: extended_enat] :
% 9.20/5.14        ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
% 9.20/5.14        = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3264_left__add__twice,axiom,
% 9.20/5.14      ! [A: real,B: real] :
% 9.20/5.14        ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.20/5.14        = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3265_left__add__twice,axiom,
% 9.20/5.14      ! [A: rat,B: rat] :
% 9.20/5.14        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.14        = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3266_left__add__twice,axiom,
% 9.20/5.14      ! [A: nat,B: nat] :
% 9.20/5.14        ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.14        = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3267_left__add__twice,axiom,
% 9.20/5.14      ! [A: int,B: int] :
% 9.20/5.14        ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.20/5.14        = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3268_div__nat__eqI,axiom,
% 9.20/5.14      ! [N2: nat,Q4: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
% 9.20/5.14       => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
% 9.20/5.14         => ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.14            = Q4 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3269_less__eq__div__iff__mult__less__eq,axiom,
% 9.20/5.14      ! [Q4: nat,M: nat,N2: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ Q4 )
% 9.20/5.14       => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q4 ) )
% 9.20/5.14          = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N2 ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3270_split__div,axiom,
% 9.20/5.14      ! [P: nat > $o,M: nat,N2: nat] :
% 9.20/5.14        ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 9.20/5.14        = ( ( ( N2 = zero_zero_nat )
% 9.20/5.14           => ( P @ zero_zero_nat ) )
% 9.20/5.14          & ( ( N2 != zero_zero_nat )
% 9.20/5.14           => ! [I2: nat,J3: nat] :
% 9.20/5.14                ( ( ord_less_nat @ J3 @ N2 )
% 9.20/5.14               => ( ( M
% 9.20/5.14                    = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
% 9.20/5.14                 => ( P @ I2 ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3271_dividend__less__div__times,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.14       => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3272_dividend__less__times__div,axiom,
% 9.20/5.14      ! [N2: nat,M: nat] :
% 9.20/5.14        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.14       => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3273_nat__bit__induct,axiom,
% 9.20/5.14      ! [P: nat > $o,N2: nat] :
% 9.20/5.14        ( ( P @ zero_zero_nat )
% 9.20/5.14       => ( ! [N3: nat] :
% 9.20/5.14              ( ( P @ N3 )
% 9.20/5.14             => ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.14               => ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.20/5.14         => ( ! [N3: nat] :
% 9.20/5.14                ( ( P @ N3 )
% 9.20/5.14               => ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
% 9.20/5.14           => ( P @ N2 ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3274_convex__bound__lt,axiom,
% 9.20/5.14      ! [X3: real,A: real,Y3: real,U: real,V: real] :
% 9.20/5.14        ( ( ord_less_real @ X3 @ A )
% 9.20/5.14       => ( ( ord_less_real @ Y3 @ A )
% 9.20/5.14         => ( ( ord_less_eq_real @ zero_zero_real @ U )
% 9.20/5.14           => ( ( ord_less_eq_real @ zero_zero_real @ V )
% 9.20/5.14             => ( ( ( plus_plus_real @ U @ V )
% 9.20/5.14                  = one_one_real )
% 9.20/5.14               => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.14  
% 9.20/5.14  thf(fact_3275_convex__bound__lt,axiom,
% 9.20/5.14      ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
% 9.20/5.14        ( ( ord_less_rat @ X3 @ A )
% 9.20/5.14       => ( ( ord_less_rat @ Y3 @ A )
% 9.20/5.14         => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
% 9.20/5.14           => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
% 9.20/5.14             => ( ( ( plus_plus_rat @ U @ V )
% 9.20/5.14                  = one_one_rat )
% 9.20/5.14               => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3276_convex__bound__lt,axiom,
% 9.20/5.15      ! [X3: int,A: int,Y3: int,U: int,V: int] :
% 9.20/5.15        ( ( ord_less_int @ X3 @ A )
% 9.20/5.15       => ( ( ord_less_int @ Y3 @ A )
% 9.20/5.15         => ( ( ord_less_eq_int @ zero_zero_int @ U )
% 9.20/5.15           => ( ( ord_less_eq_int @ zero_zero_int @ V )
% 9.20/5.15             => ( ( ( plus_plus_int @ U @ V )
% 9.20/5.15                  = one_one_int )
% 9.20/5.15               => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3277_divide__le__eq__numeral_I1_J,axiom,
% 9.20/5.15      ! [B: real,C: real,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.15        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.15           => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.20/5.15          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.15           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.15               => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.20/5.15              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.15               => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3278_divide__le__eq__numeral_I1_J,axiom,
% 9.20/5.15      ! [B: rat,C: rat,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.15        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.15           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.20/5.15          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.15           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.15               => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.20/5.15              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.15               => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3279_le__divide__eq__numeral_I1_J,axiom,
% 9.20/5.15      ! [W2: num,B: real,C: real] :
% 9.20/5.15        ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
% 9.20/5.15        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.15           => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
% 9.20/5.15          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.15           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.15               => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
% 9.20/5.15              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.15               => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3280_le__divide__eq__numeral_I1_J,axiom,
% 9.20/5.15      ! [W2: num,B: rat,C: rat] :
% 9.20/5.15        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.15        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.15           => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
% 9.20/5.15          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.15           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.15               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
% 9.20/5.15              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.15               => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3281_split__div_H,axiom,
% 9.20/5.15      ! [P: nat > $o,M: nat,N2: nat] :
% 9.20/5.15        ( ( P @ ( divide_divide_nat @ M @ N2 ) )
% 9.20/5.15        = ( ( ( N2 = zero_zero_nat )
% 9.20/5.15            & ( P @ zero_zero_nat ) )
% 9.20/5.15          | ? [Q5: nat] :
% 9.20/5.15              ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M )
% 9.20/5.15              & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
% 9.20/5.15              & ( P @ Q5 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3282_power2__sum,axiom,
% 9.20/5.15      ! [X3: complex,Y3: complex] :
% 9.20/5.15        ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3283_power2__sum,axiom,
% 9.20/5.15      ! [X3: extended_enat,Y3: extended_enat] :
% 9.20/5.15        ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3284_power2__sum,axiom,
% 9.20/5.15      ! [X3: real,Y3: real] :
% 9.20/5.15        ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3285_power2__sum,axiom,
% 9.20/5.15      ! [X3: rat,Y3: rat] :
% 9.20/5.15        ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3286_power2__sum,axiom,
% 9.20/5.15      ! [X3: nat,Y3: nat] :
% 9.20/5.15        ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3287_power2__sum,axiom,
% 9.20/5.15      ! [X3: int,Y3: int] :
% 9.20/5.15        ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3288_zero__le__even__power_H,axiom,
% 9.20/5.15      ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3289_zero__le__even__power_H,axiom,
% 9.20/5.15      ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3290_zero__le__even__power_H,axiom,
% 9.20/5.15      ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3291_triangle__def,axiom,
% 9.20/5.15      ( nat_triangle
% 9.20/5.15      = ( ^ [N: nat] : ( divide_divide_nat @ ( times_times_nat @ N @ ( suc @ N ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3292_odd__0__le__power__imp__0__le,axiom,
% 9.20/5.15      ! [A: real,N2: nat] :
% 9.20/5.15        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.15       => ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3293_odd__0__le__power__imp__0__le,axiom,
% 9.20/5.15      ! [A: rat,N2: nat] :
% 9.20/5.15        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.15       => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3294_odd__0__le__power__imp__0__le,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.15       => ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3295_odd__power__less__zero,axiom,
% 9.20/5.15      ! [A: real,N2: nat] :
% 9.20/5.15        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.15       => ( ord_less_real @ ( power_power_real @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_real ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3296_odd__power__less__zero,axiom,
% 9.20/5.15      ! [A: rat,N2: nat] :
% 9.20/5.15        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.15       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3297_odd__power__less__zero,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.15       => ( ord_less_int @ ( power_power_int @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3298_arith__geo__mean,axiom,
% 9.20/5.15      ! [U: real,X3: real,Y3: real] :
% 9.20/5.15        ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( times_times_real @ X3 @ Y3 ) )
% 9.20/5.15       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.15         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.15           => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3299_arith__geo__mean,axiom,
% 9.20/5.15      ! [U: rat,X3: rat,Y3: rat] :
% 9.20/5.15        ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( times_times_rat @ X3 @ Y3 ) )
% 9.20/5.15       => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.15         => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.15           => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3300_sum__squares__bound,axiom,
% 9.20/5.15      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3301_sum__squares__bound,axiom,
% 9.20/5.15      ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3302_set__bit__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( bit_se7882103937844011126it_nat @ zero_zero_nat @ A )
% 9.20/5.15        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3303_set__bit__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
% 9.20/5.15        = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3304_set__bit__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( bit_se7879613467334960850it_int @ zero_zero_nat @ A )
% 9.20/5.15        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3305_pos2,axiom,
% 9.20/5.15      ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3306_unset__bit__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( bit_se4205575877204974255it_nat @ zero_zero_nat @ A )
% 9.20/5.15        = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3307_unset__bit__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
% 9.20/5.15        = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3308_unset__bit__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( bit_se4203085406695923979it_int @ zero_zero_nat @ A )
% 9.20/5.15        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3309_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.20/5.15      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.20/5.15       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.15          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3310_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.20/5.15      ! [C: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.15       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.15          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3311_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
% 9.20/5.15      ! [C: int,A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.15       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.15          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3312_discrete,axiom,
% 9.20/5.15      ( ord_less_nat
% 9.20/5.15      = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3313_discrete,axiom,
% 9.20/5.15      ( ord_less_int
% 9.20/5.15      = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3314_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.20/5.15         => ( ( divide6298287555418463151nteger @ A @ B )
% 9.20/5.15            = zero_z3403309356797280102nteger ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3315_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.15       => ( ( ord_less_nat @ A @ B )
% 9.20/5.15         => ( ( divide_divide_nat @ A @ B )
% 9.20/5.15            = zero_zero_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3316_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.15       => ( ( ord_less_int @ A @ B )
% 9.20/5.15         => ( ( divide_divide_int @ A @ B )
% 9.20/5.15            = zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3317_div__positive,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.15       => ( ( ord_le3102999989581377725nteger @ B @ A )
% 9.20/5.15         => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3318_div__positive,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.15       => ( ( ord_less_eq_nat @ B @ A )
% 9.20/5.15         => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3319_div__positive,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15       => ( ( ord_less_eq_int @ B @ A )
% 9.20/5.15         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3320_zle__add1__eq__le,axiom,
% 9.20/5.15      ! [W2: int,Z2: int] :
% 9.20/5.15        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 9.20/5.15        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3321_set__bit__negative__int__iff,axiom,
% 9.20/5.15      ! [N2: nat,K: int] :
% 9.20/5.15        ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.15        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3322_unset__bit__negative__int__iff,axiom,
% 9.20/5.15      ! [N2: nat,K: int] :
% 9.20/5.15        ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.15        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3323_div__pos__pos__trivial,axiom,
% 9.20/5.15      ! [K: int,L: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.15       => ( ( ord_less_int @ K @ L )
% 9.20/5.15         => ( ( divide_divide_int @ K @ L )
% 9.20/5.15            = zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3324_div__neg__neg__trivial,axiom,
% 9.20/5.15      ! [K: int,L: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_int @ L @ K )
% 9.20/5.15         => ( ( divide_divide_int @ K @ L )
% 9.20/5.15            = zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3325_nonneg1__imp__zdiv__pos__iff,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.20/5.15          = ( ( ord_less_eq_int @ B @ A )
% 9.20/5.15            & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3326_pos__imp__zdiv__nonneg__iff,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.20/5.15          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3327_neg__imp__zdiv__nonneg__iff,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
% 9.20/5.15          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3328_int__one__le__iff__zero__less,axiom,
% 9.20/5.15      ! [Z2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ one_one_int @ Z2 )
% 9.20/5.15        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3329_pos__imp__zdiv__pos__iff,axiom,
% 9.20/5.15      ! [K: int,I: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
% 9.20/5.15          = ( ord_less_eq_int @ K @ I ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3330_div__nonpos__pos__le0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3331_div__nonneg__neg__le0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.15       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.15         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3332_div__positive__int,axiom,
% 9.20/5.15      ! [L: int,K: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ L @ K )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.15         => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3333_div__int__pos__iff,axiom,
% 9.20/5.15      ! [K: int,L: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
% 9.20/5.15        = ( ( K = zero_zero_int )
% 9.20/5.15          | ( L = zero_zero_int )
% 9.20/5.15          | ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.15            & ( ord_less_eq_int @ zero_zero_int @ L ) )
% 9.20/5.15          | ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.15            & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3334_zless__imp__add1__zle,axiom,
% 9.20/5.15      ! [W2: int,Z2: int] :
% 9.20/5.15        ( ( ord_less_int @ W2 @ Z2 )
% 9.20/5.15       => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3335_zdiv__mono2__neg,axiom,
% 9.20/5.15      ! [A: int,B6: int,B: int] :
% 9.20/5.15        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.20/5.15         => ( ( ord_less_eq_int @ B6 @ B )
% 9.20/5.15           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3336_zdiv__mono1__neg,axiom,
% 9.20/5.15      ! [A: int,A6: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ A @ A6 )
% 9.20/5.15       => ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.15         => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3337_int__div__pos__eq,axiom,
% 9.20/5.15      ! [A: int,B: int,Q4: int,R2: int] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.20/5.15         => ( ( ord_less_int @ R2 @ B )
% 9.20/5.15           => ( ( divide_divide_int @ A @ B )
% 9.20/5.15              = Q4 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3338_int__div__neg__eq,axiom,
% 9.20/5.15      ! [A: int,B: int,Q4: int,R2: int] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.15       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.20/5.15         => ( ( ord_less_int @ B @ R2 )
% 9.20/5.15           => ( ( divide_divide_int @ A @ B )
% 9.20/5.15              = Q4 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3339_zdiv__eq__0__iff,axiom,
% 9.20/5.15      ! [I: int,K: int] :
% 9.20/5.15        ( ( ( divide_divide_int @ I @ K )
% 9.20/5.15          = zero_zero_int )
% 9.20/5.15        = ( ( K = zero_zero_int )
% 9.20/5.15          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.20/5.15            & ( ord_less_int @ I @ K ) )
% 9.20/5.15          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.20/5.15            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3340_zdiv__mono2,axiom,
% 9.20/5.15      ! [A: int,B6: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.20/5.15         => ( ( ord_less_eq_int @ B6 @ B )
% 9.20/5.15           => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3341_zdiv__mono1,axiom,
% 9.20/5.15      ! [A: int,A6: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ A @ A6 )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3342_split__zdiv,axiom,
% 9.20/5.15      ! [P: int > $o,N2: int,K: int] :
% 9.20/5.15        ( ( P @ ( divide_divide_int @ N2 @ K ) )
% 9.20/5.15        = ( ( ( K = zero_zero_int )
% 9.20/5.15           => ( P @ zero_zero_int ) )
% 9.20/5.15          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.15           => ! [I2: int,J3: int] :
% 9.20/5.15                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.20/5.15                  & ( ord_less_int @ J3 @ K )
% 9.20/5.15                  & ( N2
% 9.20/5.15                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.15               => ( P @ I2 ) ) )
% 9.20/5.15          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.15           => ! [I2: int,J3: int] :
% 9.20/5.15                ( ( ( ord_less_int @ K @ J3 )
% 9.20/5.15                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.20/5.15                  & ( N2
% 9.20/5.15                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.15               => ( P @ I2 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3343_le__imp__0__less,axiom,
% 9.20/5.15      ! [Z2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.15       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3344_add1__zle__eq,axiom,
% 9.20/5.15      ! [W2: int,Z2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
% 9.20/5.15        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3345_odd__less__0__iff,axiom,
% 9.20/5.15      ! [Z2: int] :
% 9.20/5.15        ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
% 9.20/5.15        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3346_pos__zmult__eq__1__iff,axiom,
% 9.20/5.15      ! [M: int,N2: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ M )
% 9.20/5.15       => ( ( ( times_times_int @ M @ N2 )
% 9.20/5.15            = one_one_int )
% 9.20/5.15          = ( ( M = one_one_int )
% 9.20/5.15            & ( N2 = one_one_int ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3347_zless__add1__eq,axiom,
% 9.20/5.15      ! [W2: int,Z2: int] :
% 9.20/5.15        ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
% 9.20/5.15        = ( ( ord_less_int @ W2 @ Z2 )
% 9.20/5.15          | ( W2 = Z2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3348_int__gr__induct,axiom,
% 9.20/5.15      ! [K: int,I: int,P: int > $o] :
% 9.20/5.15        ( ( ord_less_int @ K @ I )
% 9.20/5.15       => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
% 9.20/5.15         => ( ! [I3: int] :
% 9.20/5.15                ( ( ord_less_int @ K @ I3 )
% 9.20/5.15               => ( ( P @ I3 )
% 9.20/5.15                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.15           => ( P @ I ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3349_pos__imp__zdiv__neg__iff,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.20/5.15          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3350_neg__imp__zdiv__neg__iff,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
% 9.20/5.15          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3351_int__div__less__self,axiom,
% 9.20/5.15      ! [X3: int,K: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.20/5.15       => ( ( ord_less_int @ one_one_int @ K )
% 9.20/5.15         => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3352_div__neg__pos__less0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15         => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3353_odd__nonzero,axiom,
% 9.20/5.15      ! [Z2: int] :
% 9.20/5.15        ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
% 9.20/5.15       != zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3354_times__int__code_I2_J,axiom,
% 9.20/5.15      ! [L: int] :
% 9.20/5.15        ( ( times_times_int @ zero_zero_int @ L )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3355_times__int__code_I1_J,axiom,
% 9.20/5.15      ! [K: int] :
% 9.20/5.15        ( ( times_times_int @ K @ zero_zero_int )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3356_bot__enat__def,axiom,
% 9.20/5.15      bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3357_incr__mult__lemma,axiom,
% 9.20/5.15      ! [D: int,P: int > $o,K: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.15       => ( ! [X4: int] :
% 9.20/5.15              ( ( P @ X4 )
% 9.20/5.15             => ( P @ ( plus_plus_int @ X4 @ D ) ) )
% 9.20/5.15         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.15           => ! [X2: int] :
% 9.20/5.15                ( ( P @ X2 )
% 9.20/5.15               => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3358_zmult__zless__mono2,axiom,
% 9.20/5.15      ! [I: int,J: int,K: int] :
% 9.20/5.15        ( ( ord_less_int @ I @ J )
% 9.20/5.15       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.15         => ( ord_less_int @ ( times_times_int @ K @ I ) @ ( times_times_int @ K @ J ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3359_less__int__code_I1_J,axiom,
% 9.20/5.15      ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3360_q__pos__lemma,axiom,
% 9.20/5.15      ! [B6: int,Q6: int,R3: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 9.20/5.15       => ( ( ord_less_int @ R3 @ B6 )
% 9.20/5.15         => ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.20/5.15           => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3361_zdiv__mono2__lemma,axiom,
% 9.20/5.15      ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
% 9.20/5.15        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 9.20/5.15          = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 9.20/5.15         => ( ( ord_less_int @ R3 @ B6 )
% 9.20/5.15           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.20/5.15             => ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.20/5.15               => ( ( ord_less_eq_int @ B6 @ B )
% 9.20/5.15                 => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3362_zdiv__mono2__neg__lemma,axiom,
% 9.20/5.15      ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
% 9.20/5.15        ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
% 9.20/5.15          = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
% 9.20/5.15       => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) @ zero_zero_int )
% 9.20/5.15         => ( ( ord_less_int @ R2 @ B )
% 9.20/5.15           => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 9.20/5.15             => ( ( ord_less_int @ zero_zero_int @ B6 )
% 9.20/5.15               => ( ( ord_less_eq_int @ B6 @ B )
% 9.20/5.15                 => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3363_unique__quotient__lemma,axiom,
% 9.20/5.15      ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
% 9.20/5.15         => ( ( ord_less_int @ R3 @ B )
% 9.20/5.15           => ( ( ord_less_int @ R2 @ B )
% 9.20/5.15             => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3364_unique__quotient__lemma__neg,axiom,
% 9.20/5.15      ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.15       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.20/5.15         => ( ( ord_less_int @ B @ R2 )
% 9.20/5.15           => ( ( ord_less_int @ B @ R3 )
% 9.20/5.15             => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3365_enat__0__less__mult__iff,axiom,
% 9.20/5.15      ! [M: extended_enat,N2: extended_enat] :
% 9.20/5.15        ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
% 9.20/5.15        = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
% 9.20/5.15          & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3366_realpow__pos__nth2,axiom,
% 9.20/5.15      ! [A: real,N2: nat] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.15       => ? [R4: real] :
% 9.20/5.15            ( ( ord_less_real @ zero_zero_real @ R4 )
% 9.20/5.15            & ( ( power_power_real @ R4 @ ( suc @ N2 ) )
% 9.20/5.15              = A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3367_realpow__pos__nth__unique,axiom,
% 9.20/5.15      ! [N2: nat,A: real] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.15         => ? [X4: real] :
% 9.20/5.15              ( ( ord_less_real @ zero_zero_real @ X4 )
% 9.20/5.15              & ( ( power_power_real @ X4 @ N2 )
% 9.20/5.15                = A )
% 9.20/5.15              & ! [Y6: real] :
% 9.20/5.15                  ( ( ( ord_less_real @ zero_zero_real @ Y6 )
% 9.20/5.15                    & ( ( power_power_real @ Y6 @ N2 )
% 9.20/5.15                      = A ) )
% 9.20/5.15                 => ( Y6 = X4 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3368_realpow__pos__nth,axiom,
% 9.20/5.15      ! [N2: nat,A: real] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.15         => ? [R4: real] :
% 9.20/5.15              ( ( ord_less_real @ zero_zero_real @ R4 )
% 9.20/5.15              & ( ( power_power_real @ R4 @ N2 )
% 9.20/5.15                = A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3369_mult__le__cancel__iff2,axiom,
% 9.20/5.15      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y3 ) )
% 9.20/5.15          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3370_mult__le__cancel__iff2,axiom,
% 9.20/5.15      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.15        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ ( times_times_rat @ Z2 @ Y3 ) )
% 9.20/5.15          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3371_mult__le__cancel__iff2,axiom,
% 9.20/5.15      ! [Z2: int,X3: int,Y3: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X3 ) @ ( times_times_int @ Z2 @ Y3 ) )
% 9.20/5.15          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3372_mult__le__cancel__iff1,axiom,
% 9.20/5.15      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3373_mult__le__cancel__iff1,axiom,
% 9.20/5.15      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.15        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3374_mult__le__cancel__iff1,axiom,
% 9.20/5.15      ! [Z2: int,X3: int,Y3: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.15       => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3375_divides__aux__eq,axiom,
% 9.20/5.15      ! [Q4: nat,R2: nat] :
% 9.20/5.15        ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.20/5.15        = ( R2 = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3376_divides__aux__eq,axiom,
% 9.20/5.15      ! [Q4: int,R2: int] :
% 9.20/5.15        ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.15        = ( R2 = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3377_low__def,axiom,
% 9.20/5.15      ( vEBT_VEBT_low
% 9.20/5.15      = ( ^ [X: nat,N: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3378_even__succ__div__exp,axiom,
% 9.20/5.15      ! [A: nat,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3379_even__succ__div__exp,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3380_even__succ__div__exp,axiom,
% 9.20/5.15      ! [A: code_integer,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3381_set__decode__Suc,axiom,
% 9.20/5.15      ! [N2: nat,X3: nat] :
% 9.20/5.15        ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) )
% 9.20/5.15        = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3382_length__product,axiom,
% 9.20/5.15      ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.20/5.15        ( ( size_s7466405169056248089T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3383_length__product,axiom,
% 9.20/5.15      ! [Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.20/5.15        ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3384_length__product,axiom,
% 9.20/5.15      ! [Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.20/5.15        ( ( size_s6152045936467909847BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3385_length__product,axiom,
% 9.20/5.15      ! [Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.20/5.15        ( ( size_s3661962791536183091BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3386_length__product,axiom,
% 9.20/5.15      ! [Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.20/5.15        ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3387_length__product,axiom,
% 9.20/5.15      ! [Xs: list_o,Ys: list_o] :
% 9.20/5.15        ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3388_length__product,axiom,
% 9.20/5.15      ! [Xs: list_o,Ys: list_nat] :
% 9.20/5.15        ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3389_length__product,axiom,
% 9.20/5.15      ! [Xs: list_o,Ys: list_int] :
% 9.20/5.15        ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3390_length__product,axiom,
% 9.20/5.15      ! [Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.20/5.15        ( ( size_s4762443039079500285T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3391_length__product,axiom,
% 9.20/5.15      ! [Xs: list_nat,Ys: list_o] :
% 9.20/5.15        ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
% 9.20/5.15        = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3392_mult__less__iff1,axiom,
% 9.20/5.15      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.20/5.15       => ( ( ord_less_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3393_mult__less__iff1,axiom,
% 9.20/5.15      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.15        ( ( ord_less_rat @ zero_zero_rat @ Z2 )
% 9.20/5.15       => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3394_mult__less__iff1,axiom,
% 9.20/5.15      ! [Z2: int,X3: int,Y3: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.15       => ( ( ord_less_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
% 9.20/5.15          = ( ord_less_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3395_num_Osize__gen_I2_J,axiom,
% 9.20/5.15      ! [X22: num] :
% 9.20/5.15        ( ( size_num @ ( bit0 @ X22 ) )
% 9.20/5.15        = ( plus_plus_nat @ ( size_num @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3396_nat__dvd__1__iff__1,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ M @ one_one_nat )
% 9.20/5.15        = ( M = one_one_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3397_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3398_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.20/5.15        = ( A = zero_zero_complex ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3399_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.20/5.15        = ( A = zero_zero_real ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3400_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.20/5.15        = ( A = zero_zero_rat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3401_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.20/5.15        = ( A = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3402_dvd__0__left__iff,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.20/5.15        = ( A = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3403_dvd__0__right,axiom,
% 9.20/5.15      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3404_dvd__0__right,axiom,
% 9.20/5.15      ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3405_dvd__0__right,axiom,
% 9.20/5.15      ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3406_dvd__0__right,axiom,
% 9.20/5.15      ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3407_dvd__0__right,axiom,
% 9.20/5.15      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3408_dvd__0__right,axiom,
% 9.20/5.15      ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3409_dvd__add__triv__right__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
% 9.20/5.15        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3410_dvd__add__triv__right__iff,axiom,
% 9.20/5.15      ! [A: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
% 9.20/5.15        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3411_dvd__add__triv__right__iff,axiom,
% 9.20/5.15      ! [A: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
% 9.20/5.15        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3412_dvd__add__triv__right__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
% 9.20/5.15        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3413_dvd__add__triv__right__iff,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
% 9.20/5.15        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3414_dvd__add__triv__left__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.15        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3415_dvd__add__triv__left__iff,axiom,
% 9.20/5.15      ! [A: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
% 9.20/5.15        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3416_dvd__add__triv__left__iff,axiom,
% 9.20/5.15      ! [A: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.15        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3417_dvd__add__triv__left__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.15        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3418_dvd__add__triv__left__iff,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
% 9.20/5.15        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3419_bits__mod__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3420_bits__mod__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3421_bits__mod__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3422_mod__self,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ A @ A )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3423_mod__self,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ A @ A )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3424_mod__self,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ A @ A )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3425_mod__by__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ A @ zero_zero_nat )
% 9.20/5.15        = A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3426_mod__by__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ A @ zero_zero_int )
% 9.20/5.15        = A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3427_mod__by__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
% 9.20/5.15        = A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3428_mod__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ zero_zero_nat @ A )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3429_mod__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ zero_zero_int @ A )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3430_mod__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3431_dvd__1__left,axiom,
% 9.20/5.15      ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3432_dvd__1__iff__1,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.20/5.15        = ( M
% 9.20/5.15          = ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3433_div__dvd__div,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.15         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
% 9.20/5.15            = ( dvd_dvd_nat @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3434_div__dvd__div,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ A @ C )
% 9.20/5.15         => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
% 9.20/5.15            = ( dvd_dvd_int @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3435_div__dvd__div,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.20/5.15         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
% 9.20/5.15            = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3436_not__real__square__gt__zero,axiom,
% 9.20/5.15      ! [X3: real] :
% 9.20/5.15        ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
% 9.20/5.15        = ( X3 = zero_zero_real ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3437_mod__add__self2,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3438_mod__add__self2,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3439_mod__add__self2,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3440_mod__add__self1,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3441_mod__add__self1,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3442_mod__add__self1,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3443_nat__mult__dvd__cancel__disj,axiom,
% 9.20/5.15      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.15        = ( ( K = zero_zero_nat )
% 9.20/5.15          | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3444_mod__less,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] :
% 9.20/5.15        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.15       => ( ( modulo_modulo_nat @ M @ N2 )
% 9.20/5.15          = M ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3445_dvd__mult__cancel__left,axiom,
% 9.20/5.15      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.20/5.15        = ( ( C = zero_z3403309356797280102nteger )
% 9.20/5.15          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3446_dvd__mult__cancel__left,axiom,
% 9.20/5.15      ! [C: complex,A: complex,B: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
% 9.20/5.15        = ( ( C = zero_zero_complex )
% 9.20/5.15          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3447_dvd__mult__cancel__left,axiom,
% 9.20/5.15      ! [C: real,A: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
% 9.20/5.15        = ( ( C = zero_zero_real )
% 9.20/5.15          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3448_dvd__mult__cancel__left,axiom,
% 9.20/5.15      ! [C: rat,A: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
% 9.20/5.15        = ( ( C = zero_zero_rat )
% 9.20/5.15          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3449_dvd__mult__cancel__left,axiom,
% 9.20/5.15      ! [C: int,A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
% 9.20/5.15        = ( ( C = zero_zero_int )
% 9.20/5.15          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3450_dvd__mult__cancel__right,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.15        = ( ( C = zero_z3403309356797280102nteger )
% 9.20/5.15          | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3451_dvd__mult__cancel__right,axiom,
% 9.20/5.15      ! [A: complex,C: complex,B: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
% 9.20/5.15        = ( ( C = zero_zero_complex )
% 9.20/5.15          | ( dvd_dvd_complex @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3452_dvd__mult__cancel__right,axiom,
% 9.20/5.15      ! [A: real,C: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
% 9.20/5.15        = ( ( C = zero_zero_real )
% 9.20/5.15          | ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3453_dvd__mult__cancel__right,axiom,
% 9.20/5.15      ! [A: rat,C: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
% 9.20/5.15        = ( ( C = zero_zero_rat )
% 9.20/5.15          | ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3454_dvd__mult__cancel__right,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
% 9.20/5.15        = ( ( C = zero_zero_int )
% 9.20/5.15          | ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3455_dvd__times__left__cancel__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3456_dvd__times__left__cancel__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( A != zero_zero_nat )
% 9.20/5.15       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
% 9.20/5.15          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3457_dvd__times__left__cancel__iff,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( A != zero_zero_int )
% 9.20/5.15       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
% 9.20/5.15          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3458_dvd__times__right__cancel__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3459_dvd__times__right__cancel__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( A != zero_zero_nat )
% 9.20/5.15       => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
% 9.20/5.15          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3460_dvd__times__right__cancel__iff,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( A != zero_zero_int )
% 9.20/5.15       => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
% 9.20/5.15          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3461_unit__prod,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3462_unit__prod,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.15         => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3463_unit__prod,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.15         => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3464_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3465_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: complex,B: complex,C: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3466_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3467_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3468_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3469_dvd__add__times__triv__right__iff,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
% 9.20/5.15        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3470_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3471_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: complex,C: complex,B: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_complex @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3472_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: real,C: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3473_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: rat,C: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_rat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3474_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3475_dvd__add__times__triv__left__iff,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
% 9.20/5.15        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3476_mod__mult__self1__is__0,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3477_mod__mult__self1__is__0,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3478_mod__mult__self1__is__0,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3479_mod__mult__self2__is__0,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3480_mod__mult__self2__is__0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3481_mod__mult__self2__is__0,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3482_bits__mod__by__1,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3483_bits__mod__by__1,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3484_bits__mod__by__1,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3485_mod__by__1,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ A @ one_one_nat )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3486_mod__by__1,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ A @ one_one_int )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3487_mod__by__1,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3488_dvd__div__mult__self,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3489_dvd__div__mult__self,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3490_dvd__div__mult__self,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3491_dvd__mult__div__cancel,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3492_dvd__mult__div__cancel,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3493_dvd__mult__div__cancel,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3494_unit__div,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.15         => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3495_unit__div,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.15         => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3496_unit__div,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3497_unit__div__1__unit,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( dvd_dvd_nat @ ( divide_divide_nat @ one_one_nat @ A ) @ one_one_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3498_unit__div__1__unit,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( dvd_dvd_int @ ( divide_divide_int @ one_one_int @ A ) @ one_one_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3499_unit__div__1__unit,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3500_unit__div__1__div__1,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( ( divide_divide_nat @ one_one_nat @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3501_unit__div__1__div__1,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( ( divide_divide_int @ one_one_int @ ( divide_divide_int @ one_one_int @ A ) )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3502_unit__div__1__div__1,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3503_div__add,axiom,
% 9.20/5.15      ! [C: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.15         => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.15            = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3504_div__add,axiom,
% 9.20/5.15      ! [C: int,A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.15         => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.15            = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3505_div__add,axiom,
% 9.20/5.15      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.15         => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.20/5.15            = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3506_bits__mod__div__trivial,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3507_bits__mod__div__trivial,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3508_bits__mod__div__trivial,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3509_mod__div__trivial,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3510_mod__div__trivial,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
% 9.20/5.15        = zero_zero_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3511_mod__div__trivial,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
% 9.20/5.15        = zero_z3403309356797280102nteger ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3512_mod__mult__self4,axiom,
% 9.20/5.15      ! [B: nat,C: nat,A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3513_mod__mult__self4,axiom,
% 9.20/5.15      ! [B: int,C: int,A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3514_mod__mult__self4,axiom,
% 9.20/5.15      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3515_mod__mult__self3,axiom,
% 9.20/5.15      ! [C: nat,B: nat,A: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3516_mod__mult__self3,axiom,
% 9.20/5.15      ! [C: int,B: int,A: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3517_mod__mult__self3,axiom,
% 9.20/5.15      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3518_mod__mult__self2,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3519_mod__mult__self2,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3520_mod__mult__self2,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3521_mod__mult__self1,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
% 9.20/5.15        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3522_mod__mult__self1,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
% 9.20/5.15        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3523_mod__mult__self1,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
% 9.20/5.15        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3524_dvd__imp__mod__0,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( modulo_modulo_nat @ B @ A )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3525_dvd__imp__mod__0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( modulo_modulo_int @ B @ A )
% 9.20/5.15          = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3526_dvd__imp__mod__0,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( modulo364778990260209775nteger @ B @ A )
% 9.20/5.15          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3527_mod__by__Suc__0,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3528_set__decode__zero,axiom,
% 9.20/5.15      ( ( nat_set_decode @ zero_zero_nat )
% 9.20/5.15      = bot_bot_set_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3529_set__encode__inverse,axiom,
% 9.20/5.15      ! [A2: set_nat] :
% 9.20/5.15        ( ( finite_finite_nat @ A2 )
% 9.20/5.15       => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
% 9.20/5.15          = A2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3530_unit__div__mult__self,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3531_unit__div__mult__self,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3532_unit__div__mult__self,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
% 9.20/5.15          = B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3533_unit__mult__div__div,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.15       => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
% 9.20/5.15          = ( divide_divide_nat @ B @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3534_unit__mult__div__div,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.15       => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
% 9.20/5.15          = ( divide_divide_int @ B @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3535_unit__mult__div__div,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.15       => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
% 9.20/5.15          = ( divide6298287555418463151nteger @ B @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3536_even__Suc,axiom,
% 9.20/5.15      ! [N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N2 ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3537_even__Suc__Suc__iff,axiom,
% 9.20/5.15      ! [N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ N2 ) ) )
% 9.20/5.15        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3538_pow__divides__pow__iff,axiom,
% 9.20/5.15      ! [N2: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15       => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
% 9.20/5.15          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3539_pow__divides__pow__iff,axiom,
% 9.20/5.15      ! [N2: nat,A: int,B: int] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15       => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
% 9.20/5.15          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3540_Suc__mod__mult__self4,axiom,
% 9.20/5.15      ! [N2: nat,K: nat,M: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3541_Suc__mod__mult__self3,axiom,
% 9.20/5.15      ! [K: nat,N2: nat,M: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3542_Suc__mod__mult__self2,axiom,
% 9.20/5.15      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3543_Suc__mod__mult__self1,axiom,
% 9.20/5.15      ! [M: nat,K: nat,N2: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3544_even__add,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.15        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3545_even__add,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.15        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3546_even__add,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
% 9.20/5.15        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3547_odd__add,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
% 9.20/5.15        = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.15         != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3548_odd__add,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
% 9.20/5.15        = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.15         != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3549_odd__add,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
% 9.20/5.15        = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.15         != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3550_odd__Suc__div__two,axiom,
% 9.20/5.15      ! [N2: nat] :
% 9.20/5.15        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.15       => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3551_even__Suc__div__two,axiom,
% 9.20/5.15      ! [N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.15       => ( ( divide_divide_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3552_mod2__Suc__Suc,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3553_Suc__times__numeral__mod__eq,axiom,
% 9.20/5.15      ! [K: num,N2: nat] :
% 9.20/5.15        ( ( ( numeral_numeral_nat @ K )
% 9.20/5.15         != one_one_nat )
% 9.20/5.15       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ K ) @ N2 ) ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.15          = one_one_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3554_set__decode__0,axiom,
% 9.20/5.15      ! [X3: nat] :
% 9.20/5.15        ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3555_zero__le__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: real,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3556_zero__le__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: rat,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3557_zero__le__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: int,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3558_power__less__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: real,W2: num] :
% 9.20/5.15        ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3559_power__less__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: rat,W2: num] :
% 9.20/5.15        ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3560_power__less__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: int,W2: num] :
% 9.20/5.15        ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3561_power__less__zero__eq,axiom,
% 9.20/5.15      ! [A: real,N2: nat] :
% 9.20/5.15        ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.15          & ( ord_less_real @ A @ zero_zero_real ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3562_power__less__zero__eq,axiom,
% 9.20/5.15      ! [A: rat,N2: nat] :
% 9.20/5.15        ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.15          & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3563_power__less__zero__eq,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.15          & ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3564_even__plus__one__iff,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3565_even__plus__one__iff,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ one_one_nat ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3566_even__plus__one__iff,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ one_one_int ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3567_not__mod__2__eq__1__eq__0,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15         != one_one_nat )
% 9.20/5.15        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3568_not__mod__2__eq__1__eq__0,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15         != one_one_int )
% 9.20/5.15        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3569_not__mod__2__eq__1__eq__0,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15         != one_one_Code_integer )
% 9.20/5.15        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3570_not__mod__2__eq__0__eq__1,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15         != zero_zero_nat )
% 9.20/5.15        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = one_one_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3571_not__mod__2__eq__0__eq__1,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15         != zero_zero_int )
% 9.20/5.15        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = one_one_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3572_not__mod__2__eq__0__eq__1,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15         != zero_z3403309356797280102nteger )
% 9.20/5.15        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = one_one_Code_integer ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3573_not__mod2__eq__Suc__0__eq__0,axiom,
% 9.20/5.15      ! [N2: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15         != ( suc @ zero_zero_nat ) )
% 9.20/5.15        = ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3574_add__self__mod__2,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15        = zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3575_zero__less__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: real,W2: num] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( ( numeral_numeral_nat @ W2 )
% 9.20/5.15            = zero_zero_nat )
% 9.20/5.15          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( A != zero_zero_real ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3576_zero__less__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: rat,W2: num] :
% 9.20/5.15        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( ( numeral_numeral_nat @ W2 )
% 9.20/5.15            = zero_zero_nat )
% 9.20/5.15          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( A != zero_zero_rat ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3577_zero__less__power__eq__numeral,axiom,
% 9.20/5.15      ! [A: int,W2: num] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) )
% 9.20/5.15        = ( ( ( numeral_numeral_nat @ W2 )
% 9.20/5.15            = zero_zero_nat )
% 9.20/5.15          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( A != zero_zero_int ) )
% 9.20/5.15          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3578_even__succ__div__two,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3579_even__succ__div__two,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3580_even__succ__div__two,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3581_odd__succ__div__two,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( plus_plus_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3582_odd__succ__div__two,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_int @ ( plus_plus_int @ A @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( plus_plus_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3583_odd__succ__div__two,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3584_even__succ__div__2,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3585_even__succ__div__2,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3586_even__succ__div__2,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3587_even__power,axiom,
% 9.20/5.15      ! [A: code_integer,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 9.20/5.15        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3588_even__power,axiom,
% 9.20/5.15      ! [A: nat,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( power_power_nat @ A @ N2 ) )
% 9.20/5.15        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3589_even__power,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( power_power_int @ A @ N2 ) )
% 9.20/5.15        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3590_mod2__gr__0,axiom,
% 9.20/5.15      ! [M: nat] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.15        = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = one_one_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3591_odd__two__times__div__two__succ,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_nat )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3592_odd__two__times__div__two__succ,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ one_one_int )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3593_odd__two__times__div__two__succ,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
% 9.20/5.15          = A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3594_power__le__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: real,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_real )
% 9.20/5.15        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.20/5.15            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3595_power__le__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: rat,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
% 9.20/5.15        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.20/5.15            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3596_power__le__zero__eq__numeral,axiom,
% 9.20/5.15      ! [A: int,W2: num] :
% 9.20/5.15        ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_int )
% 9.20/5.15        = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.20/5.15            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
% 9.20/5.15              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3597_even__succ__mod__exp,axiom,
% 9.20/5.15      ! [A: nat,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( plus_plus_nat @ one_one_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3598_even__succ__mod__exp,axiom,
% 9.20/5.15      ! [A: int,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( plus_plus_int @ one_one_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3599_even__succ__mod__exp,axiom,
% 9.20/5.15      ! [A: code_integer,N2: nat] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.15         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.15            = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3600_real__arch__pow,axiom,
% 9.20/5.15      ! [X3: real,Y3: real] :
% 9.20/5.15        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.15       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3601_real__arch__pow__inv,axiom,
% 9.20/5.15      ! [Y3: real,X3: real] :
% 9.20/5.15        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.15       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.15         => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y3 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3602_int__distrib_I2_J,axiom,
% 9.20/5.15      ! [W2: int,Z1: int,Z22: int] :
% 9.20/5.15        ( ( times_times_int @ W2 @ ( plus_plus_int @ Z1 @ Z22 ) )
% 9.20/5.15        = ( plus_plus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3603_int__distrib_I1_J,axiom,
% 9.20/5.15      ! [Z1: int,Z22: int,W2: int] :
% 9.20/5.15        ( ( times_times_int @ ( plus_plus_int @ Z1 @ Z22 ) @ W2 )
% 9.20/5.15        = ( plus_plus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3604_zdvd__period,axiom,
% 9.20/5.15      ! [A: int,D: int,X3: int,T: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ D )
% 9.20/5.15       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
% 9.20/5.15          = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3605_zdvd__reduce,axiom,
% 9.20/5.15      ! [K: int,N2: int,M: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
% 9.20/5.15        = ( dvd_dvd_int @ K @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3606_less__eq__real__def,axiom,
% 9.20/5.15      ( ord_less_eq_real
% 9.20/5.15      = ( ^ [X: real,Y: real] :
% 9.20/5.15            ( ( ord_less_real @ X @ Y )
% 9.20/5.15            | ( X = Y ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3607_mod__eq__0__iff__dvd,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ B )
% 9.20/5.15          = zero_zero_nat )
% 9.20/5.15        = ( dvd_dvd_nat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3608_mod__eq__0__iff__dvd,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ B )
% 9.20/5.15          = zero_zero_int )
% 9.20/5.15        = ( dvd_dvd_int @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3609_mod__eq__0__iff__dvd,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.20/5.15          = zero_z3403309356797280102nteger )
% 9.20/5.15        = ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3610_dvd__eq__mod__eq__0,axiom,
% 9.20/5.15      ( dvd_dvd_nat
% 9.20/5.15      = ( ^ [A3: nat,B3: nat] :
% 9.20/5.15            ( ( modulo_modulo_nat @ B3 @ A3 )
% 9.20/5.15            = zero_zero_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3611_dvd__eq__mod__eq__0,axiom,
% 9.20/5.15      ( dvd_dvd_int
% 9.20/5.15      = ( ^ [A3: int,B3: int] :
% 9.20/5.15            ( ( modulo_modulo_int @ B3 @ A3 )
% 9.20/5.15            = zero_zero_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3612_dvd__eq__mod__eq__0,axiom,
% 9.20/5.15      ( dvd_dvd_Code_integer
% 9.20/5.15      = ( ^ [A3: code_integer,B3: code_integer] :
% 9.20/5.15            ( ( modulo364778990260209775nteger @ B3 @ A3 )
% 9.20/5.15            = zero_z3403309356797280102nteger ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3613_mod__0__imp__dvd,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ B )
% 9.20/5.15          = zero_zero_nat )
% 9.20/5.15       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3614_mod__0__imp__dvd,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ B )
% 9.20/5.15          = zero_zero_int )
% 9.20/5.15       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3615_mod__0__imp__dvd,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.20/5.15          = zero_z3403309356797280102nteger )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3616_dvd__refl,axiom,
% 9.20/5.15      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3617_dvd__refl,axiom,
% 9.20/5.15      ! [A: int] : ( dvd_dvd_int @ A @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3618_dvd__refl,axiom,
% 9.20/5.15      ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3619_dvd__trans,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.15         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3620_dvd__trans,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ B @ C )
% 9.20/5.15         => ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3621_dvd__trans,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ B @ C )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3622_dvd__mod__iff,axiom,
% 9.20/5.15      ! [C: nat,B: nat,A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.15          = ( dvd_dvd_nat @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3623_dvd__mod__iff,axiom,
% 9.20/5.15      ! [C: int,B: int,A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ C @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.15          = ( dvd_dvd_int @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3624_dvd__mod__iff,axiom,
% 9.20/5.15      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3625_dvd__mod__imp__dvd,axiom,
% 9.20/5.15      ! [C: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.15         => ( dvd_dvd_nat @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3626_dvd__mod__imp__dvd,axiom,
% 9.20/5.15      ! [C: int,A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.15         => ( dvd_dvd_int @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3627_dvd__mod__imp__dvd,axiom,
% 9.20/5.15      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3628_int__ge__induct,axiom,
% 9.20/5.15      ! [K: int,I: int,P: int > $o] :
% 9.20/5.15        ( ( ord_less_eq_int @ K @ I )
% 9.20/5.15       => ( ( P @ K )
% 9.20/5.15         => ( ! [I3: int] :
% 9.20/5.15                ( ( ord_less_eq_int @ K @ I3 )
% 9.20/5.15               => ( ( P @ I3 )
% 9.20/5.15                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.15           => ( P @ I ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3629_unit__imp__mod__eq__0,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.15       => ( ( modulo_modulo_nat @ A @ B )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3630_unit__imp__mod__eq__0,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.15       => ( ( modulo_modulo_int @ A @ B )
% 9.20/5.15          = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3631_unit__imp__mod__eq__0,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.15       => ( ( modulo364778990260209775nteger @ A @ B )
% 9.20/5.15          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3632_mod__greater__zero__iff__not__dvd,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
% 9.20/5.15        = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3633_mod__add__right__eq,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3634_mod__add__right__eq,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3635_mod__add__right__eq,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3636_mod__add__left__eq,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
% 9.20/5.15        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3637_mod__add__left__eq,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
% 9.20/5.15        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3638_mod__add__left__eq,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
% 9.20/5.15        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3639_mod__add__cong,axiom,
% 9.20/5.15      ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ C )
% 9.20/5.15          = ( modulo_modulo_nat @ A6 @ C ) )
% 9.20/5.15       => ( ( ( modulo_modulo_nat @ B @ C )
% 9.20/5.15            = ( modulo_modulo_nat @ B6 @ C ) )
% 9.20/5.15         => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.15            = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3640_mod__add__cong,axiom,
% 9.20/5.15      ! [A: int,C: int,A6: int,B: int,B6: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ C )
% 9.20/5.15          = ( modulo_modulo_int @ A6 @ C ) )
% 9.20/5.15       => ( ( ( modulo_modulo_int @ B @ C )
% 9.20/5.15            = ( modulo_modulo_int @ B6 @ C ) )
% 9.20/5.15         => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.15            = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3641_mod__add__cong,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.20/5.15          = ( modulo364778990260209775nteger @ A6 @ C ) )
% 9.20/5.15       => ( ( ( modulo364778990260209775nteger @ B @ C )
% 9.20/5.15            = ( modulo364778990260209775nteger @ B6 @ C ) )
% 9.20/5.15         => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.20/5.15            = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3642_mod__add__eq,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3643_mod__add__eq,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3644_mod__add__eq,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
% 9.20/5.15        = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3645_dvd__0__left,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15       => ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3646_dvd__0__left,axiom,
% 9.20/5.15      ! [A: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ zero_zero_complex @ A )
% 9.20/5.15       => ( A = zero_zero_complex ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3647_dvd__0__left,axiom,
% 9.20/5.15      ! [A: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ zero_zero_real @ A )
% 9.20/5.15       => ( A = zero_zero_real ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3648_dvd__0__left,axiom,
% 9.20/5.15      ! [A: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ zero_zero_rat @ A )
% 9.20/5.15       => ( A = zero_zero_rat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3649_dvd__0__left,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.20/5.15       => ( A = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3650_dvd__0__left,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ zero_zero_int @ A )
% 9.20/5.15       => ( A = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3651_dvd__field__iff,axiom,
% 9.20/5.15      ( dvd_dvd_complex
% 9.20/5.15      = ( ^ [A3: complex,B3: complex] :
% 9.20/5.15            ( ( A3 = zero_zero_complex )
% 9.20/5.15           => ( B3 = zero_zero_complex ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3652_dvd__field__iff,axiom,
% 9.20/5.15      ( dvd_dvd_real
% 9.20/5.15      = ( ^ [A3: real,B3: real] :
% 9.20/5.15            ( ( A3 = zero_zero_real )
% 9.20/5.15           => ( B3 = zero_zero_real ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3653_dvd__field__iff,axiom,
% 9.20/5.15      ( dvd_dvd_rat
% 9.20/5.15      = ( ^ [A3: rat,B3: rat] :
% 9.20/5.15            ( ( A3 = zero_zero_rat )
% 9.20/5.15           => ( B3 = zero_zero_rat ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3654_mod__Suc__Suc__eq,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3655_mod__Suc__eq,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] :
% 9.20/5.15        ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
% 9.20/5.15        = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3656_dvd__productE,axiom,
% 9.20/5.15      ! [P4: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
% 9.20/5.15       => ~ ! [X4: nat,Y5: nat] :
% 9.20/5.15              ( ( P4
% 9.20/5.15                = ( times_times_nat @ X4 @ Y5 ) )
% 9.20/5.15             => ( ( dvd_dvd_nat @ X4 @ A )
% 9.20/5.15               => ~ ( dvd_dvd_nat @ Y5 @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3657_dvd__productE,axiom,
% 9.20/5.15      ! [P4: int,A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
% 9.20/5.15       => ~ ! [X4: int,Y5: int] :
% 9.20/5.15              ( ( P4
% 9.20/5.15                = ( times_times_int @ X4 @ Y5 ) )
% 9.20/5.15             => ( ( dvd_dvd_int @ X4 @ A )
% 9.20/5.15               => ~ ( dvd_dvd_int @ Y5 @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3658_division__decomp,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.15       => ? [B8: nat,C5: nat] :
% 9.20/5.15            ( ( A
% 9.20/5.15              = ( times_times_nat @ B8 @ C5 ) )
% 9.20/5.15            & ( dvd_dvd_nat @ B8 @ B )
% 9.20/5.15            & ( dvd_dvd_nat @ C5 @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3659_division__decomp,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.15       => ? [B8: int,C5: int] :
% 9.20/5.15            ( ( A
% 9.20/5.15              = ( times_times_int @ B8 @ C5 ) )
% 9.20/5.15            & ( dvd_dvd_int @ B8 @ B )
% 9.20/5.15            & ( dvd_dvd_int @ C5 @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3660_dvdE,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.15       => ~ ! [K2: code_integer] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3661_dvdE,axiom,
% 9.20/5.15      ! [B: complex,A: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ B @ A )
% 9.20/5.15       => ~ ! [K2: complex] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_times_complex @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3662_dvdE,axiom,
% 9.20/5.15      ! [B: real,A: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ B @ A )
% 9.20/5.15       => ~ ! [K2: real] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_times_real @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3663_dvdE,axiom,
% 9.20/5.15      ! [B: rat,A: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ B @ A )
% 9.20/5.15       => ~ ! [K2: rat] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_times_rat @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3664_dvdE,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.15       => ~ ! [K2: nat] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_times_nat @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3665_dvdE,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ B @ A )
% 9.20/5.15       => ~ ! [K2: int] :
% 9.20/5.15              ( A
% 9.20/5.15             != ( times_times_int @ B @ K2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3666_dvdI,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,K: code_integer] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_3573771949741848930nteger @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3667_dvdI,axiom,
% 9.20/5.15      ! [A: complex,B: complex,K: complex] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_times_complex @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_complex @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3668_dvdI,axiom,
% 9.20/5.15      ! [A: real,B: real,K: real] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_times_real @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_real @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3669_dvdI,axiom,
% 9.20/5.15      ! [A: rat,B: rat,K: rat] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_times_rat @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_rat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3670_dvdI,axiom,
% 9.20/5.15      ! [A: nat,B: nat,K: nat] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_times_nat @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3671_dvdI,axiom,
% 9.20/5.15      ! [A: int,B: int,K: int] :
% 9.20/5.15        ( ( A
% 9.20/5.15          = ( times_times_int @ B @ K ) )
% 9.20/5.15       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3672_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_Code_integer
% 9.20/5.15      = ( ^ [B3: code_integer,A3: code_integer] :
% 9.20/5.15          ? [K3: code_integer] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3673_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_complex
% 9.20/5.15      = ( ^ [B3: complex,A3: complex] :
% 9.20/5.15          ? [K3: complex] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_times_complex @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3674_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_real
% 9.20/5.15      = ( ^ [B3: real,A3: real] :
% 9.20/5.15          ? [K3: real] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_times_real @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3675_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_rat
% 9.20/5.15      = ( ^ [B3: rat,A3: rat] :
% 9.20/5.15          ? [K3: rat] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_times_rat @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3676_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_nat
% 9.20/5.15      = ( ^ [B3: nat,A3: nat] :
% 9.20/5.15          ? [K3: nat] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_times_nat @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3677_dvd__def,axiom,
% 9.20/5.15      ( dvd_dvd_int
% 9.20/5.15      = ( ^ [B3: int,A3: int] :
% 9.20/5.15          ? [K3: int] :
% 9.20/5.15            ( A3
% 9.20/5.15            = ( times_times_int @ B3 @ K3 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3678_dvd__mult,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ C )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3679_dvd__mult,axiom,
% 9.20/5.15      ! [A: complex,C: complex,B: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ A @ C )
% 9.20/5.15       => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3680_dvd__mult,axiom,
% 9.20/5.15      ! [A: real,C: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ C )
% 9.20/5.15       => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3681_dvd__mult,axiom,
% 9.20/5.15      ! [A: rat,C: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ C )
% 9.20/5.15       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3682_dvd__mult,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.15       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3683_dvd__mult,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ C )
% 9.20/5.15       => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3684_dvd__mult2,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3685_dvd__mult2,axiom,
% 9.20/5.15      ! [A: complex,B: complex,C: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ A @ B )
% 9.20/5.15       => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3686_dvd__mult2,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ B )
% 9.20/5.15       => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3687_dvd__mult2,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ B )
% 9.20/5.15       => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3688_dvd__mult2,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3689_dvd__mult2,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3690_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3691_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: complex,B: complex,C: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_complex @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3692_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_real @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3693_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_rat @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3694_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_nat @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3695_dvd__mult__left,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_int @ A @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3696_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3697_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3698_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3699_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3700_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3701_dvd__triv__left,axiom,
% 9.20/5.15      ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3702_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3703_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: complex,B: complex,C: complex,D: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_complex @ C @ D )
% 9.20/5.15         => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3704_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_real @ C @ D )
% 9.20/5.15         => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3705_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_rat @ C @ D )
% 9.20/5.15         => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3706_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ D )
% 9.20/5.15         => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3707_mult__dvd__mono,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ D )
% 9.20/5.15         => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3708_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3709_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: complex,B: complex,C: complex] :
% 9.20/5.15        ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_complex @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3710_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_real @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3711_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_rat @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3712_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_nat @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3713_dvd__mult__right,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.15       => ( dvd_dvd_int @ B @ C ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3714_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3715_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3716_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3717_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3718_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3719_dvd__triv__right,axiom,
% 9.20/5.15      ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3720_one__dvd,axiom,
% 9.20/5.15      ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3721_one__dvd,axiom,
% 9.20/5.15      ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3722_one__dvd,axiom,
% 9.20/5.15      ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3723_one__dvd,axiom,
% 9.20/5.15      ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3724_one__dvd,axiom,
% 9.20/5.15      ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3725_one__dvd,axiom,
% 9.20/5.15      ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3726_unit__imp__dvd,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.15       => ( dvd_dvd_Code_integer @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3727_unit__imp__dvd,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.15       => ( dvd_dvd_nat @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3728_unit__imp__dvd,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.15       => ( dvd_dvd_int @ B @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3729_dvd__unit__imp__unit,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3730_dvd__unit__imp__unit,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.15         => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3731_dvd__unit__imp__unit,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.15         => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3732_dvd__add__right__iff,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3733_dvd__add__right__iff,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_real @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3734_dvd__add__right__iff,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_rat @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3735_dvd__add__right__iff,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3736_dvd__add__right__iff,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3737_dvd__add__left__iff,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ C )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3738_dvd__add__left__iff,axiom,
% 9.20/5.15      ! [A: real,C: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ C )
% 9.20/5.15       => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_real @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3739_dvd__add__left__iff,axiom,
% 9.20/5.15      ! [A: rat,C: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ C )
% 9.20/5.15       => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_rat @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3740_dvd__add__left__iff,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.15       => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_nat @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3741_dvd__add__left__iff,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ C )
% 9.20/5.15       => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.20/5.15          = ( dvd_dvd_int @ A @ B ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3742_dvd__add,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ A @ C )
% 9.20/5.15         => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3743_dvd__add,axiom,
% 9.20/5.15      ! [A: real,B: real,C: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_real @ A @ C )
% 9.20/5.15         => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3744_dvd__add,axiom,
% 9.20/5.15      ! [A: rat,B: rat,C: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_rat @ A @ C )
% 9.20/5.15         => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3745_dvd__add,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.15         => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3746_dvd__add,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ A @ C )
% 9.20/5.15         => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3747_dvd__div__eq__iff,axiom,
% 9.20/5.15      ! [C: nat,A: nat,B: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.15         => ( ( ( divide_divide_nat @ A @ C )
% 9.20/5.15              = ( divide_divide_nat @ B @ C ) )
% 9.20/5.15            = ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3748_dvd__div__eq__iff,axiom,
% 9.20/5.15      ! [C: int,A: int,B: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.15         => ( ( ( divide_divide_int @ A @ C )
% 9.20/5.15              = ( divide_divide_int @ B @ C ) )
% 9.20/5.15            = ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3749_dvd__div__eq__iff,axiom,
% 9.20/5.15      ! [C: real,A: real,B: real] :
% 9.20/5.15        ( ( dvd_dvd_real @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_real @ C @ B )
% 9.20/5.15         => ( ( ( divide_divide_real @ A @ C )
% 9.20/5.15              = ( divide_divide_real @ B @ C ) )
% 9.20/5.15            = ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3750_dvd__div__eq__iff,axiom,
% 9.20/5.15      ! [C: rat,A: rat,B: rat] :
% 9.20/5.15        ( ( dvd_dvd_rat @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_rat @ C @ B )
% 9.20/5.15         => ( ( ( divide_divide_rat @ A @ C )
% 9.20/5.15              = ( divide_divide_rat @ B @ C ) )
% 9.20/5.15            = ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3751_dvd__div__eq__iff,axiom,
% 9.20/5.15      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.15         => ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.20/5.15              = ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.15            = ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3752_dvd__div__eq__cancel,axiom,
% 9.20/5.15      ! [A: nat,C: nat,B: nat] :
% 9.20/5.15        ( ( ( divide_divide_nat @ A @ C )
% 9.20/5.15          = ( divide_divide_nat @ B @ C ) )
% 9.20/5.15       => ( ( dvd_dvd_nat @ C @ A )
% 9.20/5.15         => ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.15           => ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3753_dvd__div__eq__cancel,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( ( divide_divide_int @ A @ C )
% 9.20/5.15          = ( divide_divide_int @ B @ C ) )
% 9.20/5.15       => ( ( dvd_dvd_int @ C @ A )
% 9.20/5.15         => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.15           => ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3754_dvd__div__eq__cancel,axiom,
% 9.20/5.15      ! [A: real,C: real,B: real] :
% 9.20/5.15        ( ( ( divide_divide_real @ A @ C )
% 9.20/5.15          = ( divide_divide_real @ B @ C ) )
% 9.20/5.15       => ( ( dvd_dvd_real @ C @ A )
% 9.20/5.15         => ( ( dvd_dvd_real @ C @ B )
% 9.20/5.15           => ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3755_dvd__div__eq__cancel,axiom,
% 9.20/5.15      ! [A: rat,C: rat,B: rat] :
% 9.20/5.15        ( ( ( divide_divide_rat @ A @ C )
% 9.20/5.15          = ( divide_divide_rat @ B @ C ) )
% 9.20/5.15       => ( ( dvd_dvd_rat @ C @ A )
% 9.20/5.15         => ( ( dvd_dvd_rat @ C @ B )
% 9.20/5.15           => ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3756_dvd__div__eq__cancel,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ( divide6298287555418463151nteger @ A @ C )
% 9.20/5.15          = ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ C @ A )
% 9.20/5.15         => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.15           => ( A = B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3757_div__div__div__same,axiom,
% 9.20/5.15      ! [D: nat,B: nat,A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ D @ B )
% 9.20/5.15       => ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.15         => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
% 9.20/5.15            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3758_div__div__div__same,axiom,
% 9.20/5.15      ! [D: int,B: int,A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ D @ B )
% 9.20/5.15       => ( ( dvd_dvd_int @ B @ A )
% 9.20/5.15         => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
% 9.20/5.15            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3759_div__div__div__same,axiom,
% 9.20/5.15      ! [D: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ D @ B )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.15         => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
% 9.20/5.15            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3760_mod__less__eq__dividend,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3761_gcd__nat_Oextremum__uniqueI,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.20/5.15       => ( A = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3762_gcd__nat_Onot__eq__extremum,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( A != zero_zero_nat )
% 9.20/5.15        = ( ( dvd_dvd_nat @ A @ zero_zero_nat )
% 9.20/5.15          & ( A != zero_zero_nat ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3763_gcd__nat_Oextremum__unique,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.20/5.15        = ( A = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3764_gcd__nat_Oextremum__strict,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
% 9.20/5.15          & ( zero_zero_nat != A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3765_gcd__nat_Oextremum,axiom,
% 9.20/5.15      ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3766_zdvd__antisym__nonneg,axiom,
% 9.20/5.15      ! [M: int,N2: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ M )
% 9.20/5.15       => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 9.20/5.15         => ( ( dvd_dvd_int @ M @ N2 )
% 9.20/5.15           => ( ( dvd_dvd_int @ N2 @ M )
% 9.20/5.15             => ( M = N2 ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3767_zdvd__mult__cancel,axiom,
% 9.20/5.15      ! [K: int,M: int,N2: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
% 9.20/5.15       => ( ( K != zero_zero_int )
% 9.20/5.15         => ( dvd_dvd_int @ M @ N2 ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3768_even__iff__mod__2__eq__zero,axiom,
% 9.20/5.15      ! [A: nat] :
% 9.20/5.15        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.15        = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3769_even__iff__mod__2__eq__zero,axiom,
% 9.20/5.15      ! [A: int] :
% 9.20/5.15        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.15        = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3770_even__iff__mod__2__eq__zero,axiom,
% 9.20/5.15      ! [A: code_integer] :
% 9.20/5.15        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.15        = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.15          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3771_subset__decode__imp__le,axiom,
% 9.20/5.15      ! [M: nat,N2: nat] :
% 9.20/5.15        ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
% 9.20/5.15       => ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3772_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.15       => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3773_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.15       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3774_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.15       => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3775_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.20/5.15      ! [B: nat,A: nat] :
% 9.20/5.15        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.15       => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3776_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.20/5.15      ! [B: int,A: int] :
% 9.20/5.15        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.15       => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3777_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
% 9.20/5.15      ! [B: code_integer,A: code_integer] :
% 9.20/5.15        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.15       => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3778_mod__eq__self__iff__div__eq__0,axiom,
% 9.20/5.15      ! [A: nat,B: nat] :
% 9.20/5.15        ( ( ( modulo_modulo_nat @ A @ B )
% 9.20/5.15          = A )
% 9.20/5.15        = ( ( divide_divide_nat @ A @ B )
% 9.20/5.15          = zero_zero_nat ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3779_mod__eq__self__iff__div__eq__0,axiom,
% 9.20/5.15      ! [A: int,B: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ B )
% 9.20/5.15          = A )
% 9.20/5.15        = ( ( divide_divide_int @ A @ B )
% 9.20/5.15          = zero_zero_int ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3780_mod__eq__self__iff__div__eq__0,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ B )
% 9.20/5.15          = A )
% 9.20/5.15        = ( ( divide6298287555418463151nteger @ A @ B )
% 9.20/5.15          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3781_mod__eqE,axiom,
% 9.20/5.15      ! [A: int,C: int,B: int] :
% 9.20/5.15        ( ( ( modulo_modulo_int @ A @ C )
% 9.20/5.15          = ( modulo_modulo_int @ B @ C ) )
% 9.20/5.15       => ~ ! [D4: int] :
% 9.20/5.15              ( B
% 9.20/5.15             != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3782_mod__eqE,axiom,
% 9.20/5.15      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.15        ( ( ( modulo364778990260209775nteger @ A @ C )
% 9.20/5.15          = ( modulo364778990260209775nteger @ B @ C ) )
% 9.20/5.15       => ~ ! [D4: code_integer] :
% 9.20/5.15              ( B
% 9.20/5.15             != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3783_div__add1__eq,axiom,
% 9.20/5.15      ! [A: nat,B: nat,C: nat] :
% 9.20/5.15        ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.15        = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3784_div__add1__eq,axiom,
% 9.20/5.15      ! [A: int,B: int,C: int] :
% 9.20/5.15        ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.15        = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3785_div__add1__eq,axiom,
% 9.20/5.15      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.15        ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.20/5.15        = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3786_not__is__unit__0,axiom,
% 9.20/5.15      ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3787_not__is__unit__0,axiom,
% 9.20/5.15      ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3788_not__is__unit__0,axiom,
% 9.20/5.15      ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3789_minf_I10_J,axiom,
% 9.20/5.15      ! [D: code_integer,S: code_integer] :
% 9.20/5.15      ? [Z3: code_integer] :
% 9.20/5.15      ! [X2: code_integer] :
% 9.20/5.15        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3790_minf_I10_J,axiom,
% 9.20/5.15      ! [D: real,S: real] :
% 9.20/5.15      ? [Z3: real] :
% 9.20/5.15      ! [X2: real] :
% 9.20/5.15        ( ( ord_less_real @ X2 @ Z3 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3791_minf_I10_J,axiom,
% 9.20/5.15      ! [D: rat,S: rat] :
% 9.20/5.15      ? [Z3: rat] :
% 9.20/5.15      ! [X2: rat] :
% 9.20/5.15        ( ( ord_less_rat @ X2 @ Z3 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3792_minf_I10_J,axiom,
% 9.20/5.15      ! [D: nat,S: nat] :
% 9.20/5.15      ? [Z3: nat] :
% 9.20/5.15      ! [X2: nat] :
% 9.20/5.15        ( ( ord_less_nat @ X2 @ Z3 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3793_minf_I10_J,axiom,
% 9.20/5.15      ! [D: int,S: int] :
% 9.20/5.15      ? [Z3: int] :
% 9.20/5.15      ! [X2: int] :
% 9.20/5.15        ( ( ord_less_int @ X2 @ Z3 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3794_minf_I9_J,axiom,
% 9.20/5.15      ! [D: code_integer,S: code_integer] :
% 9.20/5.15      ? [Z3: code_integer] :
% 9.20/5.15      ! [X2: code_integer] :
% 9.20/5.15        ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
% 9.20/5.15       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.20/5.15          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3795_minf_I9_J,axiom,
% 9.20/5.15      ! [D: real,S: real] :
% 9.20/5.15      ? [Z3: real] :
% 9.20/5.15      ! [X2: real] :
% 9.20/5.15        ( ( ord_less_real @ X2 @ Z3 )
% 9.20/5.15       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.20/5.15          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3796_minf_I9_J,axiom,
% 9.20/5.15      ! [D: rat,S: rat] :
% 9.20/5.15      ? [Z3: rat] :
% 9.20/5.15      ! [X2: rat] :
% 9.20/5.15        ( ( ord_less_rat @ X2 @ Z3 )
% 9.20/5.15       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.20/5.15          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3797_minf_I9_J,axiom,
% 9.20/5.15      ! [D: nat,S: nat] :
% 9.20/5.15      ? [Z3: nat] :
% 9.20/5.15      ! [X2: nat] :
% 9.20/5.15        ( ( ord_less_nat @ X2 @ Z3 )
% 9.20/5.15       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.20/5.15          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3798_minf_I9_J,axiom,
% 9.20/5.15      ! [D: int,S: int] :
% 9.20/5.15      ? [Z3: int] :
% 9.20/5.15      ! [X2: int] :
% 9.20/5.15        ( ( ord_less_int @ X2 @ Z3 )
% 9.20/5.15       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.20/5.15          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3799_pinf_I10_J,axiom,
% 9.20/5.15      ! [D: code_integer,S: code_integer] :
% 9.20/5.15      ? [Z3: code_integer] :
% 9.20/5.15      ! [X2: code_integer] :
% 9.20/5.15        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
% 9.20/5.15          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).
% 9.20/5.15  
% 9.20/5.15  thf(fact_3800_pinf_I10_J,axiom,
% 9.20/5.15      ! [D: real,S: real] :
% 9.20/5.15      ? [Z3: real] :
% 9.20/5.15      ! [X2: real] :
% 9.20/5.15        ( ( ord_less_real @ Z3 @ X2 )
% 9.20/5.15       => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
% 9.20/5.16          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3801_pinf_I10_J,axiom,
% 9.20/5.16      ! [D: rat,S: rat] :
% 9.20/5.16      ? [Z3: rat] :
% 9.20/5.16      ! [X2: rat] :
% 9.20/5.16        ( ( ord_less_rat @ Z3 @ X2 )
% 9.20/5.16       => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
% 9.20/5.16          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3802_pinf_I10_J,axiom,
% 9.20/5.16      ! [D: nat,S: nat] :
% 9.20/5.16      ? [Z3: nat] :
% 9.20/5.16      ! [X2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ Z3 @ X2 )
% 9.20/5.16       => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
% 9.20/5.16          = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3803_pinf_I10_J,axiom,
% 9.20/5.16      ! [D: int,S: int] :
% 9.20/5.16      ? [Z3: int] :
% 9.20/5.16      ! [X2: int] :
% 9.20/5.16        ( ( ord_less_int @ Z3 @ X2 )
% 9.20/5.16       => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
% 9.20/5.16          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3804_pinf_I9_J,axiom,
% 9.20/5.16      ! [D: code_integer,S: code_integer] :
% 9.20/5.16      ? [Z3: code_integer] :
% 9.20/5.16      ! [X2: code_integer] :
% 9.20/5.16        ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3805_pinf_I9_J,axiom,
% 9.20/5.16      ! [D: real,S: real] :
% 9.20/5.16      ? [Z3: real] :
% 9.20/5.16      ! [X2: real] :
% 9.20/5.16        ( ( ord_less_real @ Z3 @ X2 )
% 9.20/5.16       => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
% 9.20/5.16          = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3806_pinf_I9_J,axiom,
% 9.20/5.16      ! [D: rat,S: rat] :
% 9.20/5.16      ? [Z3: rat] :
% 9.20/5.16      ! [X2: rat] :
% 9.20/5.16        ( ( ord_less_rat @ Z3 @ X2 )
% 9.20/5.16       => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
% 9.20/5.16          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3807_pinf_I9_J,axiom,
% 9.20/5.16      ! [D: nat,S: nat] :
% 9.20/5.16      ? [Z3: nat] :
% 9.20/5.16      ! [X2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ Z3 @ X2 )
% 9.20/5.16       => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
% 9.20/5.16          = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3808_pinf_I9_J,axiom,
% 9.20/5.16      ! [D: int,S: int] :
% 9.20/5.16      ? [Z3: int] :
% 9.20/5.16      ! [X2: int] :
% 9.20/5.16        ( ( ord_less_int @ Z3 @ X2 )
% 9.20/5.16       => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
% 9.20/5.16          = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3809_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: complex,A: complex] :
% 9.20/5.16        ( ( dvd_dvd_complex @ B @ A )
% 9.20/5.16       => ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.20/5.16            = zero_zero_complex )
% 9.20/5.16          = ( A = zero_zero_complex ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3810_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.16       => ( ( ( divide_divide_nat @ A @ B )
% 9.20/5.16            = zero_zero_nat )
% 9.20/5.16          = ( A = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3811_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ A )
% 9.20/5.16       => ( ( ( divide_divide_int @ A @ B )
% 9.20/5.16            = zero_zero_int )
% 9.20/5.16          = ( A = zero_zero_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3812_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: real,A: real] :
% 9.20/5.16        ( ( dvd_dvd_real @ B @ A )
% 9.20/5.16       => ( ( ( divide_divide_real @ A @ B )
% 9.20/5.16            = zero_zero_real )
% 9.20/5.16          = ( A = zero_zero_real ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3813_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: rat,A: rat] :
% 9.20/5.16        ( ( dvd_dvd_rat @ B @ A )
% 9.20/5.16       => ( ( ( divide_divide_rat @ A @ B )
% 9.20/5.16            = zero_zero_rat )
% 9.20/5.16          = ( A = zero_zero_rat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3814_dvd__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.16       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.20/5.16            = zero_z3403309356797280102nteger )
% 9.20/5.16          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3815_unit__mult__right__cancel,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ( ( ( times_3573771949741848930nteger @ B @ A )
% 9.20/5.16            = ( times_3573771949741848930nteger @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3816_unit__mult__right__cancel,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ( ( ( times_times_nat @ B @ A )
% 9.20/5.16            = ( times_times_nat @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3817_unit__mult__right__cancel,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ( ( ( times_times_int @ B @ A )
% 9.20/5.16            = ( times_times_int @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3818_unit__mult__left__cancel,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ( ( ( times_3573771949741848930nteger @ A @ B )
% 9.20/5.16            = ( times_3573771949741848930nteger @ A @ C ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3819_unit__mult__left__cancel,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ( ( ( times_times_nat @ A @ B )
% 9.20/5.16            = ( times_times_nat @ A @ C ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3820_unit__mult__left__cancel,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ( ( ( times_times_int @ A @ B )
% 9.20/5.16            = ( times_times_int @ A @ C ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3821_mult__unit__dvd__iff_H,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3822_mult__unit__dvd__iff_H,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_nat @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3823_mult__unit__dvd__iff_H,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_int @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3824_dvd__mult__unit__iff_H,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3825_dvd__mult__unit__iff_H,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.16          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3826_dvd__mult__unit__iff_H,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.16          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3827_mult__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3828_mult__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3829_mult__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3830_dvd__mult__unit__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3831_dvd__mult__unit__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3832_dvd__mult__unit__iff,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3833_is__unit__mult__iff,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
% 9.20/5.16        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16          & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3834_is__unit__mult__iff,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
% 9.20/5.16        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16          & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3835_is__unit__mult__iff,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
% 9.20/5.16        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16          & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3836_mod__Suc,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 9.20/5.16            = N2 )
% 9.20/5.16         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 9.20/5.16            = zero_zero_nat ) )
% 9.20/5.16        & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
% 9.20/5.16           != N2 )
% 9.20/5.16         => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
% 9.20/5.16            = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3837_finite__set__decode,axiom,
% 9.20/5.16      ! [N2: nat] : ( finite_finite_nat @ ( nat_set_decode @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3838_mod__induct,axiom,
% 9.20/5.16      ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
% 9.20/5.16        ( ( P @ N2 )
% 9.20/5.16       => ( ( ord_less_nat @ N2 @ P4 )
% 9.20/5.16         => ( ( ord_less_nat @ M @ P4 )
% 9.20/5.16           => ( ! [N3: nat] :
% 9.20/5.16                  ( ( ord_less_nat @ N3 @ P4 )
% 9.20/5.16                 => ( ( P @ N3 )
% 9.20/5.16                   => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
% 9.20/5.16             => ( P @ M ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3839_dvd__div__mult,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.16       => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
% 9.20/5.16          = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3840_dvd__div__mult,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16       => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
% 9.20/5.16          = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3841_dvd__div__mult,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
% 9.20/5.16          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3842_div__mult__swap,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.16       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.20/5.16          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3843_div__mult__swap,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.20/5.16          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3844_div__mult__swap,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.16          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3845_div__div__eq__right,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.16       => ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.16         => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.20/5.16            = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3846_div__div__eq__right,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16       => ( ( dvd_dvd_int @ B @ A )
% 9.20/5.16         => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.20/5.16            = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3847_div__div__eq__right,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.16            = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3848_dvd__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: nat,C: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
% 9.20/5.16       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.16          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3849_dvd__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: int,C: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
% 9.20/5.16       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.16          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3850_dvd__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
% 9.20/5.16       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.16          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3851_dvd__mult__imp__div,axiom,
% 9.20/5.16      ! [A: nat,C: nat,B: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
% 9.20/5.16       => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3852_dvd__mult__imp__div,axiom,
% 9.20/5.16      ! [A: int,C: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
% 9.20/5.16       => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3853_dvd__mult__imp__div,axiom,
% 9.20/5.16      ! [A: code_integer,C: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
% 9.20/5.16       => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3854_div__mult__div__if__dvd,axiom,
% 9.20/5.16      ! [B: nat,A: nat,D: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.16       => ( ( dvd_dvd_nat @ D @ C )
% 9.20/5.16         => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
% 9.20/5.16            = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3855_div__mult__div__if__dvd,axiom,
% 9.20/5.16      ! [B: int,A: int,D: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ A )
% 9.20/5.16       => ( ( dvd_dvd_int @ D @ C )
% 9.20/5.16         => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
% 9.20/5.16            = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3856_div__mult__div__if__dvd,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ D @ C )
% 9.20/5.16         => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3857_gcd__nat__induct,axiom,
% 9.20/5.16      ! [P: nat > nat > $o,M: nat,N2: nat] :
% 9.20/5.16        ( ! [M3: nat] : ( P @ M3 @ zero_zero_nat )
% 9.20/5.16       => ( ! [M3: nat,N3: nat] :
% 9.20/5.16              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.16             => ( ( P @ N3 @ ( modulo_modulo_nat @ M3 @ N3 ) )
% 9.20/5.16               => ( P @ M3 @ N3 ) ) )
% 9.20/5.16         => ( P @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3858_mod__less__divisor,axiom,
% 9.20/5.16      ! [N2: nat,M: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16       => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3859_dvd__div__unit__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3860_dvd__div__unit__iff,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3861_dvd__div__unit__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3862_div__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3863_div__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_int @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3864_div__unit__dvd__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.20/5.16          = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3865_unit__div__cancel,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ( ( ( divide_divide_nat @ B @ A )
% 9.20/5.16            = ( divide_divide_nat @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3866_unit__div__cancel,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ( ( ( divide_divide_int @ B @ A )
% 9.20/5.16            = ( divide_divide_int @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3867_unit__div__cancel,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.20/5.16            = ( divide6298287555418463151nteger @ C @ A ) )
% 9.20/5.16          = ( B = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3868_div__plus__div__distrib__dvd__right,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.16       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.16          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3869_div__plus__div__distrib__dvd__right,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.16          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3870_div__plus__div__distrib__dvd__right,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.20/5.16          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3871_div__plus__div__distrib__dvd__left,axiom,
% 9.20/5.16      ! [C: nat,A: nat,B: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ A )
% 9.20/5.16       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
% 9.20/5.16          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3872_div__plus__div__distrib__dvd__left,axiom,
% 9.20/5.16      ! [C: int,A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ A )
% 9.20/5.16       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
% 9.20/5.16          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3873_div__plus__div__distrib__dvd__left,axiom,
% 9.20/5.16      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.20/5.16       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
% 9.20/5.16          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3874_mod__Suc__le__divisor,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3875_mod__eq__0D,axiom,
% 9.20/5.16      ! [M: nat,D: nat] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ M @ D )
% 9.20/5.16          = zero_zero_nat )
% 9.20/5.16       => ? [Q2: nat] :
% 9.20/5.16            ( M
% 9.20/5.16            = ( times_times_nat @ D @ Q2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3876_le__imp__power__dvd,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,A: code_integer] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3877_le__imp__power__dvd,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3878_le__imp__power__dvd,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,A: real] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3879_le__imp__power__dvd,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,A: int] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3880_le__imp__power__dvd,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,A: complex] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3881_power__le__dvd,axiom,
% 9.20/5.16      ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3882_power__le__dvd,axiom,
% 9.20/5.16      ! [A: nat,N2: nat,B: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3883_power__le__dvd,axiom,
% 9.20/5.16      ! [A: real,N2: nat,B: real,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3884_power__le__dvd,axiom,
% 9.20/5.16      ! [A: int,N2: nat,B: int,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3885_power__le__dvd,axiom,
% 9.20/5.16      ! [A: complex,N2: nat,B: complex,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3886_dvd__power__le,axiom,
% 9.20/5.16      ! [X3: code_integer,Y3: code_integer,N2: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N2 ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3887_dvd__power__le,axiom,
% 9.20/5.16      ! [X3: nat,Y3: nat,N2: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N2 ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3888_dvd__power__le,axiom,
% 9.20/5.16      ! [X3: real,Y3: real,N2: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_real @ X3 @ Y3 )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ( dvd_dvd_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3889_dvd__power__le,axiom,
% 9.20/5.16      ! [X3: int,Y3: int,N2: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ( dvd_dvd_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3890_dvd__power__le,axiom,
% 9.20/5.16      ! [X3: complex,Y3: complex,N2: nat,M: nat] :
% 9.20/5.16        ( ( dvd_dvd_complex @ X3 @ Y3 )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3891_nat__mod__eq__iff,axiom,
% 9.20/5.16      ! [X3: nat,N2: nat,Y3: nat] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ X3 @ N2 )
% 9.20/5.16          = ( modulo_modulo_nat @ Y3 @ N2 ) )
% 9.20/5.16        = ( ? [Q1: nat,Q22: nat] :
% 9.20/5.16              ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N2 @ Q1 ) )
% 9.20/5.16              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3892_nat__dvd__not__less,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.16       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.16         => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3893_dvd__pos__nat,axiom,
% 9.20/5.16      ! [N2: nat,M: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16       => ( ( dvd_dvd_nat @ M @ N2 )
% 9.20/5.16         => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3894_zdvd__imp__le,axiom,
% 9.20/5.16      ! [Z2: int,N2: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ Z2 @ N2 )
% 9.20/5.16       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 9.20/5.16         => ( ord_less_eq_int @ Z2 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3895_zdvd__not__zless,axiom,
% 9.20/5.16      ! [M: int,N2: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ M )
% 9.20/5.16       => ( ( ord_less_int @ M @ N2 )
% 9.20/5.16         => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3896_bezout__add__nat,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16      ? [D4: nat,X4: nat,Y5: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ D4 @ A )
% 9.20/5.16        & ( dvd_dvd_nat @ D4 @ B )
% 9.20/5.16        & ( ( ( times_times_nat @ A @ X4 )
% 9.20/5.16            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) )
% 9.20/5.16          | ( ( times_times_nat @ B @ X4 )
% 9.20/5.16            = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D4 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3897_bezout__lemma__nat,axiom,
% 9.20/5.16      ! [D: nat,A: nat,B: nat,X3: nat,Y3: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ D @ A )
% 9.20/5.16       => ( ( dvd_dvd_nat @ D @ B )
% 9.20/5.16         => ( ( ( ( times_times_nat @ A @ X3 )
% 9.20/5.16                = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
% 9.20/5.16              | ( ( times_times_nat @ B @ X3 )
% 9.20/5.16                = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
% 9.20/5.16           => ? [X4: nat,Y5: nat] :
% 9.20/5.16                ( ( dvd_dvd_nat @ D @ A )
% 9.20/5.16                & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.16                & ( ( ( times_times_nat @ A @ X4 )
% 9.20/5.16                    = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D ) )
% 9.20/5.16                  | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
% 9.20/5.16                    = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3898_parity__cases,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16           != zero_zero_nat ) )
% 9.20/5.16       => ~ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16           => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16             != one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3899_parity__cases,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16           != zero_zero_int ) )
% 9.20/5.16       => ~ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16           => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16             != one_one_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3900_parity__cases,axiom,
% 9.20/5.16      ! [A: code_integer] :
% 9.20/5.16        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16           != zero_z3403309356797280102nteger ) )
% 9.20/5.16       => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16           => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16             != one_one_Code_integer ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3901_mod2__eq__if,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16            = zero_zero_nat ) )
% 9.20/5.16        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16            = one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3902_mod2__eq__if,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16            = zero_zero_int ) )
% 9.20/5.16        & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16            = one_one_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3903_mod2__eq__if,axiom,
% 9.20/5.16      ! [A: code_integer] :
% 9.20/5.16        ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16            = zero_z3403309356797280102nteger ) )
% 9.20/5.16        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16            = one_one_Code_integer ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3904_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.16       => ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.20/5.16         => ( ( modulo364778990260209775nteger @ A @ B )
% 9.20/5.16            = A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3905_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.16       => ( ( ord_less_nat @ A @ B )
% 9.20/5.16         => ( ( modulo_modulo_nat @ A @ B )
% 9.20/5.16            = A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3906_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.16       => ( ( ord_less_int @ A @ B )
% 9.20/5.16         => ( ( modulo_modulo_int @ A @ B )
% 9.20/5.16            = A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3907_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.16       => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3908_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.16       => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3909_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.16       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3910_cong__exp__iff__simps_I2_J,axiom,
% 9.20/5.16      ! [N2: num,Q4: num] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.20/5.16          = zero_zero_nat )
% 9.20/5.16        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.20/5.16          = zero_zero_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3911_cong__exp__iff__simps_I2_J,axiom,
% 9.20/5.16      ! [N2: num,Q4: num] :
% 9.20/5.16        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.20/5.16          = zero_zero_int )
% 9.20/5.16        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
% 9.20/5.16          = zero_zero_int ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3912_cong__exp__iff__simps_I2_J,axiom,
% 9.20/5.16      ! [N2: num,Q4: num] :
% 9.20/5.16        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.20/5.16          = zero_z3403309356797280102nteger )
% 9.20/5.16        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.20/5.16          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3913_cong__exp__iff__simps_I1_J,axiom,
% 9.20/5.16      ! [N2: num] :
% 9.20/5.16        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3914_cong__exp__iff__simps_I1_J,axiom,
% 9.20/5.16      ! [N2: num] :
% 9.20/5.16        ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) )
% 9.20/5.16        = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3915_cong__exp__iff__simps_I1_J,axiom,
% 9.20/5.16      ! [N2: num] :
% 9.20/5.16        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
% 9.20/5.16        = zero_z3403309356797280102nteger ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3916_mult__div__mod__eq,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3917_mult__div__mod__eq,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3918_mult__div__mod__eq,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3919_mod__mult__div__eq,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3920_mod__mult__div__eq,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3921_mod__mult__div__eq,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3922_mod__div__mult__eq,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3923_mod__div__mult__eq,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3924_mod__div__mult__eq,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3925_div__mult__mod__eq,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3926_div__mult__mod__eq,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3927_div__mult__mod__eq,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3928_mod__div__decomp,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( A
% 9.20/5.16        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3929_mod__div__decomp,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( A
% 9.20/5.16        = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3930_mod__div__decomp,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( A
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3931_cancel__div__mod__rules_I1_J,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_plus_nat @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3932_cancel__div__mod__rules_I1_J,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_plus_int @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3933_cancel__div__mod__rules_I1_J,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3934_cancel__div__mod__rules_I2_J,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_plus_nat @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3935_cancel__div__mod__rules_I2_J,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_plus_int @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3936_cancel__div__mod__rules_I2_J,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ A @ C ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3937_div__mult1__eq,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
% 9.20/5.16        = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3938_div__mult1__eq,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
% 9.20/5.16        = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3939_div__mult1__eq,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3940_unit__dvdE,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16           => ! [C2: code_integer] :
% 9.20/5.16                ( B
% 9.20/5.16               != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3941_unit__dvdE,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ~ ( ( A != zero_zero_nat )
% 9.20/5.16           => ! [C2: nat] :
% 9.20/5.16                ( B
% 9.20/5.16               != ( times_times_nat @ A @ C2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3942_unit__dvdE,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ~ ( ( A != zero_zero_int )
% 9.20/5.16           => ! [C2: int] :
% 9.20/5.16                ( B
% 9.20/5.16               != ( times_times_int @ A @ C2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3943_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: code_integer > $o,L: code_integer] :
% 9.20/5.16        ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: code_integer] :
% 9.20/5.16              ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3944_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: complex > $o,L: complex] :
% 9.20/5.16        ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: complex] :
% 9.20/5.16              ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3945_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: real > $o,L: real] :
% 9.20/5.16        ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: real] :
% 9.20/5.16              ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3946_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: rat > $o,L: rat] :
% 9.20/5.16        ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: rat] :
% 9.20/5.16              ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3947_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: nat > $o,L: nat] :
% 9.20/5.16        ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: nat] :
% 9.20/5.16              ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3948_unity__coeff__ex,axiom,
% 9.20/5.16      ! [P: int > $o,L: int] :
% 9.20/5.16        ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
% 9.20/5.16        = ( ? [X: int] :
% 9.20/5.16              ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
% 9.20/5.16              & ( P @ X ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3949_dvd__div__eq__mult,axiom,
% 9.20/5.16      ! [A: nat,B: nat,C: nat] :
% 9.20/5.16        ( ( A != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.16         => ( ( ( divide_divide_nat @ B @ A )
% 9.20/5.16              = C )
% 9.20/5.16            = ( B
% 9.20/5.16              = ( times_times_nat @ C @ A ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3950_dvd__div__eq__mult,axiom,
% 9.20/5.16      ! [A: int,B: int,C: int] :
% 9.20/5.16        ( ( A != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ A @ B )
% 9.20/5.16         => ( ( ( divide_divide_int @ B @ A )
% 9.20/5.16              = C )
% 9.20/5.16            = ( B
% 9.20/5.16              = ( times_times_int @ C @ A ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3951_dvd__div__eq__mult,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer,C: code_integer] :
% 9.20/5.16        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.16         => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.20/5.16              = C )
% 9.20/5.16            = ( B
% 9.20/5.16              = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3952_div__dvd__iff__mult,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( B != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.16         => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.20/5.16            = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3953_div__dvd__iff__mult,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( B != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ B @ A )
% 9.20/5.16         => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.20/5.16            = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3954_div__dvd__iff__mult,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( B != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.16         => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.20/5.16            = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3955_dvd__div__iff__mult,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( C != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ C @ B )
% 9.20/5.16         => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.20/5.16            = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3956_dvd__div__iff__mult,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( C != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16         => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.20/5.16            = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3957_dvd__div__iff__mult,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( C != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16         => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.16            = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3958_dvd__div__div__eq__mult,axiom,
% 9.20/5.16      ! [A: nat,C: nat,B: nat,D: nat] :
% 9.20/5.16        ( ( A != zero_zero_nat )
% 9.20/5.16       => ( ( C != zero_zero_nat )
% 9.20/5.16         => ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.16           => ( ( dvd_dvd_nat @ C @ D )
% 9.20/5.16             => ( ( ( divide_divide_nat @ B @ A )
% 9.20/5.16                  = ( divide_divide_nat @ D @ C ) )
% 9.20/5.16                = ( ( times_times_nat @ B @ C )
% 9.20/5.16                  = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3959_dvd__div__div__eq__mult,axiom,
% 9.20/5.16      ! [A: int,C: int,B: int,D: int] :
% 9.20/5.16        ( ( A != zero_zero_int )
% 9.20/5.16       => ( ( C != zero_zero_int )
% 9.20/5.16         => ( ( dvd_dvd_int @ A @ B )
% 9.20/5.16           => ( ( dvd_dvd_int @ C @ D )
% 9.20/5.16             => ( ( ( divide_divide_int @ B @ A )
% 9.20/5.16                  = ( divide_divide_int @ D @ C ) )
% 9.20/5.16                = ( ( times_times_int @ B @ C )
% 9.20/5.16                  = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3960_dvd__div__div__eq__mult,axiom,
% 9.20/5.16      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.20/5.16        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( C != zero_z3403309356797280102nteger )
% 9.20/5.16         => ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.16           => ( ( dvd_dvd_Code_integer @ C @ D )
% 9.20/5.16             => ( ( ( divide6298287555418463151nteger @ B @ A )
% 9.20/5.16                  = ( divide6298287555418463151nteger @ D @ C ) )
% 9.20/5.16                = ( ( times_3573771949741848930nteger @ B @ C )
% 9.20/5.16                  = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3961_unit__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( ( divide_divide_nat @ A @ B )
% 9.20/5.16            = zero_zero_nat )
% 9.20/5.16          = ( A = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3962_unit__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( ( divide_divide_int @ A @ B )
% 9.20/5.16            = zero_zero_int )
% 9.20/5.16          = ( A = zero_zero_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3963_unit__div__eq__0__iff,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.20/5.16            = zero_z3403309356797280102nteger )
% 9.20/5.16          = ( A = zero_z3403309356797280102nteger ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3964_is__unit__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: nat,C: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.20/5.16         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.16            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3965_is__unit__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: int,C: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ C @ one_one_int )
% 9.20/5.16         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.16            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3966_is__unit__div__mult2__eq,axiom,
% 9.20/5.16      ! [B: code_integer,C: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3967_unit__div__mult__swap,axiom,
% 9.20/5.16      ! [C: nat,A: nat,B: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.20/5.16       => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
% 9.20/5.16          = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3968_unit__div__mult__swap,axiom,
% 9.20/5.16      ! [C: int,A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.20/5.16       => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
% 9.20/5.16          = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3969_unit__div__mult__swap,axiom,
% 9.20/5.16      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.20/5.16       => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
% 9.20/5.16          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3970_unit__div__commute,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
% 9.20/5.16          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3971_unit__div__commute,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
% 9.20/5.16          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3972_unit__div__commute,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
% 9.20/5.16          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3973_div__mult__unit2,axiom,
% 9.20/5.16      ! [C: nat,B: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ C @ one_one_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.16         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.16            = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3974_div__mult__unit2,axiom,
% 9.20/5.16      ! [C: int,B: int,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ one_one_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ B @ A )
% 9.20/5.16         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.16            = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3975_div__mult__unit2,axiom,
% 9.20/5.16      ! [C: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3976_unit__eq__div2,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( A
% 9.20/5.16            = ( divide_divide_nat @ C @ B ) )
% 9.20/5.16          = ( ( times_times_nat @ A @ B )
% 9.20/5.16            = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3977_unit__eq__div2,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( A
% 9.20/5.16            = ( divide_divide_int @ C @ B ) )
% 9.20/5.16          = ( ( times_times_int @ A @ B )
% 9.20/5.16            = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3978_unit__eq__div2,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( A
% 9.20/5.16            = ( divide6298287555418463151nteger @ C @ B ) )
% 9.20/5.16          = ( ( times_3573771949741848930nteger @ A @ B )
% 9.20/5.16            = C ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3979_unit__eq__div1,axiom,
% 9.20/5.16      ! [B: nat,A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16       => ( ( ( divide_divide_nat @ A @ B )
% 9.20/5.16            = C )
% 9.20/5.16          = ( A
% 9.20/5.16            = ( times_times_nat @ C @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3980_unit__eq__div1,axiom,
% 9.20/5.16      ! [B: int,A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16       => ( ( ( divide_divide_int @ A @ B )
% 9.20/5.16            = C )
% 9.20/5.16          = ( A
% 9.20/5.16            = ( times_times_int @ C @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3981_unit__eq__div1,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16       => ( ( ( divide6298287555418463151nteger @ A @ B )
% 9.20/5.16            = C )
% 9.20/5.16          = ( A
% 9.20/5.16            = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3982_mod__le__divisor,axiom,
% 9.20/5.16      ! [N2: nat,M: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16       => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3983_is__unit__power__iff,axiom,
% 9.20/5.16      ! [A: code_integer,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
% 9.20/5.16        = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3984_is__unit__power__iff,axiom,
% 9.20/5.16      ! [A: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ one_one_nat )
% 9.20/5.16        = ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3985_is__unit__power__iff,axiom,
% 9.20/5.16      ! [A: int,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ one_one_int )
% 9.20/5.16        = ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3986_div__less__mono,axiom,
% 9.20/5.16      ! [A2: nat,B2: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ A2 @ B2 )
% 9.20/5.16       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16         => ( ( ( modulo_modulo_nat @ A2 @ N2 )
% 9.20/5.16              = zero_zero_nat )
% 9.20/5.16           => ( ( ( modulo_modulo_nat @ B2 @ N2 )
% 9.20/5.16                = zero_zero_nat )
% 9.20/5.16             => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3987_mod__eq__nat1E,axiom,
% 9.20/5.16      ! [M: nat,Q4: nat,N2: nat] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.20/5.16          = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 9.20/5.16       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.16         => ~ ! [S2: nat] :
% 9.20/5.16                ( M
% 9.20/5.16               != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3988_mod__eq__nat2E,axiom,
% 9.20/5.16      ! [M: nat,Q4: nat,N2: nat] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.20/5.16          = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 9.20/5.16       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16         => ~ ! [S2: nat] :
% 9.20/5.16                ( N2
% 9.20/5.16               != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3989_nat__mod__eq__lemma,axiom,
% 9.20/5.16      ! [X3: nat,N2: nat,Y3: nat] :
% 9.20/5.16        ( ( ( modulo_modulo_nat @ X3 @ N2 )
% 9.20/5.16          = ( modulo_modulo_nat @ Y3 @ N2 ) )
% 9.20/5.16       => ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.20/5.16         => ? [Q2: nat] :
% 9.20/5.16              ( X3
% 9.20/5.16              = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3990_dvd__imp__le,axiom,
% 9.20/5.16      ! [K: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ K @ N2 )
% 9.20/5.16       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16         => ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3991_div__mod__decomp,axiom,
% 9.20/5.16      ! [A2: nat,N2: nat] :
% 9.20/5.16        ( A2
% 9.20/5.16        = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_nat @ A2 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3992_mod__mult2__eq,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.16        ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q4 ) )
% 9.20/5.16        = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3993_dvd__mult__cancel,axiom,
% 9.20/5.16      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.16       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.16         => ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3994_nat__mult__dvd__cancel1,axiom,
% 9.20/5.16      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
% 9.20/5.16          = ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3995_bezout__add__strong__nat,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( A != zero_zero_nat )
% 9.20/5.16       => ? [D4: nat,X4: nat,Y5: nat] :
% 9.20/5.16            ( ( dvd_dvd_nat @ D4 @ A )
% 9.20/5.16            & ( dvd_dvd_nat @ D4 @ B )
% 9.20/5.16            & ( ( times_times_nat @ A @ X4 )
% 9.20/5.16              = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3996_even__zero,axiom,
% 9.20/5.16      dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3997_even__zero,axiom,
% 9.20/5.16      dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3998_even__zero,axiom,
% 9.20/5.16      dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_3999_is__unitE,axiom,
% 9.20/5.16      ! [A: nat,C: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ A @ one_one_nat )
% 9.20/5.16       => ~ ( ( A != zero_zero_nat )
% 9.20/5.16           => ! [B4: nat] :
% 9.20/5.16                ( ( B4 != zero_zero_nat )
% 9.20/5.16               => ( ( dvd_dvd_nat @ B4 @ one_one_nat )
% 9.20/5.16                 => ( ( ( divide_divide_nat @ one_one_nat @ A )
% 9.20/5.16                      = B4 )
% 9.20/5.16                   => ( ( ( divide_divide_nat @ one_one_nat @ B4 )
% 9.20/5.16                        = A )
% 9.20/5.16                     => ( ( ( times_times_nat @ A @ B4 )
% 9.20/5.16                          = one_one_nat )
% 9.20/5.16                       => ( ( divide_divide_nat @ C @ A )
% 9.20/5.16                         != ( times_times_nat @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4000_is__unitE,axiom,
% 9.20/5.16      ! [A: int,C: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ A @ one_one_int )
% 9.20/5.16       => ~ ( ( A != zero_zero_int )
% 9.20/5.16           => ! [B4: int] :
% 9.20/5.16                ( ( B4 != zero_zero_int )
% 9.20/5.16               => ( ( dvd_dvd_int @ B4 @ one_one_int )
% 9.20/5.16                 => ( ( ( divide_divide_int @ one_one_int @ A )
% 9.20/5.16                      = B4 )
% 9.20/5.16                   => ( ( ( divide_divide_int @ one_one_int @ B4 )
% 9.20/5.16                        = A )
% 9.20/5.16                     => ( ( ( times_times_int @ A @ B4 )
% 9.20/5.16                          = one_one_int )
% 9.20/5.16                       => ( ( divide_divide_int @ C @ A )
% 9.20/5.16                         != ( times_times_int @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4001_is__unitE,axiom,
% 9.20/5.16      ! [A: code_integer,C: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
% 9.20/5.16       => ~ ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16           => ! [B4: code_integer] :
% 9.20/5.16                ( ( B4 != zero_z3403309356797280102nteger )
% 9.20/5.16               => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
% 9.20/5.16                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
% 9.20/5.16                      = B4 )
% 9.20/5.16                   => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
% 9.20/5.16                        = A )
% 9.20/5.16                     => ( ( ( times_3573771949741848930nteger @ A @ B4 )
% 9.20/5.16                          = one_one_Code_integer )
% 9.20/5.16                       => ( ( divide6298287555418463151nteger @ C @ A )
% 9.20/5.16                         != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4002_is__unit__div__mult__cancel__left,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( A != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16         => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
% 9.20/5.16            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4003_is__unit__div__mult__cancel__left,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( A != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16         => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
% 9.20/5.16            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4004_is__unit__div__mult__cancel__left,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4005_is__unit__div__mult__cancel__right,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( A != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ B @ one_one_nat )
% 9.20/5.16         => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
% 9.20/5.16            = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4006_is__unit__div__mult__cancel__right,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( A != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ B @ one_one_int )
% 9.20/5.16         => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
% 9.20/5.16            = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4007_is__unit__div__mult__cancel__right,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4008_odd__even__add,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
% 9.20/5.16         => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4009_odd__even__add,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.16         => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4010_odd__even__add,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
% 9.20/5.16         => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4011_dvd__power__iff,axiom,
% 9.20/5.16      ! [X3: code_integer,M: nat,N2: nat] :
% 9.20/5.16        ( ( X3 != zero_z3403309356797280102nteger )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N2 ) )
% 9.20/5.16          = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
% 9.20/5.16            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4012_dvd__power__iff,axiom,
% 9.20/5.16      ! [X3: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( X3 != zero_zero_nat )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N2 ) )
% 9.20/5.16          = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
% 9.20/5.16            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4013_dvd__power__iff,axiom,
% 9.20/5.16      ! [X3: int,M: nat,N2: nat] :
% 9.20/5.16        ( ( X3 != zero_zero_int )
% 9.20/5.16       => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N2 ) )
% 9.20/5.16          = ( ( dvd_dvd_int @ X3 @ one_one_int )
% 9.20/5.16            | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4014_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: code_integer] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_Code_integer ) )
% 9.20/5.16       => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4015_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: rat] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_rat ) )
% 9.20/5.16       => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4016_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: nat] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_nat ) )
% 9.20/5.16       => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4017_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: real] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_real ) )
% 9.20/5.16       => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4018_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: int] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_int ) )
% 9.20/5.16       => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4019_dvd__power,axiom,
% 9.20/5.16      ! [N2: nat,X3: complex] :
% 9.20/5.16        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          | ( X3 = one_one_complex ) )
% 9.20/5.16       => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4020_split__mod,axiom,
% 9.20/5.16      ! [P: nat > $o,M: nat,N2: nat] :
% 9.20/5.16        ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
% 9.20/5.16        = ( ( ( N2 = zero_zero_nat )
% 9.20/5.16           => ( P @ M ) )
% 9.20/5.16          & ( ( N2 != zero_zero_nat )
% 9.20/5.16           => ! [I2: nat,J3: nat] :
% 9.20/5.16                ( ( ord_less_nat @ J3 @ N2 )
% 9.20/5.16               => ( ( M
% 9.20/5.16                    = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
% 9.20/5.16                 => ( P @ J3 ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4021_dvd__mult__cancel1,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
% 9.20/5.16          = ( N2 = one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4022_dvd__mult__cancel2,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
% 9.20/5.16          = ( N2 = one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4023_power__dvd__imp__le,axiom,
% 9.20/5.16      ! [I: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
% 9.20/5.16       => ( ( ord_less_nat @ one_one_nat @ I )
% 9.20/5.16         => ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4024_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.20/5.16      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
% 9.20/5.16       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
% 9.20/5.16          = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4025_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.20/5.16      ! [C: nat,A: nat,B: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ zero_zero_nat @ C )
% 9.20/5.16       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
% 9.20/5.16          = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4026_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
% 9.20/5.16      ! [C: int,A: int,B: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ C )
% 9.20/5.16       => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
% 9.20/5.16          = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4027_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr4953567300277697838T_VEBT @ ( produc4743750530478302277T_VEBT @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4028_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4029_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr1791586995822124652BT_nat @ ( produc7295137177222721919BT_nat @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4030_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr6837108013167703752BT_int @ ( produc7292646706713671643BT_int @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4031_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4032_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_o,Ys: list_o] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4033_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_o,Ys: list_nat] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( product_Pair_o_nat @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) @ ( nth_nat @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_nat @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4034_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_o,Ys: list_int] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( product_Pair_o_int @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) @ ( nth_int @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_int @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4035_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr744662078594809490T_VEBT @ ( produc7156399406898700509T_VEBT @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) @ ( nth_VEBT_VEBT @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4036_product__nth,axiom,
% 9.20/5.16      ! [N2: nat,Xs: list_nat,Ys: list_o] :
% 9.20/5.16        ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
% 9.20/5.16       => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N2 )
% 9.20/5.16          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4037_power__mono__odd,axiom,
% 9.20/5.16      ! [N2: nat,A: real,B: real] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_real @ A @ B )
% 9.20/5.16         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4038_power__mono__odd,axiom,
% 9.20/5.16      ! [N2: nat,A: rat,B: rat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.16         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4039_power__mono__odd,axiom,
% 9.20/5.16      ! [N2: nat,A: int,B: int] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_int @ A @ B )
% 9.20/5.16         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4040_Suc__times__mod__eq,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
% 9.20/5.16       => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
% 9.20/5.16          = one_one_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4041_odd__pos,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4042_dvd__power__iff__le,axiom,
% 9.20/5.16      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
% 9.20/5.16          = ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4043_even__unset__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          | ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4044_even__unset__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          | ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4045_even__unset__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          | ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4046_even__set__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          & ( M != zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4047_even__set__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          & ( M != zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4048_even__set__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16          & ( M != zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4049_num_Osize__gen_I1_J,axiom,
% 9.20/5.16      ( ( size_num @ one )
% 9.20/5.16      = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4050_divmod__digit__0_I2_J,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.16       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
% 9.20/5.16            = ( modulo_modulo_nat @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4051_divmod__digit__0_I2_J,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.16       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
% 9.20/5.16            = ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4052_divmod__digit__0_I2_J,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.16       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
% 9.20/5.16            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4053_bits__stable__imp__add__self,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16          = A )
% 9.20/5.16       => ( ( plus_plus_nat @ A @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.16          = zero_zero_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4054_bits__stable__imp__add__self,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16          = A )
% 9.20/5.16       => ( ( plus_plus_int @ A @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
% 9.20/5.16          = zero_zero_int ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4055_bits__stable__imp__add__self,axiom,
% 9.20/5.16      ! [A: code_integer] :
% 9.20/5.16        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16          = A )
% 9.20/5.16       => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
% 9.20/5.16          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4056_div__exp__mod__exp__eq,axiom,
% 9.20/5.16      ! [A: nat,N2: nat,M: nat] :
% 9.20/5.16        ( ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16        = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4057_div__exp__mod__exp__eq,axiom,
% 9.20/5.16      ! [A: int,N2: nat,M: nat] :
% 9.20/5.16        ( ( modulo_modulo_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16        = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4058_div__exp__mod__exp__eq,axiom,
% 9.20/5.16      ! [A: code_integer,N2: nat,M: nat] :
% 9.20/5.16        ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16        = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4059_oddE,axiom,
% 9.20/5.16      ! [A: code_integer] :
% 9.20/5.16        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ~ ! [B4: code_integer] :
% 9.20/5.16              ( A
% 9.20/5.16             != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4060_oddE,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ~ ! [B4: nat] :
% 9.20/5.16              ( A
% 9.20/5.16             != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B4 ) @ one_one_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4061_oddE,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16       => ~ ! [B4: int] :
% 9.20/5.16              ( A
% 9.20/5.16             != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B4 ) @ one_one_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4062_zero__le__even__power,axiom,
% 9.20/5.16      ! [N2: nat,A: real] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4063_zero__le__even__power,axiom,
% 9.20/5.16      ! [N2: nat,A: rat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4064_zero__le__even__power,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4065_zero__le__odd__power,axiom,
% 9.20/5.16      ! [N2: nat,A: real] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 9.20/5.16          = ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4066_zero__le__odd__power,axiom,
% 9.20/5.16      ! [N2: nat,A: rat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 9.20/5.16          = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4067_zero__le__odd__power,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 9.20/5.16          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4068_zero__le__power__eq,axiom,
% 9.20/5.16      ! [A: real,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4069_zero__le__power__eq,axiom,
% 9.20/5.16      ! [A: rat,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4070_zero__le__power__eq,axiom,
% 9.20/5.16      ! [A: int,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4071_verit__le__mono__div,axiom,
% 9.20/5.16      ! [A2: nat,B2: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ A2 @ B2 )
% 9.20/5.16       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16         => ( ord_less_eq_nat
% 9.20/5.16            @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
% 9.20/5.16              @ ( if_nat
% 9.20/5.16                @ ( ( modulo_modulo_nat @ B2 @ N2 )
% 9.20/5.16                  = zero_zero_nat )
% 9.20/5.16                @ one_one_nat
% 9.20/5.16                @ zero_zero_nat ) )
% 9.20/5.16            @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4072_even__set__encode__iff,axiom,
% 9.20/5.16      ! [A2: set_nat] :
% 9.20/5.16        ( ( finite_finite_nat @ A2 )
% 9.20/5.16       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat_set_encode @ A2 ) )
% 9.20/5.16          = ( ~ ( member_nat @ zero_zero_nat @ A2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4073_divmod__digit__0_I1_J,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.16       => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16            = ( divide_divide_nat @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4074_divmod__digit__0_I1_J,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.16       => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16            = ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4075_divmod__digit__0_I1_J,axiom,
% 9.20/5.16      ! [B: code_integer,A: code_integer] :
% 9.20/5.16        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.16       => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.16         => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4076_zero__less__power__eq,axiom,
% 9.20/5.16      ! [A: real,N2: nat] :
% 9.20/5.16        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ N2 ) )
% 9.20/5.16        = ( ( N2 = zero_zero_nat )
% 9.20/5.16          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( A != zero_zero_real ) )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4077_zero__less__power__eq,axiom,
% 9.20/5.16      ! [A: rat,N2: nat] :
% 9.20/5.16        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
% 9.20/5.16        = ( ( N2 = zero_zero_nat )
% 9.20/5.16          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( A != zero_zero_rat ) )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4078_zero__less__power__eq,axiom,
% 9.20/5.16      ! [A: int,N2: nat] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N2 ) )
% 9.20/5.16        = ( ( N2 = zero_zero_nat )
% 9.20/5.16          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( A != zero_zero_int ) )
% 9.20/5.16          | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4079_mod__double__modulus,axiom,
% 9.20/5.16      ! [M: code_integer,X3: code_integer] :
% 9.20/5.16        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
% 9.20/5.16       => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 9.20/5.16         => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( modulo364778990260209775nteger @ X3 @ M ) )
% 9.20/5.16            | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4080_mod__double__modulus,axiom,
% 9.20/5.16      ! [M: nat,X3: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.16       => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
% 9.20/5.16         => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( modulo_modulo_nat @ X3 @ M ) )
% 9.20/5.16            | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4081_mod__double__modulus,axiom,
% 9.20/5.16      ! [M: int,X3: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ M )
% 9.20/5.16       => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.16         => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( modulo_modulo_int @ X3 @ M ) )
% 9.20/5.16            | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
% 9.20/5.16              = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4082_unset__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: nat] :
% 9.20/5.16        ( ( bit_se4205575877204974255it_nat @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4083_unset__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: code_integer] :
% 9.20/5.16        ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4084_unset__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ( bit_se4203085406695923979it_int @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4085_set__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: nat] :
% 9.20/5.16        ( ( bit_se7882103937844011126it_nat @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4086_set__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: code_integer] :
% 9.20/5.16        ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4087_set__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ( bit_se7879613467334960850it_int @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4088_power__le__zero__eq,axiom,
% 9.20/5.16      ! [A: real,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ zero_zero_real )
% 9.20/5.16        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.20/5.16            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( A = zero_zero_real ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4089_power__le__zero__eq,axiom,
% 9.20/5.16      ! [A: rat,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
% 9.20/5.16        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.20/5.16            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( A = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4090_power__le__zero__eq,axiom,
% 9.20/5.16      ! [A: int,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ zero_zero_int )
% 9.20/5.16        = ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16          & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.20/5.16            | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16              & ( A = zero_zero_int ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4091_divmod__digit__1_I1_J,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.16       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.16         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16           => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
% 9.20/5.16              = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4092_divmod__digit__1_I1_J,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.16       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.16         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16           => ( ( 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 )
% 9.20/5.16              = ( divide_divide_nat @ A @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4093_divmod__digit__1_I1_J,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.16       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.16         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.16           => ( ( 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 )
% 9.20/5.16              = ( divide_divide_int @ A @ B ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4094_flip__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: nat] :
% 9.20/5.16        ( ( bit_se2161824704523386999it_nat @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4095_flip__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: code_integer] :
% 9.20/5.16        ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4096_flip__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ( bit_se2159334234014336723it_int @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4097_pos__eucl__rel__int__mult__2,axiom,
% 9.20/5.16      ! [B: int,A: int,Q4: int,R2: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.16       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.16         => ( 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 ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4098_signed__take__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: code_integer] :
% 9.20/5.16        ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4099_signed__take__bit__Suc,axiom,
% 9.20/5.16      ! [N2: nat,A: int] :
% 9.20/5.16        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ A )
% 9.20/5.16        = ( plus_plus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4100_even__mult__exp__div__exp__iff,axiom,
% 9.20/5.16      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.16        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.16        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.16          | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            = zero_zero_nat )
% 9.20/5.16          | ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16            & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4101_even__mult__exp__div__exp__iff,axiom,
% 9.20/5.16      ! [A: int,M: nat,N2: nat] :
% 9.20/5.16        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.16        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.16          | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            = zero_zero_int )
% 9.20/5.16          | ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16            & ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4102_even__mult__exp__div__exp__iff,axiom,
% 9.20/5.16      ! [A: code_integer,M: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.16        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.16          | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16            = zero_z3403309356797280102nteger )
% 9.20/5.16          | ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16            & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4103_even__flip__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         != ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4104_even__flip__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         != ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4105_even__flip__bit__iff,axiom,
% 9.20/5.16      ! [M: nat,A: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.20/5.16        = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.16         != ( M = zero_zero_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4106_one__mod__2__pow__eq,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.16        = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4107_one__mod__2__pow__eq,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( modulo_modulo_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.16        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4108_one__mod__2__pow__eq,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( modulo_modulo_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.16        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4109_set__encode__insert,axiom,
% 9.20/5.16      ! [A2: set_nat,N2: nat] :
% 9.20/5.16        ( ( finite_finite_nat @ A2 )
% 9.20/5.16       => ( ~ ( member_nat @ N2 @ A2 )
% 9.20/5.16         => ( ( nat_set_encode @ ( insert_nat @ N2 @ A2 ) )
% 9.20/5.16            = ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( nat_set_encode @ A2 ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4110_insert__absorb2,axiom,
% 9.20/5.16      ! [X3: nat,A2: set_nat] :
% 9.20/5.16        ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.16        = ( insert_nat @ X3 @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4111_insert__absorb2,axiom,
% 9.20/5.16      ! [X3: int,A2: set_int] :
% 9.20/5.16        ( ( insert_int @ X3 @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.16        = ( insert_int @ X3 @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4112_insert__absorb2,axiom,
% 9.20/5.16      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.16        ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
% 9.20/5.16        = ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4113_insert__absorb2,axiom,
% 9.20/5.16      ! [X3: real,A2: set_real] :
% 9.20/5.16        ( ( insert_real @ X3 @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.16        = ( insert_real @ X3 @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4114_insert__absorb2,axiom,
% 9.20/5.16      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.16        ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
% 9.20/5.16        = ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4115_insert__iff,axiom,
% 9.20/5.16      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.16        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4116_insert__iff,axiom,
% 9.20/5.16      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.16        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4117_insert__iff,axiom,
% 9.20/5.16      ! [A: real,B: real,A2: set_real] :
% 9.20/5.16        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member_real @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4118_insert__iff,axiom,
% 9.20/5.16      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.20/5.16        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member_set_nat @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4119_insert__iff,axiom,
% 9.20/5.16      ! [A: nat,B: nat,A2: set_nat] :
% 9.20/5.16        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member_nat @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4120_insert__iff,axiom,
% 9.20/5.16      ! [A: int,B: int,A2: set_int] :
% 9.20/5.16        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          | ( member_int @ A @ A2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4121_insertCI,axiom,
% 9.20/5.16      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.16        ( ( ~ ( member8757157785044589968at_nat @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4122_insertCI,axiom,
% 9.20/5.16      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.20/5.16        ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4123_insertCI,axiom,
% 9.20/5.16      ! [A: real,B2: set_real,B: real] :
% 9.20/5.16        ( ( ~ ( member_real @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4124_insertCI,axiom,
% 9.20/5.16      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.20/5.16        ( ( ~ ( member_set_nat @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4125_insertCI,axiom,
% 9.20/5.16      ! [A: nat,B2: set_nat,B: nat] :
% 9.20/5.16        ( ( ~ ( member_nat @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4126_insertCI,axiom,
% 9.20/5.16      ! [A: int,B2: set_int,B: int] :
% 9.20/5.16        ( ( ~ ( member_int @ A @ B2 )
% 9.20/5.16         => ( A = B ) )
% 9.20/5.16       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4127_diff__self,axiom,
% 9.20/5.16      ! [A: complex] :
% 9.20/5.16        ( ( minus_minus_complex @ A @ A )
% 9.20/5.16        = zero_zero_complex ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4128_diff__self,axiom,
% 9.20/5.16      ! [A: real] :
% 9.20/5.16        ( ( minus_minus_real @ A @ A )
% 9.20/5.16        = zero_zero_real ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4129_diff__self,axiom,
% 9.20/5.16      ! [A: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ A @ A )
% 9.20/5.16        = zero_zero_rat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4130_diff__self,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( minus_minus_int @ A @ A )
% 9.20/5.16        = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4131_diff__0__right,axiom,
% 9.20/5.16      ! [A: complex] :
% 9.20/5.16        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4132_diff__0__right,axiom,
% 9.20/5.16      ! [A: real] :
% 9.20/5.16        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4133_diff__0__right,axiom,
% 9.20/5.16      ! [A: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4134_diff__0__right,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4135_zero__diff,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ zero_zero_nat @ A )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4136_diff__zero,axiom,
% 9.20/5.16      ! [A: complex] :
% 9.20/5.16        ( ( minus_minus_complex @ A @ zero_zero_complex )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4137_diff__zero,axiom,
% 9.20/5.16      ! [A: real] :
% 9.20/5.16        ( ( minus_minus_real @ A @ zero_zero_real )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4138_diff__zero,axiom,
% 9.20/5.16      ! [A: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ A @ zero_zero_rat )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4139_diff__zero,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ A @ zero_zero_nat )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4140_diff__zero,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( minus_minus_int @ A @ zero_zero_int )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4141_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.20/5.16      ! [A: complex] :
% 9.20/5.16        ( ( minus_minus_complex @ A @ A )
% 9.20/5.16        = zero_zero_complex ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4142_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.20/5.16      ! [A: real] :
% 9.20/5.16        ( ( minus_minus_real @ A @ A )
% 9.20/5.16        = zero_zero_real ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4143_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.20/5.16      ! [A: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ A @ A )
% 9.20/5.16        = zero_zero_rat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4144_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.20/5.16      ! [A: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ A @ A )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4145_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
% 9.20/5.16      ! [A: int] :
% 9.20/5.16        ( ( minus_minus_int @ A @ A )
% 9.20/5.16        = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4146_add__diff__cancel,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4147_add__diff__cancel,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4148_add__diff__cancel,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4149_diff__add__cancel,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4150_diff__add__cancel,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4151_diff__add__cancel,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4152_add__diff__cancel__left,axiom,
% 9.20/5.16      ! [C: real,A: real,B: real] :
% 9.20/5.16        ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
% 9.20/5.16        = ( minus_minus_real @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4153_add__diff__cancel__left,axiom,
% 9.20/5.16      ! [C: rat,A: rat,B: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
% 9.20/5.16        = ( minus_minus_rat @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4154_add__diff__cancel__left,axiom,
% 9.20/5.16      ! [C: nat,A: nat,B: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
% 9.20/5.16        = ( minus_minus_nat @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4155_add__diff__cancel__left,axiom,
% 9.20/5.16      ! [C: int,A: int,B: int] :
% 9.20/5.16        ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 9.20/5.16        = ( minus_minus_int @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4156_add__diff__cancel__left_H,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
% 9.20/5.16        = B ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4157_add__diff__cancel__left_H,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
% 9.20/5.16        = B ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4158_add__diff__cancel__left_H,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
% 9.20/5.16        = B ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4159_add__diff__cancel__left_H,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 9.20/5.16        = B ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4160_add__diff__cancel__right,axiom,
% 9.20/5.16      ! [A: real,C: real,B: real] :
% 9.20/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
% 9.20/5.16        = ( minus_minus_real @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4161_add__diff__cancel__right,axiom,
% 9.20/5.16      ! [A: rat,C: rat,B: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.16        = ( minus_minus_rat @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4162_add__diff__cancel__right,axiom,
% 9.20/5.16      ! [A: nat,C: nat,B: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
% 9.20/5.16        = ( minus_minus_nat @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4163_add__diff__cancel__right,axiom,
% 9.20/5.16      ! [A: int,C: int,B: int] :
% 9.20/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 9.20/5.16        = ( minus_minus_int @ A @ B ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4164_add__diff__cancel__right_H,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4165_add__diff__cancel__right_H,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4166_add__diff__cancel__right_H,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4167_add__diff__cancel__right_H,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 9.20/5.16        = A ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4168_singletonI,axiom,
% 9.20/5.16      ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4169_singletonI,axiom,
% 9.20/5.16      ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4170_singletonI,axiom,
% 9.20/5.16      ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4171_singletonI,axiom,
% 9.20/5.16      ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4172_singletonI,axiom,
% 9.20/5.16      ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4173_singletonI,axiom,
% 9.20/5.16      ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4174_finite__insert,axiom,
% 9.20/5.16      ! [A: real,A2: set_real] :
% 9.20/5.16        ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
% 9.20/5.16        = ( finite_finite_real @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4175_finite__insert,axiom,
% 9.20/5.16      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.16        ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
% 9.20/5.16        = ( finite4343798906461161616at_nat @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4176_finite__insert,axiom,
% 9.20/5.16      ! [A: nat,A2: set_nat] :
% 9.20/5.16        ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
% 9.20/5.16        = ( finite_finite_nat @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4177_finite__insert,axiom,
% 9.20/5.16      ! [A: int,A2: set_int] :
% 9.20/5.16        ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
% 9.20/5.16        = ( finite_finite_int @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4178_finite__insert,axiom,
% 9.20/5.16      ! [A: complex,A2: set_complex] :
% 9.20/5.16        ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
% 9.20/5.16        = ( finite3207457112153483333omplex @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4179_finite__insert,axiom,
% 9.20/5.16      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.16        ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
% 9.20/5.16        = ( finite6177210948735845034at_nat @ A2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4180_insert__subset,axiom,
% 9.20/5.16      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.16        ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.16          & ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4181_insert__subset,axiom,
% 9.20/5.16      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.16        ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.16          & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4182_insert__subset,axiom,
% 9.20/5.16      ! [X3: real,A2: set_real,B2: set_real] :
% 9.20/5.16        ( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member_real @ X3 @ B2 )
% 9.20/5.16          & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4183_insert__subset,axiom,
% 9.20/5.16      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.16        ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member_set_nat @ X3 @ B2 )
% 9.20/5.16          & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4184_insert__subset,axiom,
% 9.20/5.16      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.16        ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member_nat @ X3 @ B2 )
% 9.20/5.16          & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4185_insert__subset,axiom,
% 9.20/5.16      ! [X3: int,A2: set_int,B2: set_int] :
% 9.20/5.16        ( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 9.20/5.16        = ( ( member_int @ X3 @ B2 )
% 9.20/5.16          & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4186_diff__Suc__Suc,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 9.20/5.16        = ( minus_minus_nat @ M @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4187_Suc__diff__diff,axiom,
% 9.20/5.16      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
% 9.20/5.16        = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4188_diff__0__eq__0,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ zero_zero_nat @ N2 )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4189_diff__self__eq__0,axiom,
% 9.20/5.16      ! [M: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ M @ M )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4190_diff__diff__cancel,axiom,
% 9.20/5.16      ! [I: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ I @ N2 )
% 9.20/5.16       => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
% 9.20/5.16          = I ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4191_diff__diff__left,axiom,
% 9.20/5.16      ! [I: nat,J: nat,K: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.20/5.16        = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4192_flip__bit__negative__int__iff,axiom,
% 9.20/5.16      ! [N2: nat,K: int] :
% 9.20/5.16        ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.16        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4193_of__bool__less__eq__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.20/5.16        = ( P
% 9.20/5.16         => Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4194_of__bool__less__eq__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.20/5.16        = ( P
% 9.20/5.16         => Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4195_of__bool__less__eq__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.20/5.16        = ( P
% 9.20/5.16         => Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4196_of__bool__eq_I1_J,axiom,
% 9.20/5.16      ( ( zero_n1201886186963655149omplex @ $false )
% 9.20/5.16      = zero_zero_complex ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4197_of__bool__eq_I1_J,axiom,
% 9.20/5.16      ( ( zero_n3304061248610475627l_real @ $false )
% 9.20/5.16      = zero_zero_real ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4198_of__bool__eq_I1_J,axiom,
% 9.20/5.16      ( ( zero_n2052037380579107095ol_rat @ $false )
% 9.20/5.16      = zero_zero_rat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4199_of__bool__eq_I1_J,axiom,
% 9.20/5.16      ( ( zero_n2687167440665602831ol_nat @ $false )
% 9.20/5.16      = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4200_of__bool__eq_I1_J,axiom,
% 9.20/5.16      ( ( zero_n2684676970156552555ol_int @ $false )
% 9.20/5.16      = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4201_of__bool__eq__0__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.20/5.16          = zero_zero_complex )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4202_of__bool__eq__0__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.20/5.16          = zero_zero_real )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4203_of__bool__eq__0__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.20/5.16          = zero_zero_rat )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4204_of__bool__eq__0__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.20/5.16          = zero_zero_nat )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4205_of__bool__eq__0__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.20/5.16          = zero_zero_int )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4206_of__bool__less__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) )
% 9.20/5.16        = ( ~ P
% 9.20/5.16          & Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4207_of__bool__less__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
% 9.20/5.16        = ( ~ P
% 9.20/5.16          & Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4208_of__bool__less__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) )
% 9.20/5.16        = ( ~ P
% 9.20/5.16          & Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4209_of__bool__less__iff,axiom,
% 9.20/5.16      ! [P: $o,Q: $o] :
% 9.20/5.16        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) )
% 9.20/5.16        = ( ~ P
% 9.20/5.16          & Q ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4210_of__bool__eq__1__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n1201886186963655149omplex @ P )
% 9.20/5.16          = one_one_complex )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4211_of__bool__eq__1__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n3304061248610475627l_real @ P )
% 9.20/5.16          = one_one_real )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4212_of__bool__eq__1__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2052037380579107095ol_rat @ P )
% 9.20/5.16          = one_one_rat )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4213_of__bool__eq__1__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2687167440665602831ol_nat @ P )
% 9.20/5.16          = one_one_nat )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4214_of__bool__eq__1__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ( zero_n2684676970156552555ol_int @ P )
% 9.20/5.16          = one_one_int )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4215_of__bool__eq_I2_J,axiom,
% 9.20/5.16      ( ( zero_n1201886186963655149omplex @ $true )
% 9.20/5.16      = one_one_complex ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4216_of__bool__eq_I2_J,axiom,
% 9.20/5.16      ( ( zero_n3304061248610475627l_real @ $true )
% 9.20/5.16      = one_one_real ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4217_of__bool__eq_I2_J,axiom,
% 9.20/5.16      ( ( zero_n2052037380579107095ol_rat @ $true )
% 9.20/5.16      = one_one_rat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4218_of__bool__eq_I2_J,axiom,
% 9.20/5.16      ( ( zero_n2687167440665602831ol_nat @ $true )
% 9.20/5.16      = one_one_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4219_of__bool__eq_I2_J,axiom,
% 9.20/5.16      ( ( zero_n2684676970156552555ol_int @ $true )
% 9.20/5.16      = one_one_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4220_signed__take__bit__of__0,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( bit_ri631733984087533419it_int @ N2 @ zero_zero_int )
% 9.20/5.16        = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4221_diff__ge__0__iff__ge,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.20/5.16        = ( ord_less_eq_real @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4222_diff__ge__0__iff__ge,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.16        = ( ord_less_eq_rat @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4223_diff__ge__0__iff__ge,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.20/5.16        = ( ord_less_eq_int @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4224_diff__gt__0__iff__gt,axiom,
% 9.20/5.16      ! [A: real,B: real] :
% 9.20/5.16        ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
% 9.20/5.16        = ( ord_less_real @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4225_diff__gt__0__iff__gt,axiom,
% 9.20/5.16      ! [A: rat,B: rat] :
% 9.20/5.16        ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.16        = ( ord_less_rat @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4226_diff__gt__0__iff__gt,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
% 9.20/5.16        = ( ord_less_int @ B @ A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4227_le__add__diff__inverse2,axiom,
% 9.20/5.16      ! [B: real,A: real] :
% 9.20/5.16        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.16       => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4228_le__add__diff__inverse2,axiom,
% 9.20/5.16      ! [B: rat,A: rat] :
% 9.20/5.16        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.16       => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4229_le__add__diff__inverse2,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ B @ A )
% 9.20/5.16       => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4230_le__add__diff__inverse2,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ B @ A )
% 9.20/5.16       => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4231_le__add__diff__inverse,axiom,
% 9.20/5.16      ! [B: real,A: real] :
% 9.20/5.16        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.16       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4232_le__add__diff__inverse,axiom,
% 9.20/5.16      ! [B: rat,A: rat] :
% 9.20/5.16        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.16       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4233_le__add__diff__inverse,axiom,
% 9.20/5.16      ! [B: nat,A: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ B @ A )
% 9.20/5.16       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4234_le__add__diff__inverse,axiom,
% 9.20/5.16      ! [B: int,A: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ B @ A )
% 9.20/5.16       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.20/5.16          = A ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4235_diff__numeral__special_I9_J,axiom,
% 9.20/5.16      ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
% 9.20/5.16      = zero_zero_complex ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4236_diff__numeral__special_I9_J,axiom,
% 9.20/5.16      ( ( minus_minus_real @ one_one_real @ one_one_real )
% 9.20/5.16      = zero_zero_real ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4237_diff__numeral__special_I9_J,axiom,
% 9.20/5.16      ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
% 9.20/5.16      = zero_zero_rat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4238_diff__numeral__special_I9_J,axiom,
% 9.20/5.16      ( ( minus_minus_int @ one_one_int @ one_one_int )
% 9.20/5.16      = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4239_diff__add__zero,axiom,
% 9.20/5.16      ! [A: nat,B: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4240_div__diff,axiom,
% 9.20/5.16      ! [C: int,A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ C @ A )
% 9.20/5.16       => ( ( dvd_dvd_int @ C @ B )
% 9.20/5.16         => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.16            = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4241_div__diff,axiom,
% 9.20/5.16      ! [C: code_integer,A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ C @ A )
% 9.20/5.16       => ( ( dvd_dvd_Code_integer @ C @ B )
% 9.20/5.16         => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
% 9.20/5.16            = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4242_singleton__insert__inj__eq_H,axiom,
% 9.20/5.16      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.16        ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.20/5.16          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4243_singleton__insert__inj__eq_H,axiom,
% 9.20/5.16      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.20/5.16        ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.20/5.16          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4244_singleton__insert__inj__eq_H,axiom,
% 9.20/5.16      ! [A: real,A2: set_real,B: real] :
% 9.20/5.16        ( ( ( insert_real @ A @ A2 )
% 9.20/5.16          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4245_singleton__insert__inj__eq_H,axiom,
% 9.20/5.16      ! [A: nat,A2: set_nat,B: nat] :
% 9.20/5.16        ( ( ( insert_nat @ A @ A2 )
% 9.20/5.16          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4246_singleton__insert__inj__eq_H,axiom,
% 9.20/5.16      ! [A: int,A2: set_int,B: int] :
% 9.20/5.16        ( ( ( insert_int @ A @ A2 )
% 9.20/5.16          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4247_singleton__insert__inj__eq,axiom,
% 9.20/5.16      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.16        ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
% 9.20/5.16          = ( insert9069300056098147895at_nat @ A @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4248_singleton__insert__inj__eq,axiom,
% 9.20/5.16      ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.16        ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
% 9.20/5.16          = ( insert8211810215607154385at_nat @ A @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4249_singleton__insert__inj__eq,axiom,
% 9.20/5.16      ! [B: real,A: real,A2: set_real] :
% 9.20/5.16        ( ( ( insert_real @ B @ bot_bot_set_real )
% 9.20/5.16          = ( insert_real @ A @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4250_singleton__insert__inj__eq,axiom,
% 9.20/5.16      ! [B: nat,A: nat,A2: set_nat] :
% 9.20/5.16        ( ( ( insert_nat @ B @ bot_bot_set_nat )
% 9.20/5.16          = ( insert_nat @ A @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4251_singleton__insert__inj__eq,axiom,
% 9.20/5.16      ! [B: int,A: int,A2: set_int] :
% 9.20/5.16        ( ( ( insert_int @ B @ bot_bot_set_int )
% 9.20/5.16          = ( insert_int @ A @ A2 ) )
% 9.20/5.16        = ( ( A = B )
% 9.20/5.16          & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4252_zero__less__of__bool__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4253_zero__less__of__bool__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4254_zero__less__of__bool__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4255_zero__less__of__bool__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.16        = P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4256_zero__less__diff,axiom,
% 9.20/5.16      ! [N2: nat,M: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
% 9.20/5.16        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4257_diff__is__0__eq,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ( minus_minus_nat @ M @ N2 )
% 9.20/5.16          = zero_zero_nat )
% 9.20/5.16        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4258_diff__is__0__eq_H,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.16       => ( ( minus_minus_nat @ M @ N2 )
% 9.20/5.16          = zero_zero_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4259_of__bool__less__one__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4260_of__bool__less__one__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4261_of__bool__less__one__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4262_of__bool__less__one__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( ord_less_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int )
% 9.20/5.16        = ~ P ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4263_of__bool__not__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( zero_n1201886186963655149omplex @ ~ P )
% 9.20/5.16        = ( minus_minus_complex @ one_one_complex @ ( zero_n1201886186963655149omplex @ P ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4264_of__bool__not__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( zero_n3304061248610475627l_real @ ~ P )
% 9.20/5.16        = ( minus_minus_real @ one_one_real @ ( zero_n3304061248610475627l_real @ P ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4265_of__bool__not__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( zero_n2052037380579107095ol_rat @ ~ P )
% 9.20/5.16        = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4266_of__bool__not__iff,axiom,
% 9.20/5.16      ! [P: $o] :
% 9.20/5.16        ( ( zero_n2684676970156552555ol_int @ ~ P )
% 9.20/5.16        = ( minus_minus_int @ one_one_int @ ( zero_n2684676970156552555ol_int @ P ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4267_Nat_Odiff__diff__right,axiom,
% 9.20/5.16      ! [K: nat,J: nat,I: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.16       => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.20/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4268_Nat_Oadd__diff__assoc2,axiom,
% 9.20/5.16      ! [K: nat,J: nat,I: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.16       => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.20/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4269_Nat_Oadd__diff__assoc,axiom,
% 9.20/5.16      ! [K: nat,J: nat,I: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.16       => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.20/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4270_diff__Suc__1,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
% 9.20/5.16        = N2 ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4271_Suc__0__mod__eq,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.16        = ( zero_n2687167440665602831ol_nat
% 9.20/5.16          @ ( N2
% 9.20/5.16           != ( suc @ zero_zero_nat ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4272_mod__pos__pos__trivial,axiom,
% 9.20/5.16      ! [K: int,L: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.16       => ( ( ord_less_int @ K @ L )
% 9.20/5.16         => ( ( modulo_modulo_int @ K @ L )
% 9.20/5.16            = K ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4273_mod__neg__neg__trivial,axiom,
% 9.20/5.16      ! [K: int,L: int] :
% 9.20/5.16        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.20/5.16       => ( ( ord_less_int @ L @ K )
% 9.20/5.16         => ( ( modulo_modulo_int @ K @ L )
% 9.20/5.16            = K ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4274_signed__take__bit__Suc__1,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ one_one_int )
% 9.20/5.16        = one_one_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4275_Suc__pred,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16       => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 9.20/5.16          = N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4276_diff__Suc__diff__eq2,axiom,
% 9.20/5.16      ! [K: nat,J: nat,I: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.16       => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
% 9.20/5.16          = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4277_diff__Suc__diff__eq1,axiom,
% 9.20/5.16      ! [K: nat,J: nat,I: nat] :
% 9.20/5.16        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.16       => ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
% 9.20/5.16          = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4278_signed__take__bit__Suc__bit0,axiom,
% 9.20/5.16      ! [N2: nat,K: num] :
% 9.20/5.16        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.20/5.16        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4279_Suc__diff__1,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.16       => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
% 9.20/5.16          = N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4280_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 9.20/5.16          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4281_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: vEBT_VEBT] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 9.20/5.16          = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4282_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: product_prod_nat_nat] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 9.20/5.16          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4283_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: real] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 9.20/5.16          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4284_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: nat] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 9.20/5.16          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4285_set__replicate,axiom,
% 9.20/5.16      ! [N2: nat,X3: int] :
% 9.20/5.16        ( ( N2 != zero_zero_nat )
% 9.20/5.16       => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 9.20/5.16          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4286_even__diff,axiom,
% 9.20/5.16      ! [A: code_integer,B: code_integer] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.20/5.16        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4287_even__diff,axiom,
% 9.20/5.16      ! [A: int,B: int] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
% 9.20/5.16        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4288_of__bool__half__eq__0,axiom,
% 9.20/5.16      ! [B: $o] :
% 9.20/5.16        ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.16        = zero_z3403309356797280102nteger ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4289_of__bool__half__eq__0,axiom,
% 9.20/5.16      ! [B: $o] :
% 9.20/5.16        ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.16        = zero_zero_nat ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4290_of__bool__half__eq__0,axiom,
% 9.20/5.16      ! [B: $o] :
% 9.20/5.16        ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.16        = zero_zero_int ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4291_odd__Suc__minus__one,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.16       => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
% 9.20/5.16          = N2 ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4292_even__diff__nat,axiom,
% 9.20/5.16      ! [M: nat,N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.16        = ( ( ord_less_nat @ M @ N2 )
% 9.20/5.16          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4293_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
% 9.20/5.16        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4294_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) )
% 9.20/5.16        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.16  
% 9.20/5.16  thf(fact_4295_semiring__parity__class_Oeven__mask__iff,axiom,
% 9.20/5.16      ! [N2: nat] :
% 9.20/5.16        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
% 9.20/5.17        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4296_one__div__2__pow__eq,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4297_one__div__2__pow__eq,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4298_one__div__2__pow__eq,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4299_bits__1__div__exp,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4300_bits__1__div__exp,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4301_bits__1__div__exp,axiom,
% 9.20/5.17      ! [N2: nat] :
% 9.20/5.17        ( ( divide_divide_int @ one_one_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.17        = ( zero_n2684676970156552555ol_int @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4302_flip__bit__0,axiom,
% 9.20/5.17      ! [A: code_integer] :
% 9.20/5.17        ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
% 9.20/5.17        = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4303_flip__bit__0,axiom,
% 9.20/5.17      ! [A: nat] :
% 9.20/5.17        ( ( bit_se2161824704523386999it_nat @ zero_zero_nat @ A )
% 9.20/5.17        = ( 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 ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4304_flip__bit__0,axiom,
% 9.20/5.17      ! [A: int] :
% 9.20/5.17        ( ( bit_se2159334234014336723it_int @ zero_zero_nat @ A )
% 9.20/5.17        = ( 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 ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4305_mod__int__unique,axiom,
% 9.20/5.17      ! [K: int,L: int,Q4: int,R2: int] :
% 9.20/5.17        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.17       => ( ( modulo_modulo_int @ K @ L )
% 9.20/5.17          = R2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4306_gcd__nat_Oasym,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17       => ~ ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.17            & ( B != A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4307_gcd__nat_Orefl,axiom,
% 9.20/5.17      ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4308_gcd__nat_Otrans,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17       => ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.17         => ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4309_gcd__nat_Oeq__iff,axiom,
% 9.20/5.17      ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
% 9.20/5.17      = ( ^ [A3: nat,B3: nat] :
% 9.20/5.17            ( ( dvd_dvd_nat @ A3 @ B3 )
% 9.20/5.17            & ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4310_gcd__nat_Oirrefl,axiom,
% 9.20/5.17      ! [A: nat] :
% 9.20/5.17        ~ ( ( dvd_dvd_nat @ A @ A )
% 9.20/5.17          & ( A != A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4311_gcd__nat_Oantisym,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17       => ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.17         => ( A = B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4312_gcd__nat_Ostrict__trans,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17       => ( ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.17            & ( B != C ) )
% 9.20/5.17         => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.17            & ( A != C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4313_gcd__nat_Ostrict__trans1,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17       => ( ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.17            & ( B != C ) )
% 9.20/5.17         => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.17            & ( A != C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4314_gcd__nat_Ostrict__trans2,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17       => ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.17         => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.17            & ( A != C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4315_gcd__nat_Ostrict__iff__not,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17        = ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4316_gcd__nat_Oorder__iff__strict,axiom,
% 9.20/5.17      ( dvd_dvd_nat
% 9.20/5.17      = ( ^ [A3: nat,B3: nat] :
% 9.20/5.17            ( ( ( dvd_dvd_nat @ A3 @ B3 )
% 9.20/5.17              & ( A3 != B3 ) )
% 9.20/5.17            | ( A3 = B3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4317_gcd__nat_Ostrict__iff__order,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17        = ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4318_gcd__nat_Ostrict__implies__order,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17       => ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4319_gcd__nat_Ostrict__implies__not__eq,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17          & ( A != B ) )
% 9.20/5.17       => ( A != B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4320_gcd__nat_Onot__eq__order__implies__strict,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( A != B )
% 9.20/5.17       => ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17         => ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.17            & ( A != B ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4321_dvd__antisym,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ M @ N2 )
% 9.20/5.17       => ( ( dvd_dvd_nat @ N2 @ M )
% 9.20/5.17         => ( M = N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4322_dvd__diff__nat,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ K @ M )
% 9.20/5.17       => ( ( dvd_dvd_nat @ K @ N2 )
% 9.20/5.17         => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4323_eucl__rel__int,axiom,
% 9.20/5.17      ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4324_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.17       => ? [B9: set_Pr4329608150637261639at_nat] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert9069300056098147895at_nat @ A @ B9 ) )
% 9.20/5.17            & ~ ( member8757157785044589968at_nat @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4325_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.17       => ? [B9: set_Pr1261947904930325089at_nat] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert8211810215607154385at_nat @ A @ B9 ) )
% 9.20/5.17            & ~ ( member8440522571783428010at_nat @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4326_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: real,A2: set_real] :
% 9.20/5.17        ( ( member_real @ A @ A2 )
% 9.20/5.17       => ? [B9: set_real] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert_real @ A @ B9 ) )
% 9.20/5.17            & ~ ( member_real @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4327_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: set_nat,A2: set_set_nat] :
% 9.20/5.17        ( ( member_set_nat @ A @ A2 )
% 9.20/5.17       => ? [B9: set_set_nat] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert_set_nat @ A @ B9 ) )
% 9.20/5.17            & ~ ( member_set_nat @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4328_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: nat,A2: set_nat] :
% 9.20/5.17        ( ( member_nat @ A @ A2 )
% 9.20/5.17       => ? [B9: set_nat] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert_nat @ A @ B9 ) )
% 9.20/5.17            & ~ ( member_nat @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4329_mk__disjoint__insert,axiom,
% 9.20/5.17      ! [A: int,A2: set_int] :
% 9.20/5.17        ( ( member_int @ A @ A2 )
% 9.20/5.17       => ? [B9: set_int] :
% 9.20/5.17            ( ( A2
% 9.20/5.17              = ( insert_int @ A @ B9 ) )
% 9.20/5.17            & ~ ( member_int @ A @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4330_insert__commute,axiom,
% 9.20/5.17      ! [X3: nat,Y3: nat,A2: set_nat] :
% 9.20/5.17        ( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A2 ) )
% 9.20/5.17        = ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4331_insert__commute,axiom,
% 9.20/5.17      ! [X3: int,Y3: int,A2: set_int] :
% 9.20/5.17        ( ( insert_int @ X3 @ ( insert_int @ Y3 @ A2 ) )
% 9.20/5.17        = ( insert_int @ Y3 @ ( insert_int @ X3 @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4332_insert__commute,axiom,
% 9.20/5.17      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ A2 ) )
% 9.20/5.17        = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4333_insert__commute,axiom,
% 9.20/5.17      ! [X3: real,Y3: real,A2: set_real] :
% 9.20/5.17        ( ( insert_real @ X3 @ ( insert_real @ Y3 @ A2 ) )
% 9.20/5.17        = ( insert_real @ Y3 @ ( insert_real @ X3 @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4334_insert__commute,axiom,
% 9.20/5.17      ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y3 @ A2 ) )
% 9.20/5.17        = ( insert9069300056098147895at_nat @ Y3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4335_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.17       => ( ~ ( member8757157785044589968at_nat @ B @ B2 )
% 9.20/5.17         => ( ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.20/5.17              = ( insert9069300056098147895at_nat @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert9069300056098147895at_nat @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member8757157785044589968at_nat @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert9069300056098147895at_nat @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member8757157785044589968at_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4336_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.17       => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
% 9.20/5.17         => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.20/5.17              = ( insert8211810215607154385at_nat @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert8211810215607154385at_nat @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member8440522571783428010at_nat @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert8211810215607154385at_nat @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member8440522571783428010at_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4337_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: real,A2: set_real,B: real,B2: set_real] :
% 9.20/5.17        ( ~ ( member_real @ A @ A2 )
% 9.20/5.17       => ( ~ ( member_real @ B @ B2 )
% 9.20/5.17         => ( ( ( insert_real @ A @ A2 )
% 9.20/5.17              = ( insert_real @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_real] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert_real @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member_real @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert_real @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member_real @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4338_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
% 9.20/5.17        ( ~ ( member_set_nat @ A @ A2 )
% 9.20/5.17       => ( ~ ( member_set_nat @ B @ B2 )
% 9.20/5.17         => ( ( ( insert_set_nat @ A @ A2 )
% 9.20/5.17              = ( insert_set_nat @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_set_nat] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert_set_nat @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member_set_nat @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert_set_nat @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member_set_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4339_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
% 9.20/5.17        ( ~ ( member_nat @ A @ A2 )
% 9.20/5.17       => ( ~ ( member_nat @ B @ B2 )
% 9.20/5.17         => ( ( ( insert_nat @ A @ A2 )
% 9.20/5.17              = ( insert_nat @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_nat] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert_nat @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member_nat @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert_nat @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member_nat @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4340_insert__eq__iff,axiom,
% 9.20/5.17      ! [A: int,A2: set_int,B: int,B2: set_int] :
% 9.20/5.17        ( ~ ( member_int @ A @ A2 )
% 9.20/5.17       => ( ~ ( member_int @ B @ B2 )
% 9.20/5.17         => ( ( ( insert_int @ A @ A2 )
% 9.20/5.17              = ( insert_int @ B @ B2 ) )
% 9.20/5.17            = ( ( ( A = B )
% 9.20/5.17               => ( A2 = B2 ) )
% 9.20/5.17              & ( ( A != B )
% 9.20/5.17               => ? [C6: set_int] :
% 9.20/5.17                    ( ( A2
% 9.20/5.17                      = ( insert_int @ B @ C6 ) )
% 9.20/5.17                    & ~ ( member_int @ B @ C6 )
% 9.20/5.17                    & ( B2
% 9.20/5.17                      = ( insert_int @ A @ C6 ) )
% 9.20/5.17                    & ~ ( member_int @ A @ C6 ) ) ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4341_insert__absorb,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.17       => ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4342_insert__absorb,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.17       => ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4343_insert__absorb,axiom,
% 9.20/5.17      ! [A: real,A2: set_real] :
% 9.20/5.17        ( ( member_real @ A @ A2 )
% 9.20/5.17       => ( ( insert_real @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4344_insert__absorb,axiom,
% 9.20/5.17      ! [A: set_nat,A2: set_set_nat] :
% 9.20/5.17        ( ( member_set_nat @ A @ A2 )
% 9.20/5.17       => ( ( insert_set_nat @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4345_insert__absorb,axiom,
% 9.20/5.17      ! [A: nat,A2: set_nat] :
% 9.20/5.17        ( ( member_nat @ A @ A2 )
% 9.20/5.17       => ( ( insert_nat @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4346_insert__absorb,axiom,
% 9.20/5.17      ! [A: int,A2: set_int] :
% 9.20/5.17        ( ( member_int @ A @ A2 )
% 9.20/5.17       => ( ( insert_int @ A @ A2 )
% 9.20/5.17          = A2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4347_insert__ident,axiom,
% 9.20/5.17      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert9069300056098147895at_nat @ X3 @ A2 )
% 9.20/5.17              = ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4348_insert__ident,axiom,
% 9.20/5.17      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert8211810215607154385at_nat @ X3 @ A2 )
% 9.20/5.17              = ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4349_insert__ident,axiom,
% 9.20/5.17      ! [X3: real,A2: set_real,B2: set_real] :
% 9.20/5.17        ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member_real @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert_real @ X3 @ A2 )
% 9.20/5.17              = ( insert_real @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4350_insert__ident,axiom,
% 9.20/5.17      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.17        ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member_set_nat @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert_set_nat @ X3 @ A2 )
% 9.20/5.17              = ( insert_set_nat @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4351_insert__ident,axiom,
% 9.20/5.17      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.17        ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member_nat @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert_nat @ X3 @ A2 )
% 9.20/5.17              = ( insert_nat @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4352_insert__ident,axiom,
% 9.20/5.17      ! [X3: int,A2: set_int,B2: set_int] :
% 9.20/5.17        ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.17       => ( ~ ( member_int @ X3 @ B2 )
% 9.20/5.17         => ( ( ( insert_int @ X3 @ A2 )
% 9.20/5.17              = ( insert_int @ X3 @ B2 ) )
% 9.20/5.17            = ( A2 = B2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4353_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_Pr4329608150637261639at_nat] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert9069300056098147895at_nat @ X3 @ B9 ) )
% 9.20/5.17             => ( member8757157785044589968at_nat @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4354_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_Pr1261947904930325089at_nat] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert8211810215607154385at_nat @ X3 @ B9 ) )
% 9.20/5.17             => ( member8440522571783428010at_nat @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4355_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: real,A2: set_real] :
% 9.20/5.17        ( ( member_real @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_real] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert_real @ X3 @ B9 ) )
% 9.20/5.17             => ( member_real @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4356_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: set_nat,A2: set_set_nat] :
% 9.20/5.17        ( ( member_set_nat @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_set_nat] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert_set_nat @ X3 @ B9 ) )
% 9.20/5.17             => ( member_set_nat @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4357_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: nat,A2: set_nat] :
% 9.20/5.17        ( ( member_nat @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_nat] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert_nat @ X3 @ B9 ) )
% 9.20/5.17             => ( member_nat @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4358_Set_Oset__insert,axiom,
% 9.20/5.17      ! [X3: int,A2: set_int] :
% 9.20/5.17        ( ( member_int @ X3 @ A2 )
% 9.20/5.17       => ~ ! [B9: set_int] :
% 9.20/5.17              ( ( A2
% 9.20/5.17                = ( insert_int @ X3 @ B9 ) )
% 9.20/5.17             => ( member_int @ X3 @ B9 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4359_insertI2,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ A @ B2 )
% 9.20/5.17       => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4360_insertI2,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ A @ B2 )
% 9.20/5.17       => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4361_insertI2,axiom,
% 9.20/5.17      ! [A: real,B2: set_real,B: real] :
% 9.20/5.17        ( ( member_real @ A @ B2 )
% 9.20/5.17       => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4362_insertI2,axiom,
% 9.20/5.17      ! [A: set_nat,B2: set_set_nat,B: set_nat] :
% 9.20/5.17        ( ( member_set_nat @ A @ B2 )
% 9.20/5.17       => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4363_insertI2,axiom,
% 9.20/5.17      ! [A: nat,B2: set_nat,B: nat] :
% 9.20/5.17        ( ( member_nat @ A @ B2 )
% 9.20/5.17       => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4364_insertI2,axiom,
% 9.20/5.17      ! [A: int,B2: set_int,B: int] :
% 9.20/5.17        ( ( member_int @ A @ B2 )
% 9.20/5.17       => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4365_insertI1,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4366_insertI1,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4367_insertI1,axiom,
% 9.20/5.17      ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4368_insertI1,axiom,
% 9.20/5.17      ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4369_insertI1,axiom,
% 9.20/5.17      ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4370_insertI1,axiom,
% 9.20/5.17      ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4371_insertE,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4372_insertE,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4373_insertE,axiom,
% 9.20/5.17      ! [A: real,B: real,A2: set_real] :
% 9.20/5.17        ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member_real @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4374_insertE,axiom,
% 9.20/5.17      ! [A: set_nat,B: set_nat,A2: set_set_nat] :
% 9.20/5.17        ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member_set_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4375_insertE,axiom,
% 9.20/5.17      ! [A: nat,B: nat,A2: set_nat] :
% 9.20/5.17        ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4376_insertE,axiom,
% 9.20/5.17      ! [A: int,B: int,A2: set_int] :
% 9.20/5.17        ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
% 9.20/5.17       => ( ( A != B )
% 9.20/5.17         => ( member_int @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4377_of__bool__eq__iff,axiom,
% 9.20/5.17      ! [P4: $o,Q4: $o] :
% 9.20/5.17        ( ( ( zero_n2687167440665602831ol_nat @ P4 )
% 9.20/5.17          = ( zero_n2687167440665602831ol_nat @ Q4 ) )
% 9.20/5.17        = ( P4 = Q4 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4378_of__bool__eq__iff,axiom,
% 9.20/5.17      ! [P4: $o,Q4: $o] :
% 9.20/5.17        ( ( ( zero_n2684676970156552555ol_int @ P4 )
% 9.20/5.17          = ( zero_n2684676970156552555ol_int @ Q4 ) )
% 9.20/5.17        = ( P4 = Q4 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4379_diff__commute,axiom,
% 9.20/5.17      ! [I: nat,J: nat,K: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
% 9.20/5.17        = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4380_diff__right__commute,axiom,
% 9.20/5.17      ! [A: real,C: real,B: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
% 9.20/5.17        = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4381_diff__right__commute,axiom,
% 9.20/5.17      ! [A: rat,C: rat,B: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
% 9.20/5.17        = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4382_diff__right__commute,axiom,
% 9.20/5.17      ! [A: nat,C: nat,B: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
% 9.20/5.17        = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4383_diff__right__commute,axiom,
% 9.20/5.17      ! [A: int,C: int,B: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 9.20/5.17        = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4384_diff__eq__diff__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.17        ( ( ( minus_minus_real @ A @ B )
% 9.20/5.17          = ( minus_minus_real @ C @ D ) )
% 9.20/5.17       => ( ( A = B )
% 9.20/5.17          = ( C = D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4385_diff__eq__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.17        ( ( ( minus_minus_rat @ A @ B )
% 9.20/5.17          = ( minus_minus_rat @ C @ D ) )
% 9.20/5.17       => ( ( A = B )
% 9.20/5.17          = ( C = D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4386_diff__eq__diff__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.17        ( ( ( minus_minus_int @ A @ B )
% 9.20/5.17          = ( minus_minus_int @ C @ D ) )
% 9.20/5.17       => ( ( A = B )
% 9.20/5.17          = ( C = D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4387_full__exhaustive__int_H_Ocases,axiom,
% 9.20/5.17      ! [X3: produc2285326912895808259nt_int] :
% 9.20/5.17        ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D4: int,I3: int] :
% 9.20/5.17            ( X3
% 9.20/5.17           != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4388_exhaustive__int_H_Ocases,axiom,
% 9.20/5.17      ! [X3: produc7773217078559923341nt_int] :
% 9.20/5.17        ~ ! [F2: int > option6357759511663192854e_term,D4: int,I3: int] :
% 9.20/5.17            ( X3
% 9.20/5.17           != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4389_small__lazy_H_Ocases,axiom,
% 9.20/5.17      ! [X3: product_prod_int_int] :
% 9.20/5.17        ~ ! [D4: int,I3: int] :
% 9.20/5.17            ( X3
% 9.20/5.17           != ( product_Pair_int_int @ D4 @ I3 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4390_unique__quotient,axiom,
% 9.20/5.17      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
% 9.20/5.17        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.17       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
% 9.20/5.17         => ( Q4 = Q6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4391_unique__remainder,axiom,
% 9.20/5.17      ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
% 9.20/5.17        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.17       => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
% 9.20/5.17         => ( R2 = R3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4392_of__bool__conj,axiom,
% 9.20/5.17      ! [P: $o,Q: $o] :
% 9.20/5.17        ( ( zero_n1201886186963655149omplex
% 9.20/5.17          @ ( P
% 9.20/5.17            & Q ) )
% 9.20/5.17        = ( times_times_complex @ ( zero_n1201886186963655149omplex @ P ) @ ( zero_n1201886186963655149omplex @ Q ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4393_of__bool__conj,axiom,
% 9.20/5.17      ! [P: $o,Q: $o] :
% 9.20/5.17        ( ( zero_n3304061248610475627l_real
% 9.20/5.17          @ ( P
% 9.20/5.17            & Q ) )
% 9.20/5.17        = ( times_times_real @ ( zero_n3304061248610475627l_real @ P ) @ ( zero_n3304061248610475627l_real @ Q ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4394_of__bool__conj,axiom,
% 9.20/5.17      ! [P: $o,Q: $o] :
% 9.20/5.17        ( ( zero_n2052037380579107095ol_rat
% 9.20/5.17          @ ( P
% 9.20/5.17            & Q ) )
% 9.20/5.17        = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4395_of__bool__conj,axiom,
% 9.20/5.17      ! [P: $o,Q: $o] :
% 9.20/5.17        ( ( zero_n2687167440665602831ol_nat
% 9.20/5.17          @ ( P
% 9.20/5.17            & Q ) )
% 9.20/5.17        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ ( zero_n2687167440665602831ol_nat @ Q ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4396_of__bool__conj,axiom,
% 9.20/5.17      ! [P: $o,Q: $o] :
% 9.20/5.17        ( ( zero_n2684676970156552555ol_int
% 9.20/5.17          @ ( P
% 9.20/5.17            & Q ) )
% 9.20/5.17        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ P ) @ ( zero_n2684676970156552555ol_int @ Q ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4397_diff__mono,axiom,
% 9.20/5.17      ! [A: real,B: real,D: real,C: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_real @ D @ C )
% 9.20/5.17         => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4398_diff__mono,axiom,
% 9.20/5.17      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_rat @ D @ C )
% 9.20/5.17         => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4399_diff__mono,axiom,
% 9.20/5.17      ! [A: int,B: int,D: int,C: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_int @ D @ C )
% 9.20/5.17         => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4400_diff__left__mono,axiom,
% 9.20/5.17      ! [B: real,A: real,C: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ B @ A )
% 9.20/5.17       => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4401_diff__left__mono,axiom,
% 9.20/5.17      ! [B: rat,A: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ B @ A )
% 9.20/5.17       => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4402_diff__left__mono,axiom,
% 9.20/5.17      ! [B: int,A: int,C: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ B @ A )
% 9.20/5.17       => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4403_diff__right__mono,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.17       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4404_diff__right__mono,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.17       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4405_diff__right__mono,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.17       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4406_diff__eq__diff__less__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.17        ( ( ( minus_minus_real @ A @ B )
% 9.20/5.17          = ( minus_minus_real @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_eq_real @ A @ B )
% 9.20/5.17          = ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4407_diff__eq__diff__less__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.17        ( ( ( minus_minus_rat @ A @ B )
% 9.20/5.17          = ( minus_minus_rat @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.17          = ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4408_diff__eq__diff__less__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.17        ( ( ( minus_minus_int @ A @ B )
% 9.20/5.17          = ( minus_minus_int @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_eq_int @ A @ B )
% 9.20/5.17          = ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4409_eq__iff__diff__eq__0,axiom,
% 9.20/5.17      ( ( ^ [Y4: complex,Z: complex] : ( Y4 = Z ) )
% 9.20/5.17      = ( ^ [A3: complex,B3: complex] :
% 9.20/5.17            ( ( minus_minus_complex @ A3 @ B3 )
% 9.20/5.17            = zero_zero_complex ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4410_eq__iff__diff__eq__0,axiom,
% 9.20/5.17      ( ( ^ [Y4: real,Z: real] : ( Y4 = Z ) )
% 9.20/5.17      = ( ^ [A3: real,B3: real] :
% 9.20/5.17            ( ( minus_minus_real @ A3 @ B3 )
% 9.20/5.17            = zero_zero_real ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4411_eq__iff__diff__eq__0,axiom,
% 9.20/5.17      ( ( ^ [Y4: rat,Z: rat] : ( Y4 = Z ) )
% 9.20/5.17      = ( ^ [A3: rat,B3: rat] :
% 9.20/5.17            ( ( minus_minus_rat @ A3 @ B3 )
% 9.20/5.17            = zero_zero_rat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4412_eq__iff__diff__eq__0,axiom,
% 9.20/5.17      ( ( ^ [Y4: int,Z: int] : ( Y4 = Z ) )
% 9.20/5.17      = ( ^ [A3: int,B3: int] :
% 9.20/5.17            ( ( minus_minus_int @ A3 @ B3 )
% 9.20/5.17            = zero_zero_int ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4413_diff__strict__mono,axiom,
% 9.20/5.17      ! [A: real,B: real,D: real,C: real] :
% 9.20/5.17        ( ( ord_less_real @ A @ B )
% 9.20/5.17       => ( ( ord_less_real @ D @ C )
% 9.20/5.17         => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4414_diff__strict__mono,axiom,
% 9.20/5.17      ! [A: rat,B: rat,D: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_rat @ A @ B )
% 9.20/5.17       => ( ( ord_less_rat @ D @ C )
% 9.20/5.17         => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4415_diff__strict__mono,axiom,
% 9.20/5.17      ! [A: int,B: int,D: int,C: int] :
% 9.20/5.17        ( ( ord_less_int @ A @ B )
% 9.20/5.17       => ( ( ord_less_int @ D @ C )
% 9.20/5.17         => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4416_diff__eq__diff__less,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.17        ( ( ( minus_minus_real @ A @ B )
% 9.20/5.17          = ( minus_minus_real @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_real @ A @ B )
% 9.20/5.17          = ( ord_less_real @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4417_diff__eq__diff__less,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.17        ( ( ( minus_minus_rat @ A @ B )
% 9.20/5.17          = ( minus_minus_rat @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_rat @ A @ B )
% 9.20/5.17          = ( ord_less_rat @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4418_diff__eq__diff__less,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.17        ( ( ( minus_minus_int @ A @ B )
% 9.20/5.17          = ( minus_minus_int @ C @ D ) )
% 9.20/5.17       => ( ( ord_less_int @ A @ B )
% 9.20/5.17          = ( ord_less_int @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4419_diff__strict__left__mono,axiom,
% 9.20/5.17      ! [B: real,A: real,C: real] :
% 9.20/5.17        ( ( ord_less_real @ B @ A )
% 9.20/5.17       => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4420_diff__strict__left__mono,axiom,
% 9.20/5.17      ! [B: rat,A: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_rat @ B @ A )
% 9.20/5.17       => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4421_diff__strict__left__mono,axiom,
% 9.20/5.17      ! [B: int,A: int,C: int] :
% 9.20/5.17        ( ( ord_less_int @ B @ A )
% 9.20/5.17       => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4422_diff__strict__right__mono,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( ord_less_real @ A @ B )
% 9.20/5.17       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4423_diff__strict__right__mono,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_rat @ A @ B )
% 9.20/5.17       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4424_diff__strict__right__mono,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( ord_less_int @ A @ B )
% 9.20/5.17       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4425_right__diff__distrib_H,axiom,
% 9.20/5.17      ! [A: complex,B: complex,C: complex] :
% 9.20/5.17        ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 9.20/5.17        = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4426_right__diff__distrib_H,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.20/5.17        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4427_right__diff__distrib_H,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4428_right__diff__distrib_H,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4429_right__diff__distrib_H,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.20/5.17        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4430_left__diff__distrib_H,axiom,
% 9.20/5.17      ! [B: complex,C: complex,A: complex] :
% 9.20/5.17        ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
% 9.20/5.17        = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4431_left__diff__distrib_H,axiom,
% 9.20/5.17      ! [B: real,C: real,A: real] :
% 9.20/5.17        ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
% 9.20/5.17        = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4432_left__diff__distrib_H,axiom,
% 9.20/5.17      ! [B: rat,C: rat,A: rat] :
% 9.20/5.17        ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
% 9.20/5.17        = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4433_left__diff__distrib_H,axiom,
% 9.20/5.17      ! [B: nat,C: nat,A: nat] :
% 9.20/5.17        ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
% 9.20/5.17        = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4434_left__diff__distrib_H,axiom,
% 9.20/5.17      ! [B: int,C: int,A: int] :
% 9.20/5.17        ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
% 9.20/5.17        = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4435_right__diff__distrib,axiom,
% 9.20/5.17      ! [A: complex,B: complex,C: complex] :
% 9.20/5.17        ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
% 9.20/5.17        = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4436_right__diff__distrib,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.20/5.17        = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4437_right__diff__distrib,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4438_right__diff__distrib,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.20/5.17        = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4439_left__diff__distrib,axiom,
% 9.20/5.17      ! [A: complex,B: complex,C: complex] :
% 9.20/5.17        ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4440_left__diff__distrib,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4441_left__diff__distrib,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4442_left__diff__distrib,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4443_add__diff__add,axiom,
% 9.20/5.17      ! [A: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
% 9.20/5.17        = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4444_add__diff__add,axiom,
% 9.20/5.17      ! [A: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
% 9.20/5.17        = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4445_add__diff__add,axiom,
% 9.20/5.17      ! [A: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
% 9.20/5.17        = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4446_group__cancel_Osub1,axiom,
% 9.20/5.17      ! [A2: real,K: real,A: real,B: real] :
% 9.20/5.17        ( ( A2
% 9.20/5.17          = ( plus_plus_real @ K @ A ) )
% 9.20/5.17       => ( ( minus_minus_real @ A2 @ B )
% 9.20/5.17          = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4447_group__cancel_Osub1,axiom,
% 9.20/5.17      ! [A2: rat,K: rat,A: rat,B: rat] :
% 9.20/5.17        ( ( A2
% 9.20/5.17          = ( plus_plus_rat @ K @ A ) )
% 9.20/5.17       => ( ( minus_minus_rat @ A2 @ B )
% 9.20/5.17          = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4448_group__cancel_Osub1,axiom,
% 9.20/5.17      ! [A2: int,K: int,A: int,B: int] :
% 9.20/5.17        ( ( A2
% 9.20/5.17          = ( plus_plus_int @ K @ A ) )
% 9.20/5.17       => ( ( minus_minus_int @ A2 @ B )
% 9.20/5.17          = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4449_diff__eq__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( ( minus_minus_real @ A @ B )
% 9.20/5.17          = C )
% 9.20/5.17        = ( A
% 9.20/5.17          = ( plus_plus_real @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4450_diff__eq__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( ( minus_minus_rat @ A @ B )
% 9.20/5.17          = C )
% 9.20/5.17        = ( A
% 9.20/5.17          = ( plus_plus_rat @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4451_diff__eq__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( ( minus_minus_int @ A @ B )
% 9.20/5.17          = C )
% 9.20/5.17        = ( A
% 9.20/5.17          = ( plus_plus_int @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4452_eq__diff__eq,axiom,
% 9.20/5.17      ! [A: real,C: real,B: real] :
% 9.20/5.17        ( ( A
% 9.20/5.17          = ( minus_minus_real @ C @ B ) )
% 9.20/5.17        = ( ( plus_plus_real @ A @ B )
% 9.20/5.17          = C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4453_eq__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,C: rat,B: rat] :
% 9.20/5.17        ( ( A
% 9.20/5.17          = ( minus_minus_rat @ C @ B ) )
% 9.20/5.17        = ( ( plus_plus_rat @ A @ B )
% 9.20/5.17          = C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4454_eq__diff__eq,axiom,
% 9.20/5.17      ! [A: int,C: int,B: int] :
% 9.20/5.17        ( ( A
% 9.20/5.17          = ( minus_minus_int @ C @ B ) )
% 9.20/5.17        = ( ( plus_plus_int @ A @ B )
% 9.20/5.17          = C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4455_add__diff__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.20/5.17        = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4456_add__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4457_add__diff__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.20/5.17        = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4458_diff__diff__eq2,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
% 9.20/5.17        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4459_diff__diff__eq2,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4460_diff__diff__eq2,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 9.20/5.17        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4461_diff__add__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4462_diff__add__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4463_diff__add__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4464_diff__add__eq__diff__diff__swap,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
% 9.20/5.17        = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4465_diff__add__eq__diff__diff__swap,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
% 9.20/5.17        = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4466_diff__add__eq__diff__diff__swap,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 9.20/5.17        = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4467_add__implies__diff,axiom,
% 9.20/5.17      ! [C: real,B: real,A: real] :
% 9.20/5.17        ( ( ( plus_plus_real @ C @ B )
% 9.20/5.17          = A )
% 9.20/5.17       => ( C
% 9.20/5.17          = ( minus_minus_real @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4468_add__implies__diff,axiom,
% 9.20/5.17      ! [C: rat,B: rat,A: rat] :
% 9.20/5.17        ( ( ( plus_plus_rat @ C @ B )
% 9.20/5.17          = A )
% 9.20/5.17       => ( C
% 9.20/5.17          = ( minus_minus_rat @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4469_add__implies__diff,axiom,
% 9.20/5.17      ! [C: nat,B: nat,A: nat] :
% 9.20/5.17        ( ( ( plus_plus_nat @ C @ B )
% 9.20/5.17          = A )
% 9.20/5.17       => ( C
% 9.20/5.17          = ( minus_minus_nat @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4470_add__implies__diff,axiom,
% 9.20/5.17      ! [C: int,B: int,A: int] :
% 9.20/5.17        ( ( ( plus_plus_int @ C @ B )
% 9.20/5.17          = A )
% 9.20/5.17       => ( C
% 9.20/5.17          = ( minus_minus_int @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4471_diff__diff__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4472_diff__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4473_diff__diff__eq,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4474_diff__diff__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4475_diff__divide__distrib,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4476_diff__divide__distrib,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4477_singletonD,axiom,
% 9.20/5.17      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4478_singletonD,axiom,
% 9.20/5.17      ! [B: set_nat,A: set_nat] :
% 9.20/5.17        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4479_singletonD,axiom,
% 9.20/5.17      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4480_singletonD,axiom,
% 9.20/5.17      ! [B: real,A: real] :
% 9.20/5.17        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4481_singletonD,axiom,
% 9.20/5.17      ! [B: nat,A: nat] :
% 9.20/5.17        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4482_singletonD,axiom,
% 9.20/5.17      ! [B: int,A: int] :
% 9.20/5.17        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.20/5.17       => ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4483_singleton__iff,axiom,
% 9.20/5.17      ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4484_singleton__iff,axiom,
% 9.20/5.17      ! [B: set_nat,A: set_nat] :
% 9.20/5.17        ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4485_singleton__iff,axiom,
% 9.20/5.17      ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4486_singleton__iff,axiom,
% 9.20/5.17      ! [B: real,A: real] :
% 9.20/5.17        ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4487_singleton__iff,axiom,
% 9.20/5.17      ! [B: nat,A: nat] :
% 9.20/5.17        ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4488_singleton__iff,axiom,
% 9.20/5.17      ! [B: int,A: int] :
% 9.20/5.17        ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.20/5.17        = ( B = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4489_doubleton__eq__iff,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17          = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.17        = ( ( ( A = C )
% 9.20/5.17            & ( B = D ) )
% 9.20/5.17          | ( ( A = D )
% 9.20/5.17            & ( B = C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4490_doubleton__eq__iff,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
% 9.20/5.17        ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17          = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.17        = ( ( ( A = C )
% 9.20/5.17            & ( B = D ) )
% 9.20/5.17          | ( ( A = D )
% 9.20/5.17            & ( B = C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4491_doubleton__eq__iff,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.17        ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
% 9.20/5.17          = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
% 9.20/5.17        = ( ( ( A = C )
% 9.20/5.17            & ( B = D ) )
% 9.20/5.17          | ( ( A = D )
% 9.20/5.17            & ( B = C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4492_doubleton__eq__iff,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.17        ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.20/5.17          = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
% 9.20/5.17        = ( ( ( A = C )
% 9.20/5.17            & ( B = D ) )
% 9.20/5.17          | ( ( A = D )
% 9.20/5.17            & ( B = C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4493_doubleton__eq__iff,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.17        ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
% 9.20/5.17          = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
% 9.20/5.17        = ( ( ( A = C )
% 9.20/5.17            & ( B = D ) )
% 9.20/5.17          | ( ( A = D )
% 9.20/5.17            & ( B = C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4494_insert__not__empty,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( insert9069300056098147895at_nat @ A @ A2 )
% 9.20/5.17       != bot_bo228742789529271731at_nat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4495_insert__not__empty,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( insert8211810215607154385at_nat @ A @ A2 )
% 9.20/5.17       != bot_bo2099793752762293965at_nat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4496_insert__not__empty,axiom,
% 9.20/5.17      ! [A: real,A2: set_real] :
% 9.20/5.17        ( ( insert_real @ A @ A2 )
% 9.20/5.17       != bot_bot_set_real ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4497_insert__not__empty,axiom,
% 9.20/5.17      ! [A: nat,A2: set_nat] :
% 9.20/5.17        ( ( insert_nat @ A @ A2 )
% 9.20/5.17       != bot_bot_set_nat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4498_insert__not__empty,axiom,
% 9.20/5.17      ! [A: int,A2: set_int] :
% 9.20/5.17        ( ( insert_int @ A @ A2 )
% 9.20/5.17       != bot_bot_set_int ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4499_singleton__inject,axiom,
% 9.20/5.17      ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
% 9.20/5.17          = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17       => ( A = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4500_singleton__inject,axiom,
% 9.20/5.17      ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
% 9.20/5.17        ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
% 9.20/5.17          = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17       => ( A = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4501_singleton__inject,axiom,
% 9.20/5.17      ! [A: real,B: real] :
% 9.20/5.17        ( ( ( insert_real @ A @ bot_bot_set_real )
% 9.20/5.17          = ( insert_real @ B @ bot_bot_set_real ) )
% 9.20/5.17       => ( A = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4502_singleton__inject,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ( insert_nat @ A @ bot_bot_set_nat )
% 9.20/5.17          = ( insert_nat @ B @ bot_bot_set_nat ) )
% 9.20/5.17       => ( A = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4503_singleton__inject,axiom,
% 9.20/5.17      ! [A: int,B: int] :
% 9.20/5.17        ( ( ( insert_int @ A @ bot_bot_set_int )
% 9.20/5.17          = ( insert_int @ B @ bot_bot_set_int ) )
% 9.20/5.17       => ( A = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4504_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_real,A: real] :
% 9.20/5.17        ( ( finite_finite_real @ A2 )
% 9.20/5.17       => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4505_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ A2 )
% 9.20/5.17       => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4506_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_nat,A: nat] :
% 9.20/5.17        ( ( finite_finite_nat @ A2 )
% 9.20/5.17       => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4507_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_int,A: int] :
% 9.20/5.17        ( ( finite_finite_int @ A2 )
% 9.20/5.17       => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4508_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_complex,A: complex] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.17       => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4509_finite_OinsertI,axiom,
% 9.20/5.17      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.20/5.17        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.17       => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4510_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le1268244103169919719at_nat @ X7 @ A2 )
% 9.20/5.17         => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4511_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le3146513528884898305at_nat @ X7 @ A2 )
% 9.20/5.17         => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4512_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: real,A2: set_real,X7: set_real] :
% 9.20/5.17        ( ( member_real @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_real @ X7 @ A2 )
% 9.20/5.17         => ( ord_less_eq_set_real @ ( insert_real @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4513_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: set_nat,A2: set_set_nat,X7: set_set_nat] :
% 9.20/5.17        ( ( member_set_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le6893508408891458716et_nat @ X7 @ A2 )
% 9.20/5.17         => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4514_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: nat,A2: set_nat,X7: set_nat] :
% 9.20/5.17        ( ( member_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_nat @ X7 @ A2 )
% 9.20/5.17         => ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4515_insert__subsetI,axiom,
% 9.20/5.17      ! [X3: int,A2: set_int,X7: set_int] :
% 9.20/5.17        ( ( member_int @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_int @ X7 @ A2 )
% 9.20/5.17         => ( ord_less_eq_set_int @ ( insert_int @ X3 @ X7 ) @ A2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4516_insert__mono,axiom,
% 9.20/5.17      ! [C4: set_nat,D6: set_nat,A: nat] :
% 9.20/5.17        ( ( ord_less_eq_set_nat @ C4 @ D6 )
% 9.20/5.17       => ( ord_less_eq_set_nat @ ( insert_nat @ A @ C4 ) @ ( insert_nat @ A @ D6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4517_insert__mono,axiom,
% 9.20/5.17      ! [C4: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.20/5.17        ( ( ord_le3146513528884898305at_nat @ C4 @ D6 )
% 9.20/5.17       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C4 ) @ ( insert8211810215607154385at_nat @ A @ D6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4518_insert__mono,axiom,
% 9.20/5.17      ! [C4: set_real,D6: set_real,A: real] :
% 9.20/5.17        ( ( ord_less_eq_set_real @ C4 @ D6 )
% 9.20/5.17       => ( ord_less_eq_set_real @ ( insert_real @ A @ C4 ) @ ( insert_real @ A @ D6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4519_insert__mono,axiom,
% 9.20/5.17      ! [C4: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ord_le1268244103169919719at_nat @ C4 @ D6 )
% 9.20/5.17       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C4 ) @ ( insert9069300056098147895at_nat @ A @ D6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4520_insert__mono,axiom,
% 9.20/5.17      ! [C4: set_int,D6: set_int,A: int] :
% 9.20/5.17        ( ( ord_less_eq_set_int @ C4 @ D6 )
% 9.20/5.17       => ( ord_less_eq_set_int @ ( insert_int @ A @ C4 ) @ ( insert_int @ A @ D6 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4521_subset__insert,axiom,
% 9.20/5.17      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4522_subset__insert,axiom,
% 9.20/5.17      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4523_subset__insert,axiom,
% 9.20/5.17      ! [X3: real,A2: set_real,B2: set_real] :
% 9.20/5.17        ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4524_subset__insert,axiom,
% 9.20/5.17      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.17        ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4525_subset__insert,axiom,
% 9.20/5.17      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.17        ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4526_subset__insert,axiom,
% 9.20/5.17      ! [X3: int,A2: set_int,B2: set_int] :
% 9.20/5.17        ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.17       => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 9.20/5.17          = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4527_subset__insertI,axiom,
% 9.20/5.17      ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4528_subset__insertI,axiom,
% 9.20/5.17      ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4529_subset__insertI,axiom,
% 9.20/5.17      ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4530_subset__insertI,axiom,
% 9.20/5.17      ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4531_subset__insertI,axiom,
% 9.20/5.17      ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4532_subset__insertI2,axiom,
% 9.20/5.17      ! [A2: set_nat,B2: set_nat,B: nat] :
% 9.20/5.17        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.17       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4533_subset__insertI2,axiom,
% 9.20/5.17      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.20/5.17        ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
% 9.20/5.17       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4534_subset__insertI2,axiom,
% 9.20/5.17      ! [A2: set_real,B2: set_real,B: real] :
% 9.20/5.17        ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.17       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4535_subset__insertI2,axiom,
% 9.20/5.17      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
% 9.20/5.17       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4536_subset__insertI2,axiom,
% 9.20/5.17      ! [A2: set_int,B2: set_int,B: int] :
% 9.20/5.17        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.20/5.17       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4537_dvd__diff,axiom,
% 9.20/5.17      ! [X3: code_integer,Y3: code_integer,Z2: code_integer] :
% 9.20/5.17        ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
% 9.20/5.17       => ( ( dvd_dvd_Code_integer @ X3 @ Z2 )
% 9.20/5.17         => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4538_dvd__diff,axiom,
% 9.20/5.17      ! [X3: real,Y3: real,Z2: real] :
% 9.20/5.17        ( ( dvd_dvd_real @ X3 @ Y3 )
% 9.20/5.17       => ( ( dvd_dvd_real @ X3 @ Z2 )
% 9.20/5.17         => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4539_dvd__diff,axiom,
% 9.20/5.17      ! [X3: rat,Y3: rat,Z2: rat] :
% 9.20/5.17        ( ( dvd_dvd_rat @ X3 @ Y3 )
% 9.20/5.17       => ( ( dvd_dvd_rat @ X3 @ Z2 )
% 9.20/5.17         => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4540_dvd__diff,axiom,
% 9.20/5.17      ! [X3: int,Y3: int,Z2: int] :
% 9.20/5.17        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.20/5.17       => ( ( dvd_dvd_int @ X3 @ Z2 )
% 9.20/5.17         => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4541_zero__induct__lemma,axiom,
% 9.20/5.17      ! [P: nat > $o,K: nat,I: nat] :
% 9.20/5.17        ( ( P @ K )
% 9.20/5.17       => ( ! [N3: nat] :
% 9.20/5.17              ( ( P @ ( suc @ N3 ) )
% 9.20/5.17             => ( P @ N3 ) )
% 9.20/5.17         => ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4542_minus__nat_Odiff__0,axiom,
% 9.20/5.17      ! [M: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ M @ zero_zero_nat )
% 9.20/5.17        = M ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4543_diffs0__imp__equal,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( ( minus_minus_nat @ M @ N2 )
% 9.20/5.17          = zero_zero_nat )
% 9.20/5.17       => ( ( ( minus_minus_nat @ N2 @ M )
% 9.20/5.17            = zero_zero_nat )
% 9.20/5.17         => ( M = N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4544_diff__less__mono2,axiom,
% 9.20/5.17      ! [M: nat,N2: nat,L: nat] :
% 9.20/5.17        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.17       => ( ( ord_less_nat @ M @ L )
% 9.20/5.17         => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4545_less__imp__diff__less,axiom,
% 9.20/5.17      ! [J: nat,K: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_nat @ J @ K )
% 9.20/5.17       => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4546_dvd__minus__self,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
% 9.20/5.17        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.17          | ( dvd_dvd_nat @ M @ N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4547_eq__diff__iff,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.17       => ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.17         => ( ( ( minus_minus_nat @ M @ K )
% 9.20/5.17              = ( minus_minus_nat @ N2 @ K ) )
% 9.20/5.17            = ( M = N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4548_le__diff__iff,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.17       => ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.17         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 9.20/5.17            = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4549_Nat_Odiff__diff__eq,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.17       => ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.17         => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 9.20/5.17            = ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4550_diff__le__mono,axiom,
% 9.20/5.17      ! [M: nat,N2: nat,L: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.17       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4551_diff__le__self,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4552_le__diff__iff_H,axiom,
% 9.20/5.17      ! [A: nat,C: nat,B: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ C )
% 9.20/5.17       => ( ( ord_less_eq_nat @ B @ C )
% 9.20/5.17         => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
% 9.20/5.17            = ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4553_diff__le__mono2,axiom,
% 9.20/5.17      ! [M: nat,N2: nat,L: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.17       => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4554_dvd__diffD,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.17       => ( ( dvd_dvd_nat @ K @ N2 )
% 9.20/5.17         => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.17           => ( dvd_dvd_nat @ K @ M ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4555_dvd__diffD1,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.17       => ( ( dvd_dvd_nat @ K @ M )
% 9.20/5.17         => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.17           => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4556_less__eq__dvd__minus,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.17       => ( ( dvd_dvd_nat @ M @ N2 )
% 9.20/5.17          = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4557_Nat_Odiff__cancel,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
% 9.20/5.17        = ( minus_minus_nat @ M @ N2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4558_diff__cancel2,axiom,
% 9.20/5.17      ! [M: nat,K: nat,N2: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.17        = ( minus_minus_nat @ M @ N2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4559_diff__add__inverse,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
% 9.20/5.17        = M ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4560_diff__add__inverse2,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
% 9.20/5.17        = M ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4561_diff__mult__distrib2,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.17        = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4562_diff__mult__distrib,axiom,
% 9.20/5.17      ! [M: nat,N2: nat,K: nat] :
% 9.20/5.17        ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
% 9.20/5.17        = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4563_bezout1__nat,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17      ? [D4: nat,X4: nat,Y5: nat] :
% 9.20/5.17        ( ( dvd_dvd_nat @ D4 @ A )
% 9.20/5.17        & ( dvd_dvd_nat @ D4 @ B )
% 9.20/5.17        & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
% 9.20/5.17            = D4 )
% 9.20/5.17          | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
% 9.20/5.17            = D4 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4564_eucl__rel__int__by0,axiom,
% 9.20/5.17      ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4565_div__int__unique,axiom,
% 9.20/5.17      ! [K: int,L: int,Q4: int,R2: int] :
% 9.20/5.17        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.17       => ( ( divide_divide_int @ K @ L )
% 9.20/5.17          = Q4 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4566_zero__less__eq__of__bool,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_real @ zero_zero_real @ ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4567_zero__less__eq__of__bool,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4568_zero__less__eq__of__bool,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_nat @ zero_zero_nat @ ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4569_zero__less__eq__of__bool,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_int @ zero_zero_int @ ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4570_of__bool__less__eq__one,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_real @ ( zero_n3304061248610475627l_real @ P ) @ one_one_real ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4571_of__bool__less__eq__one,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4572_of__bool__less__eq__one,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_nat @ ( zero_n2687167440665602831ol_nat @ P ) @ one_one_nat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4573_of__bool__less__eq__one,axiom,
% 9.20/5.17      ! [P: $o] : ( ord_less_eq_int @ ( zero_n2684676970156552555ol_int @ P ) @ one_one_int ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4574_split__of__bool__asm,axiom,
% 9.20/5.17      ! [P: complex > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.20/5.17        = ( ~ ( ( P4
% 9.20/5.17                & ~ ( P @ one_one_complex ) )
% 9.20/5.17              | ( ~ P4
% 9.20/5.17                & ~ ( P @ zero_zero_complex ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4575_split__of__bool__asm,axiom,
% 9.20/5.17      ! [P: real > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.20/5.17        = ( ~ ( ( P4
% 9.20/5.17                & ~ ( P @ one_one_real ) )
% 9.20/5.17              | ( ~ P4
% 9.20/5.17                & ~ ( P @ zero_zero_real ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4576_split__of__bool__asm,axiom,
% 9.20/5.17      ! [P: rat > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.20/5.17        = ( ~ ( ( P4
% 9.20/5.17                & ~ ( P @ one_one_rat ) )
% 9.20/5.17              | ( ~ P4
% 9.20/5.17                & ~ ( P @ zero_zero_rat ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4577_split__of__bool__asm,axiom,
% 9.20/5.17      ! [P: nat > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.20/5.17        = ( ~ ( ( P4
% 9.20/5.17                & ~ ( P @ one_one_nat ) )
% 9.20/5.17              | ( ~ P4
% 9.20/5.17                & ~ ( P @ zero_zero_nat ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4578_split__of__bool__asm,axiom,
% 9.20/5.17      ! [P: int > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.20/5.17        = ( ~ ( ( P4
% 9.20/5.17                & ~ ( P @ one_one_int ) )
% 9.20/5.17              | ( ~ P4
% 9.20/5.17                & ~ ( P @ zero_zero_int ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4579_split__of__bool,axiom,
% 9.20/5.17      ! [P: complex > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
% 9.20/5.17        = ( ( P4
% 9.20/5.17           => ( P @ one_one_complex ) )
% 9.20/5.17          & ( ~ P4
% 9.20/5.17           => ( P @ zero_zero_complex ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4580_split__of__bool,axiom,
% 9.20/5.17      ! [P: real > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
% 9.20/5.17        = ( ( P4
% 9.20/5.17           => ( P @ one_one_real ) )
% 9.20/5.17          & ( ~ P4
% 9.20/5.17           => ( P @ zero_zero_real ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4581_split__of__bool,axiom,
% 9.20/5.17      ! [P: rat > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
% 9.20/5.17        = ( ( P4
% 9.20/5.17           => ( P @ one_one_rat ) )
% 9.20/5.17          & ( ~ P4
% 9.20/5.17           => ( P @ zero_zero_rat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4582_split__of__bool,axiom,
% 9.20/5.17      ! [P: nat > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
% 9.20/5.17        = ( ( P4
% 9.20/5.17           => ( P @ one_one_nat ) )
% 9.20/5.17          & ( ~ P4
% 9.20/5.17           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4583_split__of__bool,axiom,
% 9.20/5.17      ! [P: int > $o,P4: $o] :
% 9.20/5.17        ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
% 9.20/5.17        = ( ( P4
% 9.20/5.17           => ( P @ one_one_int ) )
% 9.20/5.17          & ( ~ P4
% 9.20/5.17           => ( P @ zero_zero_int ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4584_of__bool__def,axiom,
% 9.20/5.17      ( zero_n1201886186963655149omplex
% 9.20/5.17      = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4585_of__bool__def,axiom,
% 9.20/5.17      ( zero_n3304061248610475627l_real
% 9.20/5.17      = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4586_of__bool__def,axiom,
% 9.20/5.17      ( zero_n2052037380579107095ol_rat
% 9.20/5.17      = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4587_of__bool__def,axiom,
% 9.20/5.17      ( zero_n2687167440665602831ol_nat
% 9.20/5.17      = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4588_of__bool__def,axiom,
% 9.20/5.17      ( zero_n2684676970156552555ol_int
% 9.20/5.17      = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4589_le__iff__diff__le__0,axiom,
% 9.20/5.17      ( ord_less_eq_real
% 9.20/5.17      = ( ^ [A3: real,B3: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4590_le__iff__diff__le__0,axiom,
% 9.20/5.17      ( ord_less_eq_rat
% 9.20/5.17      = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4591_le__iff__diff__le__0,axiom,
% 9.20/5.17      ( ord_less_eq_int
% 9.20/5.17      = ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4592_less__iff__diff__less__0,axiom,
% 9.20/5.17      ( ord_less_real
% 9.20/5.17      = ( ^ [A3: real,B3: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4593_less__iff__diff__less__0,axiom,
% 9.20/5.17      ( ord_less_rat
% 9.20/5.17      = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4594_less__iff__diff__less__0,axiom,
% 9.20/5.17      ( ord_less_int
% 9.20/5.17      = ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4595_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17         => ( ( ( minus_minus_nat @ B @ A )
% 9.20/5.17              = C )
% 9.20/5.17            = ( B
% 9.20/5.17              = ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4596_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
% 9.20/5.17          = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4597_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.20/5.17          = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4598_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
% 9.20/5.17          = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4599_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
% 9.20/5.17          = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4600_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
% 9.20/5.17          = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4601_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.20/5.17          = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4602_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
% 9.20/5.17          = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4603_le__add__diff,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4604_diff__add,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
% 9.20/5.17          = B ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4605_le__diff__eq,axiom,
% 9.20/5.17      ! [A: real,C: real,B: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.20/5.17        = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4606_le__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,C: rat,B: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.20/5.17        = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4607_le__diff__eq,axiom,
% 9.20/5.17      ! [A: int,C: int,B: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.20/5.17        = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4608_diff__le__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4609_diff__le__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4610_diff__le__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4611_add__le__add__imp__diff__le,axiom,
% 9.20/5.17      ! [I: real,K: real,N2: real,J: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 9.20/5.17       => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 9.20/5.17         => ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 9.20/5.17           => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
% 9.20/5.17             => ( ord_less_eq_real @ ( minus_minus_real @ N2 @ K ) @ J ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4612_add__le__add__imp__diff__le,axiom,
% 9.20/5.17      ! [I: rat,K: rat,N2: rat,J: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 9.20/5.17       => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 9.20/5.17         => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 9.20/5.17           => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
% 9.20/5.17             => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4613_add__le__add__imp__diff__le,axiom,
% 9.20/5.17      ! [I: nat,K: nat,N2: nat,J: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 9.20/5.17       => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 9.20/5.17         => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 9.20/5.17           => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
% 9.20/5.17             => ( ord_less_eq_nat @ ( minus_minus_nat @ N2 @ K ) @ J ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4614_add__le__add__imp__diff__le,axiom,
% 9.20/5.17      ! [I: int,K: int,N2: int,J: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 9.20/5.17       => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 9.20/5.17         => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 9.20/5.17           => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
% 9.20/5.17             => ( ord_less_eq_int @ ( minus_minus_int @ N2 @ K ) @ J ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4615_add__le__imp__le__diff,axiom,
% 9.20/5.17      ! [I: real,K: real,N2: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ ( plus_plus_real @ I @ K ) @ N2 )
% 9.20/5.17       => ( ord_less_eq_real @ I @ ( minus_minus_real @ N2 @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4616_add__le__imp__le__diff,axiom,
% 9.20/5.17      ! [I: rat,K: rat,N2: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
% 9.20/5.17       => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4617_add__le__imp__le__diff,axiom,
% 9.20/5.17      ! [I: nat,K: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N2 )
% 9.20/5.17       => ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N2 @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4618_add__le__imp__le__diff,axiom,
% 9.20/5.17      ! [I: int,K: int,N2: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N2 )
% 9.20/5.17       => ( ord_less_eq_int @ I @ ( minus_minus_int @ N2 @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4619_less__diff__eq,axiom,
% 9.20/5.17      ! [A: real,C: real,B: real] :
% 9.20/5.17        ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
% 9.20/5.17        = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4620_less__diff__eq,axiom,
% 9.20/5.17      ! [A: rat,C: rat,B: rat] :
% 9.20/5.17        ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
% 9.20/5.17        = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4621_less__diff__eq,axiom,
% 9.20/5.17      ! [A: int,C: int,B: int] :
% 9.20/5.17        ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
% 9.20/5.17        = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4622_diff__less__eq,axiom,
% 9.20/5.17      ! [A: real,B: real,C: real] :
% 9.20/5.17        ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4623_diff__less__eq,axiom,
% 9.20/5.17      ! [A: rat,B: rat,C: rat] :
% 9.20/5.17        ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4624_diff__less__eq,axiom,
% 9.20/5.17      ! [A: int,B: int,C: int] :
% 9.20/5.17        ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
% 9.20/5.17        = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4625_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.20/5.17      ! [A: real,B: real] :
% 9.20/5.17        ( ~ ( ord_less_real @ A @ B )
% 9.20/5.17       => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
% 9.20/5.17          = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4626_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.20/5.17      ! [A: rat,B: rat] :
% 9.20/5.17        ( ~ ( ord_less_rat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.17          = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4627_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.20/5.17      ! [A: nat,B: nat] :
% 9.20/5.17        ( ~ ( ord_less_nat @ A @ B )
% 9.20/5.17       => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.17          = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4628_linordered__semidom__class_Oadd__diff__inverse,axiom,
% 9.20/5.17      ! [A: int,B: int] :
% 9.20/5.17        ( ~ ( ord_less_int @ A @ B )
% 9.20/5.17       => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 9.20/5.17          = A ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4629_mult__diff__mult,axiom,
% 9.20/5.17      ! [X3: complex,Y3: complex,A: complex,B: complex] :
% 9.20/5.17        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B ) )
% 9.20/5.17        = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4630_mult__diff__mult,axiom,
% 9.20/5.17      ! [X3: real,Y3: real,A: real,B: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B ) )
% 9.20/5.17        = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4631_mult__diff__mult,axiom,
% 9.20/5.17      ! [X3: rat,Y3: rat,A: rat,B: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B ) )
% 9.20/5.17        = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4632_mult__diff__mult,axiom,
% 9.20/5.17      ! [X3: int,Y3: int,A: int,B: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B ) )
% 9.20/5.17        = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4633_square__diff__square__factored,axiom,
% 9.20/5.17      ! [X3: complex,Y3: complex] :
% 9.20/5.17        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
% 9.20/5.17        = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4634_square__diff__square__factored,axiom,
% 9.20/5.17      ! [X3: real,Y3: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
% 9.20/5.17        = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4635_square__diff__square__factored,axiom,
% 9.20/5.17      ! [X3: rat,Y3: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
% 9.20/5.17        = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4636_square__diff__square__factored,axiom,
% 9.20/5.17      ! [X3: int,Y3: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
% 9.20/5.17        = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4637_eq__add__iff2,axiom,
% 9.20/5.17      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 9.20/5.17        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( C
% 9.20/5.17          = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4638_eq__add__iff2,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( C
% 9.20/5.17          = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4639_eq__add__iff2,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( C
% 9.20/5.17          = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4640_eq__add__iff2,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( C
% 9.20/5.17          = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4641_eq__add__iff1,axiom,
% 9.20/5.17      ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
% 9.20/5.17        ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
% 9.20/5.17          = D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4642_eq__add__iff1,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
% 9.20/5.17          = D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4643_eq__add__iff1,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
% 9.20/5.17          = D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4644_eq__add__iff1,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
% 9.20/5.17          = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
% 9.20/5.17          = D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4645_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ A )
% 9.20/5.17       => ( ( A != bot_bo228742789529271731at_nat )
% 9.20/5.17         => ~ ! [A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                ( ? [A4: produc3843707927480180839at_nat] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert9069300056098147895at_nat @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite4343798906461161616at_nat @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4646_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_complex] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ A )
% 9.20/5.17       => ( ( A != bot_bot_set_complex )
% 9.20/5.17         => ~ ! [A7: set_complex] :
% 9.20/5.17                ( ? [A4: complex] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert_complex @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite3207457112153483333omplex @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4647_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ( finite6177210948735845034at_nat @ A )
% 9.20/5.17       => ( ( A != bot_bo2099793752762293965at_nat )
% 9.20/5.17         => ~ ! [A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                ( ? [A4: product_prod_nat_nat] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert8211810215607154385at_nat @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4648_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_real] :
% 9.20/5.17        ( ( finite_finite_real @ A )
% 9.20/5.17       => ( ( A != bot_bot_set_real )
% 9.20/5.17         => ~ ! [A7: set_real] :
% 9.20/5.17                ( ? [A4: real] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert_real @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite_finite_real @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4649_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_nat] :
% 9.20/5.17        ( ( finite_finite_nat @ A )
% 9.20/5.17       => ( ( A != bot_bot_set_nat )
% 9.20/5.17         => ~ ! [A7: set_nat] :
% 9.20/5.17                ( ? [A4: nat] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert_nat @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite_finite_nat @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4650_finite_Ocases,axiom,
% 9.20/5.17      ! [A: set_int] :
% 9.20/5.17        ( ( finite_finite_int @ A )
% 9.20/5.17       => ( ( A != bot_bot_set_int )
% 9.20/5.17         => ~ ! [A7: set_int] :
% 9.20/5.17                ( ? [A4: int] :
% 9.20/5.17                    ( A
% 9.20/5.17                    = ( insert_int @ A4 @ A7 ) )
% 9.20/5.17               => ~ ( finite_finite_int @ A7 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4651_finite_Osimps,axiom,
% 9.20/5.17      ( finite4343798906461161616at_nat
% 9.20/5.17      = ( ^ [A3: set_Pr4329608150637261639at_nat] :
% 9.20/5.17            ( ( A3 = bot_bo228742789529271731at_nat )
% 9.20/5.17            | ? [A5: set_Pr4329608150637261639at_nat,B3: produc3843707927480180839at_nat] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert9069300056098147895at_nat @ B3 @ A5 ) )
% 9.20/5.17                & ( finite4343798906461161616at_nat @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4652_finite_Osimps,axiom,
% 9.20/5.17      ( finite3207457112153483333omplex
% 9.20/5.17      = ( ^ [A3: set_complex] :
% 9.20/5.17            ( ( A3 = bot_bot_set_complex )
% 9.20/5.17            | ? [A5: set_complex,B3: complex] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert_complex @ B3 @ A5 ) )
% 9.20/5.17                & ( finite3207457112153483333omplex @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4653_finite_Osimps,axiom,
% 9.20/5.17      ( finite6177210948735845034at_nat
% 9.20/5.17      = ( ^ [A3: set_Pr1261947904930325089at_nat] :
% 9.20/5.17            ( ( A3 = bot_bo2099793752762293965at_nat )
% 9.20/5.17            | ? [A5: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert8211810215607154385at_nat @ B3 @ A5 ) )
% 9.20/5.17                & ( finite6177210948735845034at_nat @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4654_finite_Osimps,axiom,
% 9.20/5.17      ( finite_finite_real
% 9.20/5.17      = ( ^ [A3: set_real] :
% 9.20/5.17            ( ( A3 = bot_bot_set_real )
% 9.20/5.17            | ? [A5: set_real,B3: real] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert_real @ B3 @ A5 ) )
% 9.20/5.17                & ( finite_finite_real @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4655_finite_Osimps,axiom,
% 9.20/5.17      ( finite_finite_nat
% 9.20/5.17      = ( ^ [A3: set_nat] :
% 9.20/5.17            ( ( A3 = bot_bot_set_nat )
% 9.20/5.17            | ? [A5: set_nat,B3: nat] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert_nat @ B3 @ A5 ) )
% 9.20/5.17                & ( finite_finite_nat @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4656_finite_Osimps,axiom,
% 9.20/5.17      ( finite_finite_int
% 9.20/5.17      = ( ^ [A3: set_int] :
% 9.20/5.17            ( ( A3 = bot_bot_set_int )
% 9.20/5.17            | ? [A5: set_int,B3: int] :
% 9.20/5.17                ( ( A3
% 9.20/5.17                  = ( insert_int @ B3 @ A5 ) )
% 9.20/5.17                & ( finite_finite_int @ A5 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4657_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ F3 )
% 9.20/5.17       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.17         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                ( ( finite4343798906461161616at_nat @ F4 )
% 9.20/5.17               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4658_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.17        ( ( finite1152437895449049373et_nat @ F3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_set_nat )
% 9.20/5.17         => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.20/5.17                ( ( finite1152437895449049373et_nat @ F4 )
% 9.20/5.17               => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4659_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_complex,P: set_complex > $o] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ F3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex,F4: set_complex] :
% 9.20/5.17                ( ( finite3207457112153483333omplex @ F4 )
% 9.20/5.17               => ( ~ ( member_complex @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4660_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.17        ( ( finite6177210948735845034at_nat @ F3 )
% 9.20/5.17       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.17         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                ( ( finite6177210948735845034at_nat @ F4 )
% 9.20/5.17               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4661_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_real,P: set_real > $o] :
% 9.20/5.17        ( ( finite_finite_real @ F3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real,F4: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ F4 )
% 9.20/5.17               => ( ~ ( member_real @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4662_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_nat,P: set_nat > $o] :
% 9.20/5.17        ( ( finite_finite_nat @ F3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [X4: nat,F4: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ F4 )
% 9.20/5.17               => ( ~ ( member_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4663_finite__induct,axiom,
% 9.20/5.17      ! [F3: set_int,P: set_int > $o] :
% 9.20/5.17        ( ( finite_finite_int @ F3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [X4: int,F4: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ F4 )
% 9.20/5.17               => ( ~ ( member_int @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ F3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4664_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bo228742789529271731at_nat )
% 9.20/5.17         => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17           => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bo228742789529271731at_nat )
% 9.20/5.17                   => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4665_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.17        ( ( finite1152437895449049373et_nat @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bot_set_set_nat )
% 9.20/5.17         => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
% 9.20/5.17           => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.20/5.17                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bot_set_set_nat )
% 9.20/5.17                   => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4666_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_complex,P: set_complex > $o] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
% 9.20/5.17           => ( ! [X4: complex,F4: set_complex] :
% 9.20/5.17                  ( ( finite3207457112153483333omplex @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bot_set_complex )
% 9.20/5.17                   => ( ~ ( member_complex @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4667_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.17        ( ( finite6177210948735845034at_nat @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bo2099793752762293965at_nat )
% 9.20/5.17         => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17           => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bo2099793752762293965at_nat )
% 9.20/5.17                   => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4668_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_real,P: set_real > $o] :
% 9.20/5.17        ( ( finite_finite_real @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
% 9.20/5.17           => ( ! [X4: real,F4: set_real] :
% 9.20/5.17                  ( ( finite_finite_real @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bot_set_real )
% 9.20/5.17                   => ( ~ ( member_real @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4669_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_nat,P: set_nat > $o] :
% 9.20/5.17        ( ( finite_finite_nat @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bot_set_nat )
% 9.20/5.17         => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
% 9.20/5.17           => ( ! [X4: nat,F4: set_nat] :
% 9.20/5.17                  ( ( finite_finite_nat @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bot_set_nat )
% 9.20/5.17                   => ( ~ ( member_nat @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4670_finite__ne__induct,axiom,
% 9.20/5.17      ! [F3: set_int,P: set_int > $o] :
% 9.20/5.17        ( ( finite_finite_int @ F3 )
% 9.20/5.17       => ( ( F3 != bot_bot_set_int )
% 9.20/5.17         => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
% 9.20/5.17           => ( ! [X4: int,F4: set_int] :
% 9.20/5.17                  ( ( finite_finite_int @ F4 )
% 9.20/5.17                 => ( ( F4 != bot_bot_set_int )
% 9.20/5.17                   => ( ~ ( member_int @ X4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4671_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.17        ( ! [A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.17            ( ~ ( finite4343798906461161616at_nat @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.17         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                ( ( finite4343798906461161616at_nat @ F4 )
% 9.20/5.17               => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4672_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_set_nat > $o,A2: set_set_nat] :
% 9.20/5.17        ( ! [A7: set_set_nat] :
% 9.20/5.17            ( ~ ( finite1152437895449049373et_nat @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bot_set_set_nat )
% 9.20/5.17         => ( ! [X4: set_nat,F4: set_set_nat] :
% 9.20/5.17                ( ( finite1152437895449049373et_nat @ F4 )
% 9.20/5.17               => ( ~ ( member_set_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4673_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_complex > $o,A2: set_complex] :
% 9.20/5.17        ( ! [A7: set_complex] :
% 9.20/5.17            ( ~ ( finite3207457112153483333omplex @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex,F4: set_complex] :
% 9.20/5.17                ( ( finite3207457112153483333omplex @ F4 )
% 9.20/5.17               => ( ~ ( member_complex @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4674_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.17        ( ! [A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.17            ( ~ ( finite6177210948735845034at_nat @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.17         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                ( ( finite6177210948735845034at_nat @ F4 )
% 9.20/5.17               => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4675_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_real > $o,A2: set_real] :
% 9.20/5.17        ( ! [A7: set_real] :
% 9.20/5.17            ( ~ ( finite_finite_real @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real,F4: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ F4 )
% 9.20/5.17               => ( ~ ( member_real @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4676_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_nat > $o,A2: set_nat] :
% 9.20/5.17        ( ! [A7: set_nat] :
% 9.20/5.17            ( ~ ( finite_finite_nat @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [X4: nat,F4: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ F4 )
% 9.20/5.17               => ( ~ ( member_nat @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4677_infinite__finite__induct,axiom,
% 9.20/5.17      ! [P: set_int > $o,A2: set_int] :
% 9.20/5.17        ( ! [A7: set_int] :
% 9.20/5.17            ( ~ ( finite_finite_int @ A7 )
% 9.20/5.17           => ( P @ A7 ) )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [X4: int,F4: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ F4 )
% 9.20/5.17               => ( ~ ( member_int @ X4 @ F4 )
% 9.20/5.17                 => ( ( P @ F4 )
% 9.20/5.17                   => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4678_Euclidean__Division_Opos__mod__bound,axiom,
% 9.20/5.17      ! [L: int,K: int] :
% 9.20/5.17        ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.17       => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4679_neg__mod__bound,axiom,
% 9.20/5.17      ! [L: int,K: int] :
% 9.20/5.17        ( ( ord_less_int @ L @ zero_zero_int )
% 9.20/5.17       => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4680_subset__singleton__iff,axiom,
% 9.20/5.17      ! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17        = ( ( X7 = bot_bo228742789529271731at_nat )
% 9.20/5.17          | ( X7
% 9.20/5.17            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4681_subset__singleton__iff,axiom,
% 9.20/5.17      ! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.20/5.17        ( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17        = ( ( X7 = bot_bo2099793752762293965at_nat )
% 9.20/5.17          | ( X7
% 9.20/5.17            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4682_subset__singleton__iff,axiom,
% 9.20/5.17      ! [X7: set_real,A: real] :
% 9.20/5.17        ( ( ord_less_eq_set_real @ X7 @ ( insert_real @ A @ bot_bot_set_real ) )
% 9.20/5.17        = ( ( X7 = bot_bot_set_real )
% 9.20/5.17          | ( X7
% 9.20/5.17            = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4683_subset__singleton__iff,axiom,
% 9.20/5.17      ! [X7: set_nat,A: nat] :
% 9.20/5.17        ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
% 9.20/5.17        = ( ( X7 = bot_bot_set_nat )
% 9.20/5.17          | ( X7
% 9.20/5.17            = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4684_subset__singleton__iff,axiom,
% 9.20/5.17      ! [X7: set_int,A: int] :
% 9.20/5.17        ( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
% 9.20/5.17        = ( ( X7 = bot_bot_set_int )
% 9.20/5.17          | ( X7
% 9.20/5.17            = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4685_subset__singletonD,axiom,
% 9.20/5.17      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
% 9.20/5.17        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 9.20/5.17       => ( ( A2 = bot_bo228742789529271731at_nat )
% 9.20/5.17          | ( A2
% 9.20/5.17            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4686_subset__singletonD,axiom,
% 9.20/5.17      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
% 9.20/5.17        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.17       => ( ( A2 = bot_bo2099793752762293965at_nat )
% 9.20/5.17          | ( A2
% 9.20/5.17            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4687_subset__singletonD,axiom,
% 9.20/5.17      ! [A2: set_real,X3: real] :
% 9.20/5.17        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 9.20/5.17       => ( ( A2 = bot_bot_set_real )
% 9.20/5.17          | ( A2
% 9.20/5.17            = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4688_subset__singletonD,axiom,
% 9.20/5.17      ! [A2: set_nat,X3: nat] :
% 9.20/5.17        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 9.20/5.17       => ( ( A2 = bot_bot_set_nat )
% 9.20/5.17          | ( A2
% 9.20/5.17            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4689_subset__singletonD,axiom,
% 9.20/5.17      ! [A2: set_int,X3: int] :
% 9.20/5.17        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 9.20/5.17       => ( ( A2 = bot_bot_set_int )
% 9.20/5.17          | ( A2
% 9.20/5.17            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4690_dvd__minus__mod,axiom,
% 9.20/5.17      ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4691_dvd__minus__mod,axiom,
% 9.20/5.17      ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4692_dvd__minus__mod,axiom,
% 9.20/5.17      ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4693_diff__less__Suc,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4694_Suc__diff__Suc,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( ord_less_nat @ N2 @ M )
% 9.20/5.17       => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.17          = ( minus_minus_nat @ M @ N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4695_diff__less,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.17       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.17         => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4696_Suc__diff__le,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.17       => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 9.20/5.17          = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4697_diff__less__mono,axiom,
% 9.20/5.17      ! [A: nat,B: nat,C: nat] :
% 9.20/5.17        ( ( ord_less_nat @ A @ B )
% 9.20/5.17       => ( ( ord_less_eq_nat @ C @ A )
% 9.20/5.17         => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4698_less__diff__iff,axiom,
% 9.20/5.17      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.17       => ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.17         => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
% 9.20/5.17            = ( ord_less_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4699_diff__add__0,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.17        = zero_zero_nat ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4700_add__diff__inverse__nat,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ~ ( ord_less_nat @ M @ N2 )
% 9.20/5.17       => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.17          = M ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4701_less__diff__conv,axiom,
% 9.20/5.17      ! [I: nat,J: nat,K: nat] :
% 9.20/5.17        ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.20/5.17        = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4702_Nat_Ole__imp__diff__is__add,axiom,
% 9.20/5.17      ! [I: nat,J: nat,K: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.17       => ( ( ( minus_minus_nat @ J @ I )
% 9.20/5.17            = K )
% 9.20/5.17          = ( J
% 9.20/5.17            = ( plus_plus_nat @ K @ I ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4703_Nat_Odiff__add__assoc2,axiom,
% 9.20/5.17      ! [K: nat,J: nat,I: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.17       => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
% 9.20/5.17          = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4704_Nat_Odiff__add__assoc,axiom,
% 9.20/5.17      ! [K: nat,J: nat,I: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.17       => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
% 9.20/5.17          = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4705_Nat_Ole__diff__conv2,axiom,
% 9.20/5.17      ! [K: nat,J: nat,I: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.17       => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
% 9.20/5.17          = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4706_le__diff__conv,axiom,
% 9.20/5.17      ! [J: nat,K: nat,I: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.20/5.17        = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4707_diff__Suc__eq__diff__pred,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 9.20/5.17        = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4708_mod__geq,axiom,
% 9.20/5.17      ! [M: nat,N2: nat] :
% 9.20/5.17        ( ~ ( ord_less_nat @ M @ N2 )
% 9.20/5.17       => ( ( modulo_modulo_nat @ M @ N2 )
% 9.20/5.17          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4709_mod__if,axiom,
% 9.20/5.17      ( modulo_modulo_nat
% 9.20/5.17      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4710_le__mod__geq,axiom,
% 9.20/5.17      ! [N2: nat,M: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.17       => ( ( modulo_modulo_nat @ M @ N2 )
% 9.20/5.17          = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4711_mod__eq__dvd__iff__nat,axiom,
% 9.20/5.17      ! [N2: nat,M: nat,Q4: nat] :
% 9.20/5.17        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.17       => ( ( ( modulo_modulo_nat @ M @ Q4 )
% 9.20/5.17            = ( modulo_modulo_nat @ N2 @ Q4 ) )
% 9.20/5.17          = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4712_ordered__ring__class_Ole__add__iff2,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4713_ordered__ring__class_Ole__add__iff2,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4714_ordered__ring__class_Ole__add__iff2,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4715_ordered__ring__class_Ole__add__iff1,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4716_ordered__ring__class_Ole__add__iff1,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4717_ordered__ring__class_Ole__add__iff1,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4718_less__add__iff2,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4719_less__add__iff2,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4720_less__add__iff2,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4721_less__add__iff1,axiom,
% 9.20/5.17      ! [A: real,E2: real,C: real,B: real,D: real] :
% 9.20/5.17        ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4722_less__add__iff1,axiom,
% 9.20/5.17      ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
% 9.20/5.17        ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4723_less__add__iff1,axiom,
% 9.20/5.17      ! [A: int,E2: int,C: int,B: int,D: int] :
% 9.20/5.17        ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
% 9.20/5.17        = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4724_divide__diff__eq__iff,axiom,
% 9.20/5.17      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.17        ( ( Z2 != zero_zero_complex )
% 9.20/5.17       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
% 9.20/5.17          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4725_divide__diff__eq__iff,axiom,
% 9.20/5.17      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.17        ( ( Z2 != zero_zero_real )
% 9.20/5.17       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
% 9.20/5.17          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4726_divide__diff__eq__iff,axiom,
% 9.20/5.17      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.17        ( ( Z2 != zero_zero_rat )
% 9.20/5.17       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
% 9.20/5.17          = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4727_diff__divide__eq__iff,axiom,
% 9.20/5.17      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.17        ( ( Z2 != zero_zero_complex )
% 9.20/5.17       => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
% 9.20/5.17          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4728_diff__divide__eq__iff,axiom,
% 9.20/5.17      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.17        ( ( Z2 != zero_zero_real )
% 9.20/5.17       => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
% 9.20/5.17          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4729_diff__divide__eq__iff,axiom,
% 9.20/5.17      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.17        ( ( Z2 != zero_zero_rat )
% 9.20/5.17       => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
% 9.20/5.17          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4730_diff__frac__eq,axiom,
% 9.20/5.17      ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
% 9.20/5.17        ( ( Y3 != zero_zero_complex )
% 9.20/5.17       => ( ( Z2 != zero_zero_complex )
% 9.20/5.17         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
% 9.20/5.17            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4731_diff__frac__eq,axiom,
% 9.20/5.17      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 9.20/5.17        ( ( Y3 != zero_zero_real )
% 9.20/5.17       => ( ( Z2 != zero_zero_real )
% 9.20/5.17         => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.20/5.17            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4732_diff__frac__eq,axiom,
% 9.20/5.17      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 9.20/5.17        ( ( Y3 != zero_zero_rat )
% 9.20/5.17       => ( ( Z2 != zero_zero_rat )
% 9.20/5.17         => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.20/5.17            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4733_add__divide__eq__if__simps_I4_J,axiom,
% 9.20/5.17      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.17        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.17         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.20/5.17            = A ) )
% 9.20/5.17        & ( ( Z2 != zero_zero_complex )
% 9.20/5.17         => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
% 9.20/5.17            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4734_add__divide__eq__if__simps_I4_J,axiom,
% 9.20/5.17      ! [Z2: real,A: real,B: real] :
% 9.20/5.17        ( ( ( Z2 = zero_zero_real )
% 9.20/5.17         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.20/5.17            = A ) )
% 9.20/5.17        & ( ( Z2 != zero_zero_real )
% 9.20/5.17         => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
% 9.20/5.17            = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4735_add__divide__eq__if__simps_I4_J,axiom,
% 9.20/5.17      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.17        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.17         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.20/5.17            = A ) )
% 9.20/5.17        & ( ( Z2 != zero_zero_rat )
% 9.20/5.17         => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
% 9.20/5.17            = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4736_square__diff__one__factored,axiom,
% 9.20/5.17      ! [X3: complex] :
% 9.20/5.17        ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
% 9.20/5.17        = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4737_square__diff__one__factored,axiom,
% 9.20/5.17      ! [X3: real] :
% 9.20/5.17        ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
% 9.20/5.17        = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4738_square__diff__one__factored,axiom,
% 9.20/5.17      ! [X3: rat] :
% 9.20/5.17        ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
% 9.20/5.17        = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4739_square__diff__one__factored,axiom,
% 9.20/5.17      ! [X3: int] :
% 9.20/5.17        ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
% 9.20/5.17        = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4740_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex,S5: set_complex] :
% 9.20/5.17                ( ( finite3207457112153483333omplex @ S5 )
% 9.20/5.17               => ( ! [Y6: complex] :
% 9.20/5.17                      ( ( member_complex @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4741_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_real,P: set_real > $o,F: real > rat] :
% 9.20/5.17        ( ( finite_finite_real @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real,S5: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ S5 )
% 9.20/5.17               => ( ! [Y6: real] :
% 9.20/5.17                      ( ( member_real @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4742_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
% 9.20/5.17        ( ( finite_finite_nat @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [X4: nat,S5: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ S5 )
% 9.20/5.17               => ( ! [Y6: nat] :
% 9.20/5.17                      ( ( member_nat @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4743_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_int,P: set_int > $o,F: int > rat] :
% 9.20/5.17        ( ( finite_finite_int @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [X4: int,S5: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ S5 )
% 9.20/5.17               => ( ! [Y6: int] :
% 9.20/5.17                      ( ( member_int @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4744_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex,S5: set_complex] :
% 9.20/5.17                ( ( finite3207457112153483333omplex @ S5 )
% 9.20/5.17               => ( ! [Y6: complex] :
% 9.20/5.17                      ( ( member_complex @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4745_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_real,P: set_real > $o,F: real > num] :
% 9.20/5.17        ( ( finite_finite_real @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real,S5: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ S5 )
% 9.20/5.17               => ( ! [Y6: real] :
% 9.20/5.17                      ( ( member_real @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4746_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
% 9.20/5.17        ( ( finite_finite_nat @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [X4: nat,S5: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ S5 )
% 9.20/5.17               => ( ! [Y6: nat] :
% 9.20/5.17                      ( ( member_nat @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4747_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_int,P: set_int > $o,F: int > num] :
% 9.20/5.17        ( ( finite_finite_int @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [X4: int,S5: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ S5 )
% 9.20/5.17               => ( ! [Y6: int] :
% 9.20/5.17                      ( ( member_int @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4748_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_complex,P: set_complex > $o,F: complex > nat] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_complex )
% 9.20/5.17         => ( ! [X4: complex,S5: set_complex] :
% 9.20/5.17                ( ( finite3207457112153483333omplex @ S5 )
% 9.20/5.17               => ( ! [Y6: complex] :
% 9.20/5.17                      ( ( member_complex @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4749_finite__ranking__induct,axiom,
% 9.20/5.17      ! [S3: set_real,P: set_real > $o,F: real > nat] :
% 9.20/5.17        ( ( finite_finite_real @ S3 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [X4: real,S5: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ S5 )
% 9.20/5.17               => ( ! [Y6: real] :
% 9.20/5.17                      ( ( member_real @ Y6 @ S5 )
% 9.20/5.17                     => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
% 9.20/5.17                 => ( ( P @ S5 )
% 9.20/5.17                   => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
% 9.20/5.17           => ( P @ S3 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4750_finite__linorder__min__induct,axiom,
% 9.20/5.17      ! [A2: set_real,P: set_real > $o] :
% 9.20/5.17        ( ( finite_finite_real @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [B4: real,A7: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ A7 )
% 9.20/5.17               => ( ! [X2: real] :
% 9.20/5.17                      ( ( member_real @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_real @ B4 @ X2 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4751_finite__linorder__min__induct,axiom,
% 9.20/5.17      ! [A2: set_rat,P: set_rat > $o] :
% 9.20/5.17        ( ( finite_finite_rat @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_rat )
% 9.20/5.17         => ( ! [B4: rat,A7: set_rat] :
% 9.20/5.17                ( ( finite_finite_rat @ A7 )
% 9.20/5.17               => ( ! [X2: rat] :
% 9.20/5.17                      ( ( member_rat @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_rat @ B4 @ X2 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4752_finite__linorder__min__induct,axiom,
% 9.20/5.17      ! [A2: set_num,P: set_num > $o] :
% 9.20/5.17        ( ( finite_finite_num @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_num )
% 9.20/5.17         => ( ! [B4: num,A7: set_num] :
% 9.20/5.17                ( ( finite_finite_num @ A7 )
% 9.20/5.17               => ( ! [X2: num] :
% 9.20/5.17                      ( ( member_num @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_num @ B4 @ X2 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4753_finite__linorder__min__induct,axiom,
% 9.20/5.17      ! [A2: set_nat,P: set_nat > $o] :
% 9.20/5.17        ( ( finite_finite_nat @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [B4: nat,A7: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ A7 )
% 9.20/5.17               => ( ! [X2: nat] :
% 9.20/5.17                      ( ( member_nat @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_nat @ B4 @ X2 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4754_finite__linorder__min__induct,axiom,
% 9.20/5.17      ! [A2: set_int,P: set_int > $o] :
% 9.20/5.17        ( ( finite_finite_int @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [B4: int,A7: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ A7 )
% 9.20/5.17               => ( ! [X2: int] :
% 9.20/5.17                      ( ( member_int @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_int @ B4 @ X2 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4755_finite__linorder__max__induct,axiom,
% 9.20/5.17      ! [A2: set_real,P: set_real > $o] :
% 9.20/5.17        ( ( finite_finite_real @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_real )
% 9.20/5.17         => ( ! [B4: real,A7: set_real] :
% 9.20/5.17                ( ( finite_finite_real @ A7 )
% 9.20/5.17               => ( ! [X2: real] :
% 9.20/5.17                      ( ( member_real @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_real @ X2 @ B4 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4756_finite__linorder__max__induct,axiom,
% 9.20/5.17      ! [A2: set_rat,P: set_rat > $o] :
% 9.20/5.17        ( ( finite_finite_rat @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_rat )
% 9.20/5.17         => ( ! [B4: rat,A7: set_rat] :
% 9.20/5.17                ( ( finite_finite_rat @ A7 )
% 9.20/5.17               => ( ! [X2: rat] :
% 9.20/5.17                      ( ( member_rat @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_rat @ X2 @ B4 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4757_finite__linorder__max__induct,axiom,
% 9.20/5.17      ! [A2: set_num,P: set_num > $o] :
% 9.20/5.17        ( ( finite_finite_num @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_num )
% 9.20/5.17         => ( ! [B4: num,A7: set_num] :
% 9.20/5.17                ( ( finite_finite_num @ A7 )
% 9.20/5.17               => ( ! [X2: num] :
% 9.20/5.17                      ( ( member_num @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_num @ X2 @ B4 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4758_finite__linorder__max__induct,axiom,
% 9.20/5.17      ! [A2: set_nat,P: set_nat > $o] :
% 9.20/5.17        ( ( finite_finite_nat @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_nat )
% 9.20/5.17         => ( ! [B4: nat,A7: set_nat] :
% 9.20/5.17                ( ( finite_finite_nat @ A7 )
% 9.20/5.17               => ( ! [X2: nat] :
% 9.20/5.17                      ( ( member_nat @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_nat @ X2 @ B4 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4759_finite__linorder__max__induct,axiom,
% 9.20/5.17      ! [A2: set_int,P: set_int > $o] :
% 9.20/5.17        ( ( finite_finite_int @ A2 )
% 9.20/5.17       => ( ( P @ bot_bot_set_int )
% 9.20/5.17         => ( ! [B4: int,A7: set_int] :
% 9.20/5.17                ( ( finite_finite_int @ A7 )
% 9.20/5.17               => ( ! [X2: int] :
% 9.20/5.17                      ( ( member_int @ X2 @ A7 )
% 9.20/5.17                     => ( ord_less_int @ X2 @ B4 ) )
% 9.20/5.17                 => ( ( P @ A7 )
% 9.20/5.17                   => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
% 9.20/5.17           => ( P @ A2 ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4760_inf__period_I4_J,axiom,
% 9.20/5.17      ! [D: code_integer,D6: code_integer,T: code_integer] :
% 9.20/5.17        ( ( dvd_dvd_Code_integer @ D @ D6 )
% 9.20/5.17       => ! [X2: code_integer,K4: code_integer] :
% 9.20/5.17            ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) ) )
% 9.20/5.17            = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4761_inf__period_I4_J,axiom,
% 9.20/5.17      ! [D: complex,D6: complex,T: complex] :
% 9.20/5.17        ( ( dvd_dvd_complex @ D @ D6 )
% 9.20/5.17       => ! [X2: complex,K4: complex] :
% 9.20/5.17            ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) ) )
% 9.20/5.17            = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4762_inf__period_I4_J,axiom,
% 9.20/5.17      ! [D: real,D6: real,T: real] :
% 9.20/5.17        ( ( dvd_dvd_real @ D @ D6 )
% 9.20/5.17       => ! [X2: real,K4: real] :
% 9.20/5.17            ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) ) )
% 9.20/5.17            = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4763_inf__period_I4_J,axiom,
% 9.20/5.17      ! [D: rat,D6: rat,T: rat] :
% 9.20/5.17        ( ( dvd_dvd_rat @ D @ D6 )
% 9.20/5.17       => ! [X2: rat,K4: rat] :
% 9.20/5.17            ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
% 9.20/5.17            = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4764_inf__period_I4_J,axiom,
% 9.20/5.17      ! [D: int,D6: int,T: int] :
% 9.20/5.17        ( ( dvd_dvd_int @ D @ D6 )
% 9.20/5.17       => ! [X2: int,K4: int] :
% 9.20/5.17            ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) ) )
% 9.20/5.17            = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4765_inf__period_I3_J,axiom,
% 9.20/5.17      ! [D: code_integer,D6: code_integer,T: code_integer] :
% 9.20/5.17        ( ( dvd_dvd_Code_integer @ D @ D6 )
% 9.20/5.17       => ! [X2: code_integer,K4: code_integer] :
% 9.20/5.17            ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) )
% 9.20/5.17            = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4766_inf__period_I3_J,axiom,
% 9.20/5.17      ! [D: complex,D6: complex,T: complex] :
% 9.20/5.17        ( ( dvd_dvd_complex @ D @ D6 )
% 9.20/5.17       => ! [X2: complex,K4: complex] :
% 9.20/5.17            ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T ) )
% 9.20/5.17            = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4767_inf__period_I3_J,axiom,
% 9.20/5.17      ! [D: real,D6: real,T: real] :
% 9.20/5.17        ( ( dvd_dvd_real @ D @ D6 )
% 9.20/5.17       => ! [X2: real,K4: real] :
% 9.20/5.17            ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T ) )
% 9.20/5.17            = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4768_inf__period_I3_J,axiom,
% 9.20/5.17      ! [D: rat,D6: rat,T: rat] :
% 9.20/5.17        ( ( dvd_dvd_rat @ D @ D6 )
% 9.20/5.17       => ! [X2: rat,K4: rat] :
% 9.20/5.17            ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
% 9.20/5.17            = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4769_inf__period_I3_J,axiom,
% 9.20/5.17      ! [D: int,D6: int,T: int] :
% 9.20/5.17        ( ( dvd_dvd_int @ D @ D6 )
% 9.20/5.17       => ! [X2: int,K4: int] :
% 9.20/5.17            ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T ) )
% 9.20/5.17            = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D6 ) ) @ T ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4770_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ F3 )
% 9.20/5.17       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.17           => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.20/5.17                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.20/5.17                 => ( ( member8757157785044589968at_nat @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4771_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.17        ( ( finite1152437895449049373et_nat @ F3 )
% 9.20/5.17       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bot_set_set_nat )
% 9.20/5.17           => ( ! [A4: set_nat,F4: set_set_nat] :
% 9.20/5.17                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.20/5.17                 => ( ( member_set_nat @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member_set_nat @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4772_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 9.20/5.17        ( ( finite3207457112153483333omplex @ F3 )
% 9.20/5.17       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bot_set_complex )
% 9.20/5.17           => ( ! [A4: complex,F4: set_complex] :
% 9.20/5.17                  ( ( finite3207457112153483333omplex @ F4 )
% 9.20/5.17                 => ( ( member_complex @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member_complex @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4773_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.17        ( ( finite6177210948735845034at_nat @ F3 )
% 9.20/5.17       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.17           => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.20/5.17                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.20/5.17                 => ( ( member8440522571783428010at_nat @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4774_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 9.20/5.17        ( ( finite_finite_real @ F3 )
% 9.20/5.17       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bot_set_real )
% 9.20/5.17           => ( ! [A4: real,F4: set_real] :
% 9.20/5.17                  ( ( finite_finite_real @ F4 )
% 9.20/5.17                 => ( ( member_real @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member_real @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4775_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 9.20/5.17        ( ( finite_finite_nat @ F3 )
% 9.20/5.17       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bot_set_nat )
% 9.20/5.17           => ( ! [A4: nat,F4: set_nat] :
% 9.20/5.17                  ( ( finite_finite_nat @ F4 )
% 9.20/5.17                 => ( ( member_nat @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member_nat @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4776_finite__subset__induct,axiom,
% 9.20/5.17      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 9.20/5.17        ( ( finite_finite_int @ F3 )
% 9.20/5.17       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bot_set_int )
% 9.20/5.17           => ( ! [A4: int,F4: set_int] :
% 9.20/5.17                  ( ( finite_finite_int @ F4 )
% 9.20/5.17                 => ( ( member_int @ A4 @ A2 )
% 9.20/5.17                   => ( ~ ( member_int @ A4 @ F4 )
% 9.20/5.17                     => ( ( P @ F4 )
% 9.20/5.17                       => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) )
% 9.20/5.17             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.17  
% 9.20/5.17  thf(fact_4777_finite__subset__induct_H,axiom,
% 9.20/5.17      ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.17        ( ( finite4343798906461161616at_nat @ F3 )
% 9.20/5.17       => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
% 9.20/5.17         => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.17           => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
% 9.20/5.18                  ( ( finite4343798906461161616at_nat @ F4 )
% 9.20/5.18                 => ( ( member8757157785044589968at_nat @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4778_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_set_nat,A2: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.18        ( ( finite1152437895449049373et_nat @ F3 )
% 9.20/5.18       => ( ( ord_le6893508408891458716et_nat @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bot_set_set_nat )
% 9.20/5.18           => ( ! [A4: set_nat,F4: set_set_nat] :
% 9.20/5.18                  ( ( finite1152437895449049373et_nat @ F4 )
% 9.20/5.18                 => ( ( member_set_nat @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_le6893508408891458716et_nat @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member_set_nat @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert_set_nat @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4779_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_complex,A2: set_complex,P: set_complex > $o] :
% 9.20/5.18        ( ( finite3207457112153483333omplex @ F3 )
% 9.20/5.18       => ( ( ord_le211207098394363844omplex @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bot_set_complex )
% 9.20/5.18           => ( ! [A4: complex,F4: set_complex] :
% 9.20/5.18                  ( ( finite3207457112153483333omplex @ F4 )
% 9.20/5.18                 => ( ( member_complex @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_le211207098394363844omplex @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member_complex @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert_complex @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4780_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.18        ( ( finite6177210948735845034at_nat @ F3 )
% 9.20/5.18       => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.18           => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
% 9.20/5.18                  ( ( finite6177210948735845034at_nat @ F4 )
% 9.20/5.18                 => ( ( member8440522571783428010at_nat @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4781_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_real,A2: set_real,P: set_real > $o] :
% 9.20/5.18        ( ( finite_finite_real @ F3 )
% 9.20/5.18       => ( ( ord_less_eq_set_real @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bot_set_real )
% 9.20/5.18           => ( ! [A4: real,F4: set_real] :
% 9.20/5.18                  ( ( finite_finite_real @ F4 )
% 9.20/5.18                 => ( ( member_real @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_less_eq_set_real @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member_real @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert_real @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4782_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_nat,A2: set_nat,P: set_nat > $o] :
% 9.20/5.18        ( ( finite_finite_nat @ F3 )
% 9.20/5.18       => ( ( ord_less_eq_set_nat @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bot_set_nat )
% 9.20/5.18           => ( ! [A4: nat,F4: set_nat] :
% 9.20/5.18                  ( ( finite_finite_nat @ F4 )
% 9.20/5.18                 => ( ( member_nat @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_less_eq_set_nat @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member_nat @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4783_finite__subset__induct_H,axiom,
% 9.20/5.18      ! [F3: set_int,A2: set_int,P: set_int > $o] :
% 9.20/5.18        ( ( finite_finite_int @ F3 )
% 9.20/5.18       => ( ( ord_less_eq_set_int @ F3 @ A2 )
% 9.20/5.18         => ( ( P @ bot_bot_set_int )
% 9.20/5.18           => ( ! [A4: int,F4: set_int] :
% 9.20/5.18                  ( ( finite_finite_int @ F4 )
% 9.20/5.18                 => ( ( member_int @ A4 @ A2 )
% 9.20/5.18                   => ( ( ord_less_eq_set_int @ F4 @ A2 )
% 9.20/5.18                     => ( ~ ( member_int @ A4 @ F4 )
% 9.20/5.18                       => ( ( P @ F4 )
% 9.20/5.18                         => ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) ) )
% 9.20/5.18             => ( P @ F3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4784_minus__mult__div__eq__mod,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
% 9.20/5.18        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4785_minus__mult__div__eq__mod,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
% 9.20/5.18        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4786_minus__mult__div__eq__mod,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
% 9.20/5.18        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4787_minus__mod__eq__mult__div,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.18        = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4788_minus__mod__eq__mult__div,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.18        = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4789_minus__mod__eq__mult__div,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.18        = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4790_minus__mod__eq__div__mult,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
% 9.20/5.18        = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4791_minus__mod__eq__div__mult,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.18        = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4792_minus__mod__eq__div__mult,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
% 9.20/5.18        = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4793_minus__div__mult__eq__mod,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
% 9.20/5.18        = ( modulo_modulo_nat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4794_minus__div__mult__eq__mod,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
% 9.20/5.18        = ( modulo_modulo_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4795_minus__div__mult__eq__mod,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
% 9.20/5.18        = ( modulo364778990260209775nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4796_eucl__rel__int__dividesI,axiom,
% 9.20/5.18      ! [L: int,K: int,Q4: int] :
% 9.20/5.18        ( ( L != zero_zero_int )
% 9.20/5.18       => ( ( K
% 9.20/5.18            = ( times_times_int @ Q4 @ L ) )
% 9.20/5.18         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4797_Euclidean__Division_Opos__mod__sign,axiom,
% 9.20/5.18      ! [L: int,K: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.18       => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4798_neg__mod__sign,axiom,
% 9.20/5.18      ! [L: int,K: int] :
% 9.20/5.18        ( ( ord_less_int @ L @ zero_zero_int )
% 9.20/5.18       => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4799_neg__mod__conj,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( ord_less_int @ B @ zero_zero_int )
% 9.20/5.18       => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
% 9.20/5.18          & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4800_pos__mod__conj,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.18       => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
% 9.20/5.18          & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4801_zmod__trivial__iff,axiom,
% 9.20/5.18      ! [I: int,K: int] :
% 9.20/5.18        ( ( ( modulo_modulo_int @ I @ K )
% 9.20/5.18          = I )
% 9.20/5.18        = ( ( K = zero_zero_int )
% 9.20/5.18          | ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.20/5.18            & ( ord_less_int @ I @ K ) )
% 9.20/5.18          | ( ( ord_less_eq_int @ I @ zero_zero_int )
% 9.20/5.18            & ( ord_less_int @ K @ I ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4802_mod__int__pos__iff,axiom,
% 9.20/5.18      ! [K: int,L: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
% 9.20/5.18        = ( ( dvd_dvd_int @ L @ K )
% 9.20/5.18          | ( ( L = zero_zero_int )
% 9.20/5.18            & ( ord_less_eq_int @ zero_zero_int @ K ) )
% 9.20/5.18          | ( ord_less_int @ zero_zero_int @ L ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4803_diff__Suc__less,axiom,
% 9.20/5.18      ! [N2: nat,I: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4804_zdiv__mono__strict,axiom,
% 9.20/5.18      ! [A2: int,B2: int,N2: int] :
% 9.20/5.18        ( ( ord_less_int @ A2 @ B2 )
% 9.20/5.18       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 9.20/5.18         => ( ( ( modulo_modulo_int @ A2 @ N2 )
% 9.20/5.18              = zero_zero_int )
% 9.20/5.18           => ( ( ( modulo_modulo_int @ B2 @ N2 )
% 9.20/5.18                = zero_zero_int )
% 9.20/5.18             => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4805_nat__diff__split__asm,axiom,
% 9.20/5.18      ! [P: nat > $o,A: nat,B: nat] :
% 9.20/5.18        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.18        = ( ~ ( ( ( ord_less_nat @ A @ B )
% 9.20/5.18                & ~ ( P @ zero_zero_nat ) )
% 9.20/5.18              | ? [D5: nat] :
% 9.20/5.18                  ( ( A
% 9.20/5.18                    = ( plus_plus_nat @ B @ D5 ) )
% 9.20/5.18                  & ~ ( P @ D5 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4806_nat__diff__split,axiom,
% 9.20/5.18      ! [P: nat > $o,A: nat,B: nat] :
% 9.20/5.18        ( ( P @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.18        = ( ( ( ord_less_nat @ A @ B )
% 9.20/5.18           => ( P @ zero_zero_nat ) )
% 9.20/5.18          & ! [D5: nat] :
% 9.20/5.18              ( ( A
% 9.20/5.18                = ( plus_plus_nat @ B @ D5 ) )
% 9.20/5.18             => ( P @ D5 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4807_less__diff__conv2,axiom,
% 9.20/5.18      ! [K: nat,J: nat,I: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ K @ J )
% 9.20/5.18       => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
% 9.20/5.18          = ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4808_nat__diff__add__eq2,axiom,
% 9.20/5.18      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.18       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4809_nat__diff__add__eq1,axiom,
% 9.20/5.18      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ J @ I )
% 9.20/5.18       => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4810_nat__le__add__iff2,axiom,
% 9.20/5.18      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.18       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4811_nat__le__add__iff1,axiom,
% 9.20/5.18      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ J @ I )
% 9.20/5.18       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4812_nat__eq__add__iff2,axiom,
% 9.20/5.18      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.18       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.20/5.18            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( M
% 9.20/5.18            = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4813_nat__eq__add__iff1,axiom,
% 9.20/5.18      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ J @ I )
% 9.20/5.18       => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
% 9.20/5.18            = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
% 9.20/5.18            = N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4814_dvd__minus__add,axiom,
% 9.20/5.18      ! [Q4: nat,N2: nat,R2: nat,M: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ Q4 @ N2 )
% 9.20/5.18       => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
% 9.20/5.18         => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q4 ) )
% 9.20/5.18            = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4815_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ ( insert8211810215607154385at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4816_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_real,I: nat,X3: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ ( insert_real @ X3 @ ( set_real2 @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4817_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_P5464809261938338413at_nat,I: nat,X3: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X3 ) ) @ ( insert9069300056098147895at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4818_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ ( insert_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4819_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_nat,I: nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ ( insert_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4820_set__update__subset__insert,axiom,
% 9.20/5.18      ! [Xs: list_int,I: nat,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ ( insert_int @ X3 @ ( set_int2 @ Xs ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4821_mod__nat__eqI,axiom,
% 9.20/5.18      ! [R2: nat,N2: nat,M: nat] :
% 9.20/5.18        ( ( ord_less_nat @ R2 @ N2 )
% 9.20/5.18       => ( ( ord_less_eq_nat @ R2 @ M )
% 9.20/5.18         => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
% 9.20/5.18           => ( ( modulo_modulo_nat @ M @ N2 )
% 9.20/5.18              = R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4822_exp__div__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_3573771949741848930nteger
% 9.20/5.18          @ ( zero_n356916108424825756nteger
% 9.20/5.18            @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
% 9.20/5.18               != zero_z3403309356797280102nteger )
% 9.20/5.18              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 9.20/5.18          @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4823_exp__div__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_nat
% 9.20/5.18          @ ( zero_n2687167440665602831ol_nat
% 9.20/5.18            @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
% 9.20/5.18               != zero_zero_nat )
% 9.20/5.18              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 9.20/5.18          @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4824_exp__div__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_int
% 9.20/5.18          @ ( zero_n2684676970156552555ol_int
% 9.20/5.18            @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
% 9.20/5.18               != zero_zero_int )
% 9.20/5.18              & ( ord_less_eq_nat @ N2 @ M ) ) )
% 9.20/5.18          @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4825_frac__le__eq,axiom,
% 9.20/5.18      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 9.20/5.18        ( ( Y3 != zero_zero_real )
% 9.20/5.18       => ( ( Z2 != zero_zero_real )
% 9.20/5.18         => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.20/5.18            = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4826_frac__le__eq,axiom,
% 9.20/5.18      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 9.20/5.18        ( ( Y3 != zero_zero_rat )
% 9.20/5.18       => ( ( Z2 != zero_zero_rat )
% 9.20/5.18         => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.20/5.18            = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4827_frac__less__eq,axiom,
% 9.20/5.18      ! [Y3: real,Z2: real,X3: real,W2: real] :
% 9.20/5.18        ( ( Y3 != zero_zero_real )
% 9.20/5.18       => ( ( Z2 != zero_zero_real )
% 9.20/5.18         => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
% 9.20/5.18            = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4828_frac__less__eq,axiom,
% 9.20/5.18      ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
% 9.20/5.18        ( ( Y3 != zero_zero_rat )
% 9.20/5.18       => ( ( Z2 != zero_zero_rat )
% 9.20/5.18         => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
% 9.20/5.18            = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4829_signed__take__bit__int__less__exp,axiom,
% 9.20/5.18      ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4830_power__diff,axiom,
% 9.20/5.18      ! [A: complex,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_complex )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4831_power__diff,axiom,
% 9.20/5.18      ! [A: nat,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_nat )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4832_power__diff,axiom,
% 9.20/5.18      ! [A: int,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_int )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4833_power__diff,axiom,
% 9.20/5.18      ! [A: real,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_real )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4834_power__diff,axiom,
% 9.20/5.18      ! [A: rat,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_rat )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4835_power__diff,axiom,
% 9.20/5.18      ! [A: code_integer,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.18            = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4836_mod__pos__neg__trivial,axiom,
% 9.20/5.18      ! [K: int,L: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.18       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.20/5.18         => ( ( modulo_modulo_int @ K @ L )
% 9.20/5.18            = ( plus_plus_int @ K @ L ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4837_Suc__diff__eq__diff__pred,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
% 9.20/5.18          = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4838_Suc__pred_H,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( N2
% 9.20/5.18          = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4839_div__geq,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ~ ( ord_less_nat @ M @ N2 )
% 9.20/5.18         => ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.18            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4840_div__if,axiom,
% 9.20/5.18      ( divide_divide_nat
% 9.20/5.18      = ( ^ [M2: nat,N: nat] :
% 9.20/5.18            ( if_nat
% 9.20/5.18            @ ( ( ord_less_nat @ M2 @ N )
% 9.20/5.18              | ( N = zero_zero_nat ) )
% 9.20/5.18            @ zero_zero_nat
% 9.20/5.18            @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4841_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 9.20/5.18        ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4842_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: vEBT_VEBT] :
% 9.20/5.18        ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4843_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: product_prod_nat_nat] :
% 9.20/5.18        ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4844_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: real] :
% 9.20/5.18        ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert_real @ X3 @ bot_bot_set_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4845_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: nat] :
% 9.20/5.18        ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4846_set__replicate__Suc,axiom,
% 9.20/5.18      ! [N2: nat,X3: int] :
% 9.20/5.18        ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X3 ) )
% 9.20/5.18        = ( insert_int @ X3 @ bot_bot_set_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4847_add__eq__if,axiom,
% 9.20/5.18      ( plus_plus_nat
% 9.20/5.18      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4848_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: produc3843707927480180839at_nat] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 9.20/5.18            = bot_bo228742789529271731at_nat ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
% 9.20/5.18            = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4849_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: vEBT_VEBT] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 9.20/5.18            = bot_bo8194388402131092736T_VEBT ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
% 9.20/5.18            = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4850_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: product_prod_nat_nat] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 9.20/5.18            = bot_bo2099793752762293965at_nat ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
% 9.20/5.18            = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4851_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: real] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 9.20/5.18            = bot_bot_set_real ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
% 9.20/5.18            = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4852_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: nat] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 9.20/5.18            = bot_bot_set_nat ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
% 9.20/5.18            = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4853_set__replicate__conv__if,axiom,
% 9.20/5.18      ! [N2: nat,X3: int] :
% 9.20/5.18        ( ( ( N2 = zero_zero_nat )
% 9.20/5.18         => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 9.20/5.18            = bot_bot_set_int ) )
% 9.20/5.18        & ( ( N2 != zero_zero_nat )
% 9.20/5.18         => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
% 9.20/5.18            = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4854_nat__less__add__iff2,axiom,
% 9.20/5.18      ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.18       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4855_nat__less__add__iff1,axiom,
% 9.20/5.18      ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ J @ I )
% 9.20/5.18       => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
% 9.20/5.18          = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4856_mult__eq__if,axiom,
% 9.20/5.18      ( times_times_nat
% 9.20/5.18      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4857_scaling__mono,axiom,
% 9.20/5.18      ! [U: real,V: real,R2: real,S: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ U @ V )
% 9.20/5.18       => ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.20/5.18         => ( ( ord_less_eq_real @ R2 @ S )
% 9.20/5.18           => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R2 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4858_scaling__mono,axiom,
% 9.20/5.18      ! [U: rat,V: rat,R2: rat,S: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ U @ V )
% 9.20/5.18       => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
% 9.20/5.18         => ( ( ord_less_eq_rat @ R2 @ S )
% 9.20/5.18           => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4859_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] :
% 9.20/5.18        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.18         != zero_zero_nat )
% 9.20/5.18       => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 9.20/5.18         != zero_zero_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4860_exp__not__zero__imp__exp__diff__not__zero,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] :
% 9.20/5.18        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.18         != zero_zero_int )
% 9.20/5.18       => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) )
% 9.20/5.18         != zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4861_signed__take__bit__int__greater__eq__self__iff,axiom,
% 9.20/5.18      ! [K: int,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 9.20/5.18        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4862_signed__take__bit__int__less__self__iff,axiom,
% 9.20/5.18      ! [N2: nat,K: int] :
% 9.20/5.18        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
% 9.20/5.18        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4863_power__diff__power__eq,axiom,
% 9.20/5.18      ! [A: nat,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_nat )
% 9.20/5.18       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 9.20/5.18              = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 9.20/5.18          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
% 9.20/5.18              = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4864_power__diff__power__eq,axiom,
% 9.20/5.18      ! [A: int,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_zero_int )
% 9.20/5.18       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 9.20/5.18              = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 9.20/5.18          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
% 9.20/5.18              = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4865_power__diff__power__eq,axiom,
% 9.20/5.18      ! [A: code_integer,N2: nat,M: nat] :
% 9.20/5.18        ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.18       => ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 9.20/5.18              = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
% 9.20/5.18          & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18           => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
% 9.20/5.18              = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4866_split__zmod,axiom,
% 9.20/5.18      ! [P: int > $o,N2: int,K: int] :
% 9.20/5.18        ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
% 9.20/5.18        = ( ( ( K = zero_zero_int )
% 9.20/5.18           => ( P @ N2 ) )
% 9.20/5.18          & ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.18           => ! [I2: int,J3: int] :
% 9.20/5.18                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.20/5.18                  & ( ord_less_int @ J3 @ K )
% 9.20/5.18                  & ( N2
% 9.20/5.18                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.18               => ( P @ J3 ) ) )
% 9.20/5.18          & ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.18           => ! [I2: int,J3: int] :
% 9.20/5.18                ( ( ( ord_less_int @ K @ J3 )
% 9.20/5.18                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.20/5.18                  & ( N2
% 9.20/5.18                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.18               => ( P @ J3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4867_int__mod__neg__eq,axiom,
% 9.20/5.18      ! [A: int,B: int,Q4: int,R2: int] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.18       => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
% 9.20/5.18         => ( ( ord_less_int @ B @ R2 )
% 9.20/5.18           => ( ( modulo_modulo_int @ A @ B )
% 9.20/5.18              = R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4868_int__mod__pos__eq,axiom,
% 9.20/5.18      ! [A: int,B: int,Q4: int,R2: int] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
% 9.20/5.18       => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.20/5.18         => ( ( ord_less_int @ R2 @ B )
% 9.20/5.18           => ( ( modulo_modulo_int @ A @ B )
% 9.20/5.18              = R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4869_power__eq__if,axiom,
% 9.20/5.18      ( power_power_complex
% 9.20/5.18      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4870_power__eq__if,axiom,
% 9.20/5.18      ( power_power_real
% 9.20/5.18      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4871_power__eq__if,axiom,
% 9.20/5.18      ( power_power_rat
% 9.20/5.18      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4872_power__eq__if,axiom,
% 9.20/5.18      ( power_power_nat
% 9.20/5.18      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4873_power__eq__if,axiom,
% 9.20/5.18      ( power_power_int
% 9.20/5.18      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4874_power__minus__mult,axiom,
% 9.20/5.18      ! [N2: nat,A: complex] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( times_times_complex @ ( power_power_complex @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 9.20/5.18          = ( power_power_complex @ A @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4875_power__minus__mult,axiom,
% 9.20/5.18      ! [N2: nat,A: real] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( times_times_real @ ( power_power_real @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 9.20/5.18          = ( power_power_real @ A @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4876_power__minus__mult,axiom,
% 9.20/5.18      ! [N2: nat,A: rat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 9.20/5.18          = ( power_power_rat @ A @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4877_power__minus__mult,axiom,
% 9.20/5.18      ! [N2: nat,A: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 9.20/5.18          = ( power_power_nat @ A @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4878_power__minus__mult,axiom,
% 9.20/5.18      ! [N2: nat,A: int] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
% 9.20/5.18          = ( power_power_int @ A @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4879_diff__le__diff__pow,axiom,
% 9.20/5.18      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.20/5.18       => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4880_le__div__geq,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.18       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.18         => ( ( divide_divide_nat @ M @ N2 )
% 9.20/5.18            = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4881_bits__induct,axiom,
% 9.20/5.18      ! [P: code_integer > $o,A: code_integer] :
% 9.20/5.18        ( ! [A4: code_integer] :
% 9.20/5.18            ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.18              = A4 )
% 9.20/5.18           => ( P @ A4 ) )
% 9.20/5.18       => ( ! [A4: code_integer,B4: $o] :
% 9.20/5.18              ( ( P @ A4 )
% 9.20/5.18             => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.18                  = A4 )
% 9.20/5.18               => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 9.20/5.18         => ( P @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4882_bits__induct,axiom,
% 9.20/5.18      ! [P: nat > $o,A: nat] :
% 9.20/5.18        ( ! [A4: nat] :
% 9.20/5.18            ( ( ( divide_divide_nat @ A4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18              = A4 )
% 9.20/5.18           => ( P @ A4 ) )
% 9.20/5.18       => ( ! [A4: nat,B4: $o] :
% 9.20/5.18              ( ( P @ A4 )
% 9.20/5.18             => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18                  = A4 )
% 9.20/5.18               => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B4 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 9.20/5.18         => ( P @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4883_bits__induct,axiom,
% 9.20/5.18      ! [P: int > $o,A: int] :
% 9.20/5.18        ( ! [A4: int] :
% 9.20/5.18            ( ( ( divide_divide_int @ A4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.18              = A4 )
% 9.20/5.18           => ( P @ A4 ) )
% 9.20/5.18       => ( ! [A4: int,B4: $o] :
% 9.20/5.18              ( ( P @ A4 )
% 9.20/5.18             => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.18                  = A4 )
% 9.20/5.18               => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B4 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
% 9.20/5.18         => ( P @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4884_verit__le__mono__div__int,axiom,
% 9.20/5.18      ! [A2: int,B2: int,N2: int] :
% 9.20/5.18        ( ( ord_less_int @ A2 @ B2 )
% 9.20/5.18       => ( ( ord_less_int @ zero_zero_int @ N2 )
% 9.20/5.18         => ( ord_less_eq_int
% 9.20/5.18            @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
% 9.20/5.18              @ ( if_int
% 9.20/5.18                @ ( ( modulo_modulo_int @ B2 @ N2 )
% 9.20/5.18                  = zero_zero_int )
% 9.20/5.18                @ one_one_int
% 9.20/5.18                @ zero_zero_int ) )
% 9.20/5.18            @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4885_split__pos__lemma,axiom,
% 9.20/5.18      ! [K: int,P: int > int > $o,N2: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.18       => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 9.20/5.18          = ( ! [I2: int,J3: int] :
% 9.20/5.18                ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
% 9.20/5.18                  & ( ord_less_int @ J3 @ K )
% 9.20/5.18                  & ( N2
% 9.20/5.18                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.18               => ( P @ I2 @ J3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4886_split__neg__lemma,axiom,
% 9.20/5.18      ! [K: int,P: int > int > $o,N2: int] :
% 9.20/5.18        ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.18       => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
% 9.20/5.18          = ( ! [I2: int,J3: int] :
% 9.20/5.18                ( ( ( ord_less_int @ K @ J3 )
% 9.20/5.18                  & ( ord_less_eq_int @ J3 @ zero_zero_int )
% 9.20/5.18                  & ( N2
% 9.20/5.18                    = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
% 9.20/5.18               => ( P @ I2 @ J3 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4887_int__power__div__base,axiom,
% 9.20/5.18      ! [M: nat,K: int] :
% 9.20/5.18        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.18       => ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.18         => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
% 9.20/5.18            = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4888_set__decode__plus__power__2,axiom,
% 9.20/5.18      ! [N2: nat,Z2: nat] :
% 9.20/5.18        ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z2 ) )
% 9.20/5.18       => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z2 ) )
% 9.20/5.18          = ( insert_nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4889_exp__mod__exp,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4890_exp__mod__exp,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4891_exp__mod__exp,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4892_power2__diff,axiom,
% 9.20/5.18      ! [X3: complex,Y3: complex] :
% 9.20/5.18        ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4893_power2__diff,axiom,
% 9.20/5.18      ! [X3: real,Y3: real] :
% 9.20/5.18        ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4894_power2__diff,axiom,
% 9.20/5.18      ! [X3: rat,Y3: rat] :
% 9.20/5.18        ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4895_power2__diff,axiom,
% 9.20/5.18      ! [X3: int,Y3: int] :
% 9.20/5.18        ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4896_mult__exp__mod__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,A: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18          = ( times_times_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4897_mult__exp__mod__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,A: int] :
% 9.20/5.18        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18       => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18          = ( times_times_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4898_mult__exp__mod__exp__eq,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,A: code_integer] :
% 9.20/5.18        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18          = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4899_even__mod__4__div__2,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.18          = ( suc @ zero_zero_nat ) )
% 9.20/5.18       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4900_eucl__rel__int__iff,axiom,
% 9.20/5.18      ! [K: int,L: int,Q4: int,R2: int] :
% 9.20/5.18        ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.18        = ( ( K
% 9.20/5.18            = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
% 9.20/5.18          & ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.18           => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
% 9.20/5.18              & ( ord_less_int @ R2 @ L ) ) )
% 9.20/5.18          & ( ~ ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.18           => ( ( ( ord_less_int @ L @ zero_zero_int )
% 9.20/5.18               => ( ( ord_less_int @ L @ R2 )
% 9.20/5.18                  & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
% 9.20/5.18              & ( ~ ( ord_less_int @ L @ zero_zero_int )
% 9.20/5.18               => ( Q4 = zero_zero_int ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4901_divmod__digit__1_I2_J,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.18       => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.18         => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.18           => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.18              = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4902_divmod__digit__1_I2_J,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ zero_zero_nat @ A )
% 9.20/5.18       => ( ( ord_less_nat @ zero_zero_nat @ B )
% 9.20/5.18         => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.18           => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.18              = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4903_divmod__digit__1_I2_J,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.18       => ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.18         => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
% 9.20/5.18           => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
% 9.20/5.18              = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4904_even__mask__div__iff_H,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4905_even__mask__div__iff_H,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4906_even__mask__div__iff_H,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4907_even__mask__div__iff,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.18            = zero_zero_nat )
% 9.20/5.18          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4908_even__mask__div__iff,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.18            = zero_zero_int )
% 9.20/5.18          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4909_even__mask__div__iff,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( 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 ) ) @ N2 ) ) )
% 9.20/5.18        = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.18            = zero_z3403309356797280102nteger )
% 9.20/5.18          | ( ord_less_eq_nat @ M @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4910_divmod__step__eq,axiom,
% 9.20/5.18      ! [L: num,R2: nat,Q4: nat] :
% 9.20/5.18        ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.20/5.18         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.20/5.18            = ( 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 ) ) ) ) )
% 9.20/5.18        & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
% 9.20/5.18         => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
% 9.20/5.18            = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4911_divmod__step__eq,axiom,
% 9.20/5.18      ! [L: num,R2: int,Q4: int] :
% 9.20/5.18        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.20/5.18         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.18            = ( 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 ) ) ) ) )
% 9.20/5.18        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
% 9.20/5.18         => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.18            = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4912_divmod__step__eq,axiom,
% 9.20/5.18      ! [L: num,R2: code_integer,Q4: code_integer] :
% 9.20/5.18        ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.20/5.18         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 9.20/5.18            = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
% 9.20/5.18        & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
% 9.20/5.18         => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
% 9.20/5.18            = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4913_inrange,axiom,
% 9.20/5.18      ! [T: vEBT_VEBT,N2: nat] :
% 9.20/5.18        ( ( vEBT_invar_vebt @ T @ N2 )
% 9.20/5.18       => ( 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 ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4914_Divides_Oadjust__div__eq,axiom,
% 9.20/5.18      ! [Q4: int,R2: int] :
% 9.20/5.18        ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.18        = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4915_signed__take__bit__rec,axiom,
% 9.20/5.18      ( bit_ri6519982836138164636nteger
% 9.20/5.18      = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4916_signed__take__bit__rec,axiom,
% 9.20/5.18      ( bit_ri631733984087533419it_int
% 9.20/5.18      = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ ( uminus_uminus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( plus_plus_int @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4917_neg__eucl__rel__int__mult__2,axiom,
% 9.20/5.18      ! [B: int,A: int,Q4: int,R2: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ B @ zero_zero_int )
% 9.20/5.18       => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.18         => ( 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 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4918_diff__shunt__var,axiom,
% 9.20/5.18      ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( ( minus_1356011639430497352at_nat @ X3 @ Y3 )
% 9.20/5.18          = bot_bo2099793752762293965at_nat )
% 9.20/5.18        = ( ord_le3146513528884898305at_nat @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4919_diff__shunt__var,axiom,
% 9.20/5.18      ! [X3: set_real,Y3: set_real] :
% 9.20/5.18        ( ( ( minus_minus_set_real @ X3 @ Y3 )
% 9.20/5.18          = bot_bot_set_real )
% 9.20/5.18        = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4920_diff__shunt__var,axiom,
% 9.20/5.18      ! [X3: set_nat,Y3: set_nat] :
% 9.20/5.18        ( ( ( minus_minus_set_nat @ X3 @ Y3 )
% 9.20/5.18          = bot_bot_set_nat )
% 9.20/5.18        = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4921_diff__shunt__var,axiom,
% 9.20/5.18      ! [X3: set_int,Y3: set_int] :
% 9.20/5.18        ( ( ( minus_minus_set_int @ X3 @ Y3 )
% 9.20/5.18          = bot_bot_set_int )
% 9.20/5.18        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4922_take__bit__rec,axiom,
% 9.20/5.18      ( bit_se1745604003318907178nteger
% 9.20/5.18      = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4923_take__bit__rec,axiom,
% 9.20/5.18      ( bit_se2925701944663578781it_nat
% 9.20/5.18      = ( ^ [N: nat,A3: nat] : ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4924_take__bit__rec,axiom,
% 9.20/5.18      ( bit_se2923211474154528505it_int
% 9.20/5.18      = ( ^ [N: nat,A3: int] : ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4925_add_Oinverse__inverse,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = A ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4926_add_Oinverse__inverse,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = A ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4927_add_Oinverse__inverse,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = A ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4928_add_Oinverse__inverse,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = A ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4929_neg__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( uminus_uminus_int @ A )
% 9.20/5.18          = ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( A = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4930_neg__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( uminus_uminus_real @ A )
% 9.20/5.18          = ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( A = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4931_neg__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( A = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4932_neg__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( A = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4933_Diff__cancel,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
% 9.20/5.18        = bot_bo2099793752762293965at_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4934_Diff__cancel,axiom,
% 9.20/5.18      ! [A2: set_real] :
% 9.20/5.18        ( ( minus_minus_set_real @ A2 @ A2 )
% 9.20/5.18        = bot_bot_set_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4935_Diff__cancel,axiom,
% 9.20/5.18      ! [A2: set_int] :
% 9.20/5.18        ( ( minus_minus_set_int @ A2 @ A2 )
% 9.20/5.18        = bot_bot_set_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4936_Diff__cancel,axiom,
% 9.20/5.18      ! [A2: set_nat] :
% 9.20/5.18        ( ( minus_minus_set_nat @ A2 @ A2 )
% 9.20/5.18        = bot_bot_set_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4937_empty__Diff,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
% 9.20/5.18        = bot_bo2099793752762293965at_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4938_empty__Diff,axiom,
% 9.20/5.18      ! [A2: set_real] :
% 9.20/5.18        ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
% 9.20/5.18        = bot_bot_set_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4939_empty__Diff,axiom,
% 9.20/5.18      ! [A2: set_int] :
% 9.20/5.18        ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
% 9.20/5.18        = bot_bot_set_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4940_empty__Diff,axiom,
% 9.20/5.18      ! [A2: set_nat] :
% 9.20/5.18        ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
% 9.20/5.18        = bot_bot_set_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4941_Diff__empty,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
% 9.20/5.18        = A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4942_Diff__empty,axiom,
% 9.20/5.18      ! [A2: set_real] :
% 9.20/5.18        ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
% 9.20/5.18        = A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4943_Diff__empty,axiom,
% 9.20/5.18      ! [A2: set_int] :
% 9.20/5.18        ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
% 9.20/5.18        = A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4944_Diff__empty,axiom,
% 9.20/5.18      ! [A2: set_nat] :
% 9.20/5.18        ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
% 9.20/5.18        = A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4945_finite__Diff2,axiom,
% 9.20/5.18      ! [B2: set_int,A2: set_int] :
% 9.20/5.18        ( ( finite_finite_int @ B2 )
% 9.20/5.18       => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.18          = ( finite_finite_int @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4946_finite__Diff2,axiom,
% 9.20/5.18      ! [B2: set_complex,A2: set_complex] :
% 9.20/5.18        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.18       => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.18          = ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4947_finite__Diff2,axiom,
% 9.20/5.18      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( finite6177210948735845034at_nat @ B2 )
% 9.20/5.18       => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.18          = ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4948_finite__Diff2,axiom,
% 9.20/5.18      ! [B2: set_nat,A2: set_nat] :
% 9.20/5.18        ( ( finite_finite_nat @ B2 )
% 9.20/5.18       => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.18          = ( finite_finite_nat @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4949_finite__Diff,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] :
% 9.20/5.18        ( ( finite_finite_int @ A2 )
% 9.20/5.18       => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4950_finite__Diff,axiom,
% 9.20/5.18      ! [A2: set_complex,B2: set_complex] :
% 9.20/5.18        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.18       => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4951_finite__Diff,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.18       => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4952_finite__Diff,axiom,
% 9.20/5.18      ! [A2: set_nat,B2: set_nat] :
% 9.20/5.18        ( ( finite_finite_nat @ A2 )
% 9.20/5.18       => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4953_Compl__subset__Compl__iff,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
% 9.20/5.18        = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4954_Compl__anti__mono,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.20/5.18       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4955_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4956_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4957_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: real,A2: set_real,B2: set_real] :
% 9.20/5.18        ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4958_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.18        ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4959_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: int,A2: set_int,B2: set_int] :
% 9.20/5.18        ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4960_Diff__insert0,axiom,
% 9.20/5.18      ! [X3: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.18        ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.18       => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 9.20/5.18          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4961_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4962_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4963_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: real,B2: set_real,A2: set_real] :
% 9.20/5.18        ( ( member_real @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4964_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.20/5.18        ( ( member_set_nat @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4965_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: int,B2: set_int,A2: set_int] :
% 9.20/5.18        ( ( member_int @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4966_insert__Diff1,axiom,
% 9.20/5.18      ! [X3: nat,B2: set_nat,A2: set_nat] :
% 9.20/5.18        ( ( member_nat @ X3 @ B2 )
% 9.20/5.18       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18          = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4967_compl__le__compl__iff,axiom,
% 9.20/5.18      ! [X3: set_int,Y3: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
% 9.20/5.18        = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4968_neg__le__iff__le,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_eq_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4969_neg__le__iff__le,axiom,
% 9.20/5.18      ! [B: code_integer,A: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4970_neg__le__iff__le,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_eq_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4971_neg__le__iff__le,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_eq_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4972_neg__equal__zero,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ( uminus_uminus_int @ A )
% 9.20/5.18          = A )
% 9.20/5.18        = ( A = zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4973_neg__equal__zero,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ( uminus_uminus_real @ A )
% 9.20/5.18          = A )
% 9.20/5.18        = ( A = zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4974_neg__equal__zero,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = A )
% 9.20/5.18        = ( A = zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4975_neg__equal__zero,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = A )
% 9.20/5.18        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4976_equal__neg__zero,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( A = zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4977_equal__neg__zero,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( A = zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4978_equal__neg__zero,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( A = zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4979_equal__neg__zero,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4980_neg__equal__0__iff__equal,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( ( uminus1482373934393186551omplex @ A )
% 9.20/5.18          = zero_zero_complex )
% 9.20/5.18        = ( A = zero_zero_complex ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4981_neg__equal__0__iff__equal,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ( uminus_uminus_int @ A )
% 9.20/5.18          = zero_zero_int )
% 9.20/5.18        = ( A = zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4982_neg__equal__0__iff__equal,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ( uminus_uminus_real @ A )
% 9.20/5.18          = zero_zero_real )
% 9.20/5.18        = ( A = zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4983_neg__equal__0__iff__equal,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = zero_zero_rat )
% 9.20/5.18        = ( A = zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4984_neg__equal__0__iff__equal,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = zero_z3403309356797280102nteger )
% 9.20/5.18        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4985_neg__0__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( zero_zero_complex
% 9.20/5.18          = ( uminus1482373934393186551omplex @ A ) )
% 9.20/5.18        = ( zero_zero_complex = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4986_neg__0__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( zero_zero_int
% 9.20/5.18          = ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( zero_zero_int = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4987_neg__0__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( zero_zero_real
% 9.20/5.18          = ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( zero_zero_real = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4988_neg__0__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( zero_zero_rat
% 9.20/5.18          = ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( zero_zero_rat = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4989_neg__0__equal__iff__equal,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( zero_z3403309356797280102nteger
% 9.20/5.18          = ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( zero_z3403309356797280102nteger = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4990_add_Oinverse__neutral,axiom,
% 9.20/5.18      ( ( uminus1482373934393186551omplex @ zero_zero_complex )
% 9.20/5.18      = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4991_add_Oinverse__neutral,axiom,
% 9.20/5.18      ( ( uminus_uminus_int @ zero_zero_int )
% 9.20/5.18      = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4992_add_Oinverse__neutral,axiom,
% 9.20/5.18      ( ( uminus_uminus_real @ zero_zero_real )
% 9.20/5.18      = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4993_add_Oinverse__neutral,axiom,
% 9.20/5.18      ( ( uminus_uminus_rat @ zero_zero_rat )
% 9.20/5.18      = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4994_add_Oinverse__neutral,axiom,
% 9.20/5.18      ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
% 9.20/5.18      = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4995_neg__less__iff__less,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4996_neg__less__iff__less,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4997_neg__less__iff__less,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4998_neg__less__iff__less,axiom,
% 9.20/5.18      ! [B: code_integer,A: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_4999_mult__minus__left,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 9.20/5.18        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5000_mult__minus__left,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.18        = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5001_mult__minus__left,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.18        = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5002_mult__minus__left,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.18        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5003_mult__minus__left,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5004_minus__mult__minus,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.20/5.18        = ( times_times_complex @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5005_minus__mult__minus,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( times_times_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5006_minus__mult__minus,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( times_times_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5007_minus__mult__minus,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( times_times_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5008_minus__mult__minus,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( times_3573771949741848930nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5009_mult__minus__right,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
% 9.20/5.18        = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5010_mult__minus__right,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5011_mult__minus__right,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5012_mult__minus__right,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5013_mult__minus__right,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5014_minus__add__distrib,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.20/5.18        = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5015_minus__add__distrib,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.20/5.18        = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5016_minus__add__distrib,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.18        = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5017_minus__add__distrib,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.18        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5018_minus__add__cancel,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5019_minus__add__cancel,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5020_minus__add__cancel,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5021_minus__add__cancel,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5022_add__minus__cancel,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5023_add__minus__cancel,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5024_add__minus__cancel,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5025_add__minus__cancel,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
% 9.20/5.18        = B ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5026_minus__diff__eq,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 9.20/5.18        = ( minus_minus_int @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5027_minus__diff__eq,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
% 9.20/5.18        = ( minus_minus_real @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5028_minus__diff__eq,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.18        = ( minus_minus_rat @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5029_minus__diff__eq,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.20/5.18        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5030_dvd__minus__iff,axiom,
% 9.20/5.18      ! [X3: int,Y3: int] :
% 9.20/5.18        ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 9.20/5.18        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5031_dvd__minus__iff,axiom,
% 9.20/5.18      ! [X3: real,Y3: real] :
% 9.20/5.18        ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
% 9.20/5.18        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5032_dvd__minus__iff,axiom,
% 9.20/5.18      ! [X3: rat,Y3: rat] :
% 9.20/5.18        ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
% 9.20/5.18        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5033_dvd__minus__iff,axiom,
% 9.20/5.18      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.18        ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
% 9.20/5.18        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5034_minus__dvd__iff,axiom,
% 9.20/5.18      ! [X3: int,Y3: int] :
% 9.20/5.18        ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 9.20/5.18        = ( dvd_dvd_int @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5035_minus__dvd__iff,axiom,
% 9.20/5.18      ! [X3: real,Y3: real] :
% 9.20/5.18        ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
% 9.20/5.18        = ( dvd_dvd_real @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5036_minus__dvd__iff,axiom,
% 9.20/5.18      ! [X3: rat,Y3: rat] :
% 9.20/5.18        ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
% 9.20/5.18        = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5037_minus__dvd__iff,axiom,
% 9.20/5.18      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.18        ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
% 9.20/5.18        = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5038_Diff__eq__empty__iff,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
% 9.20/5.18          = bot_bo2099793752762293965at_nat )
% 9.20/5.18        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5039_Diff__eq__empty__iff,axiom,
% 9.20/5.18      ! [A2: set_real,B2: set_real] :
% 9.20/5.18        ( ( ( minus_minus_set_real @ A2 @ B2 )
% 9.20/5.18          = bot_bot_set_real )
% 9.20/5.18        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5040_Diff__eq__empty__iff,axiom,
% 9.20/5.18      ! [A2: set_nat,B2: set_nat] :
% 9.20/5.18        ( ( ( minus_minus_set_nat @ A2 @ B2 )
% 9.20/5.18          = bot_bot_set_nat )
% 9.20/5.18        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5041_Diff__eq__empty__iff,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] :
% 9.20/5.18        ( ( ( minus_minus_set_int @ A2 @ B2 )
% 9.20/5.18          = bot_bot_set_int )
% 9.20/5.18        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5042_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: set_nat,L: set_nat,U: set_nat] :
% 9.20/5.18        ( ( member_set_nat @ I @ ( set_or4548717258645045905et_nat @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_set_nat @ L @ I )
% 9.20/5.18          & ( ord_less_eq_set_nat @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5043_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: set_int,L: set_int,U: set_int] :
% 9.20/5.18        ( ( member_set_int @ I @ ( set_or370866239135849197et_int @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_set_int @ L @ I )
% 9.20/5.18          & ( ord_less_eq_set_int @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5044_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: rat,L: rat,U: rat] :
% 9.20/5.18        ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_rat @ L @ I )
% 9.20/5.18          & ( ord_less_eq_rat @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5045_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: num,L: num,U: num] :
% 9.20/5.18        ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_num @ L @ I )
% 9.20/5.18          & ( ord_less_eq_num @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5046_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: nat,L: nat,U: nat] :
% 9.20/5.18        ( ( member_nat @ I @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_nat @ L @ I )
% 9.20/5.18          & ( ord_less_eq_nat @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5047_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: int,L: int,U: int] :
% 9.20/5.18        ( ( member_int @ I @ ( set_or1266510415728281911st_int @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_int @ L @ I )
% 9.20/5.18          & ( ord_less_eq_int @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5048_atLeastAtMost__iff,axiom,
% 9.20/5.18      ! [I: real,L: real,U: real] :
% 9.20/5.18        ( ( member_real @ I @ ( set_or1222579329274155063t_real @ L @ U ) )
% 9.20/5.18        = ( ( ord_less_eq_real @ L @ I )
% 9.20/5.18          & ( ord_less_eq_real @ I @ U ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5049_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: set_int,H2: set_int,L3: set_int,H3: set_int] :
% 9.20/5.18        ( ( ( set_or370866239135849197et_int @ L @ H2 )
% 9.20/5.18          = ( set_or370866239135849197et_int @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5050_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: rat,H2: rat,L3: rat,H3: rat] :
% 9.20/5.18        ( ( ( set_or633870826150836451st_rat @ L @ H2 )
% 9.20/5.18          = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5051_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: num,H2: num,L3: num,H3: num] :
% 9.20/5.18        ( ( ( set_or7049704709247886629st_num @ L @ H2 )
% 9.20/5.18          = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5052_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: nat,H2: nat,L3: nat,H3: nat] :
% 9.20/5.18        ( ( ( set_or1269000886237332187st_nat @ L @ H2 )
% 9.20/5.18          = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5053_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: int,H2: int,L3: int,H3: int] :
% 9.20/5.18        ( ( ( set_or1266510415728281911st_int @ L @ H2 )
% 9.20/5.18          = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5054_Icc__eq__Icc,axiom,
% 9.20/5.18      ! [L: real,H2: real,L3: real,H3: real] :
% 9.20/5.18        ( ( ( set_or1222579329274155063t_real @ L @ H2 )
% 9.20/5.18          = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
% 9.20/5.18        = ( ( ( L = L3 )
% 9.20/5.18            & ( H2 = H3 ) )
% 9.20/5.18          | ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.20/5.18            & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5055_insert__Diff__single,axiom,
% 9.20/5.18      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.18        = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5056_insert__Diff__single,axiom,
% 9.20/5.18      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.18        = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5057_insert__Diff__single,axiom,
% 9.20/5.18      ! [A: real,A2: set_real] :
% 9.20/5.18        ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.18        = ( insert_real @ A @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5058_insert__Diff__single,axiom,
% 9.20/5.18      ! [A: int,A2: set_int] :
% 9.20/5.18        ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.18        = ( insert_int @ A @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5059_insert__Diff__single,axiom,
% 9.20/5.18      ! [A: nat,A2: set_nat] :
% 9.20/5.18        ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.18        = ( insert_nat @ A @ A2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5060_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_real,A: real,B2: set_real] :
% 9.20/5.18        ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
% 9.20/5.18        = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5061_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
% 9.20/5.18        = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5062_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_int,A: int,B2: set_int] :
% 9.20/5.18        ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
% 9.20/5.18        = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5063_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_complex,A: complex,B2: set_complex] :
% 9.20/5.18        ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
% 9.20/5.18        = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5064_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
% 9.20/5.18        = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5065_finite__Diff__insert,axiom,
% 9.20/5.18      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.20/5.18        ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
% 9.20/5.18        = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5066_take__bit__of__0,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
% 9.20/5.18        = zero_zero_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5067_take__bit__of__0,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5068_finite__atLeastAtMost,axiom,
% 9.20/5.18      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5069_neg__less__eq__nonneg,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ A )
% 9.20/5.18        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5070_neg__less__eq__nonneg,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5071_neg__less__eq__nonneg,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.20/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5072_neg__less__eq__nonneg,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
% 9.20/5.18        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5073_less__eq__neg__nonpos,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5074_less__eq__neg__nonpos,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5075_less__eq__neg__nonpos,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5076_less__eq__neg__nonpos,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5077_neg__le__0__iff__le,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.20/5.18        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5078_neg__le__0__iff__le,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5079_neg__le__0__iff__le,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.20/5.18        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5080_neg__le__0__iff__le,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.20/5.18        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5081_neg__0__le__iff__le,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5082_neg__0__le__iff__le,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5083_neg__0__le__iff__le,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5084_neg__0__le__iff__le,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5085_less__neg__neg,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5086_less__neg__neg,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_real @ A @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5087_less__neg__neg,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5088_less__neg__neg,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5089_neg__less__pos,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
% 9.20/5.18        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5090_neg__less__pos,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ A )
% 9.20/5.18        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5091_neg__less__pos,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.20/5.18        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5092_neg__less__pos,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5093_neg__0__less__iff__less,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = ( ord_less_int @ A @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5094_neg__0__less__iff__less,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = ( ord_less_real @ A @ zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5095_neg__0__less__iff__less,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = ( ord_less_rat @ A @ zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5096_neg__0__less__iff__less,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5097_neg__less__0__iff__less,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
% 9.20/5.18        = ( ord_less_int @ zero_zero_int @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5098_neg__less__0__iff__less,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ zero_zero_real )
% 9.20/5.18        = ( ord_less_real @ zero_zero_real @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5099_neg__less__0__iff__less,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
% 9.20/5.18        = ( ord_less_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5100_neg__less__0__iff__less,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5101_add_Oright__inverse,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
% 9.20/5.18        = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5102_add_Oright__inverse,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5103_add_Oright__inverse,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
% 9.20/5.18        = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5104_add_Oright__inverse,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
% 9.20/5.18        = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5105_add_Oright__inverse,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.18        = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5106_ab__left__minus,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.20/5.18        = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5107_ab__left__minus,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5108_ab__left__minus,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.20/5.18        = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5109_ab__left__minus,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.20/5.18        = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5110_ab__left__minus,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.20/5.18        = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5111_verit__minus__simplify_I3_J,axiom,
% 9.20/5.18      ! [B: complex] :
% 9.20/5.18        ( ( minus_minus_complex @ zero_zero_complex @ B )
% 9.20/5.18        = ( uminus1482373934393186551omplex @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5112_verit__minus__simplify_I3_J,axiom,
% 9.20/5.18      ! [B: int] :
% 9.20/5.18        ( ( minus_minus_int @ zero_zero_int @ B )
% 9.20/5.18        = ( uminus_uminus_int @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5113_verit__minus__simplify_I3_J,axiom,
% 9.20/5.18      ! [B: real] :
% 9.20/5.18        ( ( minus_minus_real @ zero_zero_real @ B )
% 9.20/5.18        = ( uminus_uminus_real @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5114_verit__minus__simplify_I3_J,axiom,
% 9.20/5.18      ! [B: rat] :
% 9.20/5.18        ( ( minus_minus_rat @ zero_zero_rat @ B )
% 9.20/5.18        = ( uminus_uminus_rat @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5115_verit__minus__simplify_I3_J,axiom,
% 9.20/5.18      ! [B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5116_diff__0,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( minus_minus_complex @ zero_zero_complex @ A )
% 9.20/5.18        = ( uminus1482373934393186551omplex @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5117_diff__0,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( minus_minus_int @ zero_zero_int @ A )
% 9.20/5.18        = ( uminus_uminus_int @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5118_diff__0,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( minus_minus_real @ zero_zero_real @ A )
% 9.20/5.18        = ( uminus_uminus_real @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5119_diff__0,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( minus_minus_rat @ zero_zero_rat @ A )
% 9.20/5.18        = ( uminus_uminus_rat @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5120_diff__0,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5121_add__neg__numeral__simps_I3_J,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.18        = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5122_add__neg__numeral__simps_I3_J,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 9.20/5.18        = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5123_add__neg__numeral__simps_I3_J,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 9.20/5.18        = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5124_add__neg__numeral__simps_I3_J,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5125_uminus__add__conv__diff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.18        = ( minus_minus_int @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5126_uminus__add__conv__diff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.18        = ( minus_minus_real @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5127_uminus__add__conv__diff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.18        = ( minus_minus_rat @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5128_uminus__add__conv__diff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.18        = ( minus_8373710615458151222nteger @ B @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5129_diff__minus__eq__add,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( plus_plus_int @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5130_diff__minus__eq__add,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( plus_plus_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5131_diff__minus__eq__add,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( plus_plus_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5132_diff__minus__eq__add,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( plus_p5714425477246183910nteger @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5133_divide__minus1,axiom,
% 9.20/5.18      ! [X3: complex] :
% 9.20/5.18        ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.18        = ( uminus1482373934393186551omplex @ X3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5134_divide__minus1,axiom,
% 9.20/5.18      ! [X3: real] :
% 9.20/5.18        ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.18        = ( uminus_uminus_real @ X3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5135_divide__minus1,axiom,
% 9.20/5.18      ! [X3: rat] :
% 9.20/5.18        ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.18        = ( uminus_uminus_rat @ X3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5136_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: set_int,B: set_int] :
% 9.20/5.18        ( ( ( set_or370866239135849197et_int @ A @ B )
% 9.20/5.18          = bot_bot_set_set_int )
% 9.20/5.18        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5137_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( set_or633870826150836451st_rat @ A @ B )
% 9.20/5.18          = bot_bot_set_rat )
% 9.20/5.18        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5138_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: num,B: num] :
% 9.20/5.18        ( ( ( set_or7049704709247886629st_num @ A @ B )
% 9.20/5.18          = bot_bot_set_num )
% 9.20/5.18        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5139_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.20/5.18          = bot_bot_set_nat )
% 9.20/5.18        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5140_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 9.20/5.18          = bot_bot_set_int )
% 9.20/5.18        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5141_atLeastatMost__empty__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 9.20/5.18          = bot_bot_set_real )
% 9.20/5.18        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5142_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: set_int,B: set_int] :
% 9.20/5.18        ( ( bot_bot_set_set_int
% 9.20/5.18          = ( set_or370866239135849197et_int @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5143_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( bot_bot_set_rat
% 9.20/5.18          = ( set_or633870826150836451st_rat @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5144_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: num,B: num] :
% 9.20/5.18        ( ( bot_bot_set_num
% 9.20/5.18          = ( set_or7049704709247886629st_num @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5145_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( bot_bot_set_nat
% 9.20/5.18          = ( set_or1269000886237332187st_nat @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5146_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( bot_bot_set_int
% 9.20/5.18          = ( set_or1266510415728281911st_int @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5147_atLeastatMost__empty__iff2,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( bot_bot_set_real
% 9.20/5.18          = ( set_or1222579329274155063t_real @ A @ B ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5148_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 9.20/5.18        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_set_int @ C @ A )
% 9.20/5.18            & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5149_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.18        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_rat @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_rat @ C @ A )
% 9.20/5.18            & ( ord_less_eq_rat @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5150_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: num,B: num,C: num,D: num] :
% 9.20/5.18        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_num @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_num @ C @ A )
% 9.20/5.18            & ( ord_less_eq_num @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5151_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.18        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_nat @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_nat @ C @ A )
% 9.20/5.18            & ( ord_less_eq_nat @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5152_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_int @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_int @ C @ A )
% 9.20/5.18            & ( ord_less_eq_int @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5153_atLeastatMost__subset__iff,axiom,
% 9.20/5.18      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.18        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.20/5.18        = ( ~ ( ord_less_eq_real @ A @ B )
% 9.20/5.18          | ( ( ord_less_eq_real @ C @ A )
% 9.20/5.18            & ( ord_less_eq_real @ B @ D ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5154_atLeastatMost__empty,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ B @ A )
% 9.20/5.18       => ( ( set_or633870826150836451st_rat @ A @ B )
% 9.20/5.18          = bot_bot_set_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5155_atLeastatMost__empty,axiom,
% 9.20/5.18      ! [B: num,A: num] :
% 9.20/5.18        ( ( ord_less_num @ B @ A )
% 9.20/5.18       => ( ( set_or7049704709247886629st_num @ A @ B )
% 9.20/5.18          = bot_bot_set_num ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5156_atLeastatMost__empty,axiom,
% 9.20/5.18      ! [B: nat,A: nat] :
% 9.20/5.18        ( ( ord_less_nat @ B @ A )
% 9.20/5.18       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.20/5.18          = bot_bot_set_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5157_atLeastatMost__empty,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( ord_less_int @ B @ A )
% 9.20/5.18       => ( ( set_or1266510415728281911st_int @ A @ B )
% 9.20/5.18          = bot_bot_set_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5158_atLeastatMost__empty,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( ord_less_real @ B @ A )
% 9.20/5.18       => ( ( set_or1222579329274155063t_real @ A @ B )
% 9.20/5.18          = bot_bot_set_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5159_infinite__Icc__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
% 9.20/5.18        = ( ord_less_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5160_infinite__Icc__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
% 9.20/5.18        = ( ord_less_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5161_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
% 9.20/5.18        ( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.18        = ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5162_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_set_nat,B: set_nat] :
% 9.20/5.18        ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
% 9.20/5.18        = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5163_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
% 9.20/5.18        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.18        = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5164_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_real,B: real] :
% 9.20/5.18        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
% 9.20/5.18        = ( ~ ( member_real @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5165_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_nat,B: nat] :
% 9.20/5.18        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
% 9.20/5.18        = ( ~ ( member_nat @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5166_subset__Compl__singleton,axiom,
% 9.20/5.18      ! [A2: set_int,B: int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
% 9.20/5.18        = ( ~ ( member_int @ B @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5167_take__bit__0,axiom,
% 9.20/5.18      ! [A: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
% 9.20/5.18        = zero_zero_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5168_take__bit__0,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5169_atLeastAtMost__singleton,axiom,
% 9.20/5.18      ! [A: nat] :
% 9.20/5.18        ( ( set_or1269000886237332187st_nat @ A @ A )
% 9.20/5.18        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5170_atLeastAtMost__singleton,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( set_or1266510415728281911st_int @ A @ A )
% 9.20/5.18        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5171_atLeastAtMost__singleton,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( set_or1222579329274155063t_real @ A @ A )
% 9.20/5.18        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5172_atLeastAtMost__singleton__iff,axiom,
% 9.20/5.18      ! [A: nat,B: nat,C: nat] :
% 9.20/5.18        ( ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.20/5.18          = ( insert_nat @ C @ bot_bot_set_nat ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          & ( B = C ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5173_atLeastAtMost__singleton__iff,axiom,
% 9.20/5.18      ! [A: int,B: int,C: int] :
% 9.20/5.18        ( ( ( set_or1266510415728281911st_int @ A @ B )
% 9.20/5.18          = ( insert_int @ C @ bot_bot_set_int ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          & ( B = C ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5174_atLeastAtMost__singleton__iff,axiom,
% 9.20/5.18      ! [A: real,B: real,C: real] :
% 9.20/5.18        ( ( ( set_or1222579329274155063t_real @ A @ B )
% 9.20/5.18          = ( insert_real @ C @ bot_bot_set_real ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          & ( B = C ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5175_take__bit__Suc__1,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
% 9.20/5.18        = one_one_nat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5176_take__bit__Suc__1,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
% 9.20/5.18        = one_one_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5177_add__neg__numeral__special_I8_J,axiom,
% 9.20/5.18      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
% 9.20/5.18      = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5178_add__neg__numeral__special_I8_J,axiom,
% 9.20/5.18      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
% 9.20/5.18      = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5179_add__neg__numeral__special_I8_J,axiom,
% 9.20/5.18      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
% 9.20/5.18      = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5180_add__neg__numeral__special_I8_J,axiom,
% 9.20/5.18      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
% 9.20/5.18      = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5181_add__neg__numeral__special_I8_J,axiom,
% 9.20/5.18      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
% 9.20/5.18      = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5182_add__neg__numeral__special_I7_J,axiom,
% 9.20/5.18      ( ( plus_plus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.18      = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5183_add__neg__numeral__special_I7_J,axiom,
% 9.20/5.18      ( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.18      = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5184_add__neg__numeral__special_I7_J,axiom,
% 9.20/5.18      ( ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.18      = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5185_add__neg__numeral__special_I7_J,axiom,
% 9.20/5.18      ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.18      = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5186_add__neg__numeral__special_I7_J,axiom,
% 9.20/5.18      ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.18      = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5187_diff__numeral__special_I12_J,axiom,
% 9.20/5.18      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.18      = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5188_diff__numeral__special_I12_J,axiom,
% 9.20/5.18      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.18      = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5189_diff__numeral__special_I12_J,axiom,
% 9.20/5.18      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.18      = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5190_diff__numeral__special_I12_J,axiom,
% 9.20/5.18      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.18      = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5191_diff__numeral__special_I12_J,axiom,
% 9.20/5.18      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.18      = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5192_mod__minus1__right,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5193_mod__minus1__right,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.18        = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5194_take__bit__of__1__eq__0__iff,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
% 9.20/5.18          = zero_zero_nat )
% 9.20/5.18        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5195_take__bit__of__1__eq__0__iff,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
% 9.20/5.18          = zero_zero_int )
% 9.20/5.18        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5196_semiring__norm_I168_J,axiom,
% 9.20/5.18      ! [V: num,W2: num,Y3: int] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
% 9.20/5.18        = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5197_semiring__norm_I168_J,axiom,
% 9.20/5.18      ! [V: num,W2: num,Y3: real] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y3 ) )
% 9.20/5.18        = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5198_semiring__norm_I168_J,axiom,
% 9.20/5.18      ! [V: num,W2: num,Y3: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
% 9.20/5.18        = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5199_semiring__norm_I168_J,axiom,
% 9.20/5.18      ! [V: num,W2: num,Y3: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
% 9.20/5.18        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5200_neg__numeral__le__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5201_neg__numeral__le__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5202_neg__numeral__le__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5203_neg__numeral__le__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.18        = ( ord_less_eq_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5204_neg__numeral__less__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.18        = ( ord_less_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5205_neg__numeral__less__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 9.20/5.18        = ( ord_less_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5206_neg__numeral__less__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 9.20/5.18        = ( ord_less_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5207_neg__numeral__less__iff,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 9.20/5.18        = ( ord_less_num @ N2 @ M ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5208_zle__diff1__eq,axiom,
% 9.20/5.18      ! [W2: int,Z2: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
% 9.20/5.18        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5209_take__bit__of__Suc__0,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.18        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5210_not__neg__one__le__neg__numeral__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5211_not__neg__one__le__neg__numeral__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5212_not__neg__one__le__neg__numeral__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5213_not__neg__one__le__neg__numeral__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5214_divide__le__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: real,W2: num,A: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 9.20/5.18        = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5215_divide__le__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: rat,W2: num,A: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 9.20/5.18        = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5216_le__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: real,B: real,W2: num] :
% 9.20/5.18        ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.20/5.18        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5217_le__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: rat,B: rat,W2: num] :
% 9.20/5.18        ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.20/5.18        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5218_divide__eq__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: complex,W2: num,A: complex] :
% 9.20/5.18        ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.20/5.18          = A )
% 9.20/5.18        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.18             != zero_zero_complex )
% 9.20/5.18           => ( B
% 9.20/5.18              = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
% 9.20/5.18          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.18              = zero_zero_complex )
% 9.20/5.18           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5219_divide__eq__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: real,W2: num,A: real] :
% 9.20/5.18        ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.18          = A )
% 9.20/5.18        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.18             != zero_zero_real )
% 9.20/5.18           => ( B
% 9.20/5.18              = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
% 9.20/5.18          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.18              = zero_zero_real )
% 9.20/5.18           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5220_divide__eq__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: rat,W2: num,A: rat] :
% 9.20/5.18        ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.18          = A )
% 9.20/5.18        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.18             != zero_zero_rat )
% 9.20/5.18           => ( B
% 9.20/5.18              = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
% 9.20/5.18          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.18              = zero_zero_rat )
% 9.20/5.18           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5221_eq__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: complex,B: complex,W2: num] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
% 9.20/5.18        = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.18             != zero_zero_complex )
% 9.20/5.18           => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.20/5.18              = B ) )
% 9.20/5.18          & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.18              = zero_zero_complex )
% 9.20/5.18           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5222_eq__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: real,B: real,W2: num] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.20/5.18        = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.18             != zero_zero_real )
% 9.20/5.18           => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.18              = B ) )
% 9.20/5.18          & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.18              = zero_zero_real )
% 9.20/5.18           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5223_eq__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: rat,B: rat,W2: num] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.20/5.18        = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.18             != zero_zero_rat )
% 9.20/5.18           => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.18              = B ) )
% 9.20/5.18          & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.18              = zero_zero_rat )
% 9.20/5.18           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5224_neg__numeral__less__neg__one__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5225_neg__numeral__less__neg__one__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5226_neg__numeral__less__neg__one__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5227_neg__numeral__less__neg__one__iff,axiom,
% 9.20/5.18      ! [M: num] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.18        = ( M != one ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5228_less__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: real,B: real,W2: num] :
% 9.20/5.18        ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
% 9.20/5.18        = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5229_less__divide__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [A: rat,B: rat,W2: num] :
% 9.20/5.18        ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
% 9.20/5.18        = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5230_divide__less__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: real,W2: num,A: real] :
% 9.20/5.18        ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
% 9.20/5.18        = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5231_divide__less__eq__numeral1_I2_J,axiom,
% 9.20/5.18      ! [B: rat,W2: num,A: rat] :
% 9.20/5.18        ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
% 9.20/5.18        = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5232_take__bit__of__1,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ N2 @ one_one_nat )
% 9.20/5.18        = ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5233_take__bit__of__1,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ N2 @ one_one_int )
% 9.20/5.18        = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5234_add__neg__numeral__special_I9_J,axiom,
% 9.20/5.18      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.18      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5235_add__neg__numeral__special_I9_J,axiom,
% 9.20/5.18      ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.18      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5236_add__neg__numeral__special_I9_J,axiom,
% 9.20/5.18      ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.18      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5237_add__neg__numeral__special_I9_J,axiom,
% 9.20/5.18      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.18      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5238_add__neg__numeral__special_I9_J,axiom,
% 9.20/5.18      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.18      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5239_even__take__bit__eq,axiom,
% 9.20/5.18      ! [N2: nat,A: code_integer] :
% 9.20/5.18        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
% 9.20/5.18        = ( ( N2 = zero_zero_nat )
% 9.20/5.18          | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5240_even__take__bit__eq,axiom,
% 9.20/5.18      ! [N2: nat,A: nat] :
% 9.20/5.18        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
% 9.20/5.18        = ( ( N2 = zero_zero_nat )
% 9.20/5.18          | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5241_even__take__bit__eq,axiom,
% 9.20/5.18      ! [N2: nat,A: int] :
% 9.20/5.18        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
% 9.20/5.18        = ( ( N2 = zero_zero_nat )
% 9.20/5.18          | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5242_signed__take__bit__Suc__minus__bit0,axiom,
% 9.20/5.18      ! [N2: nat,K: num] :
% 9.20/5.18        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.20/5.18        = ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5243_take__bit__Suc__0,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
% 9.20/5.18        = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5244_take__bit__Suc__0,axiom,
% 9.20/5.18      ! [A: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ ( suc @ zero_zero_nat ) @ A )
% 9.20/5.18        = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5245_take__bit__Suc__0,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ ( suc @ zero_zero_nat ) @ A )
% 9.20/5.18        = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5246_signed__take__bit__0,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
% 9.20/5.18        = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5247_signed__take__bit__0,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( bit_ri631733984087533419it_int @ zero_zero_nat @ A )
% 9.20/5.18        = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5248_take__bit__of__exp,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5249_take__bit__of__exp,axiom,
% 9.20/5.18      ! [M: nat,N2: nat] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5250_take__bit__of__2,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5251_take__bit__of__2,axiom,
% 9.20/5.18      ! [N2: nat] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.18        = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5252_minus__int__code_I2_J,axiom,
% 9.20/5.18      ! [L: int] :
% 9.20/5.18        ( ( minus_minus_int @ zero_zero_int @ L )
% 9.20/5.18        = ( uminus_uminus_int @ L ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5253_compl__mono,axiom,
% 9.20/5.18      ! [X3: set_int,Y3: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ X3 @ Y3 )
% 9.20/5.18       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5254_compl__le__swap1,axiom,
% 9.20/5.18      ! [Y3: set_int,X3: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
% 9.20/5.18       => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5255_compl__le__swap2,axiom,
% 9.20/5.18      ! [Y3: set_int,X3: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
% 9.20/5.18       => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5256_equation__minus__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5257_equation__minus__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5258_equation__minus__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5259_equation__minus__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5260_minus__equation__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( uminus_uminus_int @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( uminus_uminus_int @ B )
% 9.20/5.18          = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5261_minus__equation__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( uminus_uminus_real @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( uminus_uminus_real @ B )
% 9.20/5.18          = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5262_minus__equation__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( uminus_uminus_rat @ B )
% 9.20/5.18          = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5263_minus__equation__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( uminus1351360451143612070nteger @ B )
% 9.20/5.18          = A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5264_Compl__insert,axiom,
% 9.20/5.18      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
% 9.20/5.18        = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5265_Compl__insert,axiom,
% 9.20/5.18      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
% 9.20/5.18        = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5266_Compl__insert,axiom,
% 9.20/5.18      ! [X3: real,A2: set_real] :
% 9.20/5.18        ( ( uminus612125837232591019t_real @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.18        = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5267_Compl__insert,axiom,
% 9.20/5.18      ! [X3: int,A2: set_int] :
% 9.20/5.18        ( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.18        = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5268_Compl__insert,axiom,
% 9.20/5.18      ! [X3: nat,A2: set_nat] :
% 9.20/5.18        ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.18        = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5269_take__bit__add,axiom,
% 9.20/5.18      ! [N2: nat,A: nat,B: nat] :
% 9.20/5.18        ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
% 9.20/5.18        = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5270_take__bit__add,axiom,
% 9.20/5.18      ! [N2: nat,A: int,B: int] :
% 9.20/5.18        ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
% 9.20/5.18        = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5271_take__bit__tightened,axiom,
% 9.20/5.18      ! [N2: nat,A: nat,B: nat,M: nat] :
% 9.20/5.18        ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 9.20/5.18          = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
% 9.20/5.18       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18         => ( ( bit_se2925701944663578781it_nat @ M @ A )
% 9.20/5.18            = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5272_take__bit__tightened,axiom,
% 9.20/5.18      ! [N2: nat,A: int,B: int,M: nat] :
% 9.20/5.18        ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 9.20/5.18          = ( bit_se2923211474154528505it_int @ N2 @ B ) )
% 9.20/5.18       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18         => ( ( bit_se2923211474154528505it_int @ M @ A )
% 9.20/5.18            = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5273_take__bit__nat__less__eq__self,axiom,
% 9.20/5.18      ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5274_take__bit__tightened__less__eq__nat,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.18        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18       => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q4 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5275_le__imp__neg__le,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.18       => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5276_le__imp__neg__le,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.20/5.18       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5277_le__imp__neg__le,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.18       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5278_le__imp__neg__le,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.18       => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5279_minus__le__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.18        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5280_minus__le__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5281_minus__le__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.18        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5282_minus__le__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.18        = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5283_le__minus__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5284_le__minus__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5285_le__minus__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5286_le__minus__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5287_minus__less__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.18        = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5288_minus__less__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.18        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5289_minus__less__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.18        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5290_minus__less__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5291_less__minus__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5292_less__minus__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5293_less__minus__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5294_less__minus__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5295_verit__negate__coefficient_I2_J,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ord_less_int @ A @ B )
% 9.20/5.18       => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5296_verit__negate__coefficient_I2_J,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_real @ A @ B )
% 9.20/5.18       => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5297_verit__negate__coefficient_I2_J,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_rat @ A @ B )
% 9.20/5.18       => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5298_verit__negate__coefficient_I2_J,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.20/5.18       => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5299_square__eq__iff,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( ( times_times_complex @ A @ A )
% 9.20/5.18          = ( times_times_complex @ B @ B ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          | ( A
% 9.20/5.18            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5300_square__eq__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( times_times_int @ A @ A )
% 9.20/5.18          = ( times_times_int @ B @ B ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          | ( A
% 9.20/5.18            = ( uminus_uminus_int @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5301_square__eq__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( times_times_real @ A @ A )
% 9.20/5.18          = ( times_times_real @ B @ B ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          | ( A
% 9.20/5.18            = ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5302_square__eq__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( times_times_rat @ A @ A )
% 9.20/5.18          = ( times_times_rat @ B @ B ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          | ( A
% 9.20/5.18            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5303_square__eq__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( times_3573771949741848930nteger @ A @ A )
% 9.20/5.18          = ( times_3573771949741848930nteger @ B @ B ) )
% 9.20/5.18        = ( ( A = B )
% 9.20/5.18          | ( A
% 9.20/5.18            = ( uminus1351360451143612070nteger @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5304_minus__mult__commute,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
% 9.20/5.18        = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5305_minus__mult__commute,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.18        = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5306_minus__mult__commute,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.18        = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5307_minus__mult__commute,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.18        = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5308_minus__mult__commute,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.18        = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5309_is__num__normalize_I8_J,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.20/5.18        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5310_is__num__normalize_I8_J,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.20/5.18        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5311_is__num__normalize_I8_J,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.18        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5312_is__num__normalize_I8_J,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.18        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5313_add_Oinverse__distrib__swap,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 9.20/5.18        = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5314_add_Oinverse__distrib__swap,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
% 9.20/5.18        = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5315_add_Oinverse__distrib__swap,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
% 9.20/5.18        = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5316_add_Oinverse__distrib__swap,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
% 9.20/5.18        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5317_group__cancel_Oneg1,axiom,
% 9.20/5.18      ! [A2: int,K: int,A: int] :
% 9.20/5.18        ( ( A2
% 9.20/5.18          = ( plus_plus_int @ K @ A ) )
% 9.20/5.18       => ( ( uminus_uminus_int @ A2 )
% 9.20/5.18          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5318_group__cancel_Oneg1,axiom,
% 9.20/5.18      ! [A2: real,K: real,A: real] :
% 9.20/5.18        ( ( A2
% 9.20/5.18          = ( plus_plus_real @ K @ A ) )
% 9.20/5.18       => ( ( uminus_uminus_real @ A2 )
% 9.20/5.18          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( uminus_uminus_real @ A ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5319_group__cancel_Oneg1,axiom,
% 9.20/5.18      ! [A2: rat,K: rat,A: rat] :
% 9.20/5.18        ( ( A2
% 9.20/5.18          = ( plus_plus_rat @ K @ A ) )
% 9.20/5.18       => ( ( uminus_uminus_rat @ A2 )
% 9.20/5.18          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5320_group__cancel_Oneg1,axiom,
% 9.20/5.18      ! [A2: code_integer,K: code_integer,A: code_integer] :
% 9.20/5.18        ( ( A2
% 9.20/5.18          = ( plus_p5714425477246183910nteger @ K @ A ) )
% 9.20/5.18       => ( ( uminus1351360451143612070nteger @ A2 )
% 9.20/5.18          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5321_minus__diff__commute,axiom,
% 9.20/5.18      ! [B: int,A: int] :
% 9.20/5.18        ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 9.20/5.18        = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5322_minus__diff__commute,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
% 9.20/5.18        = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5323_minus__diff__commute,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
% 9.20/5.18        = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5324_minus__diff__commute,axiom,
% 9.20/5.18      ! [B: code_integer,A: code_integer] :
% 9.20/5.18        ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
% 9.20/5.18        = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5325_minus__divide__right,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.18        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5326_minus__divide__right,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.18        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5327_minus__divide__divide,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( divide_divide_real @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5328_minus__divide__divide,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( divide_divide_rat @ A @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5329_minus__divide__left,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.18        = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5330_minus__divide__left,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.18        = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5331_Diff__infinite__finite,axiom,
% 9.20/5.18      ! [T3: set_int,S3: set_int] :
% 9.20/5.18        ( ( finite_finite_int @ T3 )
% 9.20/5.18       => ( ~ ( finite_finite_int @ S3 )
% 9.20/5.18         => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5332_Diff__infinite__finite,axiom,
% 9.20/5.18      ! [T3: set_complex,S3: set_complex] :
% 9.20/5.18        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.18       => ( ~ ( finite3207457112153483333omplex @ S3 )
% 9.20/5.18         => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5333_Diff__infinite__finite,axiom,
% 9.20/5.18      ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( finite6177210948735845034at_nat @ T3 )
% 9.20/5.18       => ( ~ ( finite6177210948735845034at_nat @ S3 )
% 9.20/5.18         => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ T3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5334_Diff__infinite__finite,axiom,
% 9.20/5.18      ! [T3: set_nat,S3: set_nat] :
% 9.20/5.18        ( ( finite_finite_nat @ T3 )
% 9.20/5.18       => ( ~ ( finite_finite_nat @ S3 )
% 9.20/5.18         => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5335_double__diff,axiom,
% 9.20/5.18      ! [A2: set_nat,B2: set_nat,C4: set_nat] :
% 9.20/5.18        ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.18       => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 9.20/5.18         => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 9.20/5.18            = A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5336_double__diff,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int,C4: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ A2 @ B2 )
% 9.20/5.18       => ( ( ord_less_eq_set_int @ B2 @ C4 )
% 9.20/5.18         => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C4 @ A2 ) )
% 9.20/5.18            = A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5337_Diff__subset,axiom,
% 9.20/5.18      ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5338_Diff__subset,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5339_Diff__mono,axiom,
% 9.20/5.18      ! [A2: set_nat,C4: set_nat,D6: set_nat,B2: set_nat] :
% 9.20/5.18        ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 9.20/5.18       => ( ( ord_less_eq_set_nat @ D6 @ B2 )
% 9.20/5.18         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D6 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5340_Diff__mono,axiom,
% 9.20/5.18      ! [A2: set_int,C4: set_int,D6: set_int,B2: set_int] :
% 9.20/5.18        ( ( ord_less_eq_set_int @ A2 @ C4 )
% 9.20/5.18       => ( ( ord_less_eq_set_int @ D6 @ B2 )
% 9.20/5.18         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C4 @ D6 ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5341_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.18        ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert9069300056098147895at_nat @ X3 @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5342_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert8211810215607154385at_nat @ X3 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5343_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: real,B2: set_real,A2: set_real] :
% 9.20/5.18        ( ( ( member_real @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_minus_set_real @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member_real @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert_real @ X3 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5344_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
% 9.20/5.18        ( ( ( member_set_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member_set_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert_set_nat @ X3 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5345_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: int,B2: set_int,A2: set_int] :
% 9.20/5.18        ( ( ( member_int @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_minus_set_int @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member_int @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert_int @ X3 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5346_insert__Diff__if,axiom,
% 9.20/5.18      ! [X3: nat,B2: set_nat,A2: set_nat] :
% 9.20/5.18        ( ( ( member_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( minus_minus_set_nat @ A2 @ B2 ) ) )
% 9.20/5.18        & ( ~ ( member_nat @ X3 @ B2 )
% 9.20/5.18         => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
% 9.20/5.18            = ( insert_nat @ X3 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5347_minus__int__code_I1_J,axiom,
% 9.20/5.18      ! [K: int] :
% 9.20/5.18        ( ( minus_minus_int @ K @ zero_zero_int )
% 9.20/5.18        = K ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5348_uminus__int__code_I1_J,axiom,
% 9.20/5.18      ( ( uminus_uminus_int @ zero_zero_int )
% 9.20/5.18      = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5349_int__distrib_I4_J,axiom,
% 9.20/5.18      ! [W2: int,Z1: int,Z22: int] :
% 9.20/5.18        ( ( times_times_int @ W2 @ ( minus_minus_int @ Z1 @ Z22 ) )
% 9.20/5.18        = ( minus_minus_int @ ( times_times_int @ W2 @ Z1 ) @ ( times_times_int @ W2 @ Z22 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5350_int__distrib_I3_J,axiom,
% 9.20/5.18      ! [Z1: int,Z22: int,W2: int] :
% 9.20/5.18        ( ( times_times_int @ ( minus_minus_int @ Z1 @ Z22 ) @ W2 )
% 9.20/5.18        = ( minus_minus_int @ ( times_times_int @ Z1 @ W2 ) @ ( times_times_int @ Z22 @ W2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5351_psubset__imp__ex__mem,axiom,
% 9.20/5.18      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.18        ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
% 9.20/5.18       => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5352_psubset__imp__ex__mem,axiom,
% 9.20/5.18      ! [A2: set_real,B2: set_real] :
% 9.20/5.18        ( ( ord_less_set_real @ A2 @ B2 )
% 9.20/5.18       => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5353_psubset__imp__ex__mem,axiom,
% 9.20/5.18      ! [A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.18        ( ( ord_less_set_set_nat @ A2 @ B2 )
% 9.20/5.18       => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5354_psubset__imp__ex__mem,axiom,
% 9.20/5.18      ! [A2: set_int,B2: set_int] :
% 9.20/5.18        ( ( ord_less_set_int @ A2 @ B2 )
% 9.20/5.18       => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5355_psubset__imp__ex__mem,axiom,
% 9.20/5.18      ! [A2: set_nat,B2: set_nat] :
% 9.20/5.18        ( ( ord_less_set_nat @ A2 @ B2 )
% 9.20/5.18       => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5356_zdvd__zdiffD,axiom,
% 9.20/5.18      ! [K: int,M: int,N2: int] :
% 9.20/5.18        ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
% 9.20/5.18       => ( ( dvd_dvd_int @ K @ N2 )
% 9.20/5.18         => ( dvd_dvd_int @ K @ M ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5357_zmod__minus1,axiom,
% 9.20/5.18      ! [B: int] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.18       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.20/5.18          = ( minus_minus_int @ B @ one_one_int ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5358_take__bit__minus__small__eq,axiom,
% 9.20/5.18      ! [K: int,N2: nat] :
% 9.20/5.18        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.18       => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.18         => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
% 9.20/5.18            = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5359_infinite__Icc,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ord_less_rat @ A @ B )
% 9.20/5.18       => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5360_infinite__Icc,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ord_less_real @ A @ B )
% 9.20/5.18       => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5361_atLeastAtMost__singleton_H,axiom,
% 9.20/5.18      ! [A: nat,B: nat] :
% 9.20/5.18        ( ( A = B )
% 9.20/5.18       => ( ( set_or1269000886237332187st_nat @ A @ B )
% 9.20/5.18          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5362_atLeastAtMost__singleton_H,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( A = B )
% 9.20/5.18       => ( ( set_or1266510415728281911st_int @ A @ B )
% 9.20/5.18          = ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5363_atLeastAtMost__singleton_H,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( A = B )
% 9.20/5.18       => ( ( set_or1222579329274155063t_real @ A @ B )
% 9.20/5.18          = ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5364_take__bit__tightened__less__eq__int,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,K: int] :
% 9.20/5.18        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.18       => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5365_take__bit__int__greater__self__iff,axiom,
% 9.20/5.18      ! [K: int,N2: nat] :
% 9.20/5.18        ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 9.20/5.18        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5366_not__take__bit__negative,axiom,
% 9.20/5.18      ! [N2: nat,K: int] :
% 9.20/5.18        ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5367_signed__take__bit__eq__iff__take__bit__eq,axiom,
% 9.20/5.18      ! [N2: nat,A: int,B: int] :
% 9.20/5.18        ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
% 9.20/5.18          = ( bit_ri631733984087533419it_int @ N2 @ B ) )
% 9.20/5.18        = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
% 9.20/5.18          = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5368_signed__take__bit__take__bit,axiom,
% 9.20/5.18      ! [M: nat,N2: nat,A: int] :
% 9.20/5.18        ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
% 9.20/5.18        = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5369_ex__nat__less,axiom,
% 9.20/5.18      ! [N2: nat,P: nat > $o] :
% 9.20/5.18        ( ( ? [M2: nat] :
% 9.20/5.18              ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.20/5.18              & ( P @ M2 ) ) )
% 9.20/5.18        = ( ? [X: nat] :
% 9.20/5.18              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.18              & ( P @ X ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5370_all__nat__less,axiom,
% 9.20/5.18      ! [N2: nat,P: nat > $o] :
% 9.20/5.18        ( ( ! [M2: nat] :
% 9.20/5.18              ( ( ord_less_eq_nat @ M2 @ N2 )
% 9.20/5.18             => ( P @ M2 ) ) )
% 9.20/5.18        = ( ! [X: nat] :
% 9.20/5.18              ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.18             => ( P @ X ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5371_neg__numeral__le__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5372_neg__numeral__le__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5373_neg__numeral__le__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5374_neg__numeral__le__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5375_not__numeral__le__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5376_not__numeral__le__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5377_not__numeral__le__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5378_not__numeral__le__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5379_zero__neq__neg__numeral,axiom,
% 9.20/5.18      ! [N2: num] :
% 9.20/5.18        ( zero_zero_complex
% 9.20/5.18       != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5380_zero__neq__neg__numeral,axiom,
% 9.20/5.18      ! [N2: num] :
% 9.20/5.18        ( zero_zero_int
% 9.20/5.18       != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5381_zero__neq__neg__numeral,axiom,
% 9.20/5.18      ! [N2: num] :
% 9.20/5.18        ( zero_zero_real
% 9.20/5.18       != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5382_zero__neq__neg__numeral,axiom,
% 9.20/5.18      ! [N2: num] :
% 9.20/5.18        ( zero_zero_rat
% 9.20/5.18       != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5383_zero__neq__neg__numeral,axiom,
% 9.20/5.18      ! [N2: num] :
% 9.20/5.18        ( zero_z3403309356797280102nteger
% 9.20/5.18       != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5384_not__numeral__less__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5385_not__numeral__less__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5386_not__numeral__less__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5387_not__numeral__less__neg__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] :
% 9.20/5.18        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5388_neg__numeral__less__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5389_neg__numeral__less__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5390_neg__numeral__less__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5391_neg__numeral__less__numeral,axiom,
% 9.20/5.18      ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5392_le__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5393_le__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5394_le__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5395_le__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5396_le__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5397_le__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5398_le__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5399_le__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5400_zero__neq__neg__one,axiom,
% 9.20/5.18      ( zero_zero_complex
% 9.20/5.18     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5401_zero__neq__neg__one,axiom,
% 9.20/5.18      ( zero_zero_int
% 9.20/5.18     != ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5402_zero__neq__neg__one,axiom,
% 9.20/5.18      ( zero_zero_real
% 9.20/5.18     != ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5403_zero__neq__neg__one,axiom,
% 9.20/5.18      ( zero_zero_rat
% 9.20/5.18     != ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5404_zero__neq__neg__one,axiom,
% 9.20/5.18      ( zero_z3403309356797280102nteger
% 9.20/5.18     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5405_add__eq__0__iff,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( ( plus_plus_complex @ A @ B )
% 9.20/5.18          = zero_zero_complex )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus1482373934393186551omplex @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5406_add__eq__0__iff,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( plus_plus_int @ A @ B )
% 9.20/5.18          = zero_zero_int )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5407_add__eq__0__iff,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( plus_plus_real @ A @ B )
% 9.20/5.18          = zero_zero_real )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5408_add__eq__0__iff,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( plus_plus_rat @ A @ B )
% 9.20/5.18          = zero_zero_rat )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5409_add__eq__0__iff,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.20/5.18          = zero_z3403309356797280102nteger )
% 9.20/5.18        = ( B
% 9.20/5.18          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5410_ab__group__add__class_Oab__left__minus,axiom,
% 9.20/5.18      ! [A: complex] :
% 9.20/5.18        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
% 9.20/5.18        = zero_zero_complex ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5411_ab__group__add__class_Oab__left__minus,axiom,
% 9.20/5.18      ! [A: int] :
% 9.20/5.18        ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
% 9.20/5.18        = zero_zero_int ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5412_ab__group__add__class_Oab__left__minus,axiom,
% 9.20/5.18      ! [A: real] :
% 9.20/5.18        ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
% 9.20/5.18        = zero_zero_real ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5413_ab__group__add__class_Oab__left__minus,axiom,
% 9.20/5.18      ! [A: rat] :
% 9.20/5.18        ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
% 9.20/5.18        = zero_zero_rat ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5414_ab__group__add__class_Oab__left__minus,axiom,
% 9.20/5.18      ! [A: code_integer] :
% 9.20/5.18        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
% 9.20/5.18        = zero_z3403309356797280102nteger ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5415_add_Oinverse__unique,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( ( plus_plus_complex @ A @ B )
% 9.20/5.18          = zero_zero_complex )
% 9.20/5.18       => ( ( uminus1482373934393186551omplex @ A )
% 9.20/5.18          = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5416_add_Oinverse__unique,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( plus_plus_int @ A @ B )
% 9.20/5.18          = zero_zero_int )
% 9.20/5.18       => ( ( uminus_uminus_int @ A )
% 9.20/5.18          = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5417_add_Oinverse__unique,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( plus_plus_real @ A @ B )
% 9.20/5.18          = zero_zero_real )
% 9.20/5.18       => ( ( uminus_uminus_real @ A )
% 9.20/5.18          = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5418_add_Oinverse__unique,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( plus_plus_rat @ A @ B )
% 9.20/5.18          = zero_zero_rat )
% 9.20/5.18       => ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5419_add_Oinverse__unique,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.20/5.18          = zero_z3403309356797280102nteger )
% 9.20/5.18       => ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = B ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5420_eq__neg__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus1482373934393186551omplex @ B ) )
% 9.20/5.18        = ( ( plus_plus_complex @ A @ B )
% 9.20/5.18          = zero_zero_complex ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5421_eq__neg__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_int @ B ) )
% 9.20/5.18        = ( ( plus_plus_int @ A @ B )
% 9.20/5.18          = zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5422_eq__neg__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_real @ B ) )
% 9.20/5.18        = ( ( plus_plus_real @ A @ B )
% 9.20/5.18          = zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5423_eq__neg__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus_uminus_rat @ B ) )
% 9.20/5.18        = ( ( plus_plus_rat @ A @ B )
% 9.20/5.18          = zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5424_eq__neg__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( A
% 9.20/5.18          = ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.18        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.20/5.18          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5425_neg__eq__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: complex,B: complex] :
% 9.20/5.18        ( ( ( uminus1482373934393186551omplex @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( plus_plus_complex @ A @ B )
% 9.20/5.18          = zero_zero_complex ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5426_neg__eq__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: int,B: int] :
% 9.20/5.18        ( ( ( uminus_uminus_int @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( plus_plus_int @ A @ B )
% 9.20/5.18          = zero_zero_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5427_neg__eq__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: real,B: real] :
% 9.20/5.18        ( ( ( uminus_uminus_real @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( plus_plus_real @ A @ B )
% 9.20/5.18          = zero_zero_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5428_neg__eq__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: rat,B: rat] :
% 9.20/5.18        ( ( ( uminus_uminus_rat @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( plus_plus_rat @ A @ B )
% 9.20/5.18          = zero_zero_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5429_neg__eq__iff__add__eq__0,axiom,
% 9.20/5.18      ! [A: code_integer,B: code_integer] :
% 9.20/5.18        ( ( ( uminus1351360451143612070nteger @ A )
% 9.20/5.18          = B )
% 9.20/5.18        = ( ( plus_p5714425477246183910nteger @ A @ B )
% 9.20/5.18          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5430_less__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5431_less__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5432_less__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5433_less__minus__one__simps_I4_J,axiom,
% 9.20/5.18      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5434_less__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5435_less__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5436_less__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5437_less__minus__one__simps_I2_J,axiom,
% 9.20/5.18      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5438_nonzero__minus__divide__right,axiom,
% 9.20/5.18      ! [B: complex,A: complex] :
% 9.20/5.18        ( ( B != zero_zero_complex )
% 9.20/5.18       => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.20/5.18          = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5439_nonzero__minus__divide__right,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( B != zero_zero_real )
% 9.20/5.18       => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.18          = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5440_nonzero__minus__divide__right,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( B != zero_zero_rat )
% 9.20/5.18       => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.18          = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5441_nonzero__minus__divide__divide,axiom,
% 9.20/5.18      ! [B: complex,A: complex] :
% 9.20/5.18        ( ( B != zero_zero_complex )
% 9.20/5.18       => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
% 9.20/5.18          = ( divide1717551699836669952omplex @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5442_nonzero__minus__divide__divide,axiom,
% 9.20/5.18      ! [B: real,A: real] :
% 9.20/5.18        ( ( B != zero_zero_real )
% 9.20/5.18       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
% 9.20/5.18          = ( divide_divide_real @ A @ B ) ) ) ).
% 9.20/5.18  
% 9.20/5.18  thf(fact_5443_nonzero__minus__divide__divide,axiom,
% 9.20/5.18      ! [B: rat,A: rat] :
% 9.20/5.18        ( ( B != zero_zero_rat )
% 9.20/5.18       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
% 9.20/5.19          = ( divide_divide_rat @ A @ B ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5444_square__eq__1__iff,axiom,
% 9.20/5.19      ! [X3: complex] :
% 9.20/5.19        ( ( ( times_times_complex @ X3 @ X3 )
% 9.20/5.19          = one_one_complex )
% 9.20/5.19        = ( ( X3 = one_one_complex )
% 9.20/5.19          | ( X3
% 9.20/5.19            = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5445_square__eq__1__iff,axiom,
% 9.20/5.19      ! [X3: int] :
% 9.20/5.19        ( ( ( times_times_int @ X3 @ X3 )
% 9.20/5.19          = one_one_int )
% 9.20/5.19        = ( ( X3 = one_one_int )
% 9.20/5.19          | ( X3
% 9.20/5.19            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5446_square__eq__1__iff,axiom,
% 9.20/5.19      ! [X3: real] :
% 9.20/5.19        ( ( ( times_times_real @ X3 @ X3 )
% 9.20/5.19          = one_one_real )
% 9.20/5.19        = ( ( X3 = one_one_real )
% 9.20/5.19          | ( X3
% 9.20/5.19            = ( uminus_uminus_real @ one_one_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5447_square__eq__1__iff,axiom,
% 9.20/5.19      ! [X3: rat] :
% 9.20/5.19        ( ( ( times_times_rat @ X3 @ X3 )
% 9.20/5.19          = one_one_rat )
% 9.20/5.19        = ( ( X3 = one_one_rat )
% 9.20/5.19          | ( X3
% 9.20/5.19            = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5448_square__eq__1__iff,axiom,
% 9.20/5.19      ! [X3: code_integer] :
% 9.20/5.19        ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
% 9.20/5.19          = one_one_Code_integer )
% 9.20/5.19        = ( ( X3 = one_one_Code_integer )
% 9.20/5.19          | ( X3
% 9.20/5.19            = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5449_group__cancel_Osub2,axiom,
% 9.20/5.19      ! [B2: int,K: int,B: int,A: int] :
% 9.20/5.19        ( ( B2
% 9.20/5.19          = ( plus_plus_int @ K @ B ) )
% 9.20/5.19       => ( ( minus_minus_int @ A @ B2 )
% 9.20/5.19          = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5450_group__cancel_Osub2,axiom,
% 9.20/5.19      ! [B2: real,K: real,B: real,A: real] :
% 9.20/5.19        ( ( B2
% 9.20/5.19          = ( plus_plus_real @ K @ B ) )
% 9.20/5.19       => ( ( minus_minus_real @ A @ B2 )
% 9.20/5.19          = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5451_group__cancel_Osub2,axiom,
% 9.20/5.19      ! [B2: rat,K: rat,B: rat,A: rat] :
% 9.20/5.19        ( ( B2
% 9.20/5.19          = ( plus_plus_rat @ K @ B ) )
% 9.20/5.19       => ( ( minus_minus_rat @ A @ B2 )
% 9.20/5.19          = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5452_group__cancel_Osub2,axiom,
% 9.20/5.19      ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
% 9.20/5.19        ( ( B2
% 9.20/5.19          = ( plus_p5714425477246183910nteger @ K @ B ) )
% 9.20/5.19       => ( ( minus_8373710615458151222nteger @ A @ B2 )
% 9.20/5.19          = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5453_diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_int
% 9.20/5.19      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5454_diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_real
% 9.20/5.19      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5455_diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_rat
% 9.20/5.19      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5456_diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_8373710615458151222nteger
% 9.20/5.19      = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5457_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_int
% 9.20/5.19      = ( ^ [A3: int,B3: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5458_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_real
% 9.20/5.19      = ( ^ [A3: real,B3: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5459_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_minus_rat
% 9.20/5.19      = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5460_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 9.20/5.19      ( minus_8373710615458151222nteger
% 9.20/5.19      = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5461_take__bit__unset__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: nat] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5462_take__bit__unset__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: int] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5463_take__bit__set__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: nat] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5464_take__bit__set__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: int] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5465_take__bit__flip__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: nat] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
% 9.20/5.19            = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5466_take__bit__flip__bit__eq,axiom,
% 9.20/5.19      ! [N2: nat,M: nat,A: int] :
% 9.20/5.19        ( ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
% 9.20/5.19            = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5467_dvd__neg__div,axiom,
% 9.20/5.19      ! [B: int,A: int] :
% 9.20/5.19        ( ( dvd_dvd_int @ B @ A )
% 9.20/5.19       => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.19          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5468_dvd__neg__div,axiom,
% 9.20/5.19      ! [B: real,A: real] :
% 9.20/5.19        ( ( dvd_dvd_real @ B @ A )
% 9.20/5.19       => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.19          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5469_dvd__neg__div,axiom,
% 9.20/5.19      ! [B: rat,A: rat] :
% 9.20/5.19        ( ( dvd_dvd_rat @ B @ A )
% 9.20/5.19       => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.19          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5470_dvd__neg__div,axiom,
% 9.20/5.19      ! [B: code_integer,A: code_integer] :
% 9.20/5.19        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.19       => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.19          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5471_dvd__div__neg,axiom,
% 9.20/5.19      ! [B: int,A: int] :
% 9.20/5.19        ( ( dvd_dvd_int @ B @ A )
% 9.20/5.19       => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
% 9.20/5.19          = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5472_dvd__div__neg,axiom,
% 9.20/5.19      ! [B: real,A: real] :
% 9.20/5.19        ( ( dvd_dvd_real @ B @ A )
% 9.20/5.19       => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
% 9.20/5.19          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5473_dvd__div__neg,axiom,
% 9.20/5.19      ! [B: rat,A: rat] :
% 9.20/5.19        ( ( dvd_dvd_rat @ B @ A )
% 9.20/5.19       => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
% 9.20/5.19          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5474_dvd__div__neg,axiom,
% 9.20/5.19      ! [B: code_integer,A: code_integer] :
% 9.20/5.19        ( ( dvd_dvd_Code_integer @ B @ A )
% 9.20/5.19       => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
% 9.20/5.19          = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5475_subset__Compl__self__eq,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.20/5.19        = ( A2 = bot_bo2099793752762293965at_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5476_subset__Compl__self__eq,axiom,
% 9.20/5.19      ! [A2: set_real] :
% 9.20/5.19        ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ A2 ) )
% 9.20/5.19        = ( A2 = bot_bot_set_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5477_subset__Compl__self__eq,axiom,
% 9.20/5.19      ! [A2: set_nat] :
% 9.20/5.19        ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.20/5.19        = ( A2 = bot_bot_set_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5478_subset__Compl__self__eq,axiom,
% 9.20/5.19      ! [A2: set_int] :
% 9.20/5.19        ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.20/5.19        = ( A2 = bot_bot_set_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5479_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5480_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: set_nat,A2: set_set_nat] :
% 9.20/5.19        ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5481_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5482_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: real,A2: set_real] :
% 9.20/5.19        ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5483_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: int,A2: set_int] :
% 9.20/5.19        ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5484_Diff__insert__absorb,axiom,
% 9.20/5.19      ! [X3: nat,A2: set_nat] :
% 9.20/5.19        ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.19       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5485_Diff__insert2,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5486_Diff__insert2,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5487_Diff__insert2,axiom,
% 9.20/5.19      ! [A2: set_real,A: real,B2: set_real] :
% 9.20/5.19        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5488_Diff__insert2,axiom,
% 9.20/5.19      ! [A2: set_int,A: int,B2: set_int] :
% 9.20/5.19        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5489_Diff__insert2,axiom,
% 9.20/5.19      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.20/5.19        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5490_insert__Diff,axiom,
% 9.20/5.19      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.19       => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5491_insert__Diff,axiom,
% 9.20/5.19      ! [A: set_nat,A2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ A @ A2 )
% 9.20/5.19       => ( ( insert_set_nat @ A @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5492_insert__Diff,axiom,
% 9.20/5.19      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.19       => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5493_insert__Diff,axiom,
% 9.20/5.19      ! [A: real,A2: set_real] :
% 9.20/5.19        ( ( member_real @ A @ A2 )
% 9.20/5.19       => ( ( insert_real @ A @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5494_insert__Diff,axiom,
% 9.20/5.19      ! [A: int,A2: set_int] :
% 9.20/5.19        ( ( member_int @ A @ A2 )
% 9.20/5.19       => ( ( insert_int @ A @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5495_insert__Diff,axiom,
% 9.20/5.19      ! [A: nat,A2: set_nat] :
% 9.20/5.19        ( ( member_nat @ A @ A2 )
% 9.20/5.19       => ( ( insert_nat @ A @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.19          = A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5496_Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5497_Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5498_Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_real,A: real,B2: set_real] :
% 9.20/5.19        ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5499_Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_int,A: int,B2: set_int] :
% 9.20/5.19        ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5500_Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_nat,A: nat,B2: set_nat] :
% 9.20/5.19        ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
% 9.20/5.19        = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5501_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ ( insert9069300056098147895at_nat @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member8757157785044589968at_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5502_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member8440522571783428010at_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5503_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_real,B2: set_real,X3: real,C4: set_real] :
% 9.20/5.19        ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member_real @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5504_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat,C4: set_set_nat] :
% 9.20/5.19        ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member_set_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5505_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_nat,B2: set_nat,X3: nat,C4: set_nat] :
% 9.20/5.19        ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member_nat @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5506_subset__Diff__insert,axiom,
% 9.20/5.19      ! [A2: set_int,B2: set_int,X3: int,C4: set_int] :
% 9.20/5.19        ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X3 @ C4 ) ) )
% 9.20/5.19        = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C4 ) )
% 9.20/5.19          & ~ ( member_int @ X3 @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5507_real__0__less__add__iff,axiom,
% 9.20/5.19      ! [X3: real,Y3: real] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.19        = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5508_real__add__less__0__iff,axiom,
% 9.20/5.19      ! [X3: real,Y3: real] :
% 9.20/5.19        ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
% 9.20/5.19        = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5509_zminus1__lemma,axiom,
% 9.20/5.19      ! [A: int,B: int,Q4: int,R2: int] :
% 9.20/5.19        ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
% 9.20/5.19       => ( ( B != zero_zero_int )
% 9.20/5.19         => ( 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 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5510_int__le__induct,axiom,
% 9.20/5.19      ! [I: int,K: int,P: int > $o] :
% 9.20/5.19        ( ( ord_less_eq_int @ I @ K )
% 9.20/5.19       => ( ( P @ K )
% 9.20/5.19         => ( ! [I3: int] :
% 9.20/5.19                ( ( ord_less_eq_int @ I3 @ K )
% 9.20/5.19               => ( ( P @ I3 )
% 9.20/5.19                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.19           => ( P @ I ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5511_int__less__induct,axiom,
% 9.20/5.19      ! [I: int,K: int,P: int > $o] :
% 9.20/5.19        ( ( ord_less_int @ I @ K )
% 9.20/5.19       => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
% 9.20/5.19         => ( ! [I3: int] :
% 9.20/5.19                ( ( ord_less_int @ I3 @ K )
% 9.20/5.19               => ( ( P @ I3 )
% 9.20/5.19                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.19           => ( P @ I ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5512_pos__zmult__eq__1__iff__lemma,axiom,
% 9.20/5.19      ! [M: int,N2: int] :
% 9.20/5.19        ( ( ( times_times_int @ M @ N2 )
% 9.20/5.19          = one_one_int )
% 9.20/5.19       => ( ( M = one_one_int )
% 9.20/5.19          | ( M
% 9.20/5.19            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5513_zmult__eq__1__iff,axiom,
% 9.20/5.19      ! [M: int,N2: int] :
% 9.20/5.19        ( ( ( times_times_int @ M @ N2 )
% 9.20/5.19          = one_one_int )
% 9.20/5.19        = ( ( ( M = one_one_int )
% 9.20/5.19            & ( N2 = one_one_int ) )
% 9.20/5.19          | ( ( M
% 9.20/5.19              = ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.19            & ( N2
% 9.20/5.19              = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5514_signed__take__bit__eq__take__bit__shift,axiom,
% 9.20/5.19      ( bit_ri631733984087533419it_int
% 9.20/5.19      = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5515_take__bit__Suc__minus__bit0,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.20/5.19        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5516_take__bit__int__less__eq,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 9.20/5.19       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.19         => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5517_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: set_int,B: set_int,C: set_int,D: set_int] :
% 9.20/5.19        ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_set_int @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_set_int @ C @ A )
% 9.20/5.19              & ( ord_less_eq_set_int @ B @ D )
% 9.20/5.19              & ( ( ord_less_set_int @ C @ A )
% 9.20/5.19                | ( ord_less_set_int @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_set_int @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5518_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: rat,B: rat,C: rat,D: rat] :
% 9.20/5.19        ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_rat @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_rat @ C @ A )
% 9.20/5.19              & ( ord_less_eq_rat @ B @ D )
% 9.20/5.19              & ( ( ord_less_rat @ C @ A )
% 9.20/5.19                | ( ord_less_rat @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_rat @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5519_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: num,B: num,C: num,D: num] :
% 9.20/5.19        ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_num @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_num @ C @ A )
% 9.20/5.19              & ( ord_less_eq_num @ B @ D )
% 9.20/5.19              & ( ( ord_less_num @ C @ A )
% 9.20/5.19                | ( ord_less_num @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_num @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5520_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: nat,B: nat,C: nat,D: nat] :
% 9.20/5.19        ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_nat @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_nat @ C @ A )
% 9.20/5.19              & ( ord_less_eq_nat @ B @ D )
% 9.20/5.19              & ( ( ord_less_nat @ C @ A )
% 9.20/5.19                | ( ord_less_nat @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_nat @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5521_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: int,B: int,C: int,D: int] :
% 9.20/5.19        ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_int @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_int @ C @ A )
% 9.20/5.19              & ( ord_less_eq_int @ B @ D )
% 9.20/5.19              & ( ( ord_less_int @ C @ A )
% 9.20/5.19                | ( ord_less_int @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_int @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5522_atLeastatMost__psubset__iff,axiom,
% 9.20/5.19      ! [A: real,B: real,C: real,D: real] :
% 9.20/5.19        ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
% 9.20/5.19        = ( ( ~ ( ord_less_eq_real @ A @ B )
% 9.20/5.19            | ( ( ord_less_eq_real @ C @ A )
% 9.20/5.19              & ( ord_less_eq_real @ B @ D )
% 9.20/5.19              & ( ( ord_less_real @ C @ A )
% 9.20/5.19                | ( ord_less_real @ B @ D ) ) ) )
% 9.20/5.19          & ( ord_less_eq_real @ C @ D ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5523_take__bit__signed__take__bit,axiom,
% 9.20/5.19      ! [M: nat,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.19       => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
% 9.20/5.19          = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5524_not__zero__le__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5525_not__zero__le__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5526_not__zero__le__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5527_not__zero__le__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5528_neg__numeral__le__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5529_neg__numeral__le__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5530_neg__numeral__le__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5531_neg__numeral__le__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5532_neg__numeral__less__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5533_neg__numeral__less__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) @ zero_zero_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5534_neg__numeral__less__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5535_neg__numeral__less__zero,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5536_not__zero__less__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5537_not__zero__less__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5538_not__zero__less__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5539_not__zero__less__neg__numeral,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5540_le__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5541_le__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5542_le__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5543_le__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5544_le__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5545_le__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5546_le__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5547_le__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5548_less__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5549_less__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5550_less__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5551_less__minus__one__simps_I3_J,axiom,
% 9.20/5.19      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5552_less__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5553_less__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ zero_zero_real ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5554_less__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5555_less__minus__one__simps_I1_J,axiom,
% 9.20/5.19      ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5556_neg__numeral__le__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5557_neg__numeral__le__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5558_neg__numeral__le__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5559_neg__numeral__le__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5560_neg__one__le__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5561_neg__one__le__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5562_neg__one__le__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5563_neg__one__le__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5564_neg__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5565_neg__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5566_neg__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5567_neg__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5568_not__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5569_not__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5570_not__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5571_not__numeral__le__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5572_not__one__le__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5573_not__one__le__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5574_not__one__le__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5575_not__one__le__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5576_not__neg__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5577_not__neg__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5578_not__neg__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5579_not__neg__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5580_not__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5581_not__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5582_not__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5583_not__one__less__neg__numeral,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5584_not__numeral__less__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5585_not__numeral__less__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5586_not__numeral__less__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5587_not__numeral__less__neg__one,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5588_neg__one__less__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5589_neg__one__less__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5590_neg__one__less__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5591_neg__one__less__numeral,axiom,
% 9.20/5.19      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5592_neg__numeral__less__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5593_neg__numeral__less__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5594_neg__numeral__less__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5595_neg__numeral__less__one,axiom,
% 9.20/5.19      ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5596_eq__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: complex,B: complex,C: complex] :
% 9.20/5.19        ( ( A
% 9.20/5.19          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_complex )
% 9.20/5.19           => ( ( times_times_complex @ A @ C )
% 9.20/5.19              = ( uminus1482373934393186551omplex @ B ) ) )
% 9.20/5.19          & ( ( C = zero_zero_complex )
% 9.20/5.19           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5597_eq__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: real,B: real,C: real] :
% 9.20/5.19        ( ( A
% 9.20/5.19          = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_real )
% 9.20/5.19           => ( ( times_times_real @ A @ C )
% 9.20/5.19              = ( uminus_uminus_real @ B ) ) )
% 9.20/5.19          & ( ( C = zero_zero_real )
% 9.20/5.19           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5598_eq__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: rat,B: rat,C: rat] :
% 9.20/5.19        ( ( A
% 9.20/5.19          = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_rat )
% 9.20/5.19           => ( ( times_times_rat @ A @ C )
% 9.20/5.19              = ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19          & ( ( C = zero_zero_rat )
% 9.20/5.19           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5599_minus__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: complex,C: complex,A: complex] :
% 9.20/5.19        ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.19          = A )
% 9.20/5.19        = ( ( ( C != zero_zero_complex )
% 9.20/5.19           => ( ( uminus1482373934393186551omplex @ B )
% 9.20/5.19              = ( times_times_complex @ A @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_complex )
% 9.20/5.19           => ( A = zero_zero_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5600_minus__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: real,C: real,A: real] :
% 9.20/5.19        ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
% 9.20/5.19          = A )
% 9.20/5.19        = ( ( ( C != zero_zero_real )
% 9.20/5.19           => ( ( uminus_uminus_real @ B )
% 9.20/5.19              = ( times_times_real @ A @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_real )
% 9.20/5.19           => ( A = zero_zero_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5601_minus__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: rat,C: rat,A: rat] :
% 9.20/5.19        ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.19          = A )
% 9.20/5.19        = ( ( ( C != zero_zero_rat )
% 9.20/5.19           => ( ( uminus_uminus_rat @ B )
% 9.20/5.19              = ( times_times_rat @ A @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_rat )
% 9.20/5.19           => ( A = zero_zero_rat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5602_nonzero__neg__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: complex,A: complex,C: complex] :
% 9.20/5.19        ( ( B != zero_zero_complex )
% 9.20/5.19       => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.20/5.19            = C )
% 9.20/5.19          = ( ( uminus1482373934393186551omplex @ A )
% 9.20/5.19            = ( times_times_complex @ C @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5603_nonzero__neg__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: real,A: real,C: real] :
% 9.20/5.19        ( ( B != zero_zero_real )
% 9.20/5.19       => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.19            = C )
% 9.20/5.19          = ( ( uminus_uminus_real @ A )
% 9.20/5.19            = ( times_times_real @ C @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5604_nonzero__neg__divide__eq__eq,axiom,
% 9.20/5.19      ! [B: rat,A: rat,C: rat] :
% 9.20/5.19        ( ( B != zero_zero_rat )
% 9.20/5.19       => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.19            = C )
% 9.20/5.19          = ( ( uminus_uminus_rat @ A )
% 9.20/5.19            = ( times_times_rat @ C @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5605_nonzero__neg__divide__eq__eq2,axiom,
% 9.20/5.19      ! [B: complex,C: complex,A: complex] :
% 9.20/5.19        ( ( B != zero_zero_complex )
% 9.20/5.19       => ( ( C
% 9.20/5.19            = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
% 9.20/5.19          = ( ( times_times_complex @ C @ B )
% 9.20/5.19            = ( uminus1482373934393186551omplex @ A ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5606_nonzero__neg__divide__eq__eq2,axiom,
% 9.20/5.19      ! [B: real,C: real,A: real] :
% 9.20/5.19        ( ( B != zero_zero_real )
% 9.20/5.19       => ( ( C
% 9.20/5.19            = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
% 9.20/5.19          = ( ( times_times_real @ C @ B )
% 9.20/5.19            = ( uminus_uminus_real @ A ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5607_nonzero__neg__divide__eq__eq2,axiom,
% 9.20/5.19      ! [B: rat,C: rat,A: rat] :
% 9.20/5.19        ( ( B != zero_zero_rat )
% 9.20/5.19       => ( ( C
% 9.20/5.19            = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
% 9.20/5.19          = ( ( times_times_rat @ C @ B )
% 9.20/5.19            = ( uminus_uminus_rat @ A ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5608_divide__eq__minus__1__iff,axiom,
% 9.20/5.19      ! [A: complex,B: complex] :
% 9.20/5.19        ( ( ( divide1717551699836669952omplex @ A @ B )
% 9.20/5.19          = ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.19        = ( ( B != zero_zero_complex )
% 9.20/5.19          & ( A
% 9.20/5.19            = ( uminus1482373934393186551omplex @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5609_divide__eq__minus__1__iff,axiom,
% 9.20/5.19      ! [A: real,B: real] :
% 9.20/5.19        ( ( ( divide_divide_real @ A @ B )
% 9.20/5.19          = ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.19        = ( ( B != zero_zero_real )
% 9.20/5.19          & ( A
% 9.20/5.19            = ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5610_divide__eq__minus__1__iff,axiom,
% 9.20/5.19      ! [A: rat,B: rat] :
% 9.20/5.19        ( ( ( divide_divide_rat @ A @ B )
% 9.20/5.19          = ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.19        = ( ( B != zero_zero_rat )
% 9.20/5.19          & ( A
% 9.20/5.19            = ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5611_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.19        ( ( finite4343798906461161616at_nat @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.19                ( ( finite4343798906461161616at_nat @ A7 )
% 9.20/5.19               => ( ( member8757157785044589968at_nat @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5612_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.19        ( ( finite1152437895449049373et_nat @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: set_nat,A7: set_set_nat] :
% 9.20/5.19                ( ( finite1152437895449049373et_nat @ A7 )
% 9.20/5.19               => ( ( member_set_nat @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bot_set_set_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5613_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_complex,P: set_complex > $o] :
% 9.20/5.19        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: complex,A7: set_complex] :
% 9.20/5.19                ( ( finite3207457112153483333omplex @ A7 )
% 9.20/5.19               => ( ( member_complex @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bot_set_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5614_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.19        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.19                ( ( finite6177210948735845034at_nat @ A7 )
% 9.20/5.19               => ( ( member8440522571783428010at_nat @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5615_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_real,P: set_real > $o] :
% 9.20/5.19        ( ( finite_finite_real @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: real,A7: set_real] :
% 9.20/5.19                ( ( finite_finite_real @ A7 )
% 9.20/5.19               => ( ( member_real @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5616_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_int,P: set_int > $o] :
% 9.20/5.19        ( ( finite_finite_int @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: int,A7: set_int] :
% 9.20/5.19                ( ( finite_finite_int @ A7 )
% 9.20/5.19               => ( ( member_int @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5617_finite__empty__induct,axiom,
% 9.20/5.19      ! [A2: set_nat,P: set_nat > $o] :
% 9.20/5.19        ( ( finite_finite_nat @ A2 )
% 9.20/5.19       => ( ( P @ A2 )
% 9.20/5.19         => ( ! [A4: nat,A7: set_nat] :
% 9.20/5.19                ( ( finite_finite_nat @ A7 )
% 9.20/5.19               => ( ( member_nat @ A4 @ A7 )
% 9.20/5.19                 => ( ( P @ A7 )
% 9.20/5.19                   => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
% 9.20/5.19           => ( P @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5618_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: produc3843707927480180839at_nat] :
% 9.20/5.19                  ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.19                    | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5619_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_complex > $o,A2: set_complex] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_complex] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: complex] :
% 9.20/5.19                  ( ( member_complex @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
% 9.20/5.19                    | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5620_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: product_prod_nat_nat] :
% 9.20/5.19                  ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.19                    | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5621_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_real > $o,A2: set_real] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_real] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: real] :
% 9.20/5.19                  ( ( member_real @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
% 9.20/5.19                    | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite_finite_real @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5622_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_int > $o,A2: set_int] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_int] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: int] :
% 9.20/5.19                  ( ( member_int @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
% 9.20/5.19                    | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite_finite_int @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5623_infinite__coinduct,axiom,
% 9.20/5.19      ! [X7: set_nat > $o,A2: set_nat] :
% 9.20/5.19        ( ( X7 @ A2 )
% 9.20/5.19       => ( ! [A7: set_nat] :
% 9.20/5.19              ( ( X7 @ A7 )
% 9.20/5.19             => ? [X2: nat] :
% 9.20/5.19                  ( ( member_nat @ X2 @ A7 )
% 9.20/5.19                  & ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
% 9.20/5.19                    | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
% 9.20/5.19         => ~ ( finite_finite_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5624_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
% 9.20/5.19        ( ~ ( finite4343798906461161616at_nat @ S3 )
% 9.20/5.19       => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5625_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_complex,A: complex] :
% 9.20/5.19        ( ~ ( finite3207457112153483333omplex @ S3 )
% 9.20/5.19       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5626_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
% 9.20/5.19        ( ~ ( finite6177210948735845034at_nat @ S3 )
% 9.20/5.19       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5627_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_real,A: real] :
% 9.20/5.19        ( ~ ( finite_finite_real @ S3 )
% 9.20/5.19       => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5628_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_int,A: int] :
% 9.20/5.19        ( ~ ( finite_finite_int @ S3 )
% 9.20/5.19       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5629_infinite__remove,axiom,
% 9.20/5.19      ! [S3: set_nat,A: nat] :
% 9.20/5.19        ( ~ ( finite_finite_nat @ S3 )
% 9.20/5.19       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5630_Diff__single__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 )
% 9.20/5.19       => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5631_Diff__single__insert,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
% 9.20/5.19       => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5632_Diff__single__insert,axiom,
% 9.20/5.19      ! [A2: set_real,X3: real,B2: set_real] :
% 9.20/5.19        ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 )
% 9.20/5.19       => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5633_Diff__single__insert,axiom,
% 9.20/5.19      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 9.20/5.19        ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 )
% 9.20/5.19       => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5634_Diff__single__insert,axiom,
% 9.20/5.19      ! [A2: set_int,X3: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 )
% 9.20/5.19       => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5635_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5636_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_set_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5637_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5638_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_real,X3: real,B2: set_real] :
% 9.20/5.19        ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_real @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5639_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 9.20/5.19        ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5640_subset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_int,X3: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_int @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
% 9.20/5.19          & ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.19           => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5641_take__bit__Suc__minus__1__eq,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.19        = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5642_take__bit__Suc__minus__1__eq,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.19        = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5643_atLeast0__atMost__Suc,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 9.20/5.19        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5644_mod__pos__geq,axiom,
% 9.20/5.19      ! [L: int,K: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.19       => ( ( ord_less_eq_int @ L @ K )
% 9.20/5.19         => ( ( modulo_modulo_int @ K @ L )
% 9.20/5.19            = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5645_Icc__eq__insert__lb__nat,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.19       => ( ( set_or1269000886237332187st_nat @ M @ N2 )
% 9.20/5.19          = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5646_atLeastAtMostSuc__conv,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.19       => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
% 9.20/5.19          = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5647_atLeastAtMost__insertL,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.19       => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 9.20/5.19          = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5648_plusinfinity,axiom,
% 9.20/5.19      ! [D: int,P5: int > $o,P: int > $o] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.19       => ( ! [X4: int,K2: int] :
% 9.20/5.19              ( ( P5 @ X4 )
% 9.20/5.19              = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.20/5.19         => ( ? [Z4: int] :
% 9.20/5.19              ! [X4: int] :
% 9.20/5.19                ( ( ord_less_int @ Z4 @ X4 )
% 9.20/5.19               => ( ( P @ X4 )
% 9.20/5.19                  = ( P5 @ X4 ) ) )
% 9.20/5.19           => ( ? [X_12: int] : ( P5 @ X_12 )
% 9.20/5.19             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5649_minusinfinity,axiom,
% 9.20/5.19      ! [D: int,P1: int > $o,P: int > $o] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.19       => ( ! [X4: int,K2: int] :
% 9.20/5.19              ( ( P1 @ X4 )
% 9.20/5.19              = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.20/5.19         => ( ? [Z4: int] :
% 9.20/5.19              ! [X4: int] :
% 9.20/5.19                ( ( ord_less_int @ X4 @ Z4 )
% 9.20/5.19               => ( ( P @ X4 )
% 9.20/5.19                  = ( P1 @ X4 ) ) )
% 9.20/5.19           => ( ? [X_12: int] : ( P1 @ X_12 )
% 9.20/5.19             => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5650_int__induct,axiom,
% 9.20/5.19      ! [P: int > $o,K: int,I: int] :
% 9.20/5.19        ( ( P @ K )
% 9.20/5.19       => ( ! [I3: int] :
% 9.20/5.19              ( ( ord_less_eq_int @ K @ I3 )
% 9.20/5.19             => ( ( P @ I3 )
% 9.20/5.19               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.19         => ( ! [I3: int] :
% 9.20/5.19                ( ( ord_less_eq_int @ I3 @ K )
% 9.20/5.19               => ( ( P @ I3 )
% 9.20/5.19                 => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
% 9.20/5.19           => ( P @ I ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5651_subset__eq__atLeast0__atMost__finite,axiom,
% 9.20/5.19      ! [N7: set_nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.19       => ( finite_finite_nat @ N7 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5652_take__bit__int__less__exp,axiom,
% 9.20/5.19      ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5653_pos__minus__divide__less__eq,axiom,
% 9.20/5.19      ! [C: real,B: real,A: real] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5654_pos__minus__divide__less__eq,axiom,
% 9.20/5.19      ! [C: rat,B: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5655_pos__less__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: real,A: real,B: real] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5656_pos__less__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5657_neg__minus__divide__less__eq,axiom,
% 9.20/5.19      ! [C: real,B: real,A: real] :
% 9.20/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5658_neg__minus__divide__less__eq,axiom,
% 9.20/5.19      ! [C: rat,B: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19       => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5659_neg__less__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: real,A: real,B: real] :
% 9.20/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19       => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5660_neg__less__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19       => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5661_minus__divide__less__eq,axiom,
% 9.20/5.19      ! [B: real,C: real,A: real] :
% 9.20/5.19        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5662_minus__divide__less__eq,axiom,
% 9.20/5.19      ! [B: rat,C: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5663_less__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: real,B: real,C: real] :
% 9.20/5.19        ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5664_less__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: rat,B: rat,C: rat] :
% 9.20/5.19        ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5665_divide__eq__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: complex,C: complex,W2: num] :
% 9.20/5.19        ( ( ( divide1717551699836669952omplex @ B @ C )
% 9.20/5.19          = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_complex )
% 9.20/5.19           => ( B
% 9.20/5.19              = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_complex )
% 9.20/5.19           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.19              = zero_zero_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5666_divide__eq__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: real,C: real,W2: num] :
% 9.20/5.19        ( ( ( divide_divide_real @ B @ C )
% 9.20/5.19          = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_real )
% 9.20/5.19           => ( B
% 9.20/5.19              = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_real )
% 9.20/5.19           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.19              = zero_zero_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5667_divide__eq__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: rat,C: rat,W2: num] :
% 9.20/5.19        ( ( ( divide_divide_rat @ B @ C )
% 9.20/5.19          = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.19        = ( ( ( C != zero_zero_rat )
% 9.20/5.19           => ( B
% 9.20/5.19              = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ( C = zero_zero_rat )
% 9.20/5.19           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.19              = zero_zero_rat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5668_eq__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: complex,C: complex] :
% 9.20/5.19        ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.19          = ( divide1717551699836669952omplex @ B @ C ) )
% 9.20/5.19        = ( ( ( C != zero_zero_complex )
% 9.20/5.19           => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
% 9.20/5.19              = B ) )
% 9.20/5.19          & ( ( C = zero_zero_complex )
% 9.20/5.19           => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
% 9.20/5.19              = zero_zero_complex ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5669_eq__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: real,C: real] :
% 9.20/5.19        ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.19          = ( divide_divide_real @ B @ C ) )
% 9.20/5.19        = ( ( ( C != zero_zero_real )
% 9.20/5.19           => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
% 9.20/5.19              = B ) )
% 9.20/5.19          & ( ( C = zero_zero_real )
% 9.20/5.19           => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.19              = zero_zero_real ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5670_eq__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: rat,C: rat] :
% 9.20/5.19        ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.19          = ( divide_divide_rat @ B @ C ) )
% 9.20/5.19        = ( ( ( C != zero_zero_rat )
% 9.20/5.19           => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
% 9.20/5.19              = B ) )
% 9.20/5.19          & ( ( C = zero_zero_rat )
% 9.20/5.19           => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
% 9.20/5.19              = zero_zero_rat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5671_minus__divide__add__eq__iff,axiom,
% 9.20/5.19      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.19        ( ( Z2 != zero_zero_complex )
% 9.20/5.19       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5672_minus__divide__add__eq__iff,axiom,
% 9.20/5.19      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.19        ( ( Z2 != zero_zero_real )
% 9.20/5.19       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5673_minus__divide__add__eq__iff,axiom,
% 9.20/5.19      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.19        ( ( Z2 != zero_zero_rat )
% 9.20/5.19       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5674_add__divide__eq__if__simps_I3_J,axiom,
% 9.20/5.19      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.19         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.20/5.19            = B ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_complex )
% 9.20/5.19         => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5675_add__divide__eq__if__simps_I3_J,axiom,
% 9.20/5.19      ! [Z2: real,A: real,B: real] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_real )
% 9.20/5.19         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.20/5.19            = B ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_real )
% 9.20/5.19         => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5676_add__divide__eq__if__simps_I3_J,axiom,
% 9.20/5.19      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.19         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.20/5.19            = B ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_rat )
% 9.20/5.19         => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5677_add__divide__eq__if__simps_I6_J,axiom,
% 9.20/5.19      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.19         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_complex )
% 9.20/5.19         => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5678_add__divide__eq__if__simps_I6_J,axiom,
% 9.20/5.19      ! [Z2: real,A: real,B: real] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_real )
% 9.20/5.19         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( uminus_uminus_real @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_real )
% 9.20/5.19         => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5679_add__divide__eq__if__simps_I6_J,axiom,
% 9.20/5.19      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.19         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_rat )
% 9.20/5.19         => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
% 9.20/5.19            = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5680_add__divide__eq__if__simps_I5_J,axiom,
% 9.20/5.19      ! [Z2: complex,A: complex,B: complex] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_complex )
% 9.20/5.19         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.20/5.19            = ( uminus1482373934393186551omplex @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_complex )
% 9.20/5.19         => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
% 9.20/5.19            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5681_add__divide__eq__if__simps_I5_J,axiom,
% 9.20/5.19      ! [Z2: real,A: real,B: real] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_real )
% 9.20/5.19         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.20/5.19            = ( uminus_uminus_real @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_real )
% 9.20/5.19         => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
% 9.20/5.19            = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5682_add__divide__eq__if__simps_I5_J,axiom,
% 9.20/5.19      ! [Z2: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ( Z2 = zero_zero_rat )
% 9.20/5.19         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.20/5.19            = ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19        & ( ( Z2 != zero_zero_rat )
% 9.20/5.19         => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
% 9.20/5.19            = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5683_minus__divide__diff__eq__iff,axiom,
% 9.20/5.19      ! [Z2: complex,X3: complex,Y3: complex] :
% 9.20/5.19        ( ( Z2 != zero_zero_complex )
% 9.20/5.19       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5684_minus__divide__diff__eq__iff,axiom,
% 9.20/5.19      ! [Z2: real,X3: real,Y3: real] :
% 9.20/5.19        ( ( Z2 != zero_zero_real )
% 9.20/5.19       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5685_minus__divide__diff__eq__iff,axiom,
% 9.20/5.19      ! [Z2: rat,X3: rat,Y3: rat] :
% 9.20/5.19        ( ( Z2 != zero_zero_rat )
% 9.20/5.19       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
% 9.20/5.19          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5686_remove__induct,axiom,
% 9.20/5.19      ! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.19       => ( ( ~ ( finite4343798906461161616at_nat @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.19                ( ( finite4343798906461161616at_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bo228742789529271731at_nat )
% 9.20/5.19                 => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: produc3843707927480180839at_nat] :
% 9.20/5.19                          ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5687_remove__induct,axiom,
% 9.20/5.19      ! [P: set_set_nat > $o,B2: set_set_nat] :
% 9.20/5.19        ( ( P @ bot_bot_set_set_nat )
% 9.20/5.19       => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_set_nat] :
% 9.20/5.19                ( ( finite1152437895449049373et_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_set_nat )
% 9.20/5.19                 => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: set_nat] :
% 9.20/5.19                          ( ( member_set_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5688_remove__induct,axiom,
% 9.20/5.19      ! [P: set_complex > $o,B2: set_complex] :
% 9.20/5.19        ( ( P @ bot_bot_set_complex )
% 9.20/5.19       => ( ( ~ ( finite3207457112153483333omplex @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_complex] :
% 9.20/5.19                ( ( finite3207457112153483333omplex @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_complex )
% 9.20/5.19                 => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: complex] :
% 9.20/5.19                          ( ( member_complex @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5689_remove__induct,axiom,
% 9.20/5.19      ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.19       => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.19                ( ( finite6177210948735845034at_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bo2099793752762293965at_nat )
% 9.20/5.19                 => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: product_prod_nat_nat] :
% 9.20/5.19                          ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5690_remove__induct,axiom,
% 9.20/5.19      ! [P: set_real > $o,B2: set_real] :
% 9.20/5.19        ( ( P @ bot_bot_set_real )
% 9.20/5.19       => ( ( ~ ( finite_finite_real @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_real] :
% 9.20/5.19                ( ( finite_finite_real @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_real )
% 9.20/5.19                 => ( ( ord_less_eq_set_real @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: real] :
% 9.20/5.19                          ( ( member_real @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5691_remove__induct,axiom,
% 9.20/5.19      ! [P: set_nat > $o,B2: set_nat] :
% 9.20/5.19        ( ( P @ bot_bot_set_nat )
% 9.20/5.19       => ( ( ~ ( finite_finite_nat @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_nat] :
% 9.20/5.19                ( ( finite_finite_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_nat )
% 9.20/5.19                 => ( ( ord_less_eq_set_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: nat] :
% 9.20/5.19                          ( ( member_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5692_remove__induct,axiom,
% 9.20/5.19      ! [P: set_int > $o,B2: set_int] :
% 9.20/5.19        ( ( P @ bot_bot_set_int )
% 9.20/5.19       => ( ( ~ ( finite_finite_int @ B2 )
% 9.20/5.19           => ( P @ B2 ) )
% 9.20/5.19         => ( ! [A7: set_int] :
% 9.20/5.19                ( ( finite_finite_int @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_int )
% 9.20/5.19                 => ( ( ord_less_eq_set_int @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: int] :
% 9.20/5.19                          ( ( member_int @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5693_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.19        ( ( finite4343798906461161616at_nat @ B2 )
% 9.20/5.19       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.19         => ( ! [A7: set_Pr4329608150637261639at_nat] :
% 9.20/5.19                ( ( finite4343798906461161616at_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bo228742789529271731at_nat )
% 9.20/5.19                 => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: produc3843707927480180839at_nat] :
% 9.20/5.19                          ( ( member8757157785044589968at_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5694_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_set_nat,P: set_set_nat > $o] :
% 9.20/5.19        ( ( finite1152437895449049373et_nat @ B2 )
% 9.20/5.19       => ( ( P @ bot_bot_set_set_nat )
% 9.20/5.19         => ( ! [A7: set_set_nat] :
% 9.20/5.19                ( ( finite1152437895449049373et_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_set_nat )
% 9.20/5.19                 => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: set_nat] :
% 9.20/5.19                          ( ( member_set_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5695_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_complex,P: set_complex > $o] :
% 9.20/5.19        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.19       => ( ( P @ bot_bot_set_complex )
% 9.20/5.19         => ( ! [A7: set_complex] :
% 9.20/5.19                ( ( finite3207457112153483333omplex @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_complex )
% 9.20/5.19                 => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: complex] :
% 9.20/5.19                          ( ( member_complex @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5696_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.19        ( ( finite6177210948735845034at_nat @ B2 )
% 9.20/5.19       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.19         => ( ! [A7: set_Pr1261947904930325089at_nat] :
% 9.20/5.19                ( ( finite6177210948735845034at_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bo2099793752762293965at_nat )
% 9.20/5.19                 => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: product_prod_nat_nat] :
% 9.20/5.19                          ( ( member8440522571783428010at_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5697_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_real,P: set_real > $o] :
% 9.20/5.19        ( ( finite_finite_real @ B2 )
% 9.20/5.19       => ( ( P @ bot_bot_set_real )
% 9.20/5.19         => ( ! [A7: set_real] :
% 9.20/5.19                ( ( finite_finite_real @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_real )
% 9.20/5.19                 => ( ( ord_less_eq_set_real @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: real] :
% 9.20/5.19                          ( ( member_real @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5698_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_nat,P: set_nat > $o] :
% 9.20/5.19        ( ( finite_finite_nat @ B2 )
% 9.20/5.19       => ( ( P @ bot_bot_set_nat )
% 9.20/5.19         => ( ! [A7: set_nat] :
% 9.20/5.19                ( ( finite_finite_nat @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_nat )
% 9.20/5.19                 => ( ( ord_less_eq_set_nat @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: nat] :
% 9.20/5.19                          ( ( member_nat @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5699_finite__remove__induct,axiom,
% 9.20/5.19      ! [B2: set_int,P: set_int > $o] :
% 9.20/5.19        ( ( finite_finite_int @ B2 )
% 9.20/5.19       => ( ( P @ bot_bot_set_int )
% 9.20/5.19         => ( ! [A7: set_int] :
% 9.20/5.19                ( ( finite_finite_int @ A7 )
% 9.20/5.19               => ( ( A7 != bot_bot_set_int )
% 9.20/5.19                 => ( ( ord_less_eq_set_int @ A7 @ B2 )
% 9.20/5.19                   => ( ! [X2: int] :
% 9.20/5.19                          ( ( member_int @ X2 @ A7 )
% 9.20/5.19                         => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
% 9.20/5.19                     => ( P @ A7 ) ) ) ) )
% 9.20/5.19           => ( P @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5700_signed__take__bit__int__greater__self__iff,axiom,
% 9.20/5.19      ! [K: int,N2: nat] :
% 9.20/5.19        ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
% 9.20/5.19        = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5701_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
% 9.20/5.19        ( ( finite4343798906461161616at_nat @ S3 )
% 9.20/5.19       => ( ( P @ bot_bo228742789529271731at_nat )
% 9.20/5.19         => ( ! [T4: set_Pr4329608150637261639at_nat] :
% 9.20/5.19                ( ( ord_le2604355607129572851at_nat @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: produc3843707927480180839at_nat] :
% 9.20/5.19                      ( ( member8757157785044589968at_nat @ X2 @ ( minus_3314409938677909166at_nat @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert9069300056098147895at_nat @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5702_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_complex,P: set_complex > $o] :
% 9.20/5.19        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.19       => ( ( P @ bot_bot_set_complex )
% 9.20/5.19         => ( ! [T4: set_complex] :
% 9.20/5.19                ( ( ord_less_set_complex @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: complex] :
% 9.20/5.19                      ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert_complex @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5703_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
% 9.20/5.19        ( ( finite6177210948735845034at_nat @ S3 )
% 9.20/5.19       => ( ( P @ bot_bo2099793752762293965at_nat )
% 9.20/5.19         => ( ! [T4: set_Pr1261947904930325089at_nat] :
% 9.20/5.19                ( ( ord_le7866589430770878221at_nat @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: product_prod_nat_nat] :
% 9.20/5.19                      ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert8211810215607154385at_nat @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5704_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_real,P: set_real > $o] :
% 9.20/5.19        ( ( finite_finite_real @ S3 )
% 9.20/5.19       => ( ( P @ bot_bot_set_real )
% 9.20/5.19         => ( ! [T4: set_real] :
% 9.20/5.19                ( ( ord_less_set_real @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: real] :
% 9.20/5.19                      ( ( member_real @ X2 @ ( minus_minus_set_real @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert_real @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5705_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_int,P: set_int > $o] :
% 9.20/5.19        ( ( finite_finite_int @ S3 )
% 9.20/5.19       => ( ( P @ bot_bot_set_int )
% 9.20/5.19         => ( ! [T4: set_int] :
% 9.20/5.19                ( ( ord_less_set_int @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: int] :
% 9.20/5.19                      ( ( member_int @ X2 @ ( minus_minus_set_int @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert_int @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5706_finite__induct__select,axiom,
% 9.20/5.19      ! [S3: set_nat,P: set_nat > $o] :
% 9.20/5.19        ( ( finite_finite_nat @ S3 )
% 9.20/5.19       => ( ( P @ bot_bot_set_nat )
% 9.20/5.19         => ( ! [T4: set_nat] :
% 9.20/5.19                ( ( ord_less_set_nat @ T4 @ S3 )
% 9.20/5.19               => ( ( P @ T4 )
% 9.20/5.19                 => ? [X2: nat] :
% 9.20/5.19                      ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S3 @ T4 ) )
% 9.20/5.19                      & ( P @ ( insert_nat @ X2 @ T4 ) ) ) ) )
% 9.20/5.19           => ( P @ S3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5707_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
% 9.20/5.19        ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.19           => ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5708_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_set_nat @ X3 @ B2 )
% 9.20/5.19           => ( ord_less_set_set_nat @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member_set_nat @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member_set_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member_set_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5709_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.19           => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5710_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_real,X3: real,B2: set_real] :
% 9.20/5.19        ( ( ord_less_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_real @ X3 @ B2 )
% 9.20/5.19           => ( ord_less_set_real @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member_real @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member_real @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5711_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_nat,X3: nat,B2: set_nat] :
% 9.20/5.19        ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_nat @ X3 @ B2 )
% 9.20/5.19           => ( ord_less_set_nat @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member_nat @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5712_psubset__insert__iff,axiom,
% 9.20/5.19      ! [A2: set_int,X3: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
% 9.20/5.19        = ( ( ( member_int @ X3 @ B2 )
% 9.20/5.19           => ( ord_less_set_int @ A2 @ B2 ) )
% 9.20/5.19          & ( ~ ( member_int @ X3 @ B2 )
% 9.20/5.19           => ( ( ( member_int @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
% 9.20/5.19              & ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.19               => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5713_decr__mult__lemma,axiom,
% 9.20/5.19      ! [D: int,P: int > $o,K: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.19       => ( ! [X4: int] :
% 9.20/5.19              ( ( P @ X4 )
% 9.20/5.19             => ( P @ ( minus_minus_int @ X4 @ D ) ) )
% 9.20/5.19         => ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.19           => ! [X2: int] :
% 9.20/5.19                ( ( P @ X2 )
% 9.20/5.19               => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5714_verit__less__mono__div__int2,axiom,
% 9.20/5.19      ! [A2: int,B2: int,N2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ A2 @ B2 )
% 9.20/5.19       => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
% 9.20/5.19         => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5715_div__eq__minus1,axiom,
% 9.20/5.19      ! [B: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.19       => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
% 9.20/5.19          = ( uminus_uminus_int @ one_one_int ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5716_take__bit__Suc__bit0,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
% 9.20/5.19        = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5717_take__bit__Suc__bit0,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
% 9.20/5.19        = ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5718_take__bit__nat__eq__self,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.19       => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 9.20/5.19          = M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5719_take__bit__nat__less__exp,axiom,
% 9.20/5.19      ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5720_take__bit__nat__eq__self__iff,axiom,
% 9.20/5.19      ! [N2: nat,M: nat] :
% 9.20/5.19        ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
% 9.20/5.19          = M )
% 9.20/5.19        = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5721_take__bit__int__less__self__iff,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5722_take__bit__int__greater__eq__self__iff,axiom,
% 9.20/5.19      ! [K: int,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
% 9.20/5.19        = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5723_le__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: real,B: real,C: real] :
% 9.20/5.19        ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5724_le__minus__divide__eq,axiom,
% 9.20/5.19      ! [A: rat,B: rat,C: rat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5725_minus__divide__le__eq,axiom,
% 9.20/5.19      ! [B: real,C: real,A: real] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5726_minus__divide__le__eq,axiom,
% 9.20/5.19      ! [B: rat,C: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5727_neg__le__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: real,A: real,B: real] :
% 9.20/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5728_neg__le__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5729_neg__minus__divide__le__eq,axiom,
% 9.20/5.19      ! [C: real,B: real,A: real] :
% 9.20/5.19        ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5730_neg__minus__divide__le__eq,axiom,
% 9.20/5.19      ! [C: rat,B: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5731_pos__le__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: real,A: real,B: real] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19       => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5732_pos__le__minus__divide__eq,axiom,
% 9.20/5.19      ! [C: rat,A: rat,B: rat] :
% 9.20/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19       => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
% 9.20/5.19          = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5733_pos__minus__divide__le__eq,axiom,
% 9.20/5.19      ! [C: real,B: real,A: real] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5734_pos__minus__divide__le__eq,axiom,
% 9.20/5.19      ! [C: rat,B: rat,A: rat] :
% 9.20/5.19        ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
% 9.20/5.19          = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5735_divide__less__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: real,C: real,W2: num] :
% 9.20/5.19        ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5736_divide__less__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: rat,C: rat,W2: num] :
% 9.20/5.19        ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5737_less__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: real,C: real] :
% 9.20/5.19        ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5738_less__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: rat,C: rat] :
% 9.20/5.19        ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5739_signed__take__bit__int__less__eq,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K )
% 9.20/5.19       => ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( minus_minus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5740_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.20/5.19      ! [K: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.19       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.19          = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5741_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.20/5.19      ! [K: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.19       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5742_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.20/5.19      ! [K: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.19       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.19          = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5743_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.20/5.19      ! [K: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.19       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.19          = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5744_neg__one__power__add__eq__neg__one__power__diff,axiom,
% 9.20/5.19      ! [K: nat,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.19       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
% 9.20/5.19          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5745_signed__take__bit__int__eq__self,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 9.20/5.19       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.19         => ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 9.20/5.19            = K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5746_signed__take__bit__int__eq__self__iff,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
% 9.20/5.19          = K )
% 9.20/5.19        = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
% 9.20/5.19          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5747_take__bit__eq__0__iff,axiom,
% 9.20/5.19      ! [N2: nat,A: code_integer] :
% 9.20/5.19        ( ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 9.20/5.19          = zero_z3403309356797280102nteger )
% 9.20/5.19        = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5748_take__bit__eq__0__iff,axiom,
% 9.20/5.19      ! [N2: nat,A: nat] :
% 9.20/5.19        ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 9.20/5.19          = zero_zero_nat )
% 9.20/5.19        = ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5749_take__bit__eq__0__iff,axiom,
% 9.20/5.19      ! [N2: nat,A: int] :
% 9.20/5.19        ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 9.20/5.19          = zero_zero_int )
% 9.20/5.19        = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5750_take__bit__nat__less__self__iff,axiom,
% 9.20/5.19      ! [N2: nat,M: nat] :
% 9.20/5.19        ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
% 9.20/5.19        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5751_take__bit__int__eq__self,axiom,
% 9.20/5.19      ! [K: int,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.19       => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.19         => ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 9.20/5.19            = K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5752_take__bit__int__eq__self__iff,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
% 9.20/5.19          = K )
% 9.20/5.19        = ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.19          & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5753_divide__le__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: real,C: real,W2: num] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5754_divide__le__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [B: rat,C: rat,W2: num] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5755_le__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: real,C: real] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
% 9.20/5.19        = ( ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19          & ( ~ ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.19           => ( ( ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_real @ C @ zero_zero_real )
% 9.20/5.19               => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5756_le__divide__eq__numeral_I2_J,axiom,
% 9.20/5.19      ! [W2: num,B: rat,C: rat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
% 9.20/5.19        = ( ( ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
% 9.20/5.19          & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
% 9.20/5.19           => ( ( ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
% 9.20/5.19              & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
% 9.20/5.19               => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5757_square__le__1,axiom,
% 9.20/5.19      ! [X3: real] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.19       => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.20/5.19         => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5758_square__le__1,axiom,
% 9.20/5.19      ! [X3: code_integer] :
% 9.20/5.19        ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
% 9.20/5.19       => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
% 9.20/5.19         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5759_square__le__1,axiom,
% 9.20/5.19      ! [X3: rat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
% 9.20/5.19       => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
% 9.20/5.19         => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5760_square__le__1,axiom,
% 9.20/5.19      ! [X3: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
% 9.20/5.19       => ( ( ord_less_eq_int @ X3 @ one_one_int )
% 9.20/5.19         => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5761_div__pos__geq,axiom,
% 9.20/5.19      ! [L: int,K: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ L )
% 9.20/5.19       => ( ( ord_less_eq_int @ L @ K )
% 9.20/5.19         => ( ( divide_divide_int @ K @ L )
% 9.20/5.19            = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5762_div__pos__neg__trivial,axiom,
% 9.20/5.19      ! [K: int,L: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.19       => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
% 9.20/5.19         => ( ( divide_divide_int @ K @ L )
% 9.20/5.19            = ( uminus_uminus_int @ one_one_int ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5763_take__bit__int__greater__eq,axiom,
% 9.20/5.19      ! [K: int,N2: nat] :
% 9.20/5.19        ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.19       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5764_power__minus1__odd,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5765_power__minus1__odd,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5766_power__minus1__odd,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5767_power__minus1__odd,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5768_power__minus1__odd,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5769_take__bit__Suc,axiom,
% 9.20/5.19      ! [N2: nat,A: code_integer] :
% 9.20/5.19        ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
% 9.20/5.19        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5770_take__bit__Suc,axiom,
% 9.20/5.19      ! [N2: nat,A: nat] :
% 9.20/5.19        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ A )
% 9.20/5.19        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5771_take__bit__Suc,axiom,
% 9.20/5.19      ! [N2: nat,A: int] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5772_signed__take__bit__int__greater__eq,axiom,
% 9.20/5.19      ! [K: int,N2: nat] :
% 9.20/5.19        ( ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.19       => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5773_stable__imp__take__bit__eq,axiom,
% 9.20/5.19      ! [A: code_integer,N2: nat] :
% 9.20/5.19        ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.19          = A )
% 9.20/5.19       => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 9.20/5.19              = zero_z3403309356797280102nteger ) )
% 9.20/5.19          & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se1745604003318907178nteger @ N2 @ A )
% 9.20/5.19              = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5774_stable__imp__take__bit__eq,axiom,
% 9.20/5.19      ! [A: nat,N2: nat] :
% 9.20/5.19        ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.19          = A )
% 9.20/5.19       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 9.20/5.19              = zero_zero_nat ) )
% 9.20/5.19          & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se2925701944663578781it_nat @ N2 @ A )
% 9.20/5.19              = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5775_stable__imp__take__bit__eq,axiom,
% 9.20/5.19      ! [A: int,N2: nat] :
% 9.20/5.19        ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.19          = A )
% 9.20/5.19       => ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 9.20/5.19              = zero_zero_int ) )
% 9.20/5.19          & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.19           => ( ( bit_se2923211474154528505it_int @ N2 @ A )
% 9.20/5.19              = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5776_signed__take__bit__Suc__minus__bit1,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5777_concat__bit__Suc,axiom,
% 9.20/5.19      ! [N2: nat,K: int,L: int] :
% 9.20/5.19        ( ( bit_concat_bit @ ( suc @ N2 ) @ K @ L )
% 9.20/5.19        = ( plus_plus_int @ ( modulo_modulo_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_concat_bit @ N2 @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ L ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5778_Bolzano,axiom,
% 9.20/5.19      ! [A: real,B: real,P: real > real > $o] :
% 9.20/5.19        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.19       => ( ! [A4: real,B4: real,C2: real] :
% 9.20/5.19              ( ( P @ A4 @ B4 )
% 9.20/5.19             => ( ( P @ B4 @ C2 )
% 9.20/5.19               => ( ( ord_less_eq_real @ A4 @ B4 )
% 9.20/5.19                 => ( ( ord_less_eq_real @ B4 @ C2 )
% 9.20/5.19                   => ( P @ A4 @ C2 ) ) ) ) )
% 9.20/5.19         => ( ! [X4: real] :
% 9.20/5.19                ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.19               => ( ( ord_less_eq_real @ X4 @ B )
% 9.20/5.19                 => ? [D3: real] :
% 9.20/5.19                      ( ( ord_less_real @ zero_zero_real @ D3 )
% 9.20/5.19                      & ! [A4: real,B4: real] :
% 9.20/5.19                          ( ( ( ord_less_eq_real @ A4 @ X4 )
% 9.20/5.19                            & ( ord_less_eq_real @ X4 @ B4 )
% 9.20/5.19                            & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
% 9.20/5.19                         => ( P @ A4 @ B4 ) ) ) ) )
% 9.20/5.19           => ( P @ A @ B ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5779_signed__take__bit__Suc__bit1,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_ri631733984087533419it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5780_and__int_Oelims,axiom,
% 9.20/5.19      ! [X3: int,Xa2: int,Y3: int] :
% 9.20/5.19        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 9.20/5.19          = Y3 )
% 9.20/5.19       => ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19              & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19           => ( Y3
% 9.20/5.19              = ( uminus_uminus_int
% 9.20/5.19                @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.20/5.19                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.20/5.19          & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19                & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19           => ( Y3
% 9.20/5.19              = ( plus_plus_int
% 9.20/5.19                @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.20/5.19                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.20/5.19                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5781_and__int_Osimps,axiom,
% 9.20/5.19      ( bit_se725231765392027082nd_int
% 9.20/5.19      = ( ^ [K3: int,L2: int] :
% 9.20/5.19            ( if_int
% 9.20/5.19            @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19              & ( member_int @ L2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19            @ ( uminus_uminus_int
% 9.20/5.19              @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.20/5.19                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
% 9.20/5.19            @ ( plus_plus_int
% 9.20/5.19              @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
% 9.20/5.19                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
% 9.20/5.19              @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5782_Diff__idemp,axiom,
% 9.20/5.19      ! [A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
% 9.20/5.19        = ( minus_minus_set_nat @ A2 @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5783_Diff__iff,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.19        = ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.20/5.19          & ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5784_Diff__iff,axiom,
% 9.20/5.19      ! [C: real,A2: set_real,B2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.20/5.19        = ( ( member_real @ C @ A2 )
% 9.20/5.19          & ~ ( member_real @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5785_Diff__iff,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.20/5.19        = ( ( member_set_nat @ C @ A2 )
% 9.20/5.19          & ~ ( member_set_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5786_Diff__iff,axiom,
% 9.20/5.19      ! [C: int,A2: set_int,B2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.19        = ( ( member_int @ C @ A2 )
% 9.20/5.19          & ~ ( member_int @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5787_Diff__iff,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.19        = ( ( member_nat @ C @ A2 )
% 9.20/5.19          & ~ ( member_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5788_DiffI,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.20/5.19       => ( ~ ( member8440522571783428010at_nat @ C @ B2 )
% 9.20/5.19         => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5789_DiffI,axiom,
% 9.20/5.19      ! [C: real,A2: set_real,B2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ A2 )
% 9.20/5.19       => ( ~ ( member_real @ C @ B2 )
% 9.20/5.19         => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5790_DiffI,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ A2 )
% 9.20/5.19       => ( ~ ( member_set_nat @ C @ B2 )
% 9.20/5.19         => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5791_DiffI,axiom,
% 9.20/5.19      ! [C: int,A2: set_int,B2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ A2 )
% 9.20/5.19       => ( ~ ( member_int @ C @ B2 )
% 9.20/5.19         => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5792_DiffI,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ A2 )
% 9.20/5.19       => ( ~ ( member_nat @ C @ B2 )
% 9.20/5.19         => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5793_ComplI,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ~ ( member8440522571783428010at_nat @ C @ A2 )
% 9.20/5.19       => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5794_ComplI,axiom,
% 9.20/5.19      ! [C: real,A2: set_real] :
% 9.20/5.19        ( ~ ( member_real @ C @ A2 )
% 9.20/5.19       => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5795_ComplI,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat] :
% 9.20/5.19        ( ~ ( member_set_nat @ C @ A2 )
% 9.20/5.19       => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5796_ComplI,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat] :
% 9.20/5.19        ( ~ ( member_nat @ C @ A2 )
% 9.20/5.19       => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5797_ComplI,axiom,
% 9.20/5.19      ! [C: int,A2: set_int] :
% 9.20/5.19        ( ~ ( member_int @ C @ A2 )
% 9.20/5.19       => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5798_Compl__iff,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.20/5.19        = ( ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5799_Compl__iff,axiom,
% 9.20/5.19      ! [C: real,A2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.20/5.19        = ( ~ ( member_real @ C @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5800_Compl__iff,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.20/5.19        = ( ~ ( member_set_nat @ C @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5801_Compl__iff,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.20/5.19        = ( ~ ( member_nat @ C @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5802_Compl__iff,axiom,
% 9.20/5.19      ! [C: int,A2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.20/5.19        = ( ~ ( member_int @ C @ A2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5803_and__zero__eq,axiom,
% 9.20/5.19      ! [A: int] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5804_and__zero__eq,axiom,
% 9.20/5.19      ! [A: nat] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5805_zero__and__eq,axiom,
% 9.20/5.19      ! [A: int] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5806_zero__and__eq,axiom,
% 9.20/5.19      ! [A: nat] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5807_bit_Oconj__zero__left,axiom,
% 9.20/5.19      ! [X3: int] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X3 )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5808_bit_Oconj__zero__right,axiom,
% 9.20/5.19      ! [X3: int] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ X3 @ zero_zero_int )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5809_concat__bit__0,axiom,
% 9.20/5.19      ! [K: int,L: int] :
% 9.20/5.19        ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
% 9.20/5.19        = L ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5810_semiring__norm_I80_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( ord_less_num @ M @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5811_and__negative__int__iff,axiom,
% 9.20/5.19      ! [K: int,L: int] :
% 9.20/5.19        ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
% 9.20/5.19        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.19          & ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5812_semiring__norm_I81_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.19        = ( ord_less_num @ M @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5813_semiring__norm_I77_J,axiom,
% 9.20/5.19      ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5814_concat__bit__negative__iff,axiom,
% 9.20/5.19      ! [N2: nat,K: int,L: int] :
% 9.20/5.19        ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
% 9.20/5.19        = ( ord_less_int @ L @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5815_semiring__norm_I79_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_num @ M @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5816_semiring__norm_I74_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.19        = ( ord_less_num @ M @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5817_and__numerals_I5_J,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ one_one_int )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5818_and__numerals_I5_J,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ one_one_nat )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5819_and__numerals_I1_J,axiom,
% 9.20/5.19      ! [Y3: num] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 9.20/5.19        = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5820_and__numerals_I1_J,axiom,
% 9.20/5.19      ! [Y3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5821_div__Suc__eq__div__add3,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 9.20/5.19        = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5822_Suc__div__eq__add3__div__numeral,axiom,
% 9.20/5.19      ! [M: nat,V: num] :
% 9.20/5.19        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.20/5.19        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5823_mod__Suc__eq__mod__add3,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
% 9.20/5.19        = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5824_Suc__mod__eq__add3__mod__numeral,axiom,
% 9.20/5.19      ! [M: nat,V: num] :
% 9.20/5.19        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
% 9.20/5.19        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5825_and__numerals_I7_J,axiom,
% 9.20/5.19      ! [X3: num,Y3: num] :
% 9.20/5.19        ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 9.20/5.19        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5826_and__numerals_I7_J,axiom,
% 9.20/5.19      ! [X3: num,Y3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.20/5.19        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5827_DiffD2,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5828_DiffD2,axiom,
% 9.20/5.19      ! [C: real,A2: set_real,B2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( member_real @ C @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5829_DiffD2,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( member_set_nat @ C @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5830_DiffD2,axiom,
% 9.20/5.19      ! [C: int,A2: set_int,B2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( member_int @ C @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5831_DiffD2,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( member_nat @ C @ B2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5832_DiffD1,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.19       => ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5833_DiffD1,axiom,
% 9.20/5.19      ! [C: real,A2: set_real,B2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.20/5.19       => ( member_real @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5834_DiffD1,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.20/5.19       => ( member_set_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5835_DiffD1,axiom,
% 9.20/5.19      ! [C: int,A2: set_int,B2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.19       => ( member_int @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5836_DiffD1,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.19       => ( member_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5837_DiffE,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
% 9.20/5.19           => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5838_DiffE,axiom,
% 9.20/5.19      ! [C: real,A2: set_real,B2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( ( member_real @ C @ A2 )
% 9.20/5.19           => ( member_real @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5839_DiffE,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( ( member_set_nat @ C @ A2 )
% 9.20/5.19           => ( member_set_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5840_DiffE,axiom,
% 9.20/5.19      ! [C: int,A2: set_int,B2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( ( member_int @ C @ A2 )
% 9.20/5.19           => ( member_int @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5841_DiffE,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat,B2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.19       => ~ ( ( member_nat @ C @ A2 )
% 9.20/5.19           => ( member_nat @ C @ B2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5842_ComplD,axiom,
% 9.20/5.19      ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.19        ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
% 9.20/5.19       => ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5843_ComplD,axiom,
% 9.20/5.19      ! [C: real,A2: set_real] :
% 9.20/5.19        ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
% 9.20/5.19       => ~ ( member_real @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5844_ComplD,axiom,
% 9.20/5.19      ! [C: set_nat,A2: set_set_nat] :
% 9.20/5.19        ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
% 9.20/5.19       => ~ ( member_set_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5845_ComplD,axiom,
% 9.20/5.19      ! [C: nat,A2: set_nat] :
% 9.20/5.19        ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
% 9.20/5.19       => ~ ( member_nat @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5846_ComplD,axiom,
% 9.20/5.19      ! [C: int,A2: set_int] :
% 9.20/5.19        ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
% 9.20/5.19       => ~ ( member_int @ C @ A2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5847_concat__bit__assoc,axiom,
% 9.20/5.19      ! [N2: nat,K: int,M: nat,L: int,R2: int] :
% 9.20/5.19        ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
% 9.20/5.19        = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5848_xor__num_Ocases,axiom,
% 9.20/5.19      ! [X3: product_prod_num_num] :
% 9.20/5.19        ( ( X3
% 9.20/5.19         != ( product_Pair_num_num @ one @ one ) )
% 9.20/5.19       => ( ! [N3: num] :
% 9.20/5.19              ( X3
% 9.20/5.19             != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
% 9.20/5.19         => ( ! [N3: num] :
% 9.20/5.19                ( X3
% 9.20/5.19               != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
% 9.20/5.19           => ( ! [M3: num] :
% 9.20/5.19                  ( X3
% 9.20/5.19                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
% 9.20/5.19             => ( ! [M3: num,N3: num] :
% 9.20/5.19                    ( X3
% 9.20/5.19                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
% 9.20/5.19               => ( ! [M3: num,N3: num] :
% 9.20/5.19                      ( X3
% 9.20/5.19                     != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
% 9.20/5.19                 => ( ! [M3: num] :
% 9.20/5.19                        ( X3
% 9.20/5.19                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
% 9.20/5.19                   => ( ! [M3: num,N3: num] :
% 9.20/5.19                          ( X3
% 9.20/5.19                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
% 9.20/5.19                     => ~ ! [M3: num,N3: num] :
% 9.20/5.19                            ( X3
% 9.20/5.19                           != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5849_and__less__eq,axiom,
% 9.20/5.19      ! [L: int,K: int] :
% 9.20/5.19        ( ( ord_less_int @ L @ zero_zero_int )
% 9.20/5.19       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5850_AND__upper1_H_H,axiom,
% 9.20/5.19      ! [Y3: int,Z2: int,Ya: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.19       => ( ( ord_less_int @ Y3 @ Z2 )
% 9.20/5.19         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5851_AND__upper2_H_H,axiom,
% 9.20/5.19      ! [Y3: int,Z2: int,X3: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.19       => ( ( ord_less_int @ Y3 @ Z2 )
% 9.20/5.19         => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5852_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5853_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5854_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5855_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5856_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5857_numeral__Bit1,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5858_simp__from__to,axiom,
% 9.20/5.19      ( set_or1266510415728281911st_int
% 9.20/5.19      = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5859_eval__nat__numeral_I3_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( suc @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5860_cong__exp__iff__simps_I3_J,axiom,
% 9.20/5.19      ! [N2: num,Q4: num] :
% 9.20/5.19        ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.20/5.19       != zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5861_cong__exp__iff__simps_I3_J,axiom,
% 9.20/5.19      ! [N2: num,Q4: num] :
% 9.20/5.19        ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.20/5.19       != zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5862_cong__exp__iff__simps_I3_J,axiom,
% 9.20/5.19      ! [N2: num,Q4: num] :
% 9.20/5.19        ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.20/5.19       != zero_z3403309356797280102nteger ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5863_numeral__3__eq__3,axiom,
% 9.20/5.19      ( ( numeral_numeral_nat @ ( bit1 @ one ) )
% 9.20/5.19      = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5864_Suc3__eq__add__3,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( suc @ ( suc @ ( suc @ N2 ) ) )
% 9.20/5.19        = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5865_num_Osize_I6_J,axiom,
% 9.20/5.19      ! [X32: num] :
% 9.20/5.19        ( ( size_size_num @ ( bit1 @ X32 ) )
% 9.20/5.19        = ( plus_plus_nat @ ( size_size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5866_num_Osize__gen_I3_J,axiom,
% 9.20/5.19      ! [X32: num] :
% 9.20/5.19        ( ( size_num @ ( bit1 @ X32 ) )
% 9.20/5.19        = ( plus_plus_nat @ ( size_num @ X32 ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5867_cong__exp__iff__simps_I11_J,axiom,
% 9.20/5.19      ! [M: num,Q4: num] :
% 9.20/5.19        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.20/5.19          = zero_zero_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5868_cong__exp__iff__simps_I11_J,axiom,
% 9.20/5.19      ! [M: num,Q4: num] :
% 9.20/5.19        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
% 9.20/5.19          = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5869_cong__exp__iff__simps_I11_J,axiom,
% 9.20/5.19      ! [M: num,Q4: num] :
% 9.20/5.19        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.20/5.19          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5870_cong__exp__iff__simps_I7_J,axiom,
% 9.20/5.19      ! [Q4: num,N2: num] :
% 9.20/5.19        ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
% 9.20/5.19          = zero_zero_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5871_cong__exp__iff__simps_I7_J,axiom,
% 9.20/5.19      ! [Q4: num,N2: num] :
% 9.20/5.19        ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
% 9.20/5.19          = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5872_cong__exp__iff__simps_I7_J,axiom,
% 9.20/5.19      ! [Q4: num,N2: num] :
% 9.20/5.19        ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
% 9.20/5.19          = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
% 9.20/5.19        = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
% 9.20/5.19          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5873_periodic__finite__ex,axiom,
% 9.20/5.19      ! [D: int,P: int > $o] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.19       => ( ! [X4: int,K2: int] :
% 9.20/5.19              ( ( P @ X4 )
% 9.20/5.19              = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
% 9.20/5.19         => ( ( ? [X5: int] : ( P @ X5 ) )
% 9.20/5.19            = ( ? [X: int] :
% 9.20/5.19                  ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
% 9.20/5.19                  & ( P @ X ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5874_bset_I3_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( X2 = T )
% 9.20/5.19               => ( ( minus_minus_int @ X2 @ D6 )
% 9.20/5.19                  = T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5875_bset_I4_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ T @ B2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( X2 != T )
% 9.20/5.19               => ( ( minus_minus_int @ X2 @ D6 )
% 9.20/5.19                 != T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5876_bset_I5_J,axiom,
% 9.20/5.19      ! [D6: int,B2: set_int,T: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ! [X2: int] :
% 9.20/5.19            ( ! [Xa3: int] :
% 9.20/5.19                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19               => ! [Xb2: int] :
% 9.20/5.19                    ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                   => ( X2
% 9.20/5.19                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19           => ( ( ord_less_int @ X2 @ T )
% 9.20/5.19             => ( ord_less_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5877_bset_I7_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ T @ B2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( ord_less_int @ T @ X2 )
% 9.20/5.19               => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5878_aset_I3_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,A2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( X2 = T )
% 9.20/5.19               => ( ( plus_plus_int @ X2 @ D6 )
% 9.20/5.19                  = T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5879_aset_I4_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,A2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ T @ A2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( X2 != T )
% 9.20/5.19               => ( ( plus_plus_int @ X2 @ D6 )
% 9.20/5.19                 != T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5880_aset_I5_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,A2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ T @ A2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( ord_less_int @ X2 @ T )
% 9.20/5.19               => ( ord_less_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5881_aset_I7_J,axiom,
% 9.20/5.19      ! [D6: int,A2: set_int,T: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ! [X2: int] :
% 9.20/5.19            ( ! [Xa3: int] :
% 9.20/5.19                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19               => ! [Xb2: int] :
% 9.20/5.19                    ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                   => ( X2
% 9.20/5.19                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19           => ( ( ord_less_int @ T @ X2 )
% 9.20/5.19             => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5882_Suc__div__eq__add3__div,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 9.20/5.19        = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5883_Suc__mod__eq__add3__mod,axiom,
% 9.20/5.19      ! [M: nat,N2: nat] :
% 9.20/5.19        ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
% 9.20/5.19        = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5884_aset_I8_J,axiom,
% 9.20/5.19      ! [D6: int,A2: set_int,T: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ! [X2: int] :
% 9.20/5.19            ( ! [Xa3: int] :
% 9.20/5.19                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19               => ! [Xb2: int] :
% 9.20/5.19                    ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                   => ( X2
% 9.20/5.19                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19           => ( ( ord_less_eq_int @ T @ X2 )
% 9.20/5.19             => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5885_aset_I6_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,A2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ A2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( ord_less_eq_int @ X2 @ T )
% 9.20/5.19               => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5886_bset_I8_J,axiom,
% 9.20/5.19      ! [D6: int,T: int,B2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
% 9.20/5.19         => ! [X2: int] :
% 9.20/5.19              ( ! [Xa3: int] :
% 9.20/5.19                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                 => ! [Xb2: int] :
% 9.20/5.19                      ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                     => ( X2
% 9.20/5.19                       != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19             => ( ( ord_less_eq_int @ T @ X2 )
% 9.20/5.19               => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5887_bset_I6_J,axiom,
% 9.20/5.19      ! [D6: int,B2: set_int,T: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ! [X2: int] :
% 9.20/5.19            ( ! [Xa3: int] :
% 9.20/5.19                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19               => ! [Xb2: int] :
% 9.20/5.19                    ( ( member_int @ Xb2 @ B2 )
% 9.20/5.19                   => ( X2
% 9.20/5.19                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
% 9.20/5.19           => ( ( ord_less_eq_int @ X2 @ T )
% 9.20/5.19             => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5888_cppi,axiom,
% 9.20/5.19      ! [D6: int,P: int > $o,P5: int > $o,A2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ? [Z4: int] :
% 9.20/5.19            ! [X4: int] :
% 9.20/5.19              ( ( ord_less_int @ Z4 @ X4 )
% 9.20/5.19             => ( ( P @ X4 )
% 9.20/5.19                = ( P5 @ X4 ) ) )
% 9.20/5.19         => ( ! [X4: int] :
% 9.20/5.19                ( ! [Xa: int] :
% 9.20/5.19                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                   => ! [Xb3: int] :
% 9.20/5.19                        ( ( member_int @ Xb3 @ A2 )
% 9.20/5.19                       => ( X4
% 9.20/5.19                         != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
% 9.20/5.19               => ( ( P @ X4 )
% 9.20/5.19                 => ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
% 9.20/5.19           => ( ! [X4: int,K2: int] :
% 9.20/5.19                  ( ( P5 @ X4 )
% 9.20/5.19                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 9.20/5.19             => ( ( ? [X5: int] : ( P @ X5 ) )
% 9.20/5.19                = ( ? [X: int] :
% 9.20/5.19                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                      & ( P5 @ X ) )
% 9.20/5.19                  | ? [X: int] :
% 9.20/5.19                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                      & ? [Y: int] :
% 9.20/5.19                          ( ( member_int @ Y @ A2 )
% 9.20/5.19                          & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5889_cpmi,axiom,
% 9.20/5.19      ! [D6: int,P: int > $o,P5: int > $o,B2: set_int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ D6 )
% 9.20/5.19       => ( ? [Z4: int] :
% 9.20/5.19            ! [X4: int] :
% 9.20/5.19              ( ( ord_less_int @ X4 @ Z4 )
% 9.20/5.19             => ( ( P @ X4 )
% 9.20/5.19                = ( P5 @ X4 ) ) )
% 9.20/5.19         => ( ! [X4: int] :
% 9.20/5.19                ( ! [Xa: int] :
% 9.20/5.19                    ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                   => ! [Xb3: int] :
% 9.20/5.19                        ( ( member_int @ Xb3 @ B2 )
% 9.20/5.19                       => ( X4
% 9.20/5.19                         != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
% 9.20/5.19               => ( ( P @ X4 )
% 9.20/5.19                 => ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
% 9.20/5.19           => ( ! [X4: int,K2: int] :
% 9.20/5.19                  ( ( P5 @ X4 )
% 9.20/5.19                  = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
% 9.20/5.19             => ( ( ? [X5: int] : ( P @ X5 ) )
% 9.20/5.19                = ( ? [X: int] :
% 9.20/5.19                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                      & ( P5 @ X ) )
% 9.20/5.19                  | ? [X: int] :
% 9.20/5.19                      ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
% 9.20/5.19                      & ? [Y: int] :
% 9.20/5.19                          ( ( member_int @ Y @ B2 )
% 9.20/5.19                          & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5890_take__bit__Suc__bit1,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.20/5.19        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ N2 @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5891_take__bit__Suc__bit1,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5892_odd__mod__4__div__2,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.19          = ( numeral_numeral_nat @ ( bit1 @ one ) ) )
% 9.20/5.19       => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5893_mod__exhaust__less__4,axiom,
% 9.20/5.19      ! [M: nat] :
% 9.20/5.19        ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.19          = zero_zero_nat )
% 9.20/5.19        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.19          = one_one_nat )
% 9.20/5.19        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.19          = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.19        | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.19          = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5894_divmod__algorithm__code_I7_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5895_divmod__algorithm__code_I7_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5896_divmod__algorithm__code_I7_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_eq_num @ M @ N2 )
% 9.20/5.19         => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5897_divmod__algorithm__code_I8_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5898_divmod__algorithm__code_I8_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5899_divmod__algorithm__code_I8_J,axiom,
% 9.20/5.19      ! [M: num,N2: num] :
% 9.20/5.19        ( ( ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
% 9.20/5.19        & ( ~ ( ord_less_num @ M @ N2 )
% 9.20/5.19         => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
% 9.20/5.19            = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5900_take__bit__Suc__minus__bit1,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5901_and__int_Opsimps,axiom,
% 9.20/5.19      ! [K: int,L: int] :
% 9.20/5.19        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K @ L ) )
% 9.20/5.19       => ( ( ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19              & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.20/5.19              = ( uminus_uminus_int
% 9.20/5.19                @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.20/5.19                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ) ) )
% 9.20/5.19          & ( ~ ( ( member_int @ K @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19                & ( member_int @ L @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19           => ( ( bit_se725231765392027082nd_int @ K @ L )
% 9.20/5.19              = ( plus_plus_int
% 9.20/5.19                @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                  @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
% 9.20/5.19                    & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) )
% 9.20/5.19                @ ( 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 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5902_and__int_Opelims,axiom,
% 9.20/5.19      ! [X3: int,Xa2: int,Y3: int] :
% 9.20/5.19        ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
% 9.20/5.19          = Y3 )
% 9.20/5.19       => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 9.20/5.19         => ~ ( ( ( ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19                    & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19                 => ( Y3
% 9.20/5.19                    = ( uminus_uminus_int
% 9.20/5.19                      @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.20/5.19                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
% 9.20/5.19                & ( ~ ( ( member_int @ X3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19                      & ( member_int @ Xa2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19                 => ( Y3
% 9.20/5.19                    = ( plus_plus_int
% 9.20/5.19                      @ ( zero_n2684676970156552555ol_int
% 9.20/5.19                        @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
% 9.20/5.19                          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
% 9.20/5.19                      @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
% 9.20/5.19             => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5903_finite__atLeastAtMost__int,axiom,
% 9.20/5.19      ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5904_and__nat__numerals_I1_J,axiom,
% 9.20/5.19      ! [Y3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5905_and__nat__numerals_I3_J,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.19        = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5906_and__nat__numerals_I2_J,axiom,
% 9.20/5.19      ! [Y3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.20/5.19        = one_one_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5907_and__nat__numerals_I4_J,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.19        = one_one_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5908_divmod__algorithm__code_I2_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( unique5052692396658037445od_int @ M @ one )
% 9.20/5.19        = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5909_divmod__algorithm__code_I2_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( unique5055182867167087721od_nat @ M @ one )
% 9.20/5.19        = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5910_divmod__algorithm__code_I2_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( unique3479559517661332726nteger @ M @ one )
% 9.20/5.19        = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5911_add__neg__numeral__special_I6_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
% 9.20/5.19        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5912_add__neg__numeral__special_I6_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
% 9.20/5.19        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5913_add__neg__numeral__special_I6_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.19        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5914_add__neg__numeral__special_I6_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.19        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5915_add__neg__numeral__special_I6_J,axiom,
% 9.20/5.19      ! [M: num] :
% 9.20/5.19        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
% 9.20/5.19        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5916_add__neg__numeral__special_I5_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
% 9.20/5.19        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5917_add__neg__numeral__special_I5_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5918_add__neg__numeral__special_I5_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( plus_plus_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5919_add__neg__numeral__special_I5_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
% 9.20/5.19        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5920_add__neg__numeral__special_I5_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
% 9.20/5.19        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5921_and__Suc__0__eq,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.19        = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5922_Suc__0__and__eq,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.19        = ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5923_divmod__algorithm__code_I3_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique5052692396658037445od_int @ one @ ( bit0 @ N2 ) )
% 9.20/5.19        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5924_divmod__algorithm__code_I3_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique5055182867167087721od_nat @ one @ ( bit0 @ N2 ) )
% 9.20/5.19        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5925_divmod__algorithm__code_I3_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
% 9.20/5.19        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5926_divmod__algorithm__code_I4_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique5052692396658037445od_int @ one @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5927_divmod__algorithm__code_I4_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique5055182867167087721od_nat @ one @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5928_divmod__algorithm__code_I4_J,axiom,
% 9.20/5.19      ! [N2: num] :
% 9.20/5.19        ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
% 9.20/5.19        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5929_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numera6690914467698888265omplex @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5930_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numera1916890842035813515d_enat @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5931_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numeral_numeral_real @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5932_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numeral_numeral_rat @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5933_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numeral_numeral_nat @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5934_numeral__inc,axiom,
% 9.20/5.19      ! [X3: num] :
% 9.20/5.19        ( ( numeral_numeral_int @ ( inc @ X3 ) )
% 9.20/5.19        = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5935_divmod__int__def,axiom,
% 9.20/5.19      ( unique5052692396658037445od_int
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5936_divmod__def,axiom,
% 9.20/5.19      ( unique5052692396658037445od_int
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5937_divmod__def,axiom,
% 9.20/5.19      ( unique5055182867167087721od_nat
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5938_divmod__def,axiom,
% 9.20/5.19      ( unique3479559517661332726nteger
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5939_divmod_H__nat__def,axiom,
% 9.20/5.19      ( unique5055182867167087721od_nat
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5940_and__nat__unfold,axiom,
% 9.20/5.19      ( bit_se727722235901077358nd_nat
% 9.20/5.19      = ( ^ [M2: nat,N: nat] :
% 9.20/5.19            ( if_nat
% 9.20/5.19            @ ( ( M2 = zero_zero_nat )
% 9.20/5.19              | ( N = zero_zero_nat ) )
% 9.20/5.19            @ zero_zero_nat
% 9.20/5.19            @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5941_and__nat__rec,axiom,
% 9.20/5.19      ( bit_se727722235901077358nd_nat
% 9.20/5.19      = ( ^ [M2: nat,N: nat] :
% 9.20/5.19            ( plus_plus_nat
% 9.20/5.19            @ ( zero_n2687167440665602831ol_nat
% 9.20/5.19              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
% 9.20/5.19                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.20/5.19            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5942_and__int_Opinduct,axiom,
% 9.20/5.19      ! [A0: int,A12: int,P: int > int > $o] :
% 9.20/5.19        ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.20/5.19       => ( ! [K2: int,L4: int] :
% 9.20/5.19              ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
% 9.20/5.19             => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
% 9.20/5.19                      & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
% 9.20/5.19                 => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.19               => ( P @ K2 @ L4 ) ) )
% 9.20/5.19         => ( P @ A0 @ A12 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5943_divmod__divmod__step,axiom,
% 9.20/5.19      ( unique5055182867167087721od_nat
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5944_divmod__divmod__step,axiom,
% 9.20/5.19      ( unique5052692396658037445od_int
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5945_divmod__divmod__step,axiom,
% 9.20/5.19      ( unique3479559517661332726nteger
% 9.20/5.19      = ( ^ [M2: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5946_upto_Opinduct,axiom,
% 9.20/5.19      ! [A0: int,A12: int,P: int > int > $o] :
% 9.20/5.19        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
% 9.20/5.19       => ( ! [I3: int,J2: int] :
% 9.20/5.19              ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
% 9.20/5.19             => ( ( ( ord_less_eq_int @ I3 @ J2 )
% 9.20/5.19                 => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
% 9.20/5.19               => ( P @ I3 @ J2 ) ) )
% 9.20/5.19         => ( P @ A0 @ A12 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5947_signed__take__bit__eq__take__bit__minus,axiom,
% 9.20/5.19      ( bit_ri631733984087533419it_int
% 9.20/5.19      = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5948_take__bit__numeral__bit1,axiom,
% 9.20/5.19      ! [L: num,K: num] :
% 9.20/5.19        ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
% 9.20/5.19        = ( plus_plus_nat @ ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5949_take__bit__numeral__bit1,axiom,
% 9.20/5.19      ! [L: num,K: num] :
% 9.20/5.19        ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
% 9.20/5.19        = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5950_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5951_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5952_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5953_of__int__less__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5954_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5955_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5956_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5957_neg__numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5958_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5959_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5960_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5961_of__int__le__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5962_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5963_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5964_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5965_neg__numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5966_of__int__eq__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ W2 )
% 9.20/5.19          = ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( W2 = Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5967_of__int__eq__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ W2 )
% 9.20/5.19          = ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( W2 = Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5968_bit__0__eq,axiom,
% 9.20/5.19      ( ( bit_se1146084159140164899it_int @ zero_zero_int )
% 9.20/5.19      = bot_bot_nat_o ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5969_bit__0__eq,axiom,
% 9.20/5.19      ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
% 9.20/5.19      = bot_bot_nat_o ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5970_of__int__0,axiom,
% 9.20/5.19      ( ( ring_17405671764205052669omplex @ zero_zero_int )
% 9.20/5.19      = zero_zero_complex ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5971_of__int__0,axiom,
% 9.20/5.19      ( ( ring_1_of_int_int @ zero_zero_int )
% 9.20/5.19      = zero_zero_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5972_of__int__0,axiom,
% 9.20/5.19      ( ( ring_1_of_int_real @ zero_zero_int )
% 9.20/5.19      = zero_zero_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5973_of__int__0,axiom,
% 9.20/5.19      ( ( ring_1_of_int_rat @ zero_zero_int )
% 9.20/5.19      = zero_zero_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5974_of__int__0__eq__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( zero_zero_complex
% 9.20/5.19          = ( ring_17405671764205052669omplex @ Z2 ) )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5975_of__int__0__eq__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( zero_zero_int
% 9.20/5.19          = ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5976_of__int__0__eq__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( zero_zero_real
% 9.20/5.19          = ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5977_of__int__0__eq__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( zero_zero_rat
% 9.20/5.19          = ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5978_of__int__eq__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 9.20/5.19          = zero_zero_complex )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5979_of__int__eq__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ Z2 )
% 9.20/5.19          = zero_zero_int )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5980_of__int__eq__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ Z2 )
% 9.20/5.19          = zero_zero_real )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5981_of__int__eq__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ Z2 )
% 9.20/5.19          = zero_zero_rat )
% 9.20/5.19        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5982_of__int__le__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5983_of__int__le__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5984_of__int__le__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5985_of__int__eq__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ Z2 )
% 9.20/5.19          = ( numeral_numeral_real @ N2 ) )
% 9.20/5.19        = ( Z2
% 9.20/5.19          = ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5986_of__int__eq__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ Z2 )
% 9.20/5.19          = ( numeral_numeral_rat @ N2 ) )
% 9.20/5.19        = ( Z2
% 9.20/5.19          = ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5987_of__int__eq__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ Z2 )
% 9.20/5.19          = ( numeral_numeral_int @ N2 ) )
% 9.20/5.19        = ( Z2
% 9.20/5.19          = ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5988_of__int__numeral,axiom,
% 9.20/5.19      ! [K: num] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
% 9.20/5.19        = ( numeral_numeral_real @ K ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5989_of__int__numeral,axiom,
% 9.20/5.19      ! [K: num] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
% 9.20/5.19        = ( numeral_numeral_rat @ K ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5990_of__int__numeral,axiom,
% 9.20/5.19      ! [K: num] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
% 9.20/5.19        = ( numeral_numeral_int @ K ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5991_of__int__less__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5992_of__int__less__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5993_of__int__less__iff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ W2 @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5994_of__int__eq__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_17405671764205052669omplex @ Z2 )
% 9.20/5.19          = one_one_complex )
% 9.20/5.19        = ( Z2 = one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5995_of__int__eq__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ Z2 )
% 9.20/5.19          = one_one_int )
% 9.20/5.19        = ( Z2 = one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5996_of__int__eq__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ Z2 )
% 9.20/5.19          = one_one_real )
% 9.20/5.19        = ( Z2 = one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5997_of__int__eq__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ Z2 )
% 9.20/5.19          = one_one_rat )
% 9.20/5.19        = ( Z2 = one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5998_of__int__1,axiom,
% 9.20/5.19      ( ( ring_17405671764205052669omplex @ one_one_int )
% 9.20/5.19      = one_one_complex ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_5999_of__int__1,axiom,
% 9.20/5.19      ( ( ring_1_of_int_int @ one_one_int )
% 9.20/5.19      = one_one_int ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6000_of__int__1,axiom,
% 9.20/5.19      ( ( ring_1_of_int_real @ one_one_int )
% 9.20/5.19      = one_one_real ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6001_of__int__1,axiom,
% 9.20/5.19      ( ( ring_1_of_int_rat @ one_one_int )
% 9.20/5.19      = one_one_rat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6002_of__int__mult,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6003_of__int__mult,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6004_of__int__mult,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6005_of__int__mult,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6006_of__int__add,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6007_of__int__add,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6008_of__int__add,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6009_of__int__minus,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
% 9.20/5.19        = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6010_of__int__minus,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
% 9.20/5.19        = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6011_of__int__minus,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
% 9.20/5.19        = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6012_of__int__minus,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
% 9.20/5.19        = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6013_of__int__diff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6014_of__int__diff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6015_of__int__diff,axiom,
% 9.20/5.19      ! [W2: int,Z2: int] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
% 9.20/5.19        = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6016_pred__numeral__simps_I1_J,axiom,
% 9.20/5.19      ( ( pred_numeral @ one )
% 9.20/5.19      = zero_zero_nat ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6017_eq__numeral__Suc,axiom,
% 9.20/5.19      ! [K: num,N2: nat] :
% 9.20/5.19        ( ( ( numeral_numeral_nat @ K )
% 9.20/5.19          = ( suc @ N2 ) )
% 9.20/5.19        = ( ( pred_numeral @ K )
% 9.20/5.19          = N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6018_Suc__eq__numeral,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( ( suc @ N2 )
% 9.20/5.19          = ( numeral_numeral_nat @ K ) )
% 9.20/5.19        = ( N2
% 9.20/5.19          = ( pred_numeral @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6019_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ X3 )
% 9.20/5.19          = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 9.20/5.19        = ( X3
% 9.20/5.19          = ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6020_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ X3 )
% 9.20/5.19          = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 9.20/5.19        = ( X3
% 9.20/5.19          = ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6021_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ X3 )
% 9.20/5.19          = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 9.20/5.19        = ( X3
% 9.20/5.19          = ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6022_of__int__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ( ring_17405671764205052669omplex @ X3 )
% 9.20/5.19          = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
% 9.20/5.19        = ( X3
% 9.20/5.19          = ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6023_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
% 9.20/5.19          = ( ring_1_of_int_rat @ X3 ) )
% 9.20/5.19        = ( ( power_power_int @ B @ W2 )
% 9.20/5.19          = X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6024_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
% 9.20/5.19          = ( ring_1_of_int_real @ X3 ) )
% 9.20/5.19        = ( ( power_power_int @ B @ W2 )
% 9.20/5.19          = X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6025_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
% 9.20/5.19          = ( ring_1_of_int_int @ X3 ) )
% 9.20/5.19        = ( ( power_power_int @ B @ W2 )
% 9.20/5.19          = X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6026_of__int__eq__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
% 9.20/5.19          = ( ring_17405671764205052669omplex @ X3 ) )
% 9.20/5.19        = ( ( power_power_int @ B @ W2 )
% 9.20/5.19          = X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6027_of__int__power,axiom,
% 9.20/5.19      ! [Z2: int,N2: nat] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N2 ) )
% 9.20/5.19        = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6028_of__int__power,axiom,
% 9.20/5.19      ! [Z2: int,N2: nat] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N2 ) )
% 9.20/5.19        = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6029_of__int__power,axiom,
% 9.20/5.19      ! [Z2: int,N2: nat] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N2 ) )
% 9.20/5.19        = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6030_of__int__power,axiom,
% 9.20/5.19      ! [Z2: int,N2: nat] :
% 9.20/5.19        ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N2 ) )
% 9.20/5.19        = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6031_of__int__of__bool,axiom,
% 9.20/5.19      ! [P: $o] :
% 9.20/5.19        ( ( ring_1_of_int_real @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.19        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6032_of__int__of__bool,axiom,
% 9.20/5.19      ! [P: $o] :
% 9.20/5.19        ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.19        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6033_of__int__of__bool,axiom,
% 9.20/5.19      ! [P: $o] :
% 9.20/5.19        ( ( ring_1_of_int_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.19        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6034_bit__numeral__Bit0__Suc__iff,axiom,
% 9.20/5.19      ! [M: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6035_bit__numeral__Bit0__Suc__iff,axiom,
% 9.20/5.19      ! [M: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6036_bit__numeral__Bit1__Suc__iff,axiom,
% 9.20/5.19      ! [M: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6037_bit__numeral__Bit1__Suc__iff,axiom,
% 9.20/5.19      ! [M: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6038_less__numeral__Suc,axiom,
% 9.20/5.19      ! [K: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( ord_less_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6039_less__Suc__numeral,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( ord_less_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.19        = ( ord_less_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6040_le__numeral__Suc,axiom,
% 9.20/5.19      ! [K: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6041_le__Suc__numeral,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.19        = ( ord_less_eq_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6042_diff__Suc__numeral,axiom,
% 9.20/5.19      ! [N2: nat,K: num] :
% 9.20/5.19        ( ( minus_minus_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.19        = ( minus_minus_nat @ N2 @ ( pred_numeral @ K ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6043_diff__numeral__Suc,axiom,
% 9.20/5.19      ! [K: num,N2: nat] :
% 9.20/5.19        ( ( minus_minus_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( minus_minus_nat @ ( pred_numeral @ K ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6044_signed__take__bit__negative__iff,axiom,
% 9.20/5.19      ! [N2: nat,K: int] :
% 9.20/5.19        ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.19        = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6045_of__int__0__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6046_of__int__0__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6047_of__int__0__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6048_of__int__le__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6049_of__int__le__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6050_of__int__le__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6051_of__int__less__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
% 9.20/5.19        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6052_of__int__less__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
% 9.20/5.19        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6053_of__int__less__0__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
% 9.20/5.19        = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6054_of__int__0__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6055_of__int__0__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6056_of__int__0__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6057_of__int__le__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6058_of__int__le__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6059_of__int__le__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6060_of__int__numeral__le__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6061_of__int__numeral__le__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6062_of__int__numeral__le__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6063_of__int__less__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6064_of__int__less__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6065_of__int__less__numeral__iff,axiom,
% 9.20/5.19      ! [Z2: int,N2: num] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6066_of__int__numeral__less__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6067_of__int__numeral__less__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6068_of__int__numeral__less__iff,axiom,
% 9.20/5.19      ! [N2: num,Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6069_of__int__le__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6070_of__int__le__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6071_of__int__le__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 9.20/5.19        = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6072_of__int__1__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6073_of__int__1__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6074_of__int__1__le__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6075_of__int__less__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
% 9.20/5.19        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6076_of__int__less__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
% 9.20/5.19        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6077_of__int__less__1__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
% 9.20/5.19        = ( ord_less_int @ Z2 @ one_one_int ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6078_of__int__1__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6079_of__int__1__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6080_of__int__1__less__iff,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
% 9.20/5.19        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6081_of__int__le__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6082_of__int__le__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6083_of__int__le__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6084_of__int__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6085_of__int__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6086_of__int__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6087_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_17405671764205052669omplex @ Y3 )
% 9.20/5.19          = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6088_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ Y3 )
% 9.20/5.19          = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6089_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ Y3 )
% 9.20/5.19          = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6090_of__int__eq__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ Y3 )
% 9.20/5.19          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6091_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 )
% 9.20/5.19          = ( ring_17405671764205052669omplex @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6092_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_real @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6093_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_rat @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6094_numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_int @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6095_of__int__less__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6096_of__int__less__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6097_of__int__less__of__int__power__cancel__iff,axiom,
% 9.20/5.19      ! [B: int,W2: nat,X3: int] :
% 9.20/5.19        ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6098_of__int__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6099_of__int__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6100_of__int__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: int,B: int,W2: nat] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
% 9.20/5.19        = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6101_bit__minus__numeral__Bit0__Suc__iff,axiom,
% 9.20/5.19      ! [W2: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W2 ) ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6102_bit__minus__numeral__Bit1__Suc__iff,axiom,
% 9.20/5.19      ! [W2: num,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W2 ) ) ) @ ( suc @ N2 ) )
% 9.20/5.19        = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W2 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6103_bit__0,axiom,
% 9.20/5.19      ! [A: code_integer] :
% 9.20/5.19        ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
% 9.20/5.19        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6104_bit__0,axiom,
% 9.20/5.19      ! [A: int] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ A @ zero_zero_nat )
% 9.20/5.19        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6105_bit__0,axiom,
% 9.20/5.19      ! [A: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ A @ zero_zero_nat )
% 9.20/5.19        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6106_bit__mod__2__iff,axiom,
% 9.20/5.19      ! [A: code_integer,N2: nat] :
% 9.20/5.19        ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 )
% 9.20/5.19        = ( ( N2 = zero_zero_nat )
% 9.20/5.19          & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6107_bit__mod__2__iff,axiom,
% 9.20/5.19      ! [A: int,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 )
% 9.20/5.19        = ( ( N2 = zero_zero_nat )
% 9.20/5.19          & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6108_bit__mod__2__iff,axiom,
% 9.20/5.19      ! [A: nat,N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
% 9.20/5.19        = ( ( N2 = zero_zero_nat )
% 9.20/5.19          & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6109_numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6110_numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6111_numeral__power__le__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 9.20/5.19        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6112_of__int__le__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6113_of__int__le__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6114_of__int__le__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6115_numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6116_numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6117_numeral__power__less__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,A: int] :
% 9.20/5.19        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
% 9.20/5.19        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6118_of__int__less__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6119_of__int__less__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6120_of__int__less__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [A: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
% 9.20/5.19        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6121_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_17405671764205052669omplex @ Y3 )
% 9.20/5.19          = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6122_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_int @ Y3 )
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6123_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_real @ Y3 )
% 9.20/5.19          = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6124_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_1_of_int_rat @ Y3 )
% 9.20/5.19          = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6125_of__int__eq__neg__numeral__power__cancel__iff,axiom,
% 9.20/5.19      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.19        ( ( ( ring_18347121197199848620nteger @ Y3 )
% 9.20/5.19          = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
% 9.20/5.19        = ( Y3
% 9.20/5.19          = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6126_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 )
% 9.20/5.19          = ( ring_17405671764205052669omplex @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6127_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_int @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6128_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_real @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6129_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 )
% 9.20/5.19          = ( ring_1_of_int_rat @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6130_neg__numeral__power__eq__of__int__cancel__iff,axiom,
% 9.20/5.19      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.19        ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 )
% 9.20/5.19          = ( ring_18347121197199848620nteger @ Y3 ) )
% 9.20/5.19        = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
% 9.20/5.19          = Y3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6131_bit__disjunctive__add__iff,axiom,
% 9.20/5.19      ! [A: int,B: int,N2: nat] :
% 9.20/5.19        ( ! [N3: nat] :
% 9.20/5.19            ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
% 9.20/5.19            | ~ ( bit_se1146084159140164899it_int @ B @ N3 ) )
% 9.20/5.19       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
% 9.20/5.19          = ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 9.20/5.19            | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6132_bit__disjunctive__add__iff,axiom,
% 9.20/5.19      ! [A: nat,B: nat,N2: nat] :
% 9.20/5.19        ( ! [N3: nat] :
% 9.20/5.19            ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
% 9.20/5.19            | ~ ( bit_se1148574629649215175it_nat @ B @ N3 ) )
% 9.20/5.19       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 9.20/5.19          = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 9.20/5.19            | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6133_ex__le__of__int,axiom,
% 9.20/5.19      ! [X3: real] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6134_ex__le__of__int,axiom,
% 9.20/5.19      ! [X3: rat] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6135_ex__less__of__int,axiom,
% 9.20/5.19      ! [X3: real] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6136_ex__less__of__int,axiom,
% 9.20/5.19      ! [X3: rat] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6137_ex__of__int__less,axiom,
% 9.20/5.19      ! [X3: real] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X3 ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6138_ex__of__int__less,axiom,
% 9.20/5.19      ! [X3: rat] :
% 9.20/5.19      ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6139_mult__of__int__commute,axiom,
% 9.20/5.19      ! [X3: int,Y3: complex] :
% 9.20/5.19        ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X3 ) @ Y3 )
% 9.20/5.19        = ( times_times_complex @ Y3 @ ( ring_17405671764205052669omplex @ X3 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6140_mult__of__int__commute,axiom,
% 9.20/5.19      ! [X3: int,Y3: real] :
% 9.20/5.19        ( ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ Y3 )
% 9.20/5.19        = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6141_mult__of__int__commute,axiom,
% 9.20/5.19      ! [X3: int,Y3: rat] :
% 9.20/5.19        ( ( times_times_rat @ ( ring_1_of_int_rat @ X3 ) @ Y3 )
% 9.20/5.19        = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6142_mult__of__int__commute,axiom,
% 9.20/5.19      ! [X3: int,Y3: int] :
% 9.20/5.19        ( ( times_times_int @ ( ring_1_of_int_int @ X3 ) @ Y3 )
% 9.20/5.19        = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X3 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6143_not__bit__1__Suc,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6144_not__bit__1__Suc,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6145_bit__1__iff,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
% 9.20/5.19        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6146_bit__1__iff,axiom,
% 9.20/5.19      ! [N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
% 9.20/5.19        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6147_bit__take__bit__iff,axiom,
% 9.20/5.19      ! [M: nat,A: nat,N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
% 9.20/5.19        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.19          & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6148_bit__take__bit__iff,axiom,
% 9.20/5.19      ! [M: nat,A: int,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
% 9.20/5.19        = ( ( ord_less_nat @ N2 @ M )
% 9.20/5.19          & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6149_bit__of__bool__iff,axiom,
% 9.20/5.19      ! [B: $o,N2: nat] :
% 9.20/5.19        ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N2 )
% 9.20/5.19        = ( B
% 9.20/5.19          & ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6150_bit__of__bool__iff,axiom,
% 9.20/5.19      ! [B: $o,N2: nat] :
% 9.20/5.19        ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N2 )
% 9.20/5.19        = ( B
% 9.20/5.19          & ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6151_numeral__eq__Suc,axiom,
% 9.20/5.19      ( numeral_numeral_nat
% 9.20/5.19      = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6152_of__int__nonneg,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.19       => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6153_of__int__nonneg,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.19       => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6154_of__int__nonneg,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.19       => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.19  
% 9.20/5.19  thf(fact_6155_of__int__pos,axiom,
% 9.20/5.19      ! [Z2: int] :
% 9.20/5.19        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.19       => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6156_of__int__pos,axiom,
% 9.20/5.20      ! [Z2: int] :
% 9.20/5.20        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.20       => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6157_of__int__pos,axiom,
% 9.20/5.20      ! [Z2: int] :
% 9.20/5.20        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.20       => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6158_floor__exists,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20      ? [Z3: int] :
% 9.20/5.20        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
% 9.20/5.20        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6159_floor__exists,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20      ? [Z3: int] :
% 9.20/5.20        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
% 9.20/5.20        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6160_floor__exists1,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20      ? [X4: int] :
% 9.20/5.20        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X3 )
% 9.20/5.20        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 9.20/5.20        & ! [Y6: int] :
% 9.20/5.20            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X3 )
% 9.20/5.20              & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
% 9.20/5.20           => ( Y6 = X4 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6161_floor__exists1,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20      ? [X4: int] :
% 9.20/5.20        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X3 )
% 9.20/5.20        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
% 9.20/5.20        & ! [Y6: int] :
% 9.20/5.20            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X3 )
% 9.20/5.20              & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
% 9.20/5.20           => ( Y6 = X4 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6162_of__int__neg__numeral,axiom,
% 9.20/5.20      ! [K: num] :
% 9.20/5.20        ( ( ring_1_of_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.20        = ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6163_of__int__neg__numeral,axiom,
% 9.20/5.20      ! [K: num] :
% 9.20/5.20        ( ( ring_1_of_int_real @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.20        = ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6164_of__int__neg__numeral,axiom,
% 9.20/5.20      ! [K: num] :
% 9.20/5.20        ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.20        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6165_of__int__neg__numeral,axiom,
% 9.20/5.20      ! [K: num] :
% 9.20/5.20        ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.20        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6166_bit__imp__take__bit__positive,axiom,
% 9.20/5.20      ! [N2: nat,M: nat,K: int] :
% 9.20/5.20        ( ( ord_less_nat @ N2 @ M )
% 9.20/5.20       => ( ( bit_se1146084159140164899it_int @ K @ N2 )
% 9.20/5.20         => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6167_int__le__real__less,axiom,
% 9.20/5.20      ( ord_less_eq_int
% 9.20/5.20      = ( ^ [N: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6168_int__less__real__le,axiom,
% 9.20/5.20      ( ord_less_int
% 9.20/5.20      = ( ^ [N: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6169_bit__concat__bit__iff,axiom,
% 9.20/5.20      ! [M: nat,K: int,L: int,N2: nat] :
% 9.20/5.20        ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
% 9.20/5.20        = ( ( ( ord_less_nat @ N2 @ M )
% 9.20/5.20            & ( bit_se1146084159140164899it_int @ K @ N2 ) )
% 9.20/5.20          | ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.20            & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6170_exp__eq__0__imp__not__bit,axiom,
% 9.20/5.20      ! [N2: nat,A: int] :
% 9.20/5.20        ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.20          = zero_zero_int )
% 9.20/5.20       => ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6171_exp__eq__0__imp__not__bit,axiom,
% 9.20/5.20      ! [N2: nat,A: nat] :
% 9.20/5.20        ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.20          = zero_zero_nat )
% 9.20/5.20       => ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6172_bit__Suc,axiom,
% 9.20/5.20      ! [A: code_integer,N2: nat] :
% 9.20/5.20        ( ( bit_se9216721137139052372nteger @ A @ ( suc @ N2 ) )
% 9.20/5.20        = ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6173_bit__Suc,axiom,
% 9.20/5.20      ! [A: int,N2: nat] :
% 9.20/5.20        ( ( bit_se1146084159140164899it_int @ A @ ( suc @ N2 ) )
% 9.20/5.20        = ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6174_bit__Suc,axiom,
% 9.20/5.20      ! [A: nat,N2: nat] :
% 9.20/5.20        ( ( bit_se1148574629649215175it_nat @ A @ ( suc @ N2 ) )
% 9.20/5.20        = ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6175_int__bit__bound,axiom,
% 9.20/5.20      ! [K: int] :
% 9.20/5.20        ~ ! [N3: nat] :
% 9.20/5.20            ( ! [M6: nat] :
% 9.20/5.20                ( ( ord_less_eq_nat @ N3 @ M6 )
% 9.20/5.20               => ( ( bit_se1146084159140164899it_int @ K @ M6 )
% 9.20/5.20                  = ( bit_se1146084159140164899it_int @ K @ N3 ) ) )
% 9.20/5.20           => ~ ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.20               => ( ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N3 @ one_one_nat ) )
% 9.20/5.20                  = ( ~ ( bit_se1146084159140164899it_int @ K @ N3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6176_and__exp__eq__0__iff__not__bit,axiom,
% 9.20/5.20      ! [A: int,N2: nat] :
% 9.20/5.20        ( ( ( bit_se725231765392027082nd_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.20          = zero_zero_int )
% 9.20/5.20        = ( ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6177_and__exp__eq__0__iff__not__bit,axiom,
% 9.20/5.20      ! [A: nat,N2: nat] :
% 9.20/5.20        ( ( ( bit_se727722235901077358nd_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.20          = zero_zero_nat )
% 9.20/5.20        = ( ~ ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6178_even__bit__succ__iff,axiom,
% 9.20/5.20      ! [A: code_integer,N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
% 9.20/5.20          = ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6179_even__bit__succ__iff,axiom,
% 9.20/5.20      ! [A: int,N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ one_one_int @ A ) @ N2 )
% 9.20/5.20          = ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6180_even__bit__succ__iff,axiom,
% 9.20/5.20      ! [A: nat,N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ one_one_nat @ A ) @ N2 )
% 9.20/5.20          = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6181_odd__bit__iff__bit__pred,axiom,
% 9.20/5.20      ! [A: code_integer,N2: nat] :
% 9.20/5.20        ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se9216721137139052372nteger @ A @ N2 )
% 9.20/5.20          = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6182_odd__bit__iff__bit__pred,axiom,
% 9.20/5.20      ! [A: int,N2: nat] :
% 9.20/5.20        ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se1146084159140164899it_int @ A @ N2 )
% 9.20/5.20          = ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ A @ one_one_int ) @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6183_odd__bit__iff__bit__pred,axiom,
% 9.20/5.20      ! [A: nat,N2: nat] :
% 9.20/5.20        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
% 9.20/5.20       => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
% 9.20/5.20          = ( ( bit_se1148574629649215175it_nat @ ( minus_minus_nat @ A @ one_one_nat ) @ N2 )
% 9.20/5.20            | ( N2 = zero_zero_nat ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6184_bit__sum__mult__2__cases,axiom,
% 9.20/5.20      ! [A: code_integer,B: code_integer,N2: nat] :
% 9.20/5.20        ( ! [J2: nat] :
% 9.20/5.20            ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
% 9.20/5.20       => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 9.20/5.20          = ( ( ( N2 = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.20            & ( ( N2 != zero_zero_nat )
% 9.20/5.20             => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6185_bit__sum__mult__2__cases,axiom,
% 9.20/5.20      ! [A: int,B: int,N2: nat] :
% 9.20/5.20        ( ! [J2: nat] :
% 9.20/5.20            ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
% 9.20/5.20       => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 9.20/5.20          = ( ( ( N2 = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.20            & ( ( N2 != zero_zero_nat )
% 9.20/5.20             => ( bit_se1146084159140164899it_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6186_bit__sum__mult__2__cases,axiom,
% 9.20/5.20      ! [A: nat,B: nat,N2: nat] :
% 9.20/5.20        ( ! [J2: nat] :
% 9.20/5.20            ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
% 9.20/5.20       => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ N2 )
% 9.20/5.20          = ( ( ( N2 = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
% 9.20/5.20            & ( ( N2 != zero_zero_nat )
% 9.20/5.20             => ( bit_se1148574629649215175it_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6187_bit__rec,axiom,
% 9.20/5.20      ( bit_se9216721137139052372nteger
% 9.20/5.20      = ( ^ [A3: code_integer,N: nat] :
% 9.20/5.20            ( ( ( N = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
% 9.20/5.20            & ( ( N != zero_zero_nat )
% 9.20/5.20             => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6188_bit__rec,axiom,
% 9.20/5.20      ( bit_se1146084159140164899it_int
% 9.20/5.20      = ( ^ [A3: int,N: nat] :
% 9.20/5.20            ( ( ( N = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 ) )
% 9.20/5.20            & ( ( N != zero_zero_nat )
% 9.20/5.20             => ( bit_se1146084159140164899it_int @ ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6189_bit__rec,axiom,
% 9.20/5.20      ( bit_se1148574629649215175it_nat
% 9.20/5.20      = ( ^ [A3: nat,N: nat] :
% 9.20/5.20            ( ( ( N = zero_zero_nat )
% 9.20/5.20             => ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 ) )
% 9.20/5.20            & ( ( N != zero_zero_nat )
% 9.20/5.20             => ( bit_se1148574629649215175it_nat @ ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6190_take__bit__Suc__from__most,axiom,
% 9.20/5.20      ! [N2: nat,K: int] :
% 9.20/5.20        ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ K )
% 9.20/5.20        = ( plus_plus_int @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6191_round__unique,axiom,
% 9.20/5.20      ! [X3: real,Y3: int] :
% 9.20/5.20        ( ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
% 9.20/5.20       => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20         => ( ( archim8280529875227126926d_real @ X3 )
% 9.20/5.20            = Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6192_round__unique,axiom,
% 9.20/5.20      ! [X3: rat,Y3: int] :
% 9.20/5.20        ( ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
% 9.20/5.20       => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20         => ( ( archim7778729529865785530nd_rat @ X3 )
% 9.20/5.20            = Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6193_of__int__round__gt,axiom,
% 9.20/5.20      ! [X3: real] : ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6194_of__int__round__gt,axiom,
% 9.20/5.20      ! [X3: rat] : ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6195_of__int__round__ge,axiom,
% 9.20/5.20      ! [X3: real] : ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6196_of__int__round__ge,axiom,
% 9.20/5.20      ! [X3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6197_of__int__round__le,axiom,
% 9.20/5.20      ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6198_of__int__round__le,axiom,
% 9.20/5.20      ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6199_divmod__BitM__2__eq,axiom,
% 9.20/5.20      ! [M: num] :
% 9.20/5.20        ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
% 9.20/5.20        = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6200_mask__numeral,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( bit_se2002935070580805687sk_nat @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.20        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ ( pred_numeral @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6201_mask__numeral,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( bit_se2000444600071755411sk_int @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.20        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ ( pred_numeral @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6202_neg__numeral__le__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6203_neg__numeral__le__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6204_mask__nat__positive__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 9.20/5.20        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6205_ceiling__zero,axiom,
% 9.20/5.20      ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
% 9.20/5.20      = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6206_ceiling__zero,axiom,
% 9.20/5.20      ( ( archim7802044766580827645g_real @ zero_zero_real )
% 9.20/5.20      = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6207_mask__eq__0__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ( bit_se2002935070580805687sk_nat @ N2 )
% 9.20/5.20          = zero_zero_nat )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6208_mask__eq__0__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ( bit_se2000444600071755411sk_int @ N2 )
% 9.20/5.20          = zero_zero_int )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6209_mask__0,axiom,
% 9.20/5.20      ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
% 9.20/5.20      = zero_zero_nat ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6210_mask__0,axiom,
% 9.20/5.20      ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
% 9.20/5.20      = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6211_round__0,axiom,
% 9.20/5.20      ( ( archim8280529875227126926d_real @ zero_zero_real )
% 9.20/5.20      = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6212_round__0,axiom,
% 9.20/5.20      ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
% 9.20/5.20      = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6213_mask__Suc__0,axiom,
% 9.20/5.20      ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
% 9.20/5.20      = one_one_nat ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6214_mask__Suc__0,axiom,
% 9.20/5.20      ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
% 9.20/5.20      = one_one_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6215_ceiling__add__of__int,axiom,
% 9.20/5.20      ! [X3: rat,Z2: int] :
% 9.20/5.20        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6216_ceiling__add__of__int,axiom,
% 9.20/5.20      ! [X3: real,Z2: int] :
% 9.20/5.20        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6217_ceiling__le__zero,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6218_ceiling__le__zero,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6219_zero__less__ceiling,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6220_zero__less__ceiling,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6221_ceiling__le__numeral,axiom,
% 9.20/5.20      ! [X3: real,V: num] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6222_ceiling__le__numeral,axiom,
% 9.20/5.20      ! [X3: rat,V: num] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6223_ceiling__less__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6224_ceiling__less__one,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6225_one__le__ceiling,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6226_one__le__ceiling,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6227_numeral__less__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: real] :
% 9.20/5.20        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6228_numeral__less__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6229_ceiling__le__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6230_ceiling__le__one,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6231_one__less__ceiling,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ one_one_rat @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6232_one__less__ceiling,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ one_one_real @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6233_ceiling__add__numeral,axiom,
% 9.20/5.20      ! [X3: real,V: num] :
% 9.20/5.20        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ V ) ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6234_ceiling__add__numeral,axiom,
% 9.20/5.20      ! [X3: rat,V: num] :
% 9.20/5.20        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( numeral_numeral_rat @ V ) ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6235_ceiling__add__one,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6236_ceiling__add__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ one_one_real ) )
% 9.20/5.20        = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6237_ceiling__less__zero,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6238_ceiling__less__zero,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6239_zero__le__ceiling,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6240_zero__le__ceiling,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6241_ceiling__less__numeral,axiom,
% 9.20/5.20      ! [X3: real,V: num] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6242_ceiling__less__numeral,axiom,
% 9.20/5.20      ! [X3: rat,V: num] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6243_numeral__le__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6244_numeral__le__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6245_ceiling__le__neg__numeral,axiom,
% 9.20/5.20      ! [X3: real,V: num] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6246_ceiling__le__neg__numeral,axiom,
% 9.20/5.20      ! [X3: rat,V: num] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6247_neg__numeral__less__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: real] :
% 9.20/5.20        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6248_neg__numeral__less__ceiling,axiom,
% 9.20/5.20      ! [V: num,X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6249_ceiling__less__neg__numeral,axiom,
% 9.20/5.20      ! [X3: real,V: num] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6250_ceiling__less__neg__numeral,axiom,
% 9.20/5.20      ! [X3: rat,V: num] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6251_less__eq__mask,axiom,
% 9.20/5.20      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6252_bit__Suc__0__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6253_not__bit__Suc__0__Suc,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6254_ceiling__mono,axiom,
% 9.20/5.20      ! [Y3: real,X3: real] :
% 9.20/5.20        ( ( ord_less_eq_real @ Y3 @ X3 )
% 9.20/5.20       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6255_ceiling__mono,axiom,
% 9.20/5.20      ! [Y3: rat,X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_rat @ Y3 @ X3 )
% 9.20/5.20       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6256_le__of__int__ceiling,axiom,
% 9.20/5.20      ! [X3: real] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6257_le__of__int__ceiling,axiom,
% 9.20/5.20      ! [X3: rat] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6258_ceiling__less__cancel,axiom,
% 9.20/5.20      ! [X3: rat,Y3: rat] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
% 9.20/5.20       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6259_ceiling__less__cancel,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) )
% 9.20/5.20       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6260_not__mask__negative__int,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6261_not__bit__Suc__0__numeral,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6262_ceiling__le__iff,axiom,
% 9.20/5.20      ! [X3: real,Z2: int] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6263_ceiling__le__iff,axiom,
% 9.20/5.20      ! [X3: rat,Z2: int] :
% 9.20/5.20        ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6264_ceiling__le,axiom,
% 9.20/5.20      ! [X3: real,A: int] :
% 9.20/5.20        ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) )
% 9.20/5.20       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6265_ceiling__le,axiom,
% 9.20/5.20      ! [X3: rat,A: int] :
% 9.20/5.20        ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) )
% 9.20/5.20       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6266_less__ceiling__iff,axiom,
% 9.20/5.20      ! [Z2: int,X3: rat] :
% 9.20/5.20        ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6267_less__ceiling__iff,axiom,
% 9.20/5.20      ! [Z2: int,X3: real] :
% 9.20/5.20        ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6268_ceiling__add__le,axiom,
% 9.20/5.20      ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6269_ceiling__add__le,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6270_round__mono,axiom,
% 9.20/5.20      ! [X3: rat,Y3: rat] :
% 9.20/5.20        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.20/5.20       => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X3 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6271_less__mask,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.20       => ( ord_less_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6272_eval__nat__numeral_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6273_of__int__ceiling__le__add__one,axiom,
% 9.20/5.20      ! [R2: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ ( plus_plus_real @ R2 @ one_one_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6274_of__int__ceiling__le__add__one,axiom,
% 9.20/5.20      ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6275_of__int__ceiling__diff__one__le,axiom,
% 9.20/5.20      ! [R2: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R2 ) ) @ one_one_real ) @ R2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6276_of__int__ceiling__diff__one__le,axiom,
% 9.20/5.20      ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6277_ceiling__split,axiom,
% 9.20/5.20      ! [P: int > $o,T: real] :
% 9.20/5.20        ( ( P @ ( archim7802044766580827645g_real @ T ) )
% 9.20/5.20        = ( ! [I2: int] :
% 9.20/5.20              ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
% 9.20/5.20                & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
% 9.20/5.20             => ( P @ I2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6278_ceiling__split,axiom,
% 9.20/5.20      ! [P: int > $o,T: rat] :
% 9.20/5.20        ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
% 9.20/5.20        = ( ! [I2: int] :
% 9.20/5.20              ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
% 9.20/5.20                & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
% 9.20/5.20             => ( P @ I2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6279_ceiling__eq__iff,axiom,
% 9.20/5.20      ! [X3: real,A: int] :
% 9.20/5.20        ( ( ( archim7802044766580827645g_real @ X3 )
% 9.20/5.20          = A )
% 9.20/5.20        = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X3 )
% 9.20/5.20          & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6280_ceiling__eq__iff,axiom,
% 9.20/5.20      ! [X3: rat,A: int] :
% 9.20/5.20        ( ( ( archim2889992004027027881ng_rat @ X3 )
% 9.20/5.20          = A )
% 9.20/5.20        = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X3 )
% 9.20/5.20          & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6281_ceiling__unique,axiom,
% 9.20/5.20      ! [Z2: int,X3: real] :
% 9.20/5.20        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.20         => ( ( archim7802044766580827645g_real @ X3 )
% 9.20/5.20            = Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6282_ceiling__unique,axiom,
% 9.20/5.20      ! [Z2: int,X3: rat] :
% 9.20/5.20        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.20         => ( ( archim2889992004027027881ng_rat @ X3 )
% 9.20/5.20            = Z2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6283_ceiling__correct,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) @ one_one_real ) @ X3 )
% 9.20/5.20        & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6284_ceiling__correct,axiom,
% 9.20/5.20      ! [X3: rat] :
% 9.20/5.20        ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) @ one_one_rat ) @ X3 )
% 9.20/5.20        & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6285_mult__ceiling__le,axiom,
% 9.20/5.20      ! [A: real,B: real] :
% 9.20/5.20        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.20         => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6286_mult__ceiling__le,axiom,
% 9.20/5.20      ! [A: rat,B: rat] :
% 9.20/5.20        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.20       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.20         => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6287_ceiling__less__iff,axiom,
% 9.20/5.20      ! [X3: real,Z2: int] :
% 9.20/5.20        ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
% 9.20/5.20        = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6288_ceiling__less__iff,axiom,
% 9.20/5.20      ! [X3: rat,Z2: int] :
% 9.20/5.20        ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
% 9.20/5.20        = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6289_le__ceiling__iff,axiom,
% 9.20/5.20      ! [Z2: int,X3: rat] :
% 9.20/5.20        ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
% 9.20/5.20        = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6290_le__ceiling__iff,axiom,
% 9.20/5.20      ! [Z2: int,X3: real] :
% 9.20/5.20        ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
% 9.20/5.20        = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6291_ceiling__divide__upper,axiom,
% 9.20/5.20      ! [Q4: real,P4: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.20/5.20       => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6292_ceiling__divide__upper,axiom,
% 9.20/5.20      ! [Q4: rat,P4: rat] :
% 9.20/5.20        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.20/5.20       => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6293_Suc__mask__eq__exp,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( suc @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 9.20/5.20        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6294_mask__nat__less__exp,axiom,
% 9.20/5.20      ! [N2: nat] : ( ord_less_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6295_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6296_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N2 ) )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6297_semiring__bit__operations__class_Oeven__mask__iff,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2000444600071755411sk_int @ N2 ) )
% 9.20/5.20        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6298_ceiling__divide__lower,axiom,
% 9.20/5.20      ! [Q4: real,P4: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.20/5.20       => ( 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 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6299_ceiling__divide__lower,axiom,
% 9.20/5.20      ! [Q4: rat,P4: rat] :
% 9.20/5.20        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.20/5.20       => ( 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 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6300_ceiling__eq,axiom,
% 9.20/5.20      ! [N2: int,X3: real] :
% 9.20/5.20        ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 9.20/5.20         => ( ( archim7802044766580827645g_real @ X3 )
% 9.20/5.20            = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6301_ceiling__eq,axiom,
% 9.20/5.20      ! [N2: int,X3: rat] :
% 9.20/5.20        ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
% 9.20/5.20         => ( ( archim2889992004027027881ng_rat @ X3 )
% 9.20/5.20            = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6302_Suc__0__xor__eq,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.20        = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.20          @ ( zero_n2687167440665602831ol_nat
% 9.20/5.20            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6303_xor__Suc__0__eq,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.20        = ( minus_minus_nat @ ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.20          @ ( zero_n2687167440665602831ol_nat
% 9.20/5.20            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6304_horner__sum__of__bool__2__less,axiom,
% 9.20/5.20      ! [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 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6305_dbl__dec__simps_I2_J,axiom,
% 9.20/5.20      ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
% 9.20/5.20      = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6306_dbl__dec__simps_I2_J,axiom,
% 9.20/5.20      ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
% 9.20/5.20      = ( uminus_uminus_int @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6307_dbl__dec__simps_I2_J,axiom,
% 9.20/5.20      ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
% 9.20/5.20      = ( uminus_uminus_real @ one_one_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6308_dbl__dec__simps_I2_J,axiom,
% 9.20/5.20      ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
% 9.20/5.20      = ( uminus_uminus_rat @ one_one_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6309_dbl__dec__simps_I2_J,axiom,
% 9.20/5.20      ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
% 9.20/5.20      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6310_the__elem__eq,axiom,
% 9.20/5.20      ! [X3: produc3843707927480180839at_nat] :
% 9.20/5.20        ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
% 9.20/5.20        = X3 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6311_the__elem__eq,axiom,
% 9.20/5.20      ! [X3: product_prod_nat_nat] :
% 9.20/5.20        ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
% 9.20/5.20        = X3 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6312_the__elem__eq,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( the_elem_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
% 9.20/5.20        = X3 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6313_the__elem__eq,axiom,
% 9.20/5.20      ! [X3: nat] :
% 9.20/5.20        ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
% 9.20/5.20        = X3 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6314_the__elem__eq,axiom,
% 9.20/5.20      ! [X3: int] :
% 9.20/5.20        ( ( the_elem_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
% 9.20/5.20        = X3 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6315_vebt__member_Oelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ( Y3
% 9.20/5.20                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.20/5.20         => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( X3
% 9.20/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.20             => Y3 )
% 9.20/5.20           => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.20                    ( X3
% 9.20/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.20/5.20               => Y3 )
% 9.20/5.20             => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.20/5.20                 => Y3 )
% 9.20/5.20               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                      ( ? [Summary2: vEBT_VEBT] :
% 9.20/5.20                          ( X3
% 9.20/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20                     => ( Y3
% 9.20/5.20                        = ( ~ ( ( Xa2 != Mi2 )
% 9.20/5.20                             => ( ( Xa2 != Ma2 )
% 9.20/5.20                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6316_set__vebt_H__def,axiom,
% 9.20/5.20      ( vEBT_VEBT_set_vebt
% 9.20/5.20      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6317_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o] :
% 9.20/5.20        ( ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20          | ( finite_finite_real @ ( collect_real @ Q ) ) )
% 9.20/5.20       => ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [X: real] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6318_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.20/5.20        ( ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.20/5.20          | ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) )
% 9.20/5.20       => ( finite8100373058378681591st_nat
% 9.20/5.20          @ ( collect_list_nat
% 9.20/5.20            @ ^ [X: list_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6319_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.20/5.20        ( ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.20/5.20          | ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) )
% 9.20/5.20       => ( finite1152437895449049373et_nat
% 9.20/5.20          @ ( collect_set_nat
% 9.20/5.20            @ ^ [X: set_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6320_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: nat > $o] :
% 9.20/5.20        ( ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20          | ( finite_finite_nat @ ( collect_nat @ Q ) ) )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [X: nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6321_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: int > $o,Q: int > $o] :
% 9.20/5.20        ( ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20          | ( finite_finite_int @ ( collect_int @ Q ) ) )
% 9.20/5.20       => ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [X: int] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6322_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: complex > $o,Q: complex > $o] :
% 9.20/5.20        ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.20/5.20          | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
% 9.20/5.20       => ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [X: complex] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6323_finite__Collect__conjI,axiom,
% 9.20/5.20      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.20/5.20          | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
% 9.20/5.20       => ( finite6177210948735845034at_nat
% 9.20/5.20          @ ( collec3392354462482085612at_nat
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                & ( Q @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6324_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [X: real] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20          & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6325_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: list_nat > $o,Q: list_nat > $o] :
% 9.20/5.20        ( ( finite8100373058378681591st_nat
% 9.20/5.20          @ ( collect_list_nat
% 9.20/5.20            @ ^ [X: list_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.20/5.20          & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6326_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: set_nat > $o,Q: set_nat > $o] :
% 9.20/5.20        ( ( finite1152437895449049373et_nat
% 9.20/5.20          @ ( collect_set_nat
% 9.20/5.20            @ ^ [X: set_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.20/5.20          & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6327_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: nat > $o] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [X: nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20          & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6328_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: int > $o,Q: int > $o] :
% 9.20/5.20        ( ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [X: int] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20          & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6329_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: complex > $o,Q: complex > $o] :
% 9.20/5.20        ( ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [X: complex] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.20/5.20          & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6330_finite__Collect__disjI,axiom,
% 9.20/5.20      ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ( finite6177210948735845034at_nat
% 9.20/5.20          @ ( collec3392354462482085612at_nat
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                ( ( P @ X )
% 9.20/5.20                | ( Q @ X ) ) ) )
% 9.20/5.20        = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.20/5.20          & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6331_bit_Oxor__self,axiom,
% 9.20/5.20      ! [X3: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ X3 @ X3 )
% 9.20/5.20        = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6332_xor__self__eq,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ A @ A )
% 9.20/5.20        = zero_zero_nat ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6333_xor__self__eq,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ A @ A )
% 9.20/5.20        = zero_zero_int ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6334_xor_Oleft__neutral,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
% 9.20/5.20        = A ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6335_xor_Oleft__neutral,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
% 9.20/5.20        = A ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6336_xor_Oright__neutral,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
% 9.20/5.20        = A ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6337_xor_Oright__neutral,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
% 9.20/5.20        = A ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6338_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > real > $o] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                ? [Y: real] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: real] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_real
% 9.20/5.20                  @ ( collect_real
% 9.20/5.20                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6339_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: real > $o,Q: nat > real > $o] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                ? [Y: real] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: real] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_nat
% 9.20/5.20                  @ ( collect_nat
% 9.20/5.20                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6340_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: real > $o,Q: int > real > $o] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                ? [Y: real] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: real] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_int
% 9.20/5.20                  @ ( collect_int
% 9.20/5.20                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6341_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: real > $o,Q: complex > real > $o] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [X: complex] :
% 9.20/5.20                ? [Y: real] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: real] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite3207457112153483333omplex
% 9.20/5.20                  @ ( collect_complex
% 9.20/5.20                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6342_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: real > nat > $o] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                ? [Y: nat] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: nat] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_real
% 9.20/5.20                  @ ( collect_real
% 9.20/5.20                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6343_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: nat > nat > $o] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                ? [Y: nat] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: nat] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_nat
% 9.20/5.20                  @ ( collect_nat
% 9.20/5.20                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6344_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: int > nat > $o] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                ? [Y: nat] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: nat] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_int
% 9.20/5.20                  @ ( collect_int
% 9.20/5.20                    @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6345_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: nat > $o,Q: complex > nat > $o] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [X: complex] :
% 9.20/5.20                ? [Y: nat] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: nat] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite3207457112153483333omplex
% 9.20/5.20                  @ ( collect_complex
% 9.20/5.20                    @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6346_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: int > $o,Q: real > int > $o] :
% 9.20/5.20        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                ? [Y: int] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: int] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_real
% 9.20/5.20                  @ ( collect_real
% 9.20/5.20                    @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6347_finite__Collect__bounded__ex,axiom,
% 9.20/5.20      ! [P: int > $o,Q: nat > int > $o] :
% 9.20/5.20        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                ? [Y: int] :
% 9.20/5.20                  ( ( P @ Y )
% 9.20/5.20                  & ( Q @ X @ Y ) ) ) )
% 9.20/5.20          = ( ! [Y: int] :
% 9.20/5.20                ( ( P @ Y )
% 9.20/5.20               => ( finite_finite_nat
% 9.20/5.20                  @ ( collect_nat
% 9.20/5.20                    @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6348_finite__Collect__subsets,axiom,
% 9.20/5.20      ! [A2: set_nat] :
% 9.20/5.20        ( ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( finite1152437895449049373et_nat
% 9.20/5.20          @ ( collect_set_nat
% 9.20/5.20            @ ^ [B5: set_nat] : ( ord_less_eq_set_nat @ B5 @ A2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6349_finite__Collect__subsets,axiom,
% 9.20/5.20      ! [A2: set_complex] :
% 9.20/5.20        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.20       => ( finite6551019134538273531omplex
% 9.20/5.20          @ ( collect_set_complex
% 9.20/5.20            @ ^ [B5: set_complex] : ( ord_le211207098394363844omplex @ B5 @ A2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6350_finite__Collect__subsets,axiom,
% 9.20/5.20      ! [A2: set_Pr1261947904930325089at_nat] :
% 9.20/5.20        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.20       => ( finite9047747110432174090at_nat
% 9.20/5.20          @ ( collec5514110066124741708at_nat
% 9.20/5.20            @ ^ [B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B5 @ A2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6351_finite__Collect__subsets,axiom,
% 9.20/5.20      ! [A2: set_int] :
% 9.20/5.20        ( ( finite_finite_int @ A2 )
% 9.20/5.20       => ( finite6197958912794628473et_int
% 9.20/5.20          @ ( collect_set_int
% 9.20/5.20            @ ^ [B5: set_int] : ( ord_less_eq_set_int @ B5 @ A2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6352_singleton__conv2,axiom,
% 9.20/5.20      ! [A: produc3843707927480180839at_nat] :
% 9.20/5.20        ( ( collec6321179662152712658at_nat
% 9.20/5.20          @ ( ^ [Y4: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6353_singleton__conv2,axiom,
% 9.20/5.20      ! [A: list_nat] :
% 9.20/5.20        ( ( collect_list_nat
% 9.20/5.20          @ ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6354_singleton__conv2,axiom,
% 9.20/5.20      ! [A: set_nat] :
% 9.20/5.20        ( ( collect_set_nat
% 9.20/5.20          @ ( ^ [Y4: set_nat,Z: set_nat] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6355_singleton__conv2,axiom,
% 9.20/5.20      ! [A: product_prod_nat_nat] :
% 9.20/5.20        ( ( collec3392354462482085612at_nat
% 9.20/5.20          @ ( ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6356_singleton__conv2,axiom,
% 9.20/5.20      ! [A: real] :
% 9.20/5.20        ( ( collect_real
% 9.20/5.20          @ ( ^ [Y4: real,Z: real] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6357_singleton__conv2,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( collect_nat
% 9.20/5.20          @ ( ^ [Y4: nat,Z: nat] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6358_singleton__conv2,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( collect_int
% 9.20/5.20          @ ( ^ [Y4: int,Z: int] : ( Y4 = Z )
% 9.20/5.20            @ A ) )
% 9.20/5.20        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6359_singleton__conv,axiom,
% 9.20/5.20      ! [A: produc3843707927480180839at_nat] :
% 9.20/5.20        ( ( collec6321179662152712658at_nat
% 9.20/5.20          @ ^ [X: produc3843707927480180839at_nat] : ( X = A ) )
% 9.20/5.20        = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6360_singleton__conv,axiom,
% 9.20/5.20      ! [A: list_nat] :
% 9.20/5.20        ( ( collect_list_nat
% 9.20/5.20          @ ^ [X: list_nat] : ( X = A ) )
% 9.20/5.20        = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6361_singleton__conv,axiom,
% 9.20/5.20      ! [A: set_nat] :
% 9.20/5.20        ( ( collect_set_nat
% 9.20/5.20          @ ^ [X: set_nat] : ( X = A ) )
% 9.20/5.20        = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6362_singleton__conv,axiom,
% 9.20/5.20      ! [A: product_prod_nat_nat] :
% 9.20/5.20        ( ( collec3392354462482085612at_nat
% 9.20/5.20          @ ^ [X: product_prod_nat_nat] : ( X = A ) )
% 9.20/5.20        = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6363_singleton__conv,axiom,
% 9.20/5.20      ! [A: real] :
% 9.20/5.20        ( ( collect_real
% 9.20/5.20          @ ^ [X: real] : ( X = A ) )
% 9.20/5.20        = ( insert_real @ A @ bot_bot_set_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6364_singleton__conv,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( collect_nat
% 9.20/5.20          @ ^ [X: nat] : ( X = A ) )
% 9.20/5.20        = ( insert_nat @ A @ bot_bot_set_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6365_singleton__conv,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( collect_int
% 9.20/5.20          @ ^ [X: int] : ( X = A ) )
% 9.20/5.20        = ( insert_int @ A @ bot_bot_set_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6366_finite__Collect__less__nat,axiom,
% 9.20/5.20      ! [K: nat] :
% 9.20/5.20        ( finite_finite_nat
% 9.20/5.20        @ ( collect_nat
% 9.20/5.20          @ ^ [N: nat] : ( ord_less_nat @ N @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6367_finite__Collect__le__nat,axiom,
% 9.20/5.20      ! [K: nat] :
% 9.20/5.20        ( finite_finite_nat
% 9.20/5.20        @ ( collect_nat
% 9.20/5.20          @ ^ [N: nat] : ( ord_less_eq_nat @ N @ K ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6368_finite__interval__int1,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( finite_finite_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [I2: int] :
% 9.20/5.20              ( ( ord_less_eq_int @ A @ I2 )
% 9.20/5.20              & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6369_finite__interval__int4,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( finite_finite_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [I2: int] :
% 9.20/5.20              ( ( ord_less_int @ A @ I2 )
% 9.20/5.20              & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6370_finite__interval__int2,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( finite_finite_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [I2: int] :
% 9.20/5.20              ( ( ord_less_eq_int @ A @ I2 )
% 9.20/5.20              & ( ord_less_int @ I2 @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6371_finite__interval__int3,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( finite_finite_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [I2: int] :
% 9.20/5.20              ( ( ord_less_int @ A @ I2 )
% 9.20/5.20              & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6372_xor__nat__numerals_I4_J,axiom,
% 9.20/5.20      ! [X3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.20        = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6373_xor__nat__numerals_I3_J,axiom,
% 9.20/5.20      ! [X3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.20        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6374_xor__nat__numerals_I2_J,axiom,
% 9.20/5.20      ! [Y3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.20/5.20        = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6375_xor__nat__numerals_I1_J,axiom,
% 9.20/5.20      ! [Y3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.20/5.20        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6376_xor__numerals_I6_J,axiom,
% 9.20/5.20      ! [X3: num,Y3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.20/5.20        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6377_xor__numerals_I6_J,axiom,
% 9.20/5.20      ! [X3: num,Y3: num] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
% 9.20/5.20        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6378_xor__numerals_I4_J,axiom,
% 9.20/5.20      ! [X3: num,Y3: num] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.20/5.20        = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6379_xor__numerals_I4_J,axiom,
% 9.20/5.20      ! [X3: num,Y3: num] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
% 9.20/5.20        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6380_empty__def,axiom,
% 9.20/5.20      ( bot_bot_set_list_nat
% 9.20/5.20      = ( collect_list_nat
% 9.20/5.20        @ ^ [X: list_nat] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6381_empty__def,axiom,
% 9.20/5.20      ( bot_bot_set_set_nat
% 9.20/5.20      = ( collect_set_nat
% 9.20/5.20        @ ^ [X: set_nat] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6382_empty__def,axiom,
% 9.20/5.20      ( bot_bo2099793752762293965at_nat
% 9.20/5.20      = ( collec3392354462482085612at_nat
% 9.20/5.20        @ ^ [X: product_prod_nat_nat] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6383_empty__def,axiom,
% 9.20/5.20      ( bot_bot_set_real
% 9.20/5.20      = ( collect_real
% 9.20/5.20        @ ^ [X: real] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6384_empty__def,axiom,
% 9.20/5.20      ( bot_bot_set_nat
% 9.20/5.20      = ( collect_nat
% 9.20/5.20        @ ^ [X: nat] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6385_empty__def,axiom,
% 9.20/5.20      ( bot_bot_set_int
% 9.20/5.20      = ( collect_int
% 9.20/5.20        @ ^ [X: int] : $false ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6386_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collec6321179662152712658at_nat
% 9.20/5.20              @ ^ [X: produc3843707927480180839at_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collec6321179662152712658at_nat
% 9.20/5.20              @ ^ [X: produc3843707927480180839at_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6387_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: list_nat > $o,A: list_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_list_nat
% 9.20/5.20              @ ^ [X: list_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_list_nat
% 9.20/5.20              @ ^ [X: list_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_list_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6388_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: set_nat > $o,A: set_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_set_nat
% 9.20/5.20              @ ^ [X: set_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_set_nat
% 9.20/5.20              @ ^ [X: set_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_set_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6389_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collec3392354462482085612at_nat
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collec3392354462482085612at_nat
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6390_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: real > $o,A: real] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6391_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: nat > $o,A: nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6392_Collect__conv__if2,axiom,
% 9.20/5.20      ! [P: int > $o,A: int] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                  ( ( A = X )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6393_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collec6321179662152712658at_nat
% 9.20/5.20              @ ^ [X: produc3843707927480180839at_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collec6321179662152712658at_nat
% 9.20/5.20              @ ^ [X: produc3843707927480180839at_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6394_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: list_nat > $o,A: list_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_list_nat
% 9.20/5.20              @ ^ [X: list_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_list_nat
% 9.20/5.20              @ ^ [X: list_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_list_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6395_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: set_nat > $o,A: set_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_set_nat
% 9.20/5.20              @ ^ [X: set_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_set_nat
% 9.20/5.20              @ ^ [X: set_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_set_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6396_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collec3392354462482085612at_nat
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collec3392354462482085612at_nat
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6397_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: real > $o,A: real] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6398_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: nat > $o,A: nat] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6399_Collect__conv__if,axiom,
% 9.20/5.20      ! [P: int > $o,A: int] :
% 9.20/5.20        ( ( ( P @ A )
% 9.20/5.20         => ( ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.20        & ( ~ ( P @ A )
% 9.20/5.20         => ( ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                  ( ( X = A )
% 9.20/5.20                  & ( P @ X ) ) )
% 9.20/5.20            = bot_bot_set_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6400_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 9.20/5.20        ( ( ord_le2646555220125990790_nat_o
% 9.20/5.20          @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6401_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 9.20/5.20        ( ( ord_le6741204236512500942_int_o
% 9.20/5.20          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6402_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
% 9.20/5.20        ( ( ord_le3636971675376928563eger_o
% 9.20/5.20          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le3216752416896350996nteger @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6403_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
% 9.20/5.20        ( ( ord_le4340812435750786203eger_o
% 9.20/5.20          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le653643898420964396nteger @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6404_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
% 9.20/5.20        ( ( ord_le5643404153117327598_int_o
% 9.20/5.20          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le8725513860283290265nt_int @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6405_pred__subset__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
% 9.20/5.20        ( ( ord_le2124322318746777828_int_o
% 9.20/5.20          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R )
% 9.20/5.20          @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) )
% 9.20/5.20        = ( ord_le135402666524580259nt_int @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6406_strict__subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: real,B: real] :
% 9.20/5.20        ( ( ord_less_set_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 9.20/5.20        = ( ( dvd_dvd_real @ A @ B )
% 9.20/5.20          & ~ ( dvd_dvd_real @ B @ A ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6407_strict__subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: nat,B: nat] :
% 9.20/5.20        ( ( ord_less_set_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 9.20/5.20        = ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.20          & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6408_strict__subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( ( ord_less_set_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 9.20/5.20        = ( ( dvd_dvd_int @ A @ B )
% 9.20/5.20          & ~ ( dvd_dvd_int @ B @ A ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6409_strict__subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: code_integer,B: code_integer] :
% 9.20/5.20        ( ( ord_le1307284697595431911nteger
% 9.20/5.20          @ ( collect_Code_integer
% 9.20/5.20            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 9.20/5.20          @ ( collect_Code_integer
% 9.20/5.20            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 9.20/5.20        = ( ( dvd_dvd_Code_integer @ A @ B )
% 9.20/5.20          & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6410_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 9.20/5.20        ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6411_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
% 9.20/5.20        ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6412_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
% 9.20/5.20        ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6413_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
% 9.20/5.20        ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6414_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
% 9.20/5.20        ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6415_pred__equals__eq2,axiom,
% 9.20/5.20      ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
% 9.20/5.20        ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R ) )
% 9.20/5.20          = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) ) )
% 9.20/5.20        = ( R = S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6416_subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: real,B: real] :
% 9.20/5.20        ( ( ord_less_eq_set_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
% 9.20/5.20        = ( dvd_dvd_real @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6417_subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: nat,B: nat] :
% 9.20/5.20        ( ( ord_less_eq_set_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
% 9.20/5.20        = ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6418_subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: code_integer,B: code_integer] :
% 9.20/5.20        ( ( ord_le7084787975880047091nteger
% 9.20/5.20          @ ( collect_Code_integer
% 9.20/5.20            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
% 9.20/5.20          @ ( collect_Code_integer
% 9.20/5.20            @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
% 9.20/5.20        = ( dvd_dvd_Code_integer @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6419_subset__divisors__dvd,axiom,
% 9.20/5.20      ! [A: int,B: int] :
% 9.20/5.20        ( ( ord_less_eq_set_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
% 9.20/5.20        = ( dvd_dvd_int @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6420_less__set__def,axiom,
% 9.20/5.20      ( ord_le7866589430770878221at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( ord_le549003669493604880_nat_o
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6421_less__set__def,axiom,
% 9.20/5.20      ( ord_less_set_real
% 9.20/5.20      = ( ^ [A5: set_real,B5: set_real] :
% 9.20/5.20            ( ord_less_real_o
% 9.20/5.20            @ ^ [X: real] : ( member_real @ X @ A5 )
% 9.20/5.20            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6422_less__set__def,axiom,
% 9.20/5.20      ( ord_less_set_set_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.20/5.20            ( ord_less_set_nat_o
% 9.20/5.20            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6423_less__set__def,axiom,
% 9.20/5.20      ( ord_less_set_nat
% 9.20/5.20      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.20/5.20            ( ord_less_nat_o
% 9.20/5.20            @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6424_less__set__def,axiom,
% 9.20/5.20      ( ord_less_set_int
% 9.20/5.20      = ( ^ [A5: set_int,B5: set_int] :
% 9.20/5.20            ( ord_less_int_o
% 9.20/5.20            @ ^ [X: int] : ( member_int @ X @ A5 )
% 9.20/5.20            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6425_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ord_le3146513528884898305at_nat
% 9.20/5.20        @ ( collec3392354462482085612at_nat
% 9.20/5.20          @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20              ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6426_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_real,P: real > $o] :
% 9.20/5.20        ( ord_less_eq_set_real
% 9.20/5.20        @ ( collect_real
% 9.20/5.20          @ ^ [X: real] :
% 9.20/5.20              ( ( member_real @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6427_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_list_nat,P: list_nat > $o] :
% 9.20/5.20        ( ord_le6045566169113846134st_nat
% 9.20/5.20        @ ( collect_list_nat
% 9.20/5.20          @ ^ [X: list_nat] :
% 9.20/5.20              ( ( member_list_nat @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6428_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_set_nat,P: set_nat > $o] :
% 9.20/5.20        ( ord_le6893508408891458716et_nat
% 9.20/5.20        @ ( collect_set_nat
% 9.20/5.20          @ ^ [X: set_nat] :
% 9.20/5.20              ( ( member_set_nat @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6429_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_nat,P: nat > $o] :
% 9.20/5.20        ( ord_less_eq_set_nat
% 9.20/5.20        @ ( collect_nat
% 9.20/5.20          @ ^ [X: nat] :
% 9.20/5.20              ( ( member_nat @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6430_Collect__subset,axiom,
% 9.20/5.20      ! [A2: set_int,P: int > $o] :
% 9.20/5.20        ( ord_less_eq_set_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [X: int] :
% 9.20/5.20              ( ( member_int @ X @ A2 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ A2 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6431_less__eq__set__def,axiom,
% 9.20/5.20      ( ord_le3146513528884898305at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( ord_le704812498762024988_nat_o
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6432_less__eq__set__def,axiom,
% 9.20/5.20      ( ord_less_eq_set_real
% 9.20/5.20      = ( ^ [A5: set_real,B5: set_real] :
% 9.20/5.20            ( ord_less_eq_real_o
% 9.20/5.20            @ ^ [X: real] : ( member_real @ X @ A5 )
% 9.20/5.20            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6433_less__eq__set__def,axiom,
% 9.20/5.20      ( ord_le6893508408891458716et_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.20/5.20            ( ord_le3964352015994296041_nat_o
% 9.20/5.20            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6434_less__eq__set__def,axiom,
% 9.20/5.20      ( ord_less_eq_set_nat
% 9.20/5.20      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.20/5.20            ( ord_less_eq_nat_o
% 9.20/5.20            @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 9.20/5.20            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6435_less__eq__set__def,axiom,
% 9.20/5.20      ( ord_less_eq_set_int
% 9.20/5.20      = ( ^ [A5: set_int,B5: set_int] :
% 9.20/5.20            ( ord_less_eq_int_o
% 9.20/5.20            @ ^ [X: int] : ( member_int @ X @ A5 )
% 9.20/5.20            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6436_prop__restrict,axiom,
% 9.20/5.20      ! [X3: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ( member8440522571783428010at_nat @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_le3146513528884898305at_nat @ Z5
% 9.20/5.20            @ ( collec3392354462482085612at_nat
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                  ( ( member8440522571783428010at_nat @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6437_prop__restrict,axiom,
% 9.20/5.20      ! [X3: real,Z5: set_real,X7: set_real,P: real > $o] :
% 9.20/5.20        ( ( member_real @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_less_eq_set_real @ Z5
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [X: real] :
% 9.20/5.20                  ( ( member_real @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6438_prop__restrict,axiom,
% 9.20/5.20      ! [X3: list_nat,Z5: set_list_nat,X7: set_list_nat,P: list_nat > $o] :
% 9.20/5.20        ( ( member_list_nat @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_le6045566169113846134st_nat @ Z5
% 9.20/5.20            @ ( collect_list_nat
% 9.20/5.20              @ ^ [X: list_nat] :
% 9.20/5.20                  ( ( member_list_nat @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6439_prop__restrict,axiom,
% 9.20/5.20      ! [X3: set_nat,Z5: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
% 9.20/5.20        ( ( member_set_nat @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_le6893508408891458716et_nat @ Z5
% 9.20/5.20            @ ( collect_set_nat
% 9.20/5.20              @ ^ [X: set_nat] :
% 9.20/5.20                  ( ( member_set_nat @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6440_prop__restrict,axiom,
% 9.20/5.20      ! [X3: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
% 9.20/5.20        ( ( member_nat @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_less_eq_set_nat @ Z5
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [X: nat] :
% 9.20/5.20                  ( ( member_nat @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6441_prop__restrict,axiom,
% 9.20/5.20      ! [X3: int,Z5: set_int,X7: set_int,P: int > $o] :
% 9.20/5.20        ( ( member_int @ X3 @ Z5 )
% 9.20/5.20       => ( ( ord_less_eq_set_int @ Z5
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [X: int] :
% 9.20/5.20                  ( ( member_int @ X @ X7 )
% 9.20/5.20                  & ( P @ X ) ) ) )
% 9.20/5.20         => ( P @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6442_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ord_le3146513528884898305at_nat
% 9.20/5.20        @ ( collec3392354462482085612at_nat
% 9.20/5.20          @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20              ( ( member8440522571783428010at_nat @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6443_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_real,P: real > $o] :
% 9.20/5.20        ( ord_less_eq_set_real
% 9.20/5.20        @ ( collect_real
% 9.20/5.20          @ ^ [X: real] :
% 9.20/5.20              ( ( member_real @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6444_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_list_nat,P: list_nat > $o] :
% 9.20/5.20        ( ord_le6045566169113846134st_nat
% 9.20/5.20        @ ( collect_list_nat
% 9.20/5.20          @ ^ [X: list_nat] :
% 9.20/5.20              ( ( member_list_nat @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6445_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_set_nat,P: set_nat > $o] :
% 9.20/5.20        ( ord_le6893508408891458716et_nat
% 9.20/5.20        @ ( collect_set_nat
% 9.20/5.20          @ ^ [X: set_nat] :
% 9.20/5.20              ( ( member_set_nat @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6446_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_nat,P: nat > $o] :
% 9.20/5.20        ( ord_less_eq_set_nat
% 9.20/5.20        @ ( collect_nat
% 9.20/5.20          @ ^ [X: nat] :
% 9.20/5.20              ( ( member_nat @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6447_Collect__restrict,axiom,
% 9.20/5.20      ! [X7: set_int,P: int > $o] :
% 9.20/5.20        ( ord_less_eq_set_int
% 9.20/5.20        @ ( collect_int
% 9.20/5.20          @ ^ [X: int] :
% 9.20/5.20              ( ( member_int @ X @ X7 )
% 9.20/5.20              & ( P @ X ) ) )
% 9.20/5.20        @ X7 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6448_pred__subset__eq,axiom,
% 9.20/5.20      ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
% 9.20/5.20        ( ( ord_le704812498762024988_nat_o
% 9.20/5.20          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
% 9.20/5.20          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S3 ) )
% 9.20/5.20        = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6449_pred__subset__eq,axiom,
% 9.20/5.20      ! [R: set_real,S3: set_real] :
% 9.20/5.20        ( ( ord_less_eq_real_o
% 9.20/5.20          @ ^ [X: real] : ( member_real @ X @ R )
% 9.20/5.20          @ ^ [X: real] : ( member_real @ X @ S3 ) )
% 9.20/5.20        = ( ord_less_eq_set_real @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6450_pred__subset__eq,axiom,
% 9.20/5.20      ! [R: set_set_nat,S3: set_set_nat] :
% 9.20/5.20        ( ( ord_le3964352015994296041_nat_o
% 9.20/5.20          @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
% 9.20/5.20          @ ^ [X: set_nat] : ( member_set_nat @ X @ S3 ) )
% 9.20/5.20        = ( ord_le6893508408891458716et_nat @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6451_pred__subset__eq,axiom,
% 9.20/5.20      ! [R: set_nat,S3: set_nat] :
% 9.20/5.20        ( ( ord_less_eq_nat_o
% 9.20/5.20          @ ^ [X: nat] : ( member_nat @ X @ R )
% 9.20/5.20          @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
% 9.20/5.20        = ( ord_less_eq_set_nat @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6452_pred__subset__eq,axiom,
% 9.20/5.20      ! [R: set_int,S3: set_int] :
% 9.20/5.20        ( ( ord_less_eq_int_o
% 9.20/5.20          @ ^ [X: int] : ( member_int @ X @ R )
% 9.20/5.20          @ ^ [X: int] : ( member_int @ X @ S3 ) )
% 9.20/5.20        = ( ord_less_eq_set_int @ R @ S3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6453_lambda__one,axiom,
% 9.20/5.20      ( ( ^ [X: complex] : X )
% 9.20/5.20      = ( times_times_complex @ one_one_complex ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6454_lambda__one,axiom,
% 9.20/5.20      ( ( ^ [X: real] : X )
% 9.20/5.20      = ( times_times_real @ one_one_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6455_lambda__one,axiom,
% 9.20/5.20      ( ( ^ [X: rat] : X )
% 9.20/5.20      = ( times_times_rat @ one_one_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6456_lambda__one,axiom,
% 9.20/5.20      ( ( ^ [X: nat] : X )
% 9.20/5.20      = ( times_times_nat @ one_one_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6457_lambda__one,axiom,
% 9.20/5.20      ( ( ^ [X: int] : X )
% 9.20/5.20      = ( times_times_int @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6458_lambda__zero,axiom,
% 9.20/5.20      ( ( ^ [H: complex] : zero_zero_complex )
% 9.20/5.20      = ( times_times_complex @ zero_zero_complex ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6459_lambda__zero,axiom,
% 9.20/5.20      ( ( ^ [H: real] : zero_zero_real )
% 9.20/5.20      = ( times_times_real @ zero_zero_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6460_lambda__zero,axiom,
% 9.20/5.20      ( ( ^ [H: rat] : zero_zero_rat )
% 9.20/5.20      = ( times_times_rat @ zero_zero_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6461_lambda__zero,axiom,
% 9.20/5.20      ( ( ^ [H: nat] : zero_zero_nat )
% 9.20/5.20      = ( times_times_nat @ zero_zero_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6462_lambda__zero,axiom,
% 9.20/5.20      ( ( ^ [H: int] : zero_zero_int )
% 9.20/5.20      = ( times_times_int @ zero_zero_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6463_uminus__set__def,axiom,
% 9.20/5.20      ( uminus6524753893492686040at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( collec3392354462482085612at_nat
% 9.20/5.20            @ ( uminus8676089048583255045_nat_o
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6464_uminus__set__def,axiom,
% 9.20/5.20      ( uminus612125837232591019t_real
% 9.20/5.20      = ( ^ [A5: set_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ( uminus_uminus_real_o
% 9.20/5.20              @ ^ [X: real] : ( member_real @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6465_uminus__set__def,axiom,
% 9.20/5.20      ( uminus3195874150345416415st_nat
% 9.20/5.20      = ( ^ [A5: set_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ( uminus5770388063884162150_nat_o
% 9.20/5.20              @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6466_uminus__set__def,axiom,
% 9.20/5.20      ( uminus613421341184616069et_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ( uminus6401447641752708672_nat_o
% 9.20/5.20              @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6467_uminus__set__def,axiom,
% 9.20/5.20      ( uminus5710092332889474511et_nat
% 9.20/5.20      = ( ^ [A5: set_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ( uminus_uminus_nat_o
% 9.20/5.20              @ ^ [X: nat] : ( member_nat @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6468_uminus__set__def,axiom,
% 9.20/5.20      ( uminus1532241313380277803et_int
% 9.20/5.20      = ( ^ [A5: set_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ( uminus_uminus_int_o
% 9.20/5.20              @ ^ [X: int] : ( member_int @ X @ A5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6469_Collect__neg__eq,axiom,
% 9.20/5.20      ! [P: real > $o] :
% 9.20/5.20        ( ( collect_real
% 9.20/5.20          @ ^ [X: real] :
% 9.20/5.20              ~ ( P @ X ) )
% 9.20/5.20        = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6470_Collect__neg__eq,axiom,
% 9.20/5.20      ! [P: list_nat > $o] :
% 9.20/5.20        ( ( collect_list_nat
% 9.20/5.20          @ ^ [X: list_nat] :
% 9.20/5.20              ~ ( P @ X ) )
% 9.20/5.20        = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6471_Collect__neg__eq,axiom,
% 9.20/5.20      ! [P: set_nat > $o] :
% 9.20/5.20        ( ( collect_set_nat
% 9.20/5.20          @ ^ [X: set_nat] :
% 9.20/5.20              ~ ( P @ X ) )
% 9.20/5.20        = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6472_Collect__neg__eq,axiom,
% 9.20/5.20      ! [P: nat > $o] :
% 9.20/5.20        ( ( collect_nat
% 9.20/5.20          @ ^ [X: nat] :
% 9.20/5.20              ~ ( P @ X ) )
% 9.20/5.20        = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6473_Collect__neg__eq,axiom,
% 9.20/5.20      ! [P: int > $o] :
% 9.20/5.20        ( ( collect_int
% 9.20/5.20          @ ^ [X: int] :
% 9.20/5.20              ~ ( P @ X ) )
% 9.20/5.20        = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6474_Compl__eq,axiom,
% 9.20/5.20      ( uminus6524753893492686040at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( collec3392354462482085612at_nat
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                ~ ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6475_Compl__eq,axiom,
% 9.20/5.20      ( uminus612125837232591019t_real
% 9.20/5.20      = ( ^ [A5: set_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ^ [X: real] :
% 9.20/5.20                ~ ( member_real @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6476_Compl__eq,axiom,
% 9.20/5.20      ( uminus3195874150345416415st_nat
% 9.20/5.20      = ( ^ [A5: set_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ^ [X: list_nat] :
% 9.20/5.20                ~ ( member_list_nat @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6477_Compl__eq,axiom,
% 9.20/5.20      ( uminus613421341184616069et_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ^ [X: set_nat] :
% 9.20/5.20                ~ ( member_set_nat @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6478_Compl__eq,axiom,
% 9.20/5.20      ( uminus5710092332889474511et_nat
% 9.20/5.20      = ( ^ [A5: set_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ^ [X: nat] :
% 9.20/5.20                ~ ( member_nat @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6479_Compl__eq,axiom,
% 9.20/5.20      ( uminus1532241313380277803et_int
% 9.20/5.20      = ( ^ [A5: set_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ^ [X: int] :
% 9.20/5.20                ~ ( member_int @ X @ A5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6480_set__diff__eq,axiom,
% 9.20/5.20      ( minus_1356011639430497352at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( collec3392354462482085612at_nat
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                ( ( member8440522571783428010at_nat @ X @ A5 )
% 9.20/5.20                & ~ ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6481_set__diff__eq,axiom,
% 9.20/5.20      ( minus_minus_set_real
% 9.20/5.20      = ( ^ [A5: set_real,B5: set_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ^ [X: real] :
% 9.20/5.20                ( ( member_real @ X @ A5 )
% 9.20/5.20                & ~ ( member_real @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6482_set__diff__eq,axiom,
% 9.20/5.20      ( minus_7954133019191499631st_nat
% 9.20/5.20      = ( ^ [A5: set_list_nat,B5: set_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ^ [X: list_nat] :
% 9.20/5.20                ( ( member_list_nat @ X @ A5 )
% 9.20/5.20                & ~ ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6483_set__diff__eq,axiom,
% 9.20/5.20      ( minus_2163939370556025621et_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ^ [X: set_nat] :
% 9.20/5.20                ( ( member_set_nat @ X @ A5 )
% 9.20/5.20                & ~ ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6484_set__diff__eq,axiom,
% 9.20/5.20      ( minus_minus_set_int
% 9.20/5.20      = ( ^ [A5: set_int,B5: set_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ^ [X: int] :
% 9.20/5.20                ( ( member_int @ X @ A5 )
% 9.20/5.20                & ~ ( member_int @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6485_set__diff__eq,axiom,
% 9.20/5.20      ( minus_minus_set_nat
% 9.20/5.20      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ^ [X: nat] :
% 9.20/5.20                ( ( member_nat @ X @ A5 )
% 9.20/5.20                & ~ ( member_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6486_insert__compr,axiom,
% 9.20/5.20      ( insert9069300056098147895at_nat
% 9.20/5.20      = ( ^ [A3: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
% 9.20/5.20            ( collec6321179662152712658at_nat
% 9.20/5.20            @ ^ [X: produc3843707927480180839at_nat] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member8757157785044589968at_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6487_insert__compr,axiom,
% 9.20/5.20      ( insert8211810215607154385at_nat
% 9.20/5.20      = ( ^ [A3: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( collec3392354462482085612at_nat
% 9.20/5.20            @ ^ [X: product_prod_nat_nat] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6488_insert__compr,axiom,
% 9.20/5.20      ( insert_real
% 9.20/5.20      = ( ^ [A3: real,B5: set_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ^ [X: real] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member_real @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6489_insert__compr,axiom,
% 9.20/5.20      ( insert_list_nat
% 9.20/5.20      = ( ^ [A3: list_nat,B5: set_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ^ [X: list_nat] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6490_insert__compr,axiom,
% 9.20/5.20      ( insert_set_nat
% 9.20/5.20      = ( ^ [A3: set_nat,B5: set_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ^ [X: set_nat] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6491_insert__compr,axiom,
% 9.20/5.20      ( insert_nat
% 9.20/5.20      = ( ^ [A3: nat,B5: set_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ^ [X: nat] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6492_insert__compr,axiom,
% 9.20/5.20      ( insert_int
% 9.20/5.20      = ( ^ [A3: int,B5: set_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ^ [X: int] :
% 9.20/5.20                ( ( X = A3 )
% 9.20/5.20                | ( member_int @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6493_minus__set__def,axiom,
% 9.20/5.20      ( minus_1356011639430497352at_nat
% 9.20/5.20      = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
% 9.20/5.20            ( collec3392354462482085612at_nat
% 9.20/5.20            @ ( minus_2270307095948843157_nat_o
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
% 9.20/5.20              @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6494_minus__set__def,axiom,
% 9.20/5.20      ( minus_minus_set_real
% 9.20/5.20      = ( ^ [A5: set_real,B5: set_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ( minus_minus_real_o
% 9.20/5.20              @ ^ [X: real] : ( member_real @ X @ A5 )
% 9.20/5.20              @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6495_minus__set__def,axiom,
% 9.20/5.20      ( minus_7954133019191499631st_nat
% 9.20/5.20      = ( ^ [A5: set_list_nat,B5: set_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ( minus_1139252259498527702_nat_o
% 9.20/5.20              @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
% 9.20/5.20              @ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6496_minus__set__def,axiom,
% 9.20/5.20      ( minus_2163939370556025621et_nat
% 9.20/5.20      = ( ^ [A5: set_set_nat,B5: set_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ( minus_6910147592129066416_nat_o
% 9.20/5.20              @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
% 9.20/5.20              @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6497_minus__set__def,axiom,
% 9.20/5.20      ( minus_minus_set_int
% 9.20/5.20      = ( ^ [A5: set_int,B5: set_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ( minus_minus_int_o
% 9.20/5.20              @ ^ [X: int] : ( member_int @ X @ A5 )
% 9.20/5.20              @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6498_minus__set__def,axiom,
% 9.20/5.20      ( minus_minus_set_nat
% 9.20/5.20      = ( ^ [A5: set_nat,B5: set_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ( minus_minus_nat_o
% 9.20/5.20              @ ^ [X: nat] : ( member_nat @ X @ A5 )
% 9.20/5.20              @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6499_insert__Collect,axiom,
% 9.20/5.20      ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
% 9.20/5.20        = ( collec3392354462482085612at_nat
% 9.20/5.20          @ ^ [U2: product_prod_nat_nat] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6500_insert__Collect,axiom,
% 9.20/5.20      ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
% 9.20/5.20        ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
% 9.20/5.20        = ( collec6321179662152712658at_nat
% 9.20/5.20          @ ^ [U2: produc3843707927480180839at_nat] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6501_insert__Collect,axiom,
% 9.20/5.20      ! [A: real,P: real > $o] :
% 9.20/5.20        ( ( insert_real @ A @ ( collect_real @ P ) )
% 9.20/5.20        = ( collect_real
% 9.20/5.20          @ ^ [U2: real] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6502_insert__Collect,axiom,
% 9.20/5.20      ! [A: list_nat,P: list_nat > $o] :
% 9.20/5.20        ( ( insert_list_nat @ A @ ( collect_list_nat @ P ) )
% 9.20/5.20        = ( collect_list_nat
% 9.20/5.20          @ ^ [U2: list_nat] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6503_insert__Collect,axiom,
% 9.20/5.20      ! [A: set_nat,P: set_nat > $o] :
% 9.20/5.20        ( ( insert_set_nat @ A @ ( collect_set_nat @ P ) )
% 9.20/5.20        = ( collect_set_nat
% 9.20/5.20          @ ^ [U2: set_nat] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6504_insert__Collect,axiom,
% 9.20/5.20      ! [A: nat,P: nat > $o] :
% 9.20/5.20        ( ( insert_nat @ A @ ( collect_nat @ P ) )
% 9.20/5.20        = ( collect_nat
% 9.20/5.20          @ ^ [U2: nat] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6505_insert__Collect,axiom,
% 9.20/5.20      ! [A: int,P: int > $o] :
% 9.20/5.20        ( ( insert_int @ A @ ( collect_int @ P ) )
% 9.20/5.20        = ( collect_int
% 9.20/5.20          @ ^ [U2: int] :
% 9.20/5.20              ( ( U2 != A )
% 9.20/5.20             => ( P @ U2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6506_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: real > $o] :
% 9.20/5.20        ( ~ ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ? [X_1: real] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6507_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: list_nat > $o] :
% 9.20/5.20        ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
% 9.20/5.20       => ? [X_1: list_nat] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6508_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: set_nat > $o] :
% 9.20/5.20        ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
% 9.20/5.20       => ? [X_1: set_nat] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6509_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: nat > $o] :
% 9.20/5.20        ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ? [X_1: nat] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6510_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: int > $o] :
% 9.20/5.20        ( ~ ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20       => ? [X_1: int] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6511_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: complex > $o] :
% 9.20/5.20        ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
% 9.20/5.20       => ? [X_1: complex] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6512_not__finite__existsD,axiom,
% 9.20/5.20      ! [P: product_prod_nat_nat > $o] :
% 9.20/5.20        ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
% 9.20/5.20       => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6513_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_real,B2: set_nat,R: real > nat > $o] :
% 9.20/5.20        ( ~ ( finite_finite_real @ A2 )
% 9.20/5.20       => ( ( finite_finite_nat @ B2 )
% 9.20/5.20         => ( ! [X4: real] :
% 9.20/5.20                ( ( member_real @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: nat] :
% 9.20/5.20                    ( ( member_nat @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_real
% 9.20/5.20                    @ ( collect_real
% 9.20/5.20                      @ ^ [A3: real] :
% 9.20/5.20                          ( ( member_real @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6514_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_real,B2: set_int,R: real > int > $o] :
% 9.20/5.20        ( ~ ( finite_finite_real @ A2 )
% 9.20/5.20       => ( ( finite_finite_int @ B2 )
% 9.20/5.20         => ( ! [X4: real] :
% 9.20/5.20                ( ( member_real @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: int] :
% 9.20/5.20                    ( ( member_int @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_real
% 9.20/5.20                    @ ( collect_real
% 9.20/5.20                      @ ^ [A3: real] :
% 9.20/5.20                          ( ( member_real @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6515_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_real,B2: set_complex,R: real > complex > $o] :
% 9.20/5.20        ( ~ ( finite_finite_real @ A2 )
% 9.20/5.20       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.20         => ( ! [X4: real] :
% 9.20/5.20                ( ( member_real @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: complex] :
% 9.20/5.20                    ( ( member_complex @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: complex] :
% 9.20/5.20                ( ( member_complex @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_real
% 9.20/5.20                    @ ( collect_real
% 9.20/5.20                      @ ^ [A3: real] :
% 9.20/5.20                          ( ( member_real @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6516_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
% 9.20/5.20        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( ( finite_finite_nat @ B2 )
% 9.20/5.20         => ( ! [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: nat] :
% 9.20/5.20                    ( ( member_nat @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_nat
% 9.20/5.20                    @ ( collect_nat
% 9.20/5.20                      @ ^ [A3: nat] :
% 9.20/5.20                          ( ( member_nat @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6517_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
% 9.20/5.20        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( ( finite_finite_int @ B2 )
% 9.20/5.20         => ( ! [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: int] :
% 9.20/5.20                    ( ( member_int @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_nat
% 9.20/5.20                    @ ( collect_nat
% 9.20/5.20                      @ ^ [A3: nat] :
% 9.20/5.20                          ( ( member_nat @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6518_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
% 9.20/5.20        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.20         => ( ! [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: complex] :
% 9.20/5.20                    ( ( member_complex @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: complex] :
% 9.20/5.20                ( ( member_complex @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_nat
% 9.20/5.20                    @ ( collect_nat
% 9.20/5.20                      @ ^ [A3: nat] :
% 9.20/5.20                          ( ( member_nat @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6519_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
% 9.20/5.20        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.20       => ( ( finite_finite_nat @ B2 )
% 9.20/5.20         => ( ! [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: nat] :
% 9.20/5.20                    ( ( member_nat @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_int
% 9.20/5.20                    @ ( collect_int
% 9.20/5.20                      @ ^ [A3: int] :
% 9.20/5.20                          ( ( member_int @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6520_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_int,B2: set_int,R: int > int > $o] :
% 9.20/5.20        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.20       => ( ( finite_finite_int @ B2 )
% 9.20/5.20         => ( ! [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: int] :
% 9.20/5.20                    ( ( member_int @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_int
% 9.20/5.20                    @ ( collect_int
% 9.20/5.20                      @ ^ [A3: int] :
% 9.20/5.20                          ( ( member_int @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6521_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_int,B2: set_complex,R: int > complex > $o] :
% 9.20/5.20        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.20       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.20         => ( ! [X4: int] :
% 9.20/5.20                ( ( member_int @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: complex] :
% 9.20/5.20                    ( ( member_complex @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: complex] :
% 9.20/5.20                ( ( member_complex @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite_finite_int
% 9.20/5.20                    @ ( collect_int
% 9.20/5.20                      @ ^ [A3: int] :
% 9.20/5.20                          ( ( member_int @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6522_pigeonhole__infinite__rel,axiom,
% 9.20/5.20      ! [A2: set_complex,B2: set_nat,R: complex > nat > $o] :
% 9.20/5.20        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.20       => ( ( finite_finite_nat @ B2 )
% 9.20/5.20         => ( ! [X4: complex] :
% 9.20/5.20                ( ( member_complex @ X4 @ A2 )
% 9.20/5.20               => ? [Xa: nat] :
% 9.20/5.20                    ( ( member_nat @ Xa @ B2 )
% 9.20/5.20                    & ( R @ X4 @ Xa ) ) )
% 9.20/5.20           => ? [X4: nat] :
% 9.20/5.20                ( ( member_nat @ X4 @ B2 )
% 9.20/5.20                & ~ ( finite3207457112153483333omplex
% 9.20/5.20                    @ ( collect_complex
% 9.20/5.20                      @ ^ [A3: complex] :
% 9.20/5.20                          ( ( member_complex @ A3 @ A2 )
% 9.20/5.20                          & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6523_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o,F: real > real > real] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [Uu3: real] :
% 9.20/5.20                ? [X: real,Y: real] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6524_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o,F: real > real > nat] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [Uu3: nat] :
% 9.20/5.20                ? [X: real,Y: real] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6525_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o,F: real > real > int] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [Uu3: int] :
% 9.20/5.20                ? [X: real,Y: real] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6526_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: real > $o,F: real > real > complex] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_real @ ( collect_real @ Q ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [Uu3: complex] :
% 9.20/5.20                ? [X: real,Y: real] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6527_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [Uu3: real] :
% 9.20/5.20                ? [X: real,Y: nat] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6528_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [Uu3: nat] :
% 9.20/5.20                ? [X: real,Y: nat] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6529_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [Uu3: int] :
% 9.20/5.20                ? [X: real,Y: nat] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6530_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [Uu3: complex] :
% 9.20/5.20                ? [X: real,Y: nat] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6531_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: int > $o,F: real > int > real] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [Uu3: real] :
% 9.20/5.20                ? [X: real,Y: int] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6532_finite__image__set2,axiom,
% 9.20/5.20      ! [P: real > $o,Q: int > $o,F: real > int > nat] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( ( finite_finite_int @ ( collect_int @ Q ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [Uu3: nat] :
% 9.20/5.20                ? [X: real,Y: int] :
% 9.20/5.20                  ( ( Uu3
% 9.20/5.20                    = ( F @ X @ Y ) )
% 9.20/5.20                  & ( P @ X )
% 9.20/5.20                  & ( Q @ Y ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6533_finite__image__set,axiom,
% 9.20/5.20      ! [P: real > $o,F: real > real] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [Uu3: real] :
% 9.20/5.20              ? [X: real] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6534_finite__image__set,axiom,
% 9.20/5.20      ! [P: real > $o,F: real > nat] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [Uu3: nat] :
% 9.20/5.20              ? [X: real] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6535_finite__image__set,axiom,
% 9.20/5.20      ! [P: real > $o,F: real > int] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [Uu3: int] :
% 9.20/5.20              ? [X: real] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6536_finite__image__set,axiom,
% 9.20/5.20      ! [P: real > $o,F: real > complex] :
% 9.20/5.20        ( ( finite_finite_real @ ( collect_real @ P ) )
% 9.20/5.20       => ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [Uu3: complex] :
% 9.20/5.20              ? [X: real] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6537_finite__image__set,axiom,
% 9.20/5.20      ! [P: nat > $o,F: nat > real] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [Uu3: real] :
% 9.20/5.20              ? [X: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6538_finite__image__set,axiom,
% 9.20/5.20      ! [P: nat > $o,F: nat > nat] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [Uu3: nat] :
% 9.20/5.20              ? [X: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6539_finite__image__set,axiom,
% 9.20/5.20      ! [P: nat > $o,F: nat > int] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [Uu3: int] :
% 9.20/5.20              ? [X: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6540_finite__image__set,axiom,
% 9.20/5.20      ! [P: nat > $o,F: nat > complex] :
% 9.20/5.20        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.20       => ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [Uu3: complex] :
% 9.20/5.20              ? [X: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6541_finite__image__set,axiom,
% 9.20/5.20      ! [P: int > $o,F: int > real] :
% 9.20/5.20        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20       => ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [Uu3: real] :
% 9.20/5.20              ? [X: int] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6542_finite__image__set,axiom,
% 9.20/5.20      ! [P: int > $o,F: int > nat] :
% 9.20/5.20        ( ( finite_finite_int @ ( collect_int @ P ) )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [Uu3: nat] :
% 9.20/5.20              ? [X: int] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( F @ X ) )
% 9.20/5.20                & ( P @ X ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6543_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bot_int_int_o
% 9.20/5.20      = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6544_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bo5358457235160185703eger_o
% 9.20/5.20      = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6545_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bo3000040243691356879eger_o
% 9.20/5.20      = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6546_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bo8662317086119403298_int_o
% 9.20/5.20      = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6547_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bo1403522918969695512_int_o
% 9.20/5.20      = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6548_bot__empty__eq2,axiom,
% 9.20/5.20      ( bot_bot_nat_nat_o
% 9.20/5.20      = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6549_finite__M__bounded__by__nat,axiom,
% 9.20/5.20      ! [P: nat > $o,I: nat] :
% 9.20/5.20        ( finite_finite_nat
% 9.20/5.20        @ ( collect_nat
% 9.20/5.20          @ ^ [K3: nat] :
% 9.20/5.20              ( ( P @ K3 )
% 9.20/5.20              & ( ord_less_nat @ K3 @ I ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6550_finite__less__ub,axiom,
% 9.20/5.20      ! [F: nat > nat,U: nat] :
% 9.20/5.20        ( ! [N3: nat] : ( ord_less_eq_nat @ N3 @ ( F @ N3 ) )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ U ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6551_numeral__code_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6552_numeral__code_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_real @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6553_numeral__code_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6554_numeral__code_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6555_numeral__code_I2_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_int @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6556_set__vebt__def,axiom,
% 9.20/5.20      ( vEBT_set_vebt
% 9.20/5.20      = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6557_finite__divisors__int,axiom,
% 9.20/5.20      ! [I: int] :
% 9.20/5.20        ( ( I != zero_zero_int )
% 9.20/5.20       => ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6558_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numera6690914467698888265omplex @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_plus_complex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) @ one_one_complex ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6559_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numera1916890842035813515d_enat @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) @ one_on7984719198319812577d_enat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6560_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_real @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_plus_real @ ( plus_plus_real @ ( numeral_numeral_real @ N2 ) @ ( numeral_numeral_real @ N2 ) ) @ one_one_real ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6561_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6562_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_nat @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ N2 ) ) @ one_one_nat ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6563_numeral__code_I3_J,axiom,
% 9.20/5.20      ! [N2: num] :
% 9.20/5.20        ( ( numeral_numeral_int @ ( bit1 @ N2 ) )
% 9.20/5.20        = ( plus_plus_int @ ( plus_plus_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ N2 ) ) @ one_one_int ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6564_finite__divisors__nat,axiom,
% 9.20/5.20      ! [M: nat] :
% 9.20/5.20        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.20       => ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6565_set__conv__nth,axiom,
% 9.20/5.20      ( set_real2
% 9.20/5.20      = ( ^ [Xs2: list_real] :
% 9.20/5.20            ( collect_real
% 9.20/5.20            @ ^ [Uu3: real] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_real @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6566_set__conv__nth,axiom,
% 9.20/5.20      ( set_list_nat2
% 9.20/5.20      = ( ^ [Xs2: list_list_nat] :
% 9.20/5.20            ( collect_list_nat
% 9.20/5.20            @ ^ [Uu3: list_nat] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_list_nat @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6567_set__conv__nth,axiom,
% 9.20/5.20      ( set_set_nat2
% 9.20/5.20      = ( ^ [Xs2: list_set_nat] :
% 9.20/5.20            ( collect_set_nat
% 9.20/5.20            @ ^ [Uu3: set_nat] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_set_nat @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6568_set__conv__nth,axiom,
% 9.20/5.20      ( set_VEBT_VEBT2
% 9.20/5.20      = ( ^ [Xs2: list_VEBT_VEBT] :
% 9.20/5.20            ( collect_VEBT_VEBT
% 9.20/5.20            @ ^ [Uu3: vEBT_VEBT] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6569_set__conv__nth,axiom,
% 9.20/5.20      ( set_o2
% 9.20/5.20      = ( ^ [Xs2: list_o] :
% 9.20/5.20            ( collect_o
% 9.20/5.20            @ ^ [Uu3: $o] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_o @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6570_set__conv__nth,axiom,
% 9.20/5.20      ( set_nat2
% 9.20/5.20      = ( ^ [Xs2: list_nat] :
% 9.20/5.20            ( collect_nat
% 9.20/5.20            @ ^ [Uu3: nat] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_nat @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6571_set__conv__nth,axiom,
% 9.20/5.20      ( set_int2
% 9.20/5.20      = ( ^ [Xs2: list_int] :
% 9.20/5.20            ( collect_int
% 9.20/5.20            @ ^ [Uu3: int] :
% 9.20/5.20              ? [I2: nat] :
% 9.20/5.20                ( ( Uu3
% 9.20/5.20                  = ( nth_int @ Xs2 @ I2 ) )
% 9.20/5.20                & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6572_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_complex,N2: nat] :
% 9.20/5.20        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.20       => ( finite8712137658972009173omplex
% 9.20/5.20          @ ( collect_list_complex
% 9.20/5.20            @ ^ [Xs2: list_complex] :
% 9.20/5.20                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_s3451745648224563538omplex @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6573_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
% 9.20/5.20        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.20       => ( finite500796754983035824at_nat
% 9.20/5.20          @ ( collec3343600615725829874at_nat
% 9.20/5.20            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.20/5.20                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_s5460976970255530739at_nat @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6574_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_VEBT_VEBT,N2: nat] :
% 9.20/5.20        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.20/5.20       => ( finite3004134309566078307T_VEBT
% 9.20/5.20          @ ( collec5608196760682091941T_VEBT
% 9.20/5.20            @ ^ [Xs2: list_VEBT_VEBT] :
% 9.20/5.20                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6575_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_o,N2: nat] :
% 9.20/5.20        ( ( finite_finite_o @ A2 )
% 9.20/5.20       => ( finite_finite_list_o
% 9.20/5.20          @ ( collect_list_o
% 9.20/5.20            @ ^ [Xs2: list_o] :
% 9.20/5.20                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_size_list_o @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6576_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_nat,N2: nat] :
% 9.20/5.20        ( ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( finite8100373058378681591st_nat
% 9.20/5.20          @ ( collect_list_nat
% 9.20/5.20            @ ^ [Xs2: list_nat] :
% 9.20/5.20                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_size_list_nat @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6577_finite__lists__length__eq,axiom,
% 9.20/5.20      ! [A2: set_int,N2: nat] :
% 9.20/5.20        ( ( finite_finite_int @ A2 )
% 9.20/5.20       => ( finite3922522038869484883st_int
% 9.20/5.20          @ ( collect_list_int
% 9.20/5.20            @ ^ [Xs2: list_int] :
% 9.20/5.20                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ( size_size_list_int @ Xs2 )
% 9.20/5.20                  = N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6578_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_complex,N2: nat] :
% 9.20/5.20        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.20       => ( finite8712137658972009173omplex
% 9.20/5.20          @ ( collect_list_complex
% 9.20/5.20            @ ^ [Xs2: list_complex] :
% 9.20/5.20                ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6579_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
% 9.20/5.20        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.20       => ( finite500796754983035824at_nat
% 9.20/5.20          @ ( collec3343600615725829874at_nat
% 9.20/5.20            @ ^ [Xs2: list_P6011104703257516679at_nat] :
% 9.20/5.20                ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6580_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_VEBT_VEBT,N2: nat] :
% 9.20/5.20        ( ( finite5795047828879050333T_VEBT @ A2 )
% 9.20/5.20       => ( finite3004134309566078307T_VEBT
% 9.20/5.20          @ ( collec5608196760682091941T_VEBT
% 9.20/5.20            @ ^ [Xs2: list_VEBT_VEBT] :
% 9.20/5.20                ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6581_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_o,N2: nat] :
% 9.20/5.20        ( ( finite_finite_o @ A2 )
% 9.20/5.20       => ( finite_finite_list_o
% 9.20/5.20          @ ( collect_list_o
% 9.20/5.20            @ ^ [Xs2: list_o] :
% 9.20/5.20                ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6582_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_nat,N2: nat] :
% 9.20/5.20        ( ( finite_finite_nat @ A2 )
% 9.20/5.20       => ( finite8100373058378681591st_nat
% 9.20/5.20          @ ( collect_list_nat
% 9.20/5.20            @ ^ [Xs2: list_nat] :
% 9.20/5.20                ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6583_finite__lists__length__le,axiom,
% 9.20/5.20      ! [A2: set_int,N2: nat] :
% 9.20/5.20        ( ( finite_finite_int @ A2 )
% 9.20/5.20       => ( finite3922522038869484883st_int
% 9.20/5.20          @ ( collect_list_int
% 9.20/5.20            @ ^ [Xs2: list_int] :
% 9.20/5.20                ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
% 9.20/5.20                & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6584_vebt__buildup_Osimps_I3_J,axiom,
% 9.20/5.20      ! [Va: nat] :
% 9.20/5.20        ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.20/5.20         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.20/5.20            = ( 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 ) ) ) ) ) ) )
% 9.20/5.20        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
% 9.20/5.20         => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
% 9.20/5.20            = ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6585_vebt__buildup_Oelims,axiom,
% 9.20/5.20      ! [X3: nat,Y3: vEBT_VEBT] :
% 9.20/5.20        ( ( ( vEBT_vebt_buildup @ X3 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( ( X3 = zero_zero_nat )
% 9.20/5.20           => ( Y3
% 9.20/5.20             != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.20/5.20         => ( ( ( X3
% 9.20/5.20                = ( suc @ zero_zero_nat ) )
% 9.20/5.20             => ( Y3
% 9.20/5.20               != ( vEBT_Leaf @ $false @ $false ) ) )
% 9.20/5.20           => ~ ! [Va2: nat] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                 => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                       => ( Y3
% 9.20/5.20                          = ( 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 ) ) ) ) ) ) )
% 9.20/5.20                      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                       => ( Y3
% 9.20/5.20                          = ( 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 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6586_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
% 9.20/5.20      ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
% 9.20/5.20        ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
% 9.20/5.20        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.20/5.20           => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.20          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6587_dbl__dec__def,axiom,
% 9.20/5.20      ( neg_nu6511756317524482435omplex
% 9.20/5.20      = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6588_dbl__dec__def,axiom,
% 9.20/5.20      ( neg_nu6075765906172075777c_real
% 9.20/5.20      = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6589_dbl__dec__def,axiom,
% 9.20/5.20      ( neg_nu3179335615603231917ec_rat
% 9.20/5.20      = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6590_dbl__dec__def,axiom,
% 9.20/5.20      ( neg_nu3811975205180677377ec_int
% 9.20/5.20      = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6591_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
% 9.20/5.20      ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
% 9.20/5.20        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X3 )
% 9.20/5.20        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.20/5.20           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.20          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6592_vebt__member_Osimps_I5_J,axiom,
% 9.20/5.20      ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
% 9.20/5.20        ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
% 9.20/5.20        = ( ( X3 != Mi )
% 9.20/5.20         => ( ( X3 != Ma )
% 9.20/5.20           => ( ~ ( ord_less_nat @ X3 @ Mi )
% 9.20/5.20              & ( ~ ( ord_less_nat @ X3 @ Mi )
% 9.20/5.20               => ( ~ ( ord_less_nat @ Ma @ X3 )
% 9.20/5.20                  & ( ~ ( ord_less_nat @ Ma @ X3 )
% 9.20/5.20                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.20/5.20                       => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.20                      & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6593_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
% 9.20/5.20      ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
% 9.20/5.20        ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
% 9.20/5.20        = ( ( X3 = Mi )
% 9.20/5.20          | ( X3 = Ma )
% 9.20/5.20          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
% 9.20/5.20             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.20            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6594_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ( Y3
% 9.20/5.20                = ( ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) ) )
% 9.20/5.20         => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( X3
% 9.20/5.20                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.20/5.20             => Y3 )
% 9.20/5.20           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                  ( ? [S2: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20                 => ( Y3
% 9.20/5.20                    = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6595_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                   => A4 )
% 9.20/5.20                  & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                   => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                       => B4 )
% 9.20/5.20                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                ( ? [S2: vEBT_VEBT] :
% 9.20/5.20                    ( X3
% 9.20/5.20                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                     => ( 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 ) ) ) ) ) )
% 9.20/5.20                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6596_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                 => A4 )
% 9.20/5.20                & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                 => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                     => B4 )
% 9.20/5.20                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                ( X3
% 9.20/5.20               != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.20/5.20           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                  ( ? [S2: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                     => ( 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 ) ) ) ) ) )
% 9.20/5.20                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6597_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [Mi2: nat,Ma2: nat] :
% 9.20/5.20              ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                  ( X3
% 9.20/5.20                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20             => ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                  | ( Xa2 = Ma2 ) ) )
% 9.20/5.20         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                ( ? [Vc2: vEBT_VEBT] :
% 9.20/5.20                    ( X3
% 9.20/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20               => ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                    | ( Xa2 = Ma2 )
% 9.20/5.20                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                       => ( 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 ) ) ) ) ) )
% 9.20/5.20                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.20/5.20           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                  ( ? [Vd: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                       => ( 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 ) ) ) ) ) )
% 9.20/5.20                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6598_xor__nat__unfold,axiom,
% 9.20/5.20      ( bit_se6528837805403552850or_nat
% 9.20/5.20      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = 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 @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6599_xor__nat__rec,axiom,
% 9.20/5.20      ( bit_se6528837805403552850or_nat
% 9.20/5.20      = ( ^ [M2: nat,N: nat] :
% 9.20/5.20            ( plus_plus_nat
% 9.20/5.20            @ ( zero_n2687167440665602831ol_nat
% 9.20/5.20              @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
% 9.20/5.20               != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
% 9.20/5.20            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6600_one__xor__eq,axiom,
% 9.20/5.20      ! [A: code_integer] :
% 9.20/5.20        ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
% 9.20/5.20        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n356916108424825756nteger
% 9.20/5.20            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6601_one__xor__eq,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
% 9.20/5.20        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n2687167440665602831ol_nat
% 9.20/5.20            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6602_one__xor__eq,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
% 9.20/5.20        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n2684676970156552555ol_int
% 9.20/5.20            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6603_xor__one__eq,axiom,
% 9.20/5.20      ! [A: code_integer] :
% 9.20/5.20        ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
% 9.20/5.20        = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n356916108424825756nteger
% 9.20/5.20            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6604_xor__one__eq,axiom,
% 9.20/5.20      ! [A: nat] :
% 9.20/5.20        ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
% 9.20/5.20        = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n2687167440665602831ol_nat
% 9.20/5.20            @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6605_xor__one__eq,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
% 9.20/5.20        = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
% 9.20/5.20          @ ( zero_n2684676970156552555ol_int
% 9.20/5.20            @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6606_vebt__member_Oelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                   => A4 )
% 9.20/5.20                  & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                   => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                       => B4 )
% 9.20/5.20                      & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20         => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                ( ? [Summary2: vEBT_VEBT] :
% 9.20/5.20                    ( X3
% 9.20/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20               => ~ ( ( Xa2 != Mi2 )
% 9.20/5.20                   => ( ( Xa2 != Ma2 )
% 9.20/5.20                     => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                        & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                         => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                            & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                             => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                 => ( 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 ) ) ) ) ) )
% 9.20/5.20                                & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6607_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.20              ( X3
% 9.20/5.20             != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.20         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.20                ( X3
% 9.20/5.20               != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.20/5.20           => ( ! [Mi2: nat,Ma2: nat] :
% 9.20/5.20                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20                 => ( ( Xa2 = Mi2 )
% 9.20/5.20                    | ( Xa2 = Ma2 ) ) )
% 9.20/5.20             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                    ( ? [Vc2: vEBT_VEBT] :
% 9.20/5.20                        ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20                   => ( ( Xa2 = Mi2 )
% 9.20/5.20                      | ( Xa2 = Ma2 )
% 9.20/5.20                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                         => ( 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 ) ) ) ) ) )
% 9.20/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.20/5.20               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                      ( ? [Vd: vEBT_VEBT] :
% 9.20/5.20                          ( X3
% 9.20/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                         => ( 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 ) ) ) ) ) )
% 9.20/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6608_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.20/5.20                ( X3
% 9.20/5.20                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.20           => Y3 )
% 9.20/5.20         => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.20                  ( X3
% 9.20/5.20                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.20/5.20             => Y3 )
% 9.20/5.20           => ( ! [Mi2: nat,Ma2: nat] :
% 9.20/5.20                  ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                      ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20                 => ( Y3
% 9.20/5.20                    = ( ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                          | ( Xa2 = Ma2 ) ) ) ) )
% 9.20/5.20             => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                    ( ? [Vc2: vEBT_VEBT] :
% 9.20/5.20                        ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20                   => ( Y3
% 9.20/5.20                      = ( ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                            | ( Xa2 = Ma2 )
% 9.20/5.20                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                               => ( 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 ) ) ) ) ) )
% 9.20/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
% 9.20/5.20               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                      ( ? [Vd: vEBT_VEBT] :
% 9.20/5.20                          ( X3
% 9.20/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                     => ( Y3
% 9.20/5.20                        = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                               => ( 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 ) ) ) ) ) )
% 9.20/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6609_bit__horner__sum__bit__iff,axiom,
% 9.20/5.20      ! [Bs: list_o,N2: nat] :
% 9.20/5.20        ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
% 9.20/5.20        = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
% 9.20/5.20          & ( nth_o @ Bs @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6610_bit__horner__sum__bit__iff,axiom,
% 9.20/5.20      ! [Bs: list_o,N2: nat] :
% 9.20/5.20        ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
% 9.20/5.20        = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
% 9.20/5.20          & ( nth_o @ Bs @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6611_vebt__member_Oelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20       => ( ! [A4: $o,B4: $o] :
% 9.20/5.20              ( ( X3
% 9.20/5.20                = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20             => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                 => A4 )
% 9.20/5.20                & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                 => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                     => B4 )
% 9.20/5.20                    & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                ( X3
% 9.20/5.20               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.20           => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.20                  ( X3
% 9.20/5.20                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.20/5.20             => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                    ( X3
% 9.20/5.20                   != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.20/5.20               => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
% 9.20/5.20                      ( ? [Summary2: vEBT_VEBT] :
% 9.20/5.20                          ( X3
% 9.20/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20                     => ( ( Xa2 != Mi2 )
% 9.20/5.20                       => ( ( Xa2 != Ma2 )
% 9.20/5.20                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                     => ( 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 ) ) ) ) ) )
% 9.20/5.20                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6612_of__int__code__if,axiom,
% 9.20/5.20      ( ring_17405671764205052669omplex
% 9.20/5.20      = ( ^ [K3: int] :
% 9.20/5.20            ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
% 9.20/5.20            @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.20              @ ( if_complex
% 9.20/5.20                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.20                  = zero_zero_int )
% 9.20/5.20                @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20                @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6613_of__int__code__if,axiom,
% 9.20/5.20      ( ring_1_of_int_int
% 9.20/5.20      = ( ^ [K3: int] :
% 9.20/5.20            ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
% 9.20/5.20            @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.20              @ ( if_int
% 9.20/5.20                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.20                  = zero_zero_int )
% 9.20/5.20                @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20                @ ( 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 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6614_of__int__code__if,axiom,
% 9.20/5.20      ( ring_1_of_int_real
% 9.20/5.20      = ( ^ [K3: int] :
% 9.20/5.20            ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
% 9.20/5.20            @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.20              @ ( if_real
% 9.20/5.20                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.20                  = zero_zero_int )
% 9.20/5.20                @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20                @ ( 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 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6615_of__int__code__if,axiom,
% 9.20/5.20      ( ring_1_of_int_rat
% 9.20/5.20      = ( ^ [K3: int] :
% 9.20/5.20            ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
% 9.20/5.20            @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.20              @ ( if_rat
% 9.20/5.20                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.20                  = zero_zero_int )
% 9.20/5.20                @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20                @ ( 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 ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6616_of__int__code__if,axiom,
% 9.20/5.20      ( ring_18347121197199848620nteger
% 9.20/5.20      = ( ^ [K3: int] :
% 9.20/5.20            ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.20/5.20            @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.20              @ ( if_Code_integer
% 9.20/5.20                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.20                  = zero_zero_int )
% 9.20/5.20                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.20                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6617_finite__nth__roots,axiom,
% 9.20/5.20      ! [N2: nat,C: complex] :
% 9.20/5.20        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.20       => ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [Z6: complex] :
% 9.20/5.20                ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.20                = C ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6618_finite__roots__unity,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 9.20/5.20       => ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [Z6: real] :
% 9.20/5.20                ( ( power_power_real @ Z6 @ N2 )
% 9.20/5.20                = one_one_real ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6619_finite__roots__unity,axiom,
% 9.20/5.20      ! [N2: nat] :
% 9.20/5.20        ( ( ord_less_eq_nat @ one_one_nat @ N2 )
% 9.20/5.20       => ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [Z6: complex] :
% 9.20/5.20                ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.20                = one_one_complex ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6620_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6621_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6622_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 9.20/5.20        ( ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [I2: int] :
% 9.20/5.20                ( ( member_int @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6623_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 9.20/5.20        ( ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [I2: complex] :
% 9.20/5.20                ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_complex ) ) ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_complex ) ) ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6624_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_real ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6625_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_real ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6626_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 9.20/5.20        ( ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [I2: int] :
% 9.20/5.20                ( ( member_int @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_real ) ) ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6627_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 9.20/5.20        ( ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [I2: complex] :
% 9.20/5.20                ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_real ) ) ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_real ) ) ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6628_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_rat ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_rat ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_rat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6629_prod_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != one_one_rat ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != one_one_rat ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != one_one_rat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6630_vebt__member_Opelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( Y3
% 9.20/5.20                    = ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 9.20/5.20           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.20                 => ( ~ Y3
% 9.20/5.20                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.20/5.20             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.20/5.20                   => ( ~ Y3
% 9.20/5.20                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
% 9.20/5.20               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                      ( ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.20/5.20                     => ( ~ Y3
% 9.20/5.20                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
% 9.20/5.20                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.20                        ( ( X3
% 9.20/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20                       => ( ( Y3
% 9.20/5.20                            = ( ( Xa2 != Mi2 )
% 9.20/5.20                             => ( ( Xa2 != Ma2 )
% 9.20/5.20                               => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                  & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                   => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                      & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) )
% 9.20/5.20                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6631_vebt__member_Opelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 9.20/5.20                 => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                     => A4 )
% 9.20/5.20                    & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                     => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                         => B4 )
% 9.20/5.20                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.20/5.20           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.20/5.20             => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
% 9.20/5.20                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ Xa2 ) ) )
% 9.20/5.20               => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                      ( ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
% 9.20/5.20                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
% 9.20/5.20                 => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.20                        ( ( X3
% 9.20/5.20                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.20/5.20                         => ( ( Xa2 != Mi2 )
% 9.20/5.20                           => ( ( Xa2 != Ma2 )
% 9.20/5.20                             => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                                 => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                    & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                         => ( 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 ) ) ) ) ) )
% 9.20/5.20                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6632_xor__negative__int__iff,axiom,
% 9.20/5.20      ! [K: int,L: int] :
% 9.20/5.20        ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
% 9.20/5.20        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.20         != ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6633_XOR__upper,axiom,
% 9.20/5.20      ! [X3: int,N2: nat,Y3: int] :
% 9.20/5.20        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.20       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.20         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.20           => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6634_vebt__member_Opelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_vebt_member @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 9.20/5.20                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.20/5.20           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
% 9.20/5.20                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.20/5.20                   => ~ ( ( Xa2 != Mi2 )
% 9.20/5.20                       => ( ( Xa2 != Ma2 )
% 9.20/5.20                         => ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                            & ( ~ ( ord_less_nat @ Xa2 @ Mi2 )
% 9.20/5.20                             => ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                & ( ~ ( ord_less_nat @ Ma2 @ Xa2 )
% 9.20/5.20                                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                                     => ( 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 ) ) ) ) ) )
% 9.20/5.20                                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6635_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > complex,Y3: real > complex] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6636_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6637_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_int,X3: int > complex,Y3: int > complex] :
% 9.20/5.20        ( ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [I2: int] :
% 9.20/5.20                ( ( member_int @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_complex ) ) ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_complex ) ) ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6638_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
% 9.20/5.20        ( ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [I2: complex] :
% 9.20/5.20                ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_complex ) ) ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_complex ) ) ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_complex ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6639_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > real,Y3: real > real] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_real ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6640_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_real ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6641_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_int,X3: int > real,Y3: int > real] :
% 9.20/5.20        ( ( finite_finite_int
% 9.20/5.20          @ ( collect_int
% 9.20/5.20            @ ^ [I2: int] :
% 9.20/5.20                ( ( member_int @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_real ) ) ) )
% 9.20/5.20       => ( ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_real ) ) ) )
% 9.20/5.20         => ( finite_finite_int
% 9.20/5.20            @ ( collect_int
% 9.20/5.20              @ ^ [I2: int] :
% 9.20/5.20                  ( ( member_int @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6642_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
% 9.20/5.20        ( ( finite3207457112153483333omplex
% 9.20/5.20          @ ( collect_complex
% 9.20/5.20            @ ^ [I2: complex] :
% 9.20/5.20                ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_real ) ) ) )
% 9.20/5.20       => ( ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_real ) ) ) )
% 9.20/5.20         => ( finite3207457112153483333omplex
% 9.20/5.20            @ ( collect_complex
% 9.20/5.20              @ ^ [I2: complex] :
% 9.20/5.20                  ( ( member_complex @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6643_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_real,X3: real > rat,Y3: real > rat] :
% 9.20/5.20        ( ( finite_finite_real
% 9.20/5.20          @ ( collect_real
% 9.20/5.20            @ ^ [I2: real] :
% 9.20/5.20                ( ( member_real @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_rat ) ) ) )
% 9.20/5.20       => ( ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_rat ) ) ) )
% 9.20/5.20         => ( finite_finite_real
% 9.20/5.20            @ ( collect_real
% 9.20/5.20              @ ^ [I2: real] :
% 9.20/5.20                  ( ( member_real @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6644_sum_Ofinite__Collect__op,axiom,
% 9.20/5.20      ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
% 9.20/5.20        ( ( finite_finite_nat
% 9.20/5.20          @ ( collect_nat
% 9.20/5.20            @ ^ [I2: nat] :
% 9.20/5.20                ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                & ( ( X3 @ I2 )
% 9.20/5.20                 != zero_zero_rat ) ) ) )
% 9.20/5.20       => ( ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( Y3 @ I2 )
% 9.20/5.20                   != zero_zero_rat ) ) ) )
% 9.20/5.20         => ( finite_finite_nat
% 9.20/5.20            @ ( collect_nat
% 9.20/5.20              @ ^ [I2: nat] :
% 9.20/5.20                  ( ( member_nat @ I2 @ I5 )
% 9.20/5.20                  & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
% 9.20/5.20                   != zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6645_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( Y3
% 9.20/5.20                    = ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) )
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
% 9.20/5.20           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.20/5.20                 => ( ~ Y3
% 9.20/5.20                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
% 9.20/5.20             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20                   => ( ( Y3
% 9.20/5.20                        = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.20/5.20                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6646_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 9.20/5.20                 => ~ ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                       => A4 )
% 9.20/5.20                      & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                       => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                           => B4 )
% 9.20/5.20                          & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.20/5.20           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 9.20/5.20                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                         => ( 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 ) ) ) ) ) )
% 9.20/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6647_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [A4: $o,B4: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.20               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
% 9.20/5.20                 => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.20                     => A4 )
% 9.20/5.20                    & ( ( Xa2 != zero_zero_nat )
% 9.20/5.20                     => ( ( ( Xa2 = one_one_nat )
% 9.20/5.20                         => B4 )
% 9.20/5.20                        & ( Xa2 = one_one_nat ) ) ) ) ) )
% 9.20/5.20           => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa2 ) ) )
% 9.20/5.20             => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
% 9.20/5.20                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ Xa2 ) )
% 9.20/5.20                     => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                         => ( 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 ) ) ) ) ) )
% 9.20/5.20                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6648_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.20               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
% 9.20/5.20           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) )
% 9.20/5.20             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.20/5.20                     => ( ( Xa2 = Mi2 )
% 9.20/5.20                        | ( Xa2 = Ma2 ) ) ) )
% 9.20/5.20               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                      ( ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.20/5.20                       => ( ( Xa2 = Mi2 )
% 9.20/5.20                          | ( Xa2 = Ma2 )
% 9.20/5.20                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                             => ( 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 ) ) ) ) ) )
% 9.20/5.20                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.20/5.20                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.20/5.20                        ( ( X3
% 9.20/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
% 9.20/5.20                         => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                             => ( 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 ) ) ) ) ) )
% 9.20/5.20                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6649_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.20        ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.20               => ( ~ Y3
% 9.20/5.20                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.20/5.20           => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
% 9.20/5.20                 => ( ~ Y3
% 9.20/5.20                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa2 ) ) ) )
% 9.20/5.20             => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20                   => ( ( Y3
% 9.20/5.20                        = ( ( Xa2 = Mi2 )
% 9.20/5.20                          | ( Xa2 = Ma2 ) ) )
% 9.20/5.20                     => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
% 9.20/5.20               => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                      ( ( X3
% 9.20/5.20                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20                     => ( ( Y3
% 9.20/5.20                          = ( ( Xa2 = Mi2 )
% 9.20/5.20                            | ( Xa2 = Ma2 )
% 9.20/5.20                            | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                               => ( 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 ) ) ) ) ) )
% 9.20/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
% 9.20/5.20                       => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) ) ) )
% 9.20/5.20                 => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.20/5.20                        ( ( X3
% 9.20/5.20                          = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                       => ( ( Y3
% 9.20/5.20                            = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                               => ( 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 ) ) ) ) ) )
% 9.20/5.20                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
% 9.20/5.20                         => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6650_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
% 9.20/5.20      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.20        ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
% 9.20/5.20       => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.20         => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
% 9.20/5.20                ( ( X3
% 9.20/5.20                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
% 9.20/5.20               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa2 ) )
% 9.20/5.20                 => ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                      | ( Xa2 = Ma2 ) ) ) )
% 9.20/5.20           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.20                  ( ( X3
% 9.20/5.20                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
% 9.20/5.20                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ Xa2 ) )
% 9.20/5.20                   => ~ ( ( Xa2 = Mi2 )
% 9.20/5.20                        | ( Xa2 = Ma2 )
% 9.20/5.20                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
% 9.20/5.20             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
% 9.20/5.20                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ Xa2 ) )
% 9.20/5.20                     => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
% 9.20/5.20                           => ( 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 ) ) ) ) ) )
% 9.20/5.20                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6651_arcosh__1,axiom,
% 9.20/5.20      ( ( arcosh_real @ one_one_real )
% 9.20/5.20      = zero_zero_real ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6652_arsinh__0,axiom,
% 9.20/5.20      ( ( arsinh_real @ zero_zero_real )
% 9.20/5.20      = zero_zero_real ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6653_artanh__0,axiom,
% 9.20/5.20      ( ( artanh_real @ zero_zero_real )
% 9.20/5.20      = zero_zero_real ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6654_vebt__buildup_Opelims,axiom,
% 9.20/5.20      ! [X3: nat,Y3: vEBT_VEBT] :
% 9.20/5.20        ( ( ( vEBT_vebt_buildup @ X3 )
% 9.20/5.20          = Y3 )
% 9.20/5.20       => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
% 9.20/5.20         => ( ( ( X3 = zero_zero_nat )
% 9.20/5.20             => ( ( Y3
% 9.20/5.20                  = ( vEBT_Leaf @ $false @ $false ) )
% 9.20/5.20               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
% 9.20/5.20           => ( ( ( X3
% 9.20/5.20                  = ( suc @ zero_zero_nat ) )
% 9.20/5.20               => ( ( Y3
% 9.20/5.20                    = ( vEBT_Leaf @ $false @ $false ) )
% 9.20/5.20                 => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
% 9.20/5.20             => ~ ! [Va2: nat] :
% 9.20/5.20                    ( ( X3
% 9.20/5.20                      = ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                   => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                         => ( Y3
% 9.20/5.20                            = ( 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 ) ) ) ) ) ) )
% 9.20/5.20                        & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
% 9.20/5.20                         => ( Y3
% 9.20/5.20                            = ( 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 ) ) ) ) ) ) ) ) )
% 9.20/5.20                     => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6655_divmod__algorithm__code_I6_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc4245557441103728435nt_int
% 9.20/5.20          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
% 9.20/5.20          @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6656_divmod__algorithm__code_I6_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc2626176000494625587at_nat
% 9.20/5.20          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
% 9.20/5.20          @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6657_divmod__algorithm__code_I6_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc6916734918728496179nteger
% 9.20/5.20          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
% 9.20/5.20          @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6658_artanh__def,axiom,
% 9.20/5.20      ( artanh_real
% 9.20/5.20      = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6659_ln__inj__iff,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ( ( ln_ln_real @ X3 )
% 9.20/5.20              = ( ln_ln_real @ Y3 ) )
% 9.20/5.20            = ( X3 = Y3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6660_ln__less__cancel__iff,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 9.20/5.20            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6661_case__prod__conv,axiom,
% 9.20/5.20      ! [F: nat > nat > nat,A: nat,B: nat] :
% 9.20/5.20        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.20/5.20        = ( F @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6662_case__prod__conv,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
% 9.20/5.20        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.20/5.20        = ( F @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6663_case__prod__conv,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
% 9.20/5.20        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
% 9.20/5.20        = ( F @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6664_case__prod__conv,axiom,
% 9.20/5.20      ! [F: int > int > product_prod_int_int,A: int,B: int] :
% 9.20/5.20        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.20/5.20        = ( F @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6665_case__prod__conv,axiom,
% 9.20/5.20      ! [F: int > int > $o,A: int,B: int] :
% 9.20/5.20        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.20/5.20        = ( F @ A @ B ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6666_ln__le__cancel__iff,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
% 9.20/5.20            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6667_ln__less__zero__iff,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 9.20/5.20          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6668_ln__gt__zero__iff,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.20          = ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6669_ln__eq__zero__iff,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ( ln_ln_real @ X3 )
% 9.20/5.20            = zero_zero_real )
% 9.20/5.20          = ( X3 = one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6670_ln__one,axiom,
% 9.20/5.20      ( ( ln_ln_real @ one_one_real )
% 9.20/5.20      = zero_zero_real ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6671_ln__le__zero__iff,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
% 9.20/5.20          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6672_ln__ge__zero__iff,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.20          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6673_divmod__algorithm__code_I5_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc4245557441103728435nt_int
% 9.20/5.20          @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
% 9.20/5.20          @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6674_divmod__algorithm__code_I5_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc2626176000494625587at_nat
% 9.20/5.20          @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
% 9.20/5.20          @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6675_divmod__algorithm__code_I5_J,axiom,
% 9.20/5.20      ! [M: num,N2: num] :
% 9.20/5.20        ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
% 9.20/5.20        = ( produc6916734918728496179nteger
% 9.20/5.20          @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
% 9.20/5.20          @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6676_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 9.20/5.20        = ( produc6842872674320459806at_nat
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6677_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
% 9.20/5.20        ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 9.20/5.20        = ( produc4947309494688390418_int_o
% 9.20/5.20          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6678_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 9.20/5.20        ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 9.20/5.20        = ( produc4947309494688390418_int_o
% 9.20/5.20          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6679_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
% 9.20/5.20        ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
% 9.20/5.20        = ( produc4245557441103728435nt_int
% 9.20/5.20          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6680_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 9.20/5.20        = ( produc8739625826339149834_nat_o
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6681_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 9.20/5.20        = ( produc6842872674320459806at_nat
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6682_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
% 9.20/5.20        ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
% 9.20/5.20        = ( produc4245557441103728435nt_int
% 9.20/5.20          @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6683_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
% 9.20/5.20        = ( produc27273713700761075at_nat
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6684_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
% 9.20/5.20        = ( produc6842872674320459806at_nat
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6685_prod_Ocase__distrib,axiom,
% 9.20/5.20      ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
% 9.20/5.20        ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
% 9.20/5.20        = ( produc8739625826339149834_nat_o
% 9.20/5.20          @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
% 9.20/5.20          @ Prod ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6686_old_Oprod_Ocase,axiom,
% 9.20/5.20      ! [F: nat > nat > nat,X1: nat,X22: nat] :
% 9.20/5.20        ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.20/5.20        = ( F @ X1 @ X22 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6687_old_Oprod_Ocase,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
% 9.20/5.20        ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.20/5.20        = ( F @ X1 @ X22 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6688_old_Oprod_Ocase,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
% 9.20/5.20        ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
% 9.20/5.20        = ( F @ X1 @ X22 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6689_old_Oprod_Ocase,axiom,
% 9.20/5.20      ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
% 9.20/5.20        ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 9.20/5.20        = ( F @ X1 @ X22 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6690_old_Oprod_Ocase,axiom,
% 9.20/5.20      ! [F: int > int > $o,X1: int,X22: int] :
% 9.20/5.20        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
% 9.20/5.20        = ( F @ X1 @ X22 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6691_split__cong,axiom,
% 9.20/5.20      ! [Q4: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P4: product_prod_nat_nat] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 9.20/5.20              = Q4 )
% 9.20/5.20           => ( ( F @ X4 @ Y5 )
% 9.20/5.20              = ( G @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( P4 = Q4 )
% 9.20/5.20         => ( ( produc6842872674320459806at_nat @ F @ P4 )
% 9.20/5.20            = ( produc6842872674320459806at_nat @ G @ Q4 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6692_split__cong,axiom,
% 9.20/5.20      ! [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] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 9.20/5.20              = Q4 )
% 9.20/5.20           => ( ( F @ X4 @ Y5 )
% 9.20/5.20              = ( G @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( P4 = Q4 )
% 9.20/5.20         => ( ( produc27273713700761075at_nat @ F @ P4 )
% 9.20/5.20            = ( produc27273713700761075at_nat @ G @ Q4 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6693_split__cong,axiom,
% 9.20/5.20      ! [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] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
% 9.20/5.20              = Q4 )
% 9.20/5.20           => ( ( F @ X4 @ Y5 )
% 9.20/5.20              = ( G @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( P4 = Q4 )
% 9.20/5.20         => ( ( produc8739625826339149834_nat_o @ F @ P4 )
% 9.20/5.20            = ( produc8739625826339149834_nat_o @ G @ Q4 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6694_split__cong,axiom,
% 9.20/5.20      ! [Q4: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
% 9.20/5.20        ( ! [X4: int,Y5: int] :
% 9.20/5.20            ( ( ( product_Pair_int_int @ X4 @ Y5 )
% 9.20/5.20              = Q4 )
% 9.20/5.20           => ( ( F @ X4 @ Y5 )
% 9.20/5.20              = ( G @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( P4 = Q4 )
% 9.20/5.20         => ( ( produc4245557441103728435nt_int @ F @ P4 )
% 9.20/5.20            = ( produc4245557441103728435nt_int @ G @ Q4 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6695_split__cong,axiom,
% 9.20/5.20      ! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
% 9.20/5.20        ( ! [X4: int,Y5: int] :
% 9.20/5.20            ( ( ( product_Pair_int_int @ X4 @ Y5 )
% 9.20/5.20              = Q4 )
% 9.20/5.20           => ( ( F @ X4 @ Y5 )
% 9.20/5.20              = ( G @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( P4 = Q4 )
% 9.20/5.20         => ( ( produc4947309494688390418_int_o @ F @ P4 )
% 9.20/5.20            = ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6696_cond__case__prod__eta,axiom,
% 9.20/5.20      ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( F @ X4 @ Y5 )
% 9.20/5.20            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( produc6842872674320459806at_nat @ F )
% 9.20/5.20          = G ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6697_cond__case__prod__eta,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( F @ X4 @ Y5 )
% 9.20/5.20            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( produc27273713700761075at_nat @ F )
% 9.20/5.20          = G ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6698_cond__case__prod__eta,axiom,
% 9.20/5.20      ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.20/5.20        ( ! [X4: nat,Y5: nat] :
% 9.20/5.20            ( ( F @ X4 @ Y5 )
% 9.20/5.20            = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( produc8739625826339149834_nat_o @ F )
% 9.20/5.20          = G ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6699_cond__case__prod__eta,axiom,
% 9.20/5.20      ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
% 9.20/5.20        ( ! [X4: int,Y5: int] :
% 9.20/5.20            ( ( F @ X4 @ Y5 )
% 9.20/5.20            = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( produc4245557441103728435nt_int @ F )
% 9.20/5.20          = G ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6700_cond__case__prod__eta,axiom,
% 9.20/5.20      ! [F: int > int > $o,G: product_prod_int_int > $o] :
% 9.20/5.20        ( ! [X4: int,Y5: int] :
% 9.20/5.20            ( ( F @ X4 @ Y5 )
% 9.20/5.20            = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
% 9.20/5.20       => ( ( produc4947309494688390418_int_o @ F )
% 9.20/5.20          = G ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6701_case__prod__eta,axiom,
% 9.20/5.20      ! [F: product_prod_nat_nat > nat] :
% 9.20/5.20        ( ( produc6842872674320459806at_nat
% 9.20/5.20          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.20/5.20        = F ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6702_case__prod__eta,axiom,
% 9.20/5.20      ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
% 9.20/5.20        ( ( produc27273713700761075at_nat
% 9.20/5.20          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.20/5.20        = F ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6703_case__prod__eta,axiom,
% 9.20/5.20      ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
% 9.20/5.20        ( ( produc8739625826339149834_nat_o
% 9.20/5.20          @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
% 9.20/5.20        = F ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6704_case__prod__eta,axiom,
% 9.20/5.20      ! [F: product_prod_int_int > product_prod_int_int] :
% 9.20/5.20        ( ( produc4245557441103728435nt_int
% 9.20/5.20          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 9.20/5.20        = F ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6705_case__prod__eta,axiom,
% 9.20/5.20      ! [F: product_prod_int_int > $o] :
% 9.20/5.20        ( ( produc4947309494688390418_int_o
% 9.20/5.20          @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
% 9.20/5.20        = F ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6706_case__prodE2,axiom,
% 9.20/5.20      ! [Q: nat > $o,P: nat > nat > nat,Z2: product_prod_nat_nat] :
% 9.20/5.20        ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z2 ) )
% 9.20/5.20       => ~ ! [X4: nat,Y5: nat] :
% 9.20/5.20              ( ( Z2
% 9.20/5.20                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 9.20/5.20             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6707_case__prodE2,axiom,
% 9.20/5.20      ! [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] :
% 9.20/5.20        ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
% 9.20/5.20       => ~ ! [X4: nat,Y5: nat] :
% 9.20/5.20              ( ( Z2
% 9.20/5.20                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 9.20/5.20             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6708_case__prodE2,axiom,
% 9.20/5.20      ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
% 9.20/5.20        ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
% 9.20/5.20       => ~ ! [X4: nat,Y5: nat] :
% 9.20/5.20              ( ( Z2
% 9.20/5.20                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 9.20/5.20             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6709_case__prodE2,axiom,
% 9.20/5.20      ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
% 9.20/5.20        ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
% 9.20/5.20       => ~ ! [X4: int,Y5: int] :
% 9.20/5.20              ( ( Z2
% 9.20/5.20                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.20             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6710_case__prodE2,axiom,
% 9.20/5.20      ! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
% 9.20/5.20        ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
% 9.20/5.20       => ~ ! [X4: int,Y5: int] :
% 9.20/5.20              ( ( Z2
% 9.20/5.20                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.20             => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6711_ln__less__self,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6712_ln__bound,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6713_ln__gt__zero__imp__gt__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20         => ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6714_ln__less__zero,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.20         => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6715_ln__gt__zero,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.20       => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6716_ln__div,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.20/5.20            = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6717_ln__2__less__1,axiom,
% 9.20/5.20      ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6718_ln__ge__zero__imp__ge__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20         => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6719_ln__diff__le,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6720_ln__mult,axiom,
% 9.20/5.20      ! [X3: real,Y3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.20         => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
% 9.20/5.20            = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6721_ln__eq__minus__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ( ln_ln_real @ X3 )
% 9.20/5.20            = ( minus_minus_real @ X3 @ one_one_real ) )
% 9.20/5.20         => ( X3 = one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6722_ln__le__minus__one,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6723_ln__add__one__self__le__self2,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.20       => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6724_ln__one__minus__pos__upper__bound,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.20         => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6725_divmod__step__nat__def,axiom,
% 9.20/5.20      ( unique5026877609467782581ep_nat
% 9.20/5.20      = ( ^ [L2: num] :
% 9.20/5.20            ( produc2626176000494625587at_nat
% 9.20/5.20            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6726_divmod__step__int__def,axiom,
% 9.20/5.20      ( unique5024387138958732305ep_int
% 9.20/5.20      = ( ^ [L2: num] :
% 9.20/5.20            ( produc4245557441103728435nt_int
% 9.20/5.20            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6727_divmod__step__def,axiom,
% 9.20/5.20      ( unique5026877609467782581ep_nat
% 9.20/5.20      = ( ^ [L2: num] :
% 9.20/5.20            ( produc2626176000494625587at_nat
% 9.20/5.20            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6728_divmod__step__def,axiom,
% 9.20/5.20      ( unique5024387138958732305ep_int
% 9.20/5.20      = ( ^ [L2: num] :
% 9.20/5.20            ( produc4245557441103728435nt_int
% 9.20/5.20            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6729_divmod__step__def,axiom,
% 9.20/5.20      ( unique4921790084139445826nteger
% 9.20/5.20      = ( ^ [L2: num] :
% 9.20/5.20            ( produc6916734918728496179nteger
% 9.20/5.20            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6730_divmod__nat__if,axiom,
% 9.20/5.20      ( divmod_nat
% 9.20/5.20      = ( ^ [M2: nat,N: nat] :
% 9.20/5.20            ( if_Pro6206227464963214023at_nat
% 9.20/5.20            @ ( ( N = zero_zero_nat )
% 9.20/5.20              | ( ord_less_nat @ M2 @ N ) )
% 9.20/5.20            @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
% 9.20/5.20            @ ( produc2626176000494625587at_nat
% 9.20/5.20              @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
% 9.20/5.20              @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6731_tanh__ln__real,axiom,
% 9.20/5.20      ! [X3: real] :
% 9.20/5.20        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.20       => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.20          = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6732_Sum__Icc__nat,axiom,
% 9.20/5.20      ! [M: nat,N2: nat] :
% 9.20/5.20        ( ( groups3542108847815614940at_nat
% 9.20/5.20          @ ^ [X: nat] : X
% 9.20/5.20          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.20        = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6733_arith__series__nat,axiom,
% 9.20/5.20      ! [A: nat,D: nat,N2: nat] :
% 9.20/5.20        ( ( groups3542108847815614940at_nat
% 9.20/5.20          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ D ) )
% 9.20/5.20          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.20        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6734_case__prod__Pair__iden,axiom,
% 9.20/5.20      ! [P4: produc8763457246119570046nteger] :
% 9.20/5.20        ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P4 )
% 9.20/5.20        = P4 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6735_case__prod__Pair__iden,axiom,
% 9.20/5.20      ! [P4: produc1908205239877642774nteger] :
% 9.20/5.20        ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
% 9.20/5.20        = P4 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6736_case__prod__Pair__iden,axiom,
% 9.20/5.20      ! [P4: produc2285326912895808259nt_int] :
% 9.20/5.20        ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
% 9.20/5.20        = P4 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6737_case__prod__Pair__iden,axiom,
% 9.20/5.20      ! [P4: produc7773217078559923341nt_int] :
% 9.20/5.20        ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
% 9.20/5.20        = P4 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6738_case__prod__Pair__iden,axiom,
% 9.20/5.20      ! [P4: product_prod_int_int] :
% 9.20/5.20        ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
% 9.20/5.20        = P4 ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6739_round__unique_H,axiom,
% 9.20/5.20      ! [X3: real,N2: int] :
% 9.20/5.20        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.20       => ( ( archim8280529875227126926d_real @ X3 )
% 9.20/5.20          = N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6740_round__unique_H,axiom,
% 9.20/5.20      ! [X3: rat,N2: int] :
% 9.20/5.20        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.20/5.20       => ( ( archim7778729529865785530nd_rat @ X3 )
% 9.20/5.20          = N2 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6741_abs__abs,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.20/5.20        = ( abs_abs_int @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6742_abs__abs,axiom,
% 9.20/5.20      ! [A: real] :
% 9.20/5.20        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.20/5.20        = ( abs_abs_real @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6743_abs__abs,axiom,
% 9.20/5.20      ! [A: rat] :
% 9.20/5.20        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.20/5.20        = ( abs_abs_rat @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6744_abs__abs,axiom,
% 9.20/5.20      ! [A: code_integer] :
% 9.20/5.20        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.20/5.20        = ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6745_abs__idempotent,axiom,
% 9.20/5.20      ! [A: int] :
% 9.20/5.20        ( ( abs_abs_int @ ( abs_abs_int @ A ) )
% 9.20/5.20        = ( abs_abs_int @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6746_abs__idempotent,axiom,
% 9.20/5.20      ! [A: real] :
% 9.20/5.20        ( ( abs_abs_real @ ( abs_abs_real @ A ) )
% 9.20/5.20        = ( abs_abs_real @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6747_abs__idempotent,axiom,
% 9.20/5.20      ! [A: rat] :
% 9.20/5.20        ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
% 9.20/5.20        = ( abs_abs_rat @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6748_abs__idempotent,axiom,
% 9.20/5.20      ! [A: code_integer] :
% 9.20/5.20        ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
% 9.20/5.20        = ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6749_case__prodI,axiom,
% 9.20/5.20      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.20        ( ( F @ A @ B )
% 9.20/5.20       => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6750_case__prodI,axiom,
% 9.20/5.20      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.20        ( ( F @ A @ B )
% 9.20/5.20       => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6751_case__prodI,axiom,
% 9.20/5.20      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.20        ( ( F @ A @ B )
% 9.20/5.20       => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6752_case__prodI,axiom,
% 9.20/5.20      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.20        ( ( F @ A @ B )
% 9.20/5.20       => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6753_case__prodI,axiom,
% 9.20/5.20      ! [F: int > int > $o,A: int,B: int] :
% 9.20/5.20        ( ( F @ A @ B )
% 9.20/5.20       => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6754_case__prodI2,axiom,
% 9.20/5.20      ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.20/5.20        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.20/5.20            ( ( P4
% 9.20/5.20              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 9.20/5.20           => ( C @ A4 @ B4 ) )
% 9.20/5.20       => ( produc127349428274296955eger_o @ C @ P4 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6755_case__prodI2,axiom,
% 9.20/5.20      ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
% 9.20/5.20        ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.20/5.20            ( ( P4
% 9.20/5.20              = ( produc8603105652947943368nteger @ A4 @ B4 ) )
% 9.20/5.20           => ( C @ A4 @ B4 ) )
% 9.20/5.20       => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6756_case__prodI2,axiom,
% 9.20/5.20      ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.20/5.20        ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.20/5.20            ( ( P4
% 9.20/5.20              = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
% 9.20/5.20           => ( C @ A4 @ B4 ) )
% 9.20/5.20       => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6757_case__prodI2,axiom,
% 9.20/5.20      ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
% 9.20/5.20        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.20/5.20            ( ( P4
% 9.20/5.20              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 9.20/5.20           => ( C @ A4 @ B4 ) )
% 9.20/5.20       => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6758_case__prodI2,axiom,
% 9.20/5.20      ! [P4: product_prod_int_int,C: int > int > $o] :
% 9.20/5.20        ( ! [A4: int,B4: int] :
% 9.20/5.20            ( ( P4
% 9.20/5.20              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.20           => ( C @ A4 @ B4 ) )
% 9.20/5.20       => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6759_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: real,C: int > int > set_real,A: int,B: int] :
% 9.20/5.20        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6760_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
% 9.20/5.20        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6761_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: int,C: int > int > set_int,A: int,B: int] :
% 9.20/5.20        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6762_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: set_nat,C: int > int > set_set_nat,A: int,B: int] :
% 9.20/5.20        ( ( member_set_nat @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6763_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
% 9.20/5.20        ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.20  
% 9.20/5.20  thf(fact_6764_mem__case__prodI,axiom,
% 9.20/5.20      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.20        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.20/5.20       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6765_mem__case__prodI,axiom,
% 9.20/5.21      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.21        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.20/5.21       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6766_mem__case__prodI,axiom,
% 9.20/5.21      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.21        ( ( member_int @ Z2 @ ( C @ A @ B ) )
% 9.20/5.21       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6767_mem__case__prodI,axiom,
% 9.20/5.21      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.21        ( ( member_real @ Z2 @ ( C @ A @ B ) )
% 9.20/5.21       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6768_mem__case__prodI,axiom,
% 9.20/5.21      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.21        ( ( member_nat @ Z2 @ ( C @ A @ B ) )
% 9.20/5.21       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6769_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: product_prod_int_int,Z2: real,C: int > int > set_real] :
% 9.20/5.21        ( ! [A4: int,B4: int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6770_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
% 9.20/5.21        ( ! [A4: int,B4: int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6771_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: product_prod_int_int,Z2: int,C: int > int > set_int] :
% 9.20/5.21        ( ! [A4: int,B4: int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6772_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: product_prod_int_int,Z2: set_nat,C: int > int > set_set_nat] :
% 9.20/5.21        ( ! [A4: int,B4: int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_set_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6773_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
% 9.20/5.21        ( ! [A4: int,B4: int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( product_Pair_int_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member8440522571783428010at_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6774_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
% 9.20/5.21        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 9.20/5.21           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6775_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
% 9.20/5.21        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 9.20/5.21           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6776_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
% 9.20/5.21        ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( produc6137756002093451184nteger @ A4 @ B4 ) )
% 9.20/5.21           => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6777_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: produc7773217078559923341nt_int,Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
% 9.20/5.21        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6778_mem__case__prodI2,axiom,
% 9.20/5.21      ! [P4: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
% 9.20/5.21        ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
% 9.20/5.21            ( ( P4
% 9.20/5.21              = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
% 9.20/5.21           => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
% 9.20/5.21       => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6779_case__prodI2_H,axiom,
% 9.20/5.21      ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
% 9.20/5.21        ( ! [A4: nat,B4: nat] :
% 9.20/5.21            ( ( ( product_Pair_nat_nat @ A4 @ B4 )
% 9.20/5.21              = P4 )
% 9.20/5.21           => ( C @ A4 @ B4 @ X3 ) )
% 9.20/5.21       => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6780_abs__0__eq,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( zero_z3403309356797280102nteger
% 9.20/5.21          = ( abs_abs_Code_integer @ A ) )
% 9.20/5.21        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6781_abs__0__eq,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( zero_zero_real
% 9.20/5.21          = ( abs_abs_real @ A ) )
% 9.20/5.21        = ( A = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6782_abs__0__eq,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( zero_zero_rat
% 9.20/5.21          = ( abs_abs_rat @ A ) )
% 9.20/5.21        = ( A = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6783_abs__0__eq,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( zero_zero_int
% 9.20/5.21          = ( abs_abs_int @ A ) )
% 9.20/5.21        = ( A = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6784_abs__eq__0,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = zero_z3403309356797280102nteger )
% 9.20/5.21        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6785_abs__eq__0,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ( abs_abs_real @ A )
% 9.20/5.21          = zero_zero_real )
% 9.20/5.21        = ( A = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6786_abs__eq__0,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ( abs_abs_rat @ A )
% 9.20/5.21          = zero_zero_rat )
% 9.20/5.21        = ( A = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6787_abs__eq__0,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ A )
% 9.20/5.21          = zero_zero_int )
% 9.20/5.21        = ( A = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6788_abs__zero,axiom,
% 9.20/5.21      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.20/5.21      = zero_z3403309356797280102nteger ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6789_abs__zero,axiom,
% 9.20/5.21      ( ( abs_abs_real @ zero_zero_real )
% 9.20/5.21      = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6790_abs__zero,axiom,
% 9.20/5.21      ( ( abs_abs_rat @ zero_zero_rat )
% 9.20/5.21      = zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6791_abs__zero,axiom,
% 9.20/5.21      ( ( abs_abs_int @ zero_zero_int )
% 9.20/5.21      = zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6792_abs__0,axiom,
% 9.20/5.21      ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
% 9.20/5.21      = zero_z3403309356797280102nteger ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6793_abs__0,axiom,
% 9.20/5.21      ( ( abs_abs_complex @ zero_zero_complex )
% 9.20/5.21      = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6794_abs__0,axiom,
% 9.20/5.21      ( ( abs_abs_real @ zero_zero_real )
% 9.20/5.21      = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6795_abs__0,axiom,
% 9.20/5.21      ( ( abs_abs_rat @ zero_zero_rat )
% 9.20/5.21      = zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6796_abs__0,axiom,
% 9.20/5.21      ( ( abs_abs_int @ zero_zero_int )
% 9.20/5.21      = zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6797_abs__mult__self__eq,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
% 9.20/5.21        = ( times_3573771949741848930nteger @ A @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6798_abs__mult__self__eq,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ A ) )
% 9.20/5.21        = ( times_times_real @ A @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6799_abs__mult__self__eq,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
% 9.20/5.21        = ( times_times_rat @ A @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6800_abs__mult__self__eq,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ A ) )
% 9.20/5.21        = ( times_times_int @ A @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6801_abs__1,axiom,
% 9.20/5.21      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.20/5.21      = one_one_Code_integer ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6802_abs__1,axiom,
% 9.20/5.21      ( ( abs_abs_complex @ one_one_complex )
% 9.20/5.21      = one_one_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6803_abs__1,axiom,
% 9.20/5.21      ( ( abs_abs_real @ one_one_real )
% 9.20/5.21      = one_one_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6804_abs__1,axiom,
% 9.20/5.21      ( ( abs_abs_rat @ one_one_rat )
% 9.20/5.21      = one_one_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6805_abs__1,axiom,
% 9.20/5.21      ( ( abs_abs_int @ one_one_int )
% 9.20/5.21      = one_one_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6806_abs__add__abs,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
% 9.20/5.21        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6807_abs__add__abs,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
% 9.20/5.21        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6808_abs__add__abs,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
% 9.20/5.21        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6809_abs__add__abs,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
% 9.20/5.21        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6810_abs__divide,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.21        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6811_abs__divide,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.21        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6812_abs__minus,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.20/5.21        = ( abs_abs_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6813_abs__minus,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.20/5.21        = ( abs_abs_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6814_abs__minus,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.20/5.21        = ( abs_abs_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6815_abs__minus,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.21        = ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6816_abs__minus__cancel,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
% 9.20/5.21        = ( abs_abs_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6817_abs__minus__cancel,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
% 9.20/5.21        = ( abs_abs_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6818_abs__minus__cancel,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
% 9.20/5.21        = ( abs_abs_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6819_abs__minus__cancel,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
% 9.20/5.21        = ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6820_dvd__abs__iff,axiom,
% 9.20/5.21      ! [M: int,K: int] :
% 9.20/5.21        ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
% 9.20/5.21        = ( dvd_dvd_int @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6821_dvd__abs__iff,axiom,
% 9.20/5.21      ! [M: real,K: real] :
% 9.20/5.21        ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
% 9.20/5.21        = ( dvd_dvd_real @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6822_dvd__abs__iff,axiom,
% 9.20/5.21      ! [M: rat,K: rat] :
% 9.20/5.21        ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
% 9.20/5.21        = ( dvd_dvd_rat @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6823_dvd__abs__iff,axiom,
% 9.20/5.21      ! [M: code_integer,K: code_integer] :
% 9.20/5.21        ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
% 9.20/5.21        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6824_abs__dvd__iff,axiom,
% 9.20/5.21      ! [M: int,K: int] :
% 9.20/5.21        ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
% 9.20/5.21        = ( dvd_dvd_int @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6825_abs__dvd__iff,axiom,
% 9.20/5.21      ! [M: real,K: real] :
% 9.20/5.21        ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
% 9.20/5.21        = ( dvd_dvd_real @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6826_abs__dvd__iff,axiom,
% 9.20/5.21      ! [M: rat,K: rat] :
% 9.20/5.21        ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
% 9.20/5.21        = ( dvd_dvd_rat @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6827_abs__dvd__iff,axiom,
% 9.20/5.21      ! [M: code_integer,K: code_integer] :
% 9.20/5.21        ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
% 9.20/5.21        = ( dvd_dvd_Code_integer @ M @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6828_of__int__abs,axiom,
% 9.20/5.21      ! [X3: int] :
% 9.20/5.21        ( ( ring_1_of_int_int @ ( abs_abs_int @ X3 ) )
% 9.20/5.21        = ( abs_abs_int @ ( ring_1_of_int_int @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6829_of__int__abs,axiom,
% 9.20/5.21      ! [X3: int] :
% 9.20/5.21        ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X3 ) )
% 9.20/5.21        = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6830_of__int__abs,axiom,
% 9.20/5.21      ! [X3: int] :
% 9.20/5.21        ( ( ring_1_of_int_real @ ( abs_abs_int @ X3 ) )
% 9.20/5.21        = ( abs_abs_real @ ( ring_1_of_int_real @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6831_of__int__abs,axiom,
% 9.20/5.21      ! [X3: int] :
% 9.20/5.21        ( ( ring_1_of_int_rat @ ( abs_abs_int @ X3 ) )
% 9.20/5.21        = ( abs_abs_rat @ ( ring_1_of_int_rat @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6832_abs__bool__eq,axiom,
% 9.20/5.21      ! [P: $o] :
% 9.20/5.21        ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
% 9.20/5.21        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6833_abs__bool__eq,axiom,
% 9.20/5.21      ! [P: $o] :
% 9.20/5.21        ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
% 9.20/5.21        = ( zero_n2052037380579107095ol_rat @ P ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6834_abs__bool__eq,axiom,
% 9.20/5.21      ! [P: $o] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
% 9.20/5.21        = ( zero_n356916108424825756nteger @ P ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6835_abs__bool__eq,axiom,
% 9.20/5.21      ! [P: $o] :
% 9.20/5.21        ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.21        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6836_zdvd1__eq,axiom,
% 9.20/5.21      ! [X3: int] :
% 9.20/5.21        ( ( dvd_dvd_int @ X3 @ one_one_int )
% 9.20/5.21        = ( ( abs_abs_int @ X3 )
% 9.20/5.21          = one_one_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6837_tanh__0,axiom,
% 9.20/5.21      ( ( tanh_complex @ zero_zero_complex )
% 9.20/5.21      = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6838_tanh__0,axiom,
% 9.20/5.21      ( ( tanh_real @ zero_zero_real )
% 9.20/5.21      = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6839_tanh__real__less__iff,axiom,
% 9.20/5.21      ! [X3: real,Y3: real] :
% 9.20/5.21        ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
% 9.20/5.21        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6840_sum_Oneutral__const,axiom,
% 9.20/5.21      ! [A2: set_nat] :
% 9.20/5.21        ( ( groups3542108847815614940at_nat
% 9.20/5.21          @ ^ [Uu3: nat] : zero_zero_nat
% 9.20/5.21          @ A2 )
% 9.20/5.21        = zero_zero_nat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6841_sum_Oneutral__const,axiom,
% 9.20/5.21      ! [A2: set_complex] :
% 9.20/5.21        ( ( groups7754918857620584856omplex
% 9.20/5.21          @ ^ [Uu3: complex] : zero_zero_complex
% 9.20/5.21          @ A2 )
% 9.20/5.21        = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6842_sum_Oneutral__const,axiom,
% 9.20/5.21      ! [A2: set_nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [Uu3: nat] : zero_zero_real
% 9.20/5.21          @ A2 )
% 9.20/5.21        = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6843_of__int__sum,axiom,
% 9.20/5.21      ! [F: complex > int,A2: set_complex] :
% 9.20/5.21        ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
% 9.20/5.21        = ( groups7754918857620584856omplex
% 9.20/5.21          @ ^ [X: complex] : ( ring_17405671764205052669omplex @ ( F @ X ) )
% 9.20/5.21          @ A2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6844_of__int__sum,axiom,
% 9.20/5.21      ! [F: nat > int,A2: set_nat] :
% 9.20/5.21        ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
% 9.20/5.21        = ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
% 9.20/5.21          @ A2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6845_abs__of__nonneg,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.21       => ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6846_abs__of__nonneg,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.21       => ( ( abs_abs_real @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6847_abs__of__nonneg,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.21       => ( ( abs_abs_rat @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6848_abs__of__nonneg,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.21       => ( ( abs_abs_int @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6849_abs__le__self__iff,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
% 9.20/5.21        = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6850_abs__le__self__iff,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ A )
% 9.20/5.21        = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6851_abs__le__self__iff,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
% 9.20/5.21        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6852_abs__le__self__iff,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
% 9.20/5.21        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6853_abs__le__zero__iff,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
% 9.20/5.21        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6854_abs__le__zero__iff,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ zero_zero_real )
% 9.20/5.21        = ( A = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6855_abs__le__zero__iff,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
% 9.20/5.21        = ( A = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6856_abs__le__zero__iff,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
% 9.20/5.21        = ( A = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6857_zero__less__abs__iff,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
% 9.20/5.21        = ( A != zero_z3403309356797280102nteger ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6858_zero__less__abs__iff,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ A ) )
% 9.20/5.21        = ( A != zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6859_zero__less__abs__iff,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
% 9.20/5.21        = ( A != zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6860_zero__less__abs__iff,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
% 9.20/5.21        = ( A != zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6861_sum_Oempty,axiom,
% 9.20/5.21      ! [G: real > complex] :
% 9.20/5.21        ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
% 9.20/5.21        = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6862_sum_Oempty,axiom,
% 9.20/5.21      ! [G: real > real] :
% 9.20/5.21        ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
% 9.20/5.21        = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6863_sum_Oempty,axiom,
% 9.20/5.21      ! [G: real > rat] :
% 9.20/5.21        ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
% 9.20/5.21        = zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6864_sum_Oempty,axiom,
% 9.20/5.21      ! [G: real > nat] :
% 9.20/5.21        ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
% 9.20/5.21        = zero_zero_nat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6865_sum_Oempty,axiom,
% 9.20/5.21      ! [G: real > int] :
% 9.20/5.21        ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
% 9.20/5.21        = zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6866_sum_Oempty,axiom,
% 9.20/5.21      ! [G: nat > complex] :
% 9.20/5.21        ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
% 9.20/5.21        = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6867_sum_Oempty,axiom,
% 9.20/5.21      ! [G: nat > rat] :
% 9.20/5.21        ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
% 9.20/5.21        = zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6868_sum_Oempty,axiom,
% 9.20/5.21      ! [G: nat > int] :
% 9.20/5.21        ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
% 9.20/5.21        = zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6869_sum_Oempty,axiom,
% 9.20/5.21      ! [G: int > complex] :
% 9.20/5.21        ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
% 9.20/5.21        = zero_zero_complex ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6870_sum_Oempty,axiom,
% 9.20/5.21      ! [G: int > real] :
% 9.20/5.21        ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
% 9.20/5.21        = zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6871_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > complex] :
% 9.20/5.21        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.20/5.21          = zero_zero_complex ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6872_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > complex] :
% 9.20/5.21        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3049146728041665814omplex @ G @ A2 )
% 9.20/5.21          = zero_zero_complex ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6873_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > real] :
% 9.20/5.21        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.20/5.21          = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6874_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > real] :
% 9.20/5.21        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.20/5.21          = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6875_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > rat] :
% 9.20/5.21        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.20/5.21          = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6876_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > rat] :
% 9.20/5.21        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.20/5.21          = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6877_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > rat] :
% 9.20/5.21        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.20/5.21          = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6878_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > nat] :
% 9.20/5.21        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.20/5.21          = zero_zero_nat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6879_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > nat] :
% 9.20/5.21        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.20/5.21          = zero_zero_nat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6880_sum_Oinfinite,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > int] :
% 9.20/5.21        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( groups3539618377306564664at_int @ G @ A2 )
% 9.20/5.21          = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6881_sum__eq__0__iff,axiom,
% 9.20/5.21      ! [F3: set_int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ F3 )
% 9.20/5.21       => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
% 9.20/5.21            = zero_zero_nat )
% 9.20/5.21          = ( ! [X: int] :
% 9.20/5.21                ( ( member_int @ X @ F3 )
% 9.20/5.21               => ( ( F @ X )
% 9.20/5.21                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6882_sum__eq__0__iff,axiom,
% 9.20/5.21      ! [F3: set_complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ F3 )
% 9.20/5.21       => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
% 9.20/5.21            = zero_zero_nat )
% 9.20/5.21          = ( ! [X: complex] :
% 9.20/5.21                ( ( member_complex @ X @ F3 )
% 9.20/5.21               => ( ( F @ X )
% 9.20/5.21                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6883_sum__eq__0__iff,axiom,
% 9.20/5.21      ! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.20/5.21        ( ( finite6177210948735845034at_nat @ F3 )
% 9.20/5.21       => ( ( ( groups977919841031483927at_nat @ F @ F3 )
% 9.20/5.21            = zero_zero_nat )
% 9.20/5.21          = ( ! [X: product_prod_nat_nat] :
% 9.20/5.21                ( ( member8440522571783428010at_nat @ X @ F3 )
% 9.20/5.21               => ( ( F @ X )
% 9.20/5.21                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6884_sum__eq__0__iff,axiom,
% 9.20/5.21      ! [F3: set_nat,F: nat > nat] :
% 9.20/5.21        ( ( finite_finite_nat @ F3 )
% 9.20/5.21       => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
% 9.20/5.21            = zero_zero_nat )
% 9.20/5.21          = ( ! [X: nat] :
% 9.20/5.21                ( ( member_nat @ X @ F3 )
% 9.20/5.21               => ( ( F @ X )
% 9.20/5.21                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6885_artanh__minus__real,axiom,
% 9.20/5.21      ! [X3: real] :
% 9.20/5.21        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.21       => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
% 9.20/5.21          = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6886_zabs__less__one__iff,axiom,
% 9.20/5.21      ! [Z2: int] :
% 9.20/5.21        ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
% 9.20/5.21        = ( Z2 = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6887_tanh__real__pos__iff,axiom,
% 9.20/5.21      ! [X3: real] :
% 9.20/5.21        ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
% 9.20/5.21        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6888_tanh__real__neg__iff,axiom,
% 9.20/5.21      ! [X3: real] :
% 9.20/5.21        ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
% 9.20/5.21        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6889_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups5754745047067104278omplex
% 9.20/5.21                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups5754745047067104278omplex
% 9.20/5.21                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6890_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_nat,A: nat,B: nat > complex] :
% 9.20/5.21        ( ( finite_finite_nat @ S3 )
% 9.20/5.21       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2073611262835488442omplex
% 9.20/5.21                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2073611262835488442omplex
% 9.20/5.21                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6891_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > complex] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3049146728041665814omplex
% 9.20/5.21                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3049146728041665814omplex
% 9.20/5.21                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6892_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real
% 9.20/5.21                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real
% 9.20/5.21                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6893_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real
% 9.20/5.21                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real
% 9.20/5.21                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6894_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_complex,A: complex,B: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real
% 9.20/5.21                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real
% 9.20/5.21                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6895_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat
% 9.20/5.21                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat
% 9.20/5.21                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6896_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_nat,A: nat,B: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ S3 )
% 9.20/5.21       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat
% 9.20/5.21                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat
% 9.20/5.21                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6897_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat
% 9.20/5.21                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat
% 9.20/5.21                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6898_sum_Odelta,axiom,
% 9.20/5.21      ! [S3: set_complex,A: complex,B: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat
% 9.20/5.21                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat
% 9.20/5.21                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6899_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups5754745047067104278omplex
% 9.20/5.21                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups5754745047067104278omplex
% 9.20/5.21                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6900_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_nat,A: nat,B: nat > complex] :
% 9.20/5.21        ( ( finite_finite_nat @ S3 )
% 9.20/5.21       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2073611262835488442omplex
% 9.20/5.21                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2073611262835488442omplex
% 9.20/5.21                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6901_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > complex] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3049146728041665814omplex
% 9.20/5.21                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3049146728041665814omplex
% 9.20/5.21                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_complex ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6902_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real
% 9.20/5.21                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real
% 9.20/5.21                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6903_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real
% 9.20/5.21                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real
% 9.20/5.21                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6904_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_complex,A: complex,B: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real
% 9.20/5.21                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real
% 9.20/5.21                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_real ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6905_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_real,A: real,B: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ S3 )
% 9.20/5.21       => ( ( ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat
% 9.20/5.21                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat
% 9.20/5.21                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6906_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_nat,A: nat,B: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ S3 )
% 9.20/5.21       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat
% 9.20/5.21                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat
% 9.20/5.21                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6907_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_int,A: int,B: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ S3 )
% 9.20/5.21       => ( ( ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat
% 9.20/5.21                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat
% 9.20/5.21                @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6908_sum_Odelta_H,axiom,
% 9.20/5.21      ! [S3: set_complex,A: complex,B: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat
% 9.20/5.21                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = ( B @ A ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat
% 9.20/5.21                @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
% 9.20/5.21                @ S3 )
% 9.20/5.21              = zero_zero_rat ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6909_sum__abs,axiom,
% 9.20/5.21      ! [F: nat > real,A2: set_nat] :
% 9.20/5.21        ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
% 9.20/5.21        @ ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
% 9.20/5.21          @ A2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6910_divide__le__0__abs__iff,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
% 9.20/5.21        = ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.21          | ( B = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6911_divide__le__0__abs__iff,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
% 9.20/5.21        = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.21          | ( B = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6912_zero__le__divide__abs__iff,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
% 9.20/5.21        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.21          | ( B = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6913_zero__le__divide__abs__iff,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
% 9.20/5.21        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.21          | ( B = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6914_abs__of__nonpos,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.21       => ( ( abs_abs_real @ A )
% 9.20/5.21          = ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6915_abs__of__nonpos,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
% 9.20/5.21       => ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6916_abs__of__nonpos,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ A @ zero_zero_rat )
% 9.20/5.21       => ( ( abs_abs_rat @ A )
% 9.20/5.21          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6917_abs__of__nonpos,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ A @ zero_zero_int )
% 9.20/5.21       => ( ( abs_abs_int @ A )
% 9.20/5.21          = ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6918_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6919_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6920_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6921_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6922_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6923_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6924_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6925_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6926_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6927_sum_Oinsert,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6928_sum__abs__ge__zero,axiom,
% 9.20/5.21      ! [F: nat > real,A2: set_nat] :
% 9.20/5.21        ( ord_less_eq_real @ zero_zero_real
% 9.20/5.21        @ ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
% 9.20/5.21          @ A2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6929_zero__less__power__abs__iff,axiom,
% 9.20/5.21      ! [A: code_integer,N2: nat] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
% 9.20/5.21        = ( ( A != zero_z3403309356797280102nteger )
% 9.20/5.21          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6930_zero__less__power__abs__iff,axiom,
% 9.20/5.21      ! [A: real,N2: nat] :
% 9.20/5.21        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 9.20/5.21        = ( ( A != zero_zero_real )
% 9.20/5.21          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6931_zero__less__power__abs__iff,axiom,
% 9.20/5.21      ! [A: rat,N2: nat] :
% 9.20/5.21        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
% 9.20/5.21        = ( ( A != zero_zero_rat )
% 9.20/5.21          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6932_zero__less__power__abs__iff,axiom,
% 9.20/5.21      ! [A: int,N2: nat] :
% 9.20/5.21        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) )
% 9.20/5.21        = ( ( A != zero_zero_int )
% 9.20/5.21          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6933_sum_Ocl__ivl__Suc,axiom,
% 9.20/5.21      ! [N2: nat,M: nat,G: nat > complex] :
% 9.20/5.21        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = zero_zero_complex ) )
% 9.20/5.21        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6934_sum_Ocl__ivl__Suc,axiom,
% 9.20/5.21      ! [N2: nat,M: nat,G: nat > rat] :
% 9.20/5.21        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = zero_zero_rat ) )
% 9.20/5.21        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6935_sum_Ocl__ivl__Suc,axiom,
% 9.20/5.21      ! [N2: nat,M: nat,G: nat > int] :
% 9.20/5.21        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = zero_zero_int ) )
% 9.20/5.21        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6936_sum_Ocl__ivl__Suc,axiom,
% 9.20/5.21      ! [N2: nat,M: nat,G: nat > nat] :
% 9.20/5.21        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = zero_zero_nat ) )
% 9.20/5.21        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6937_sum_Ocl__ivl__Suc,axiom,
% 9.20/5.21      ! [N2: nat,M: nat,G: nat > real] :
% 9.20/5.21        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = zero_zero_real ) )
% 9.20/5.21        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.21         => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.21            = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6938_sum__zero__power,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > complex] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2073611262835488442omplex
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( C @ zero_zero_nat ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2073611262835488442omplex
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_complex ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6939_sum__zero__power,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > rat] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( C @ zero_zero_nat ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6940_sum__zero__power,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > real] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( C @ zero_zero_nat ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6941_sum__zero__power_H,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > complex,D: nat > complex] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2073611262835488442omplex
% 9.20/5.21              @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2073611262835488442omplex
% 9.20/5.21              @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_complex ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6942_sum__zero__power_H,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > rat,D: nat > rat] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat
% 9.20/5.21              @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups2906978787729119204at_rat
% 9.20/5.21              @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6943_sum__zero__power_H,axiom,
% 9.20/5.21      ! [A2: set_nat,C: nat > real,D: nat > real] :
% 9.20/5.21        ( ( ( ( finite_finite_nat @ A2 )
% 9.20/5.21            & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
% 9.20/5.21        & ( ~ ( ( finite_finite_nat @ A2 )
% 9.20/5.21              & ( member_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.21         => ( ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6944_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: real > complex,A2: set_real] :
% 9.20/5.21        ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.20/5.21         != zero_zero_complex )
% 9.20/5.21       => ~ ! [A4: real] :
% 9.20/5.21              ( ( member_real @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_complex ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6945_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: nat > complex,A2: set_nat] :
% 9.20/5.21        ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.20/5.21         != zero_zero_complex )
% 9.20/5.21       => ~ ! [A4: nat] :
% 9.20/5.21              ( ( member_nat @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_complex ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6946_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: int > complex,A2: set_int] :
% 9.20/5.21        ( ( ( groups3049146728041665814omplex @ G @ A2 )
% 9.20/5.21         != zero_zero_complex )
% 9.20/5.21       => ~ ! [A4: int] :
% 9.20/5.21              ( ( member_int @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_complex ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6947_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: real > real,A2: set_real] :
% 9.20/5.21        ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.20/5.21         != zero_zero_real )
% 9.20/5.21       => ~ ! [A4: real] :
% 9.20/5.21              ( ( member_real @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6948_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: int > real,A2: set_int] :
% 9.20/5.21        ( ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.20/5.21         != zero_zero_real )
% 9.20/5.21       => ~ ! [A4: int] :
% 9.20/5.21              ( ( member_int @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_real ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6949_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: real > rat,A2: set_real] :
% 9.20/5.21        ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.20/5.21         != zero_zero_rat )
% 9.20/5.21       => ~ ! [A4: real] :
% 9.20/5.21              ( ( member_real @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6950_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: nat > rat,A2: set_nat] :
% 9.20/5.21        ( ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.20/5.21         != zero_zero_rat )
% 9.20/5.21       => ~ ! [A4: nat] :
% 9.20/5.21              ( ( member_nat @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6951_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: int > rat,A2: set_int] :
% 9.20/5.21        ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.20/5.21         != zero_zero_rat )
% 9.20/5.21       => ~ ! [A4: int] :
% 9.20/5.21              ( ( member_int @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_rat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6952_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: real > nat,A2: set_real] :
% 9.20/5.21        ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.20/5.21         != zero_zero_nat )
% 9.20/5.21       => ~ ! [A4: real] :
% 9.20/5.21              ( ( member_real @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6953_sum_Onot__neutral__contains__not__neutral,axiom,
% 9.20/5.21      ! [G: int > nat,A2: set_int] :
% 9.20/5.21        ( ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.20/5.21         != zero_zero_nat )
% 9.20/5.21       => ~ ! [A4: int] :
% 9.20/5.21              ( ( member_int @ A4 @ A2 )
% 9.20/5.21             => ( ( G @ A4 )
% 9.20/5.21                = zero_zero_nat ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6954_sum_Oneutral,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > nat] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ( G @ X4 )
% 9.20/5.21              = zero_zero_nat ) )
% 9.20/5.21       => ( ( groups3542108847815614940at_nat @ G @ A2 )
% 9.20/5.21          = zero_zero_nat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6955_sum_Oneutral,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > complex] :
% 9.20/5.21        ( ! [X4: complex] :
% 9.20/5.21            ( ( member_complex @ X4 @ A2 )
% 9.20/5.21           => ( ( G @ X4 )
% 9.20/5.21              = zero_zero_complex ) )
% 9.20/5.21       => ( ( groups7754918857620584856omplex @ G @ A2 )
% 9.20/5.21          = zero_zero_complex ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6956_sum_Oneutral,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > real] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ( G @ X4 )
% 9.20/5.21              = zero_zero_real ) )
% 9.20/5.21       => ( ( groups6591440286371151544t_real @ G @ A2 )
% 9.20/5.21          = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6957_abs__le__D1,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_real @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6958_abs__le__D1,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.20/5.21       => ( ord_le3102999989581377725nteger @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6959_abs__le__D1,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_rat @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6960_abs__le__D1,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_int @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6961_abs__ge__self,axiom,
% 9.20/5.21      ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6962_abs__ge__self,axiom,
% 9.20/5.21      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6963_abs__ge__self,axiom,
% 9.20/5.21      ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6964_abs__ge__self,axiom,
% 9.20/5.21      ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6965_abs__eq__0__iff,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = zero_z3403309356797280102nteger )
% 9.20/5.21        = ( A = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6966_abs__eq__0__iff,axiom,
% 9.20/5.21      ! [A: complex] :
% 9.20/5.21        ( ( ( abs_abs_complex @ A )
% 9.20/5.21          = zero_zero_complex )
% 9.20/5.21        = ( A = zero_zero_complex ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6967_abs__eq__0__iff,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ( abs_abs_real @ A )
% 9.20/5.21          = zero_zero_real )
% 9.20/5.21        = ( A = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6968_abs__eq__0__iff,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ( abs_abs_rat @ A )
% 9.20/5.21          = zero_zero_rat )
% 9.20/5.21        = ( A = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6969_abs__eq__0__iff,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ A )
% 9.20/5.21          = zero_zero_int )
% 9.20/5.21        = ( A = zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6970_abs__mult,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.20/5.21        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6971_abs__mult,axiom,
% 9.20/5.21      ! [A: complex,B: complex] :
% 9.20/5.21        ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
% 9.20/5.21        = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6972_abs__mult,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 9.20/5.21        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6973_abs__mult,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.21        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6974_abs__mult,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 9.20/5.21        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6975_abs__one,axiom,
% 9.20/5.21      ( ( abs_abs_Code_integer @ one_one_Code_integer )
% 9.20/5.21      = one_one_Code_integer ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6976_abs__one,axiom,
% 9.20/5.21      ( ( abs_abs_real @ one_one_real )
% 9.20/5.21      = one_one_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6977_abs__one,axiom,
% 9.20/5.21      ( ( abs_abs_rat @ one_one_rat )
% 9.20/5.21      = one_one_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6978_abs__one,axiom,
% 9.20/5.21      ( ( abs_abs_int @ one_one_int )
% 9.20/5.21      = one_one_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6979_abs__minus__commute,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
% 9.20/5.21        = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6980_abs__minus__commute,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
% 9.20/5.21        = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6981_abs__minus__commute,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
% 9.20/5.21        = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6982_abs__minus__commute,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
% 9.20/5.21        = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6983_abs__eq__iff,axiom,
% 9.20/5.21      ! [X3: int,Y3: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ X3 )
% 9.20/5.21          = ( abs_abs_int @ Y3 ) )
% 9.20/5.21        = ( ( X3 = Y3 )
% 9.20/5.21          | ( X3
% 9.20/5.21            = ( uminus_uminus_int @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6984_abs__eq__iff,axiom,
% 9.20/5.21      ! [X3: real,Y3: real] :
% 9.20/5.21        ( ( ( abs_abs_real @ X3 )
% 9.20/5.21          = ( abs_abs_real @ Y3 ) )
% 9.20/5.21        = ( ( X3 = Y3 )
% 9.20/5.21          | ( X3
% 9.20/5.21            = ( uminus_uminus_real @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6985_abs__eq__iff,axiom,
% 9.20/5.21      ! [X3: rat,Y3: rat] :
% 9.20/5.21        ( ( ( abs_abs_rat @ X3 )
% 9.20/5.21          = ( abs_abs_rat @ Y3 ) )
% 9.20/5.21        = ( ( X3 = Y3 )
% 9.20/5.21          | ( X3
% 9.20/5.21            = ( uminus_uminus_rat @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6986_abs__eq__iff,axiom,
% 9.20/5.21      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.21        ( ( ( abs_abs_Code_integer @ X3 )
% 9.20/5.21          = ( abs_abs_Code_integer @ Y3 ) )
% 9.20/5.21        = ( ( X3 = Y3 )
% 9.20/5.21          | ( X3
% 9.20/5.21            = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6987_dvd__if__abs__eq,axiom,
% 9.20/5.21      ! [L: int,K: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ L )
% 9.20/5.21          = ( abs_abs_int @ K ) )
% 9.20/5.21       => ( dvd_dvd_int @ L @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6988_dvd__if__abs__eq,axiom,
% 9.20/5.21      ! [L: real,K: real] :
% 9.20/5.21        ( ( ( abs_abs_real @ L )
% 9.20/5.21          = ( abs_abs_real @ K ) )
% 9.20/5.21       => ( dvd_dvd_real @ L @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6989_dvd__if__abs__eq,axiom,
% 9.20/5.21      ! [L: rat,K: rat] :
% 9.20/5.21        ( ( ( abs_abs_rat @ L )
% 9.20/5.21          = ( abs_abs_rat @ K ) )
% 9.20/5.21       => ( dvd_dvd_rat @ L @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6990_dvd__if__abs__eq,axiom,
% 9.20/5.21      ! [L: code_integer,K: code_integer] :
% 9.20/5.21        ( ( ( abs_abs_Code_integer @ L )
% 9.20/5.21          = ( abs_abs_Code_integer @ K ) )
% 9.20/5.21       => ( dvd_dvd_Code_integer @ L @ K ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6991_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: real,C: int > int > set_real,P4: product_prod_int_int] :
% 9.20/5.21        ( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6992_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: nat,C: int > int > set_nat,P4: product_prod_int_int] :
% 9.20/5.21        ( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6993_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: int,C: int > int > set_int,P4: product_prod_int_int] :
% 9.20/5.21        ( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6994_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: set_nat,C: int > int > set_set_nat,P4: product_prod_int_int] :
% 9.20/5.21        ( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_set_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6995_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
% 9.20/5.21        ( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6996_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
% 9.20/5.21        ( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6997_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
% 9.20/5.21        ( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6998_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
% 9.20/5.21        ( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_6999_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
% 9.20/5.21        ( ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7000_mem__case__prodE,axiom,
% 9.20/5.21      ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
% 9.20/5.21        ( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) )
% 9.20/5.21       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7001_zdvd__antisym__abs,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( dvd_dvd_int @ A @ B )
% 9.20/5.21       => ( ( dvd_dvd_int @ B @ A )
% 9.20/5.21         => ( ( abs_abs_int @ A )
% 9.20/5.21            = ( abs_abs_int @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7002_case__prodD,axiom,
% 9.20/5.21      ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.21        ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
% 9.20/5.21       => ( F @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7003_case__prodD,axiom,
% 9.20/5.21      ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
% 9.20/5.21        ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
% 9.20/5.21       => ( F @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7004_case__prodD,axiom,
% 9.20/5.21      ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.21        ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
% 9.20/5.21       => ( F @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7005_case__prodD,axiom,
% 9.20/5.21      ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
% 9.20/5.21        ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
% 9.20/5.21       => ( F @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7006_case__prodD,axiom,
% 9.20/5.21      ! [F: int > int > $o,A: int,B: int] :
% 9.20/5.21        ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
% 9.20/5.21       => ( F @ A @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7007_case__prodE,axiom,
% 9.20/5.21      ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
% 9.20/5.21        ( ( produc127349428274296955eger_o @ C @ P4 )
% 9.20/5.21       => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7008_case__prodE,axiom,
% 9.20/5.21      ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
% 9.20/5.21        ( ( produc6253627499356882019eger_o @ C @ P4 )
% 9.20/5.21       => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7009_case__prodE,axiom,
% 9.20/5.21      ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
% 9.20/5.21        ( ( produc1573362020775583542_int_o @ C @ P4 )
% 9.20/5.21       => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7010_case__prodE,axiom,
% 9.20/5.21      ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
% 9.20/5.21        ( ( produc2558449545302689196_int_o @ C @ P4 )
% 9.20/5.21       => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7011_case__prodE,axiom,
% 9.20/5.21      ! [C: int > int > $o,P4: product_prod_int_int] :
% 9.20/5.21        ( ( produc4947309494688390418_int_o @ C @ P4 )
% 9.20/5.21       => ~ ! [X4: int,Y5: int] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_int_int @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7012_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_real,F: real > rat,G: real > rat] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7013_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_nat,F: nat > rat,G: nat > rat] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7014_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_int,F: int > rat,G: int > rat] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7015_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_real,F: real > nat,G: real > nat] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7016_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_int,F: int > nat,G: int > nat] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7017_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_real,F: real > int,G: real > int] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7018_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_nat,F: nat > int,G: nat > int] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7019_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_int,F: int > int,G: int > int] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7020_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_nat,F: nat > nat,G: nat > nat] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7021_sum__mono,axiom,
% 9.20/5.21      ! [K5: set_nat,F: nat > real,G: nat > real] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ K5 )
% 9.20/5.21           => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21       => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K5 ) @ ( groups6591440286371151544t_real @ G @ K5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7022_sum_Odistrib,axiom,
% 9.20/5.21      ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
% 9.20/5.21        ( ( groups3542108847815614940at_nat
% 9.20/5.21          @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
% 9.20/5.21          @ A2 )
% 9.20/5.21        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7023_sum_Odistrib,axiom,
% 9.20/5.21      ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
% 9.20/5.21        ( ( groups7754918857620584856omplex
% 9.20/5.21          @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
% 9.20/5.21          @ A2 )
% 9.20/5.21        = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7024_sum_Odistrib,axiom,
% 9.20/5.21      ! [G: nat > real,H2: nat > real,A2: set_nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
% 9.20/5.21          @ A2 )
% 9.20/5.21        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7025_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups1935376822645274424al_nat
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( groups3542108847815614940at_nat @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups3542108847815614940at_nat
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups1935376822645274424al_nat
% 9.20/5.21                  @ ^ [X: real] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_real
% 9.20/5.21                    @ ^ [X: real] :
% 9.20/5.21                        ( ( member_real @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7026_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups4541462559716669496nt_nat
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( groups3542108847815614940at_nat @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups3542108847815614940at_nat
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups4541462559716669496nt_nat
% 9.20/5.21                  @ ^ [X: int] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_int
% 9.20/5.21                    @ ^ [X: int] :
% 9.20/5.21                        ( ( member_int @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7027_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups5693394587270226106ex_nat
% 9.20/5.21              @ ^ [X: complex] :
% 9.20/5.21                  ( groups3542108847815614940at_nat @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups3542108847815614940at_nat
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups5693394587270226106ex_nat
% 9.20/5.21                  @ ^ [X: complex] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_complex
% 9.20/5.21                    @ ^ [X: complex] :
% 9.20/5.21                        ( ( member_complex @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7028_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_real,B2: set_complex,G: real > complex > complex,R: real > complex > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.21         => ( ( groups5754745047067104278omplex
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( groups7754918857620584856omplex @ ( G @ X )
% 9.20/5.21                  @ ( collect_complex
% 9.20/5.21                    @ ^ [Y: complex] :
% 9.20/5.21                        ( ( member_complex @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups7754918857620584856omplex
% 9.20/5.21              @ ^ [Y: complex] :
% 9.20/5.21                  ( groups5754745047067104278omplex
% 9.20/5.21                  @ ^ [X: real] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_real
% 9.20/5.21                    @ ^ [X: real] :
% 9.20/5.21                        ( ( member_real @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7029_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_nat,B2: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.21         => ( ( groups2073611262835488442omplex
% 9.20/5.21              @ ^ [X: nat] :
% 9.20/5.21                  ( groups7754918857620584856omplex @ ( G @ X )
% 9.20/5.21                  @ ( collect_complex
% 9.20/5.21                    @ ^ [Y: complex] :
% 9.20/5.21                        ( ( member_complex @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups7754918857620584856omplex
% 9.20/5.21              @ ^ [Y: complex] :
% 9.20/5.21                  ( groups2073611262835488442omplex
% 9.20/5.21                  @ ^ [X: nat] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [X: nat] :
% 9.20/5.21                        ( ( member_nat @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7030_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_int,B2: set_complex,G: int > complex > complex,R: int > complex > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.21         => ( ( groups3049146728041665814omplex
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( groups7754918857620584856omplex @ ( G @ X )
% 9.20/5.21                  @ ( collect_complex
% 9.20/5.21                    @ ^ [Y: complex] :
% 9.20/5.21                        ( ( member_complex @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups7754918857620584856omplex
% 9.20/5.21              @ ^ [Y: complex] :
% 9.20/5.21                  ( groups3049146728041665814omplex
% 9.20/5.21                  @ ^ [X: int] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_int
% 9.20/5.21                    @ ^ [X: int] :
% 9.20/5.21                        ( ( member_int @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7031_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_real,B2: set_nat,G: real > nat > real,R: real > nat > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups8097168146408367636l_real
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( groups6591440286371151544t_real @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups8097168146408367636l_real
% 9.20/5.21                  @ ^ [X: real] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_real
% 9.20/5.21                    @ ^ [X: real] :
% 9.20/5.21                        ( ( member_real @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7032_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups8778361861064173332t_real
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( groups6591440286371151544t_real @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups8778361861064173332t_real
% 9.20/5.21                  @ ^ [X: int] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_int
% 9.20/5.21                    @ ^ [X: int] :
% 9.20/5.21                        ( ( member_int @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7033_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( finite_finite_nat @ B2 )
% 9.20/5.21         => ( ( groups5808333547571424918x_real
% 9.20/5.21              @ ^ [X: complex] :
% 9.20/5.21                  ( groups6591440286371151544t_real @ ( G @ X )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [Y: nat] :
% 9.20/5.21                        ( ( member_nat @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups6591440286371151544t_real
% 9.20/5.21              @ ^ [Y: nat] :
% 9.20/5.21                  ( groups5808333547571424918x_real
% 9.20/5.21                  @ ^ [X: complex] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_complex
% 9.20/5.21                    @ ^ [X: complex] :
% 9.20/5.21                        ( ( member_complex @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7034_sum_Oswap__restrict,axiom,
% 9.20/5.21      ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( finite_finite_real @ B2 )
% 9.20/5.21         => ( ( groups3542108847815614940at_nat
% 9.20/5.21              @ ^ [X: nat] :
% 9.20/5.21                  ( groups1935376822645274424al_nat @ ( G @ X )
% 9.20/5.21                  @ ( collect_real
% 9.20/5.21                    @ ^ [Y: real] :
% 9.20/5.21                        ( ( member_real @ Y @ B2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ A2 )
% 9.20/5.21            = ( groups1935376822645274424al_nat
% 9.20/5.21              @ ^ [Y: real] :
% 9.20/5.21                  ( groups3542108847815614940at_nat
% 9.20/5.21                  @ ^ [X: nat] : ( G @ X @ Y )
% 9.20/5.21                  @ ( collect_nat
% 9.20/5.21                    @ ^ [X: nat] :
% 9.20/5.21                        ( ( member_nat @ X @ A2 )
% 9.20/5.21                        & ( R @ X @ Y ) ) ) )
% 9.20/5.21              @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7035_case__prodD_H,axiom,
% 9.20/5.21      ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
% 9.20/5.21        ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
% 9.20/5.21       => ( R @ A @ B @ C ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7036_case__prodE_H,axiom,
% 9.20/5.21      ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z2: product_prod_nat_nat] :
% 9.20/5.21        ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z2 )
% 9.20/5.21       => ~ ! [X4: nat,Y5: nat] :
% 9.20/5.21              ( ( P4
% 9.20/5.21                = ( product_Pair_nat_nat @ X4 @ Y5 ) )
% 9.20/5.21             => ~ ( C @ X4 @ Y5 @ Z2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7037_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > real] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7038_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > real] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7039_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > rat] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7040_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > rat] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7041_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > rat] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7042_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > nat] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7043_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7044_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > int] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7045_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > int] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7046_sum__nonneg,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > int] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7047_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > real] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 9.20/5.21       => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7048_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > real] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
% 9.20/5.21       => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7049_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > rat] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7050_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > rat] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7051_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > rat] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
% 9.20/5.21       => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7052_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > nat] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 9.20/5.21       => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7053_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
% 9.20/5.21       => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7054_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > int] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7055_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > int] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7056_sum__nonpos,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > int] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
% 9.20/5.21       => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7057_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
% 9.20/5.21        ( ( ( groups1300246762558778688al_rat @ F @ I5 )
% 9.20/5.21          = ( groups1300246762558778688al_rat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_real @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_real @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7058_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
% 9.20/5.21        ( ( ( groups2906978787729119204at_rat @ F @ I5 )
% 9.20/5.21          = ( groups2906978787729119204at_rat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: nat] :
% 9.20/5.21              ( ( member_nat @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_nat @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_nat @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7059_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
% 9.20/5.21        ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
% 9.20/5.21          = ( groups3906332499630173760nt_rat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_int @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_int @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7060_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
% 9.20/5.21        ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
% 9.20/5.21          = ( groups5058264527183730370ex_rat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_complex @ I @ I5 )
% 9.20/5.21           => ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7061_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
% 9.20/5.21        ( ( ( groups1935376822645274424al_nat @ F @ I5 )
% 9.20/5.21          = ( groups1935376822645274424al_nat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_real @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_real @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7062_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
% 9.20/5.21        ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
% 9.20/5.21          = ( groups4541462559716669496nt_nat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_int @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_int @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7063_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
% 9.20/5.21        ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
% 9.20/5.21          = ( groups5693394587270226106ex_nat @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_complex @ I @ I5 )
% 9.20/5.21           => ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7064_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: real > int,I5: set_real,G: real > int,I: real] :
% 9.20/5.21        ( ( ( groups1932886352136224148al_int @ F @ I5 )
% 9.20/5.21          = ( groups1932886352136224148al_int @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_real @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_real @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7065_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
% 9.20/5.21        ( ( ( groups3539618377306564664at_int @ F @ I5 )
% 9.20/5.21          = ( groups3539618377306564664at_int @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: nat] :
% 9.20/5.21              ( ( member_nat @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_nat @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_nat @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7066_sum__mono__inv,axiom,
% 9.20/5.21      ! [F: int > int,I5: set_int,G: int > int,I: int] :
% 9.20/5.21        ( ( ( groups4538972089207619220nt_int @ F @ I5 )
% 9.20/5.21          = ( groups4538972089207619220nt_int @ G @ I5 ) )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ I5 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
% 9.20/5.21         => ( ( member_int @ I @ I5 )
% 9.20/5.21           => ( ( finite_finite_int @ I5 )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = ( G @ I ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7067_abs__ge__zero,axiom,
% 9.20/5.21      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7068_abs__ge__zero,axiom,
% 9.20/5.21      ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7069_abs__ge__zero,axiom,
% 9.20/5.21      ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7070_abs__ge__zero,axiom,
% 9.20/5.21      ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7071_abs__not__less__zero,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7072_abs__not__less__zero,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7073_abs__not__less__zero,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7074_abs__not__less__zero,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7075_abs__of__pos,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.21       => ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7076_abs__of__pos,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.21       => ( ( abs_abs_real @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7077_abs__of__pos,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_rat @ zero_zero_rat @ A )
% 9.20/5.21       => ( ( abs_abs_rat @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7078_abs__of__pos,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.21       => ( ( abs_abs_int @ A )
% 9.20/5.21          = A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7079_abs__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7080_abs__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7081_abs__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7082_abs__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7083_abs__mult__less,axiom,
% 9.20/5.21      ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
% 9.20/5.21       => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
% 9.20/5.21         => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7084_abs__mult__less,axiom,
% 9.20/5.21      ! [A: real,C: real,B: real,D: real] :
% 9.20/5.21        ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
% 9.20/5.21       => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
% 9.20/5.21         => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7085_abs__mult__less,axiom,
% 9.20/5.21      ! [A: rat,C: rat,B: rat,D: rat] :
% 9.20/5.21        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
% 9.20/5.21       => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
% 9.20/5.21         => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7086_abs__mult__less,axiom,
% 9.20/5.21      ! [A: int,C: int,B: int,D: int] :
% 9.20/5.21        ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
% 9.20/5.21       => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
% 9.20/5.21         => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7087_abs__triangle__ineq2,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7088_abs__triangle__ineq2,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7089_abs__triangle__ineq2,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7090_abs__triangle__ineq2,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7091_abs__triangle__ineq3,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7092_abs__triangle__ineq3,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7093_abs__triangle__ineq3,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7094_abs__triangle__ineq3,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7095_abs__triangle__ineq2__sym,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7096_abs__triangle__ineq2__sym,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7097_abs__triangle__ineq2__sym,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7098_abs__triangle__ineq2__sym,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7099_nonzero__abs__divide,axiom,
% 9.20/5.21      ! [B: real,A: real] :
% 9.20/5.21        ( ( B != zero_zero_real )
% 9.20/5.21       => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.21          = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7100_nonzero__abs__divide,axiom,
% 9.20/5.21      ! [B: rat,A: rat] :
% 9.20/5.21        ( ( B != zero_zero_rat )
% 9.20/5.21       => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
% 9.20/5.21          = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7101_abs__ge__minus__self,axiom,
% 9.20/5.21      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ ( abs_abs_real @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7102_abs__ge__minus__self,axiom,
% 9.20/5.21      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7103_abs__ge__minus__self,axiom,
% 9.20/5.21      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7104_abs__ge__minus__self,axiom,
% 9.20/5.21      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7105_abs__le__iff,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_eq_real @ A @ B )
% 9.20/5.21          & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7106_abs__le__iff,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.20/5.21        = ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.20/5.21          & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7107_abs__le__iff,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.21          & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7108_abs__le__iff,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_eq_int @ A @ B )
% 9.20/5.21          & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7109_abs__le__D2,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7110_abs__le__D2,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.20/5.21       => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7111_abs__le__D2,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7112_abs__le__D2,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
% 9.20/5.21       => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7113_abs__leI,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ A @ B )
% 9.20/5.21       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
% 9.20/5.21         => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7114_abs__leI,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ A @ B )
% 9.20/5.21       => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
% 9.20/5.21         => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7115_abs__leI,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ A @ B )
% 9.20/5.21       => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
% 9.20/5.21         => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7116_abs__leI,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ A @ B )
% 9.20/5.21       => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 9.20/5.21         => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7117_abs__less__iff,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_int @ A @ B )
% 9.20/5.21          & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7118_abs__less__iff,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_real @ A @ B )
% 9.20/5.21          & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7119_abs__less__iff,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
% 9.20/5.21        = ( ( ord_less_rat @ A @ B )
% 9.20/5.21          & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7120_abs__less__iff,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
% 9.20/5.21        = ( ( ord_le6747313008572928689nteger @ A @ B )
% 9.20/5.21          & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7121_sum__cong__Suc,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > nat,G: nat > nat] :
% 9.20/5.21        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 9.20/5.21       => ( ! [X4: nat] :
% 9.20/5.21              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 9.20/5.21             => ( ( F @ ( suc @ X4 ) )
% 9.20/5.21                = ( G @ ( suc @ X4 ) ) ) )
% 9.20/5.21         => ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.20/5.21            = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7122_sum__cong__Suc,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > real,G: nat > real] :
% 9.20/5.21        ( ~ ( member_nat @ zero_zero_nat @ A2 )
% 9.20/5.21       => ( ! [X4: nat] :
% 9.20/5.21              ( ( member_nat @ ( suc @ X4 ) @ A2 )
% 9.20/5.21             => ( ( F @ ( suc @ X4 ) )
% 9.20/5.21                = ( G @ ( suc @ X4 ) ) ) )
% 9.20/5.21         => ( ( groups6591440286371151544t_real @ F @ A2 )
% 9.20/5.21            = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7123_Collect__case__prod__mono,axiom,
% 9.20/5.21      ! [A2: int > int > $o,B2: int > int > $o] :
% 9.20/5.21        ( ( ord_le6741204236512500942_int_o @ A2 @ B2 )
% 9.20/5.21       => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7124_abs__zmult__eq__1,axiom,
% 9.20/5.21      ! [M: int,N2: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
% 9.20/5.21          = one_one_int )
% 9.20/5.21       => ( ( abs_abs_int @ M )
% 9.20/5.21          = one_one_int ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7125_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > complex,P: real > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups5754745047067104278omplex @ G
% 9.20/5.21            @ ( collect_real
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups5754745047067104278omplex
% 9.20/5.21            @ ^ [X: real] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7126_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > complex,P: nat > $o] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( groups2073611262835488442omplex @ G
% 9.20/5.21            @ ( collect_nat
% 9.20/5.21              @ ^ [X: nat] :
% 9.20/5.21                  ( ( member_nat @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups2073611262835488442omplex
% 9.20/5.21            @ ^ [X: nat] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7127_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > complex,P: int > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3049146728041665814omplex @ G
% 9.20/5.21            @ ( collect_int
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups3049146728041665814omplex
% 9.20/5.21            @ ^ [X: int] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7128_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > real,P: real > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups8097168146408367636l_real @ G
% 9.20/5.21            @ ( collect_real
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups8097168146408367636l_real
% 9.20/5.21            @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7129_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > real,P: int > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups8778361861064173332t_real @ G
% 9.20/5.21            @ ( collect_int
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups8778361861064173332t_real
% 9.20/5.21            @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7130_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > real,P: complex > $o] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5808333547571424918x_real @ G
% 9.20/5.21            @ ( collect_complex
% 9.20/5.21              @ ^ [X: complex] :
% 9.20/5.21                  ( ( member_complex @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups5808333547571424918x_real
% 9.20/5.21            @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7131_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > rat,P: real > $o] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups1300246762558778688al_rat @ G
% 9.20/5.21            @ ( collect_real
% 9.20/5.21              @ ^ [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups1300246762558778688al_rat
% 9.20/5.21            @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7132_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > rat,P: nat > $o] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( groups2906978787729119204at_rat @ G
% 9.20/5.21            @ ( collect_nat
% 9.20/5.21              @ ^ [X: nat] :
% 9.20/5.21                  ( ( member_nat @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups2906978787729119204at_rat
% 9.20/5.21            @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7133_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > rat,P: int > $o] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3906332499630173760nt_rat @ G
% 9.20/5.21            @ ( collect_int
% 9.20/5.21              @ ^ [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups3906332499630173760nt_rat
% 9.20/5.21            @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7134_sum_Ointer__filter,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > rat,P: complex > $o] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5058264527183730370ex_rat @ G
% 9.20/5.21            @ ( collect_complex
% 9.20/5.21              @ ^ [X: complex] :
% 9.20/5.21                  ( ( member_complex @ X @ A2 )
% 9.20/5.21                  & ( P @ X ) ) ) )
% 9.20/5.21          = ( groups5058264527183730370ex_rat
% 9.20/5.21            @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
% 9.20/5.21            @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7135_sum__subtractf__nat,axiom,
% 9.20/5.21      ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
% 9.20/5.21        ( ! [X4: product_prod_nat_nat] :
% 9.20/5.21            ( ( member8440522571783428010at_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ( groups977919841031483927at_nat
% 9.20/5.21            @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.20/5.21            @ A2 )
% 9.20/5.21          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7136_sum__subtractf__nat,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > nat,F: real > nat] :
% 9.20/5.21        ( ! [X4: real] :
% 9.20/5.21            ( ( member_real @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ( groups1935376822645274424al_nat
% 9.20/5.21            @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.20/5.21            @ A2 )
% 9.20/5.21          = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7137_sum__subtractf__nat,axiom,
% 9.20/5.21      ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
% 9.20/5.21        ( ! [X4: set_nat] :
% 9.20/5.21            ( ( member_set_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ( groups8294997508430121362at_nat
% 9.20/5.21            @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.20/5.21            @ A2 )
% 9.20/5.21          = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7138_sum__subtractf__nat,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > nat,F: int > nat] :
% 9.20/5.21        ( ! [X4: int] :
% 9.20/5.21            ( ( member_int @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ( groups4541462559716669496nt_nat
% 9.20/5.21            @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.20/5.21            @ A2 )
% 9.20/5.21          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7139_sum__subtractf__nat,axiom,
% 9.20/5.21      ! [A2: set_nat,G: nat > nat,F: nat > nat] :
% 9.20/5.21        ( ! [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21           => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
% 9.20/5.21       => ( ( groups3542108847815614940at_nat
% 9.20/5.21            @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
% 9.20/5.21            @ A2 )
% 9.20/5.21          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7140_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ I5 )
% 9.20/5.21           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
% 9.20/5.21            = one_one_Code_integer )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_le3102999989581377725nteger
% 9.20/5.21              @ ( abs_abs_Code_integer
% 9.20/5.21                @ ( minus_8373710615458151222nteger
% 9.20/5.21                  @ ( groups7713935264441627589nteger
% 9.20/5.21                    @ ^ [I2: real] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7141_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ I5 )
% 9.20/5.21           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
% 9.20/5.21            = one_one_Code_integer )
% 9.20/5.21         => ( ! [I3: nat] :
% 9.20/5.21                ( ( member_nat @ I3 @ I5 )
% 9.20/5.21               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_le3102999989581377725nteger
% 9.20/5.21              @ ( abs_abs_Code_integer
% 9.20/5.21                @ ( minus_8373710615458151222nteger
% 9.20/5.21                  @ ( groups7501900531339628137nteger
% 9.20/5.21                    @ ^ [I2: nat] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7142_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ I5 )
% 9.20/5.21           => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
% 9.20/5.21            = one_one_Code_integer )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_le3102999989581377725nteger
% 9.20/5.21              @ ( abs_abs_Code_integer
% 9.20/5.21                @ ( minus_8373710615458151222nteger
% 9.20/5.21                  @ ( groups7873554091576472773nteger
% 9.20/5.21                    @ ^ [I2: int] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7143_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_real,X3: real > real,A: real > real,B: real,Delta: real] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
% 9.20/5.21            = one_one_real )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_real
% 9.20/5.21              @ ( abs_abs_real
% 9.20/5.21                @ ( minus_minus_real
% 9.20/5.21                  @ ( groups8097168146408367636l_real
% 9.20/5.21                    @ ^ [I2: real] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7144_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_int,X3: int > real,A: int > real,B: real,Delta: real] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
% 9.20/5.21            = one_one_real )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_real
% 9.20/5.21              @ ( abs_abs_real
% 9.20/5.21                @ ( minus_minus_real
% 9.20/5.21                  @ ( groups8778361861064173332t_real
% 9.20/5.21                    @ ^ [I2: int] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7145_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_real,X3: real > rat,A: real > rat,B: rat,Delta: rat] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
% 9.20/5.21            = one_one_rat )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_rat
% 9.20/5.21              @ ( abs_abs_rat
% 9.20/5.21                @ ( minus_minus_rat
% 9.20/5.21                  @ ( groups1300246762558778688al_rat
% 9.20/5.21                    @ ^ [I2: real] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7146_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_nat,X3: nat > rat,A: nat > rat,B: rat,Delta: rat] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
% 9.20/5.21            = one_one_rat )
% 9.20/5.21         => ( ! [I3: nat] :
% 9.20/5.21                ( ( member_nat @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_rat
% 9.20/5.21              @ ( abs_abs_rat
% 9.20/5.21                @ ( minus_minus_rat
% 9.20/5.21                  @ ( groups2906978787729119204at_rat
% 9.20/5.21                    @ ^ [I2: nat] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7147_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_int,X3: int > rat,A: int > rat,B: rat,Delta: rat] :
% 9.20/5.21        ( ! [I3: int] :
% 9.20/5.21            ( ( member_int @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
% 9.20/5.21            = one_one_rat )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_rat
% 9.20/5.21              @ ( abs_abs_rat
% 9.20/5.21                @ ( minus_minus_rat
% 9.20/5.21                  @ ( groups3906332499630173760nt_rat
% 9.20/5.21                    @ ^ [I2: int] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7148_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_real,X3: real > int,A: real > int,B: int,Delta: int] :
% 9.20/5.21        ( ! [I3: real] :
% 9.20/5.21            ( ( member_real @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups1932886352136224148al_int @ X3 @ I5 )
% 9.20/5.21            = one_one_int )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_int
% 9.20/5.21              @ ( abs_abs_int
% 9.20/5.21                @ ( minus_minus_int
% 9.20/5.21                  @ ( groups1932886352136224148al_int
% 9.20/5.21                    @ ^ [I2: real] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7149_convex__sum__bound__le,axiom,
% 9.20/5.21      ! [I5: set_nat,X3: nat > int,A: nat > int,B: int,Delta: int] :
% 9.20/5.21        ( ! [I3: nat] :
% 9.20/5.21            ( ( member_nat @ I3 @ I5 )
% 9.20/5.21           => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
% 9.20/5.21       => ( ( ( groups3539618377306564664at_int @ X3 @ I5 )
% 9.20/5.21            = one_one_int )
% 9.20/5.21         => ( ! [I3: nat] :
% 9.20/5.21                ( ( member_nat @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
% 9.20/5.21           => ( ord_less_eq_int
% 9.20/5.21              @ ( abs_abs_int
% 9.20/5.21                @ ( minus_minus_int
% 9.20/5.21                  @ ( groups3539618377306564664at_int
% 9.20/5.21                    @ ^ [I2: nat] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
% 9.20/5.21                    @ I5 )
% 9.20/5.21                  @ B ) )
% 9.20/5.21              @ Delta ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7150_infinite__int__iff__unbounded__le,axiom,
% 9.20/5.21      ! [S3: set_int] :
% 9.20/5.21        ( ( ~ ( finite_finite_int @ S3 ) )
% 9.20/5.21        = ( ! [M2: int] :
% 9.20/5.21            ? [N: int] :
% 9.20/5.21              ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N ) )
% 9.20/5.21              & ( member_int @ N @ S3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7151_infinite__int__iff__unbounded,axiom,
% 9.20/5.21      ! [S3: set_int] :
% 9.20/5.21        ( ( ~ ( finite_finite_int @ S3 ) )
% 9.20/5.21        = ( ! [M2: int] :
% 9.20/5.21            ? [N: int] :
% 9.20/5.21              ( ( ord_less_int @ M2 @ ( abs_abs_int @ N ) )
% 9.20/5.21              & ( member_int @ N @ S3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7152_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 9.20/5.21      ! [G: nat > nat,M: nat,N2: nat] :
% 9.20/5.21        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 9.20/5.21        = ( groups3542108847815614940at_nat
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7153_sum_Oshift__bounds__cl__Suc__ivl,axiom,
% 9.20/5.21      ! [G: nat > real,M: nat,N2: nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 9.20/5.21        = ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7154_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.20/5.21      ! [G: nat > nat,M: nat,K: nat,N2: nat] :
% 9.20/5.21        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.20/5.21        = ( groups3542108847815614940at_nat
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7155_sum_Oshift__bounds__cl__nat__ivl,axiom,
% 9.20/5.21      ! [G: nat > real,M: nat,K: nat,N2: nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
% 9.20/5.21        = ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7156_tanh__real__lt__1,axiom,
% 9.20/5.21      ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7157_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ( finite_finite_int @ T )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S )
% 9.20/5.21                 => ? [Xa: int] :
% 9.20/5.21                      ( ( member_int @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7158_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S )
% 9.20/5.21                 => ? [Xa: complex] :
% 9.20/5.21                      ( ( member_complex @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7159_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ( finite_finite_int @ T )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S )
% 9.20/5.21                 => ? [Xa: int] :
% 9.20/5.21                      ( ( member_int @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7160_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S )
% 9.20/5.21                 => ? [Xa: complex] :
% 9.20/5.21                      ( ( member_complex @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7161_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ S )
% 9.20/5.21       => ( ( finite_finite_nat @ T )
% 9.20/5.21         => ( ! [X4: nat] :
% 9.20/5.21                ( ( member_nat @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S )
% 9.20/5.21                 => ? [Xa: nat] :
% 9.20/5.21                      ( ( member_nat @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7162_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ S )
% 9.20/5.21       => ( ( finite_finite_int @ T )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S )
% 9.20/5.21                 => ? [Xa: int] :
% 9.20/5.21                      ( ( member_int @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7163_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ S )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S )
% 9.20/5.21                 => ? [Xa: complex] :
% 9.20/5.21                      ( ( member_complex @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7164_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ( finite_finite_nat @ T )
% 9.20/5.21         => ( ! [X4: nat] :
% 9.20/5.21                ( ( member_nat @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S )
% 9.20/5.21                 => ? [Xa: nat] :
% 9.20/5.21                      ( ( member_nat @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7165_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ( finite_finite_int @ T )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S )
% 9.20/5.21                 => ? [Xa: int] :
% 9.20/5.21                      ( ( member_int @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7166_sum__le__included,axiom,
% 9.20/5.21      ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ T )
% 9.20/5.21               => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S )
% 9.20/5.21                 => ? [Xa: complex] :
% 9.20/5.21                      ( ( member_complex @ Xa @ T )
% 9.20/5.21                      & ( ( I @ Xa )
% 9.20/5.21                        = X4 )
% 9.20/5.21                      & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
% 9.20/5.21             => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7167_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ! [X4: real] :
% 9.20/5.21              ( ( member_real @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups8097168146408367636l_real @ F @ A2 )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21            = ( ! [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7168_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups8778361861064173332t_real @ F @ A2 )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21            = ( ! [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7169_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups5808333547571424918x_real @ F @ A2 )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21            = ( ! [X: complex] :
% 9.20/5.21                  ( ( member_complex @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7170_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ! [X4: real] :
% 9.20/5.21              ( ( member_real @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21            = ( ! [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7171_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ! [X4: nat] :
% 9.20/5.21              ( ( member_nat @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21            = ( ! [X: nat] :
% 9.20/5.21                  ( ( member_nat @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7172_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21            = ( ! [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7173_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21            = ( ! [X: complex] :
% 9.20/5.21                  ( ( member_complex @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7174_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ! [X4: real] :
% 9.20/5.21              ( ( member_real @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21            = ( ! [X: real] :
% 9.20/5.21                  ( ( member_real @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7175_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21            = ( ! [X: int] :
% 9.20/5.21                  ( ( member_int @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7176_sum__nonneg__eq__0__iff,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.21         => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21            = ( ! [X: complex] :
% 9.20/5.21                  ( ( member_complex @ X @ A2 )
% 9.20/5.21                 => ( ( F @ X )
% 9.20/5.21                    = zero_zero_nat ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7177_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > real,G: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: int] :
% 9.20/5.21                ( ( member_int @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7178_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: complex] :
% 9.20/5.21                ( ( member_complex @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_real @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7179_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ! [X4: nat] :
% 9.20/5.21              ( ( member_nat @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: nat] :
% 9.20/5.21                ( ( member_nat @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7180_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > rat,G: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: int] :
% 9.20/5.21                ( ( member_int @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7181_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: complex] :
% 9.20/5.21                ( ( member_complex @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7182_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat,G: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: int] :
% 9.20/5.21                ( ( member_int @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7183_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: complex] :
% 9.20/5.21                ( ( member_complex @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_nat @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7184_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > int,G: nat > int] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ! [X4: nat] :
% 9.20/5.21              ( ( member_nat @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: nat] :
% 9.20/5.21                ( ( member_nat @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7185_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > int,G: int > int] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ! [X4: int] :
% 9.20/5.21              ( ( member_int @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: int] :
% 9.20/5.21                ( ( member_int @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7186_sum__strict__mono__ex1,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > int,G: complex > int] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ! [X4: complex] :
% 9.20/5.21              ( ( member_complex @ X4 @ A2 )
% 9.20/5.21             => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21         => ( ? [X2: complex] :
% 9.20/5.21                ( ( member_complex @ X2 @ A2 )
% 9.20/5.21                & ( ord_less_int @ ( F @ X2 ) @ ( G @ X2 ) ) )
% 9.20/5.21           => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7187_sum_Orelated,axiom,
% 9.20/5.21      ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 9.20/5.21        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 9.20/5.21       => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_nat @ S3 )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7188_sum_Orelated,axiom,
% 9.20/5.21      ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
% 9.20/5.21        ( ( R @ zero_zero_complex @ zero_zero_complex )
% 9.20/5.21       => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_int @ S3 )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7189_sum_Orelated,axiom,
% 9.20/5.21      ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
% 9.20/5.21        ( ( R @ zero_zero_real @ zero_zero_real )
% 9.20/5.21       => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_int @ S3 )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7190_sum_Orelated,axiom,
% 9.20/5.21      ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
% 9.20/5.21        ( ( R @ zero_zero_real @ zero_zero_real )
% 9.20/5.21       => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7191_sum_Orelated,axiom,
% 9.20/5.21      ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
% 9.20/5.21        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.20/5.21       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_nat @ S3 )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7192_sum_Orelated,axiom,
% 9.20/5.21      ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
% 9.20/5.21        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.20/5.21       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_int @ S3 )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7193_sum_Orelated,axiom,
% 9.20/5.21      ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 9.20/5.21        ( ( R @ zero_zero_rat @ zero_zero_rat )
% 9.20/5.21       => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7194_sum_Orelated,axiom,
% 9.20/5.21      ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
% 9.20/5.21        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.20/5.21       => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_int @ S3 )
% 9.20/5.21           => ( ! [X4: int] :
% 9.20/5.21                  ( ( member_int @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7195_sum_Orelated,axiom,
% 9.20/5.21      ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 9.20/5.21        ( ( R @ zero_zero_nat @ zero_zero_nat )
% 9.20/5.21       => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7196_sum_Orelated,axiom,
% 9.20/5.21      ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
% 9.20/5.21        ( ( R @ zero_zero_int @ zero_zero_int )
% 9.20/5.21       => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
% 9.20/5.21              ( ( ( R @ X16 @ X23 )
% 9.20/5.21                & ( R @ Y15 @ Y23 ) )
% 9.20/5.21             => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
% 9.20/5.21         => ( ( finite_finite_nat @ S3 )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S3 )
% 9.20/5.21                 => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21             => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7197_dense__eq0__I,axiom,
% 9.20/5.21      ! [X3: real] :
% 9.20/5.21        ( ! [E: real] :
% 9.20/5.21            ( ( ord_less_real @ zero_zero_real @ E )
% 9.20/5.21           => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E ) )
% 9.20/5.21       => ( X3 = zero_zero_real ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7198_dense__eq0__I,axiom,
% 9.20/5.21      ! [X3: rat] :
% 9.20/5.21        ( ! [E: rat] :
% 9.20/5.21            ( ( ord_less_rat @ zero_zero_rat @ E )
% 9.20/5.21           => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E ) )
% 9.20/5.21       => ( X3 = zero_zero_rat ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7199_abs__mult__pos,axiom,
% 9.20/5.21      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
% 9.20/5.21       => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
% 9.20/5.21          = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7200_abs__mult__pos,axiom,
% 9.20/5.21      ! [X3: real,Y3: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.21       => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
% 9.20/5.21          = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7201_abs__mult__pos,axiom,
% 9.20/5.21      ! [X3: rat,Y3: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.21       => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
% 9.20/5.21          = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7202_abs__mult__pos,axiom,
% 9.20/5.21      ! [X3: int,Y3: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.21       => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
% 9.20/5.21          = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7203_abs__eq__mult,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.21            | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
% 9.20/5.21          & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.21            | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
% 9.20/5.21       => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
% 9.20/5.21          = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7204_abs__eq__mult,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.21            | ( ord_less_eq_real @ A @ zero_zero_real ) )
% 9.20/5.21          & ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.21            | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
% 9.20/5.21       => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
% 9.20/5.21          = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7205_abs__eq__mult,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.21            | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
% 9.20/5.21          & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.21            | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
% 9.20/5.21       => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
% 9.20/5.21          = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7206_abs__eq__mult,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.21            | ( ord_less_eq_int @ A @ zero_zero_int ) )
% 9.20/5.21          & ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.21            | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
% 9.20/5.21       => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
% 9.20/5.21          = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7207_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > real,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7208_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > real,G: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_real )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7209_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > real,G: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_int )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7210_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > rat,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7211_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > rat,G: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_real )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7212_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > rat,G: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_nat )
% 9.20/5.21         => ( ! [X4: nat] :
% 9.20/5.21                ( ( member_nat @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7213_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > rat,G: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_int )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7214_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > nat,G: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7215_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_real,F: real > nat,G: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_real )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7216_sum__strict__mono,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat,G: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( A2 != bot_bot_set_int )
% 9.20/5.21         => ( ! [X4: int] :
% 9.20/5.21                ( ( member_int @ X4 @ A2 )
% 9.20/5.21               => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
% 9.20/5.21           => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7217_abs__minus__le__zero,axiom,
% 9.20/5.21      ! [A: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( abs_abs_real @ A ) ) @ zero_zero_real ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7218_abs__minus__le__zero,axiom,
% 9.20/5.21      ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7219_abs__minus__le__zero,axiom,
% 9.20/5.21      ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7220_abs__minus__le__zero,axiom,
% 9.20/5.21      ! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7221_abs__eq__iff_H,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( ( abs_abs_real @ A )
% 9.20/5.21          = B )
% 9.20/5.21        = ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.21          & ( ( A = B )
% 9.20/5.21            | ( A
% 9.20/5.21              = ( uminus_uminus_real @ B ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7222_abs__eq__iff_H,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = B )
% 9.20/5.21        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
% 9.20/5.21          & ( ( A = B )
% 9.20/5.21            | ( A
% 9.20/5.21              = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7223_abs__eq__iff_H,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( ( abs_abs_rat @ A )
% 9.20/5.21          = B )
% 9.20/5.21        = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.21          & ( ( A = B )
% 9.20/5.21            | ( A
% 9.20/5.21              = ( uminus_uminus_rat @ B ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7224_abs__eq__iff_H,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( ( abs_abs_int @ A )
% 9.20/5.21          = B )
% 9.20/5.21        = ( ( ord_less_eq_int @ zero_zero_int @ B )
% 9.20/5.21          & ( ( A = B )
% 9.20/5.21            | ( A
% 9.20/5.21              = ( uminus_uminus_int @ B ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7225_eq__abs__iff_H,axiom,
% 9.20/5.21      ! [A: real,B: real] :
% 9.20/5.21        ( ( A
% 9.20/5.21          = ( abs_abs_real @ B ) )
% 9.20/5.21        = ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.21          & ( ( B = A )
% 9.20/5.21            | ( B
% 9.20/5.21              = ( uminus_uminus_real @ A ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7226_eq__abs__iff_H,axiom,
% 9.20/5.21      ! [A: code_integer,B: code_integer] :
% 9.20/5.21        ( ( A
% 9.20/5.21          = ( abs_abs_Code_integer @ B ) )
% 9.20/5.21        = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
% 9.20/5.21          & ( ( B = A )
% 9.20/5.21            | ( B
% 9.20/5.21              = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7227_eq__abs__iff_H,axiom,
% 9.20/5.21      ! [A: rat,B: rat] :
% 9.20/5.21        ( ( A
% 9.20/5.21          = ( abs_abs_rat @ B ) )
% 9.20/5.21        = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.21          & ( ( B = A )
% 9.20/5.21            | ( B
% 9.20/5.21              = ( uminus_uminus_rat @ A ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7228_eq__abs__iff_H,axiom,
% 9.20/5.21      ! [A: int,B: int] :
% 9.20/5.21        ( ( A
% 9.20/5.21          = ( abs_abs_int @ B ) )
% 9.20/5.21        = ( ( ord_less_eq_int @ zero_zero_int @ A )
% 9.20/5.21          & ( ( B = A )
% 9.20/5.21            | ( B
% 9.20/5.21              = ( uminus_uminus_int @ A ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7229_zero__le__power__abs,axiom,
% 9.20/5.21      ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7230_zero__le__power__abs,axiom,
% 9.20/5.21      ! [A: real,N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7231_zero__le__power__abs,axiom,
% 9.20/5.21      ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7232_zero__le__power__abs,axiom,
% 9.20/5.21      ! [A: int,N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7233_abs__div__pos,axiom,
% 9.20/5.21      ! [Y3: real,X3: real] :
% 9.20/5.21        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.21       => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
% 9.20/5.21          = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7234_abs__div__pos,axiom,
% 9.20/5.21      ! [Y3: rat,X3: rat] :
% 9.20/5.21        ( ( ord_less_rat @ zero_zero_rat @ Y3 )
% 9.20/5.21       => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
% 9.20/5.21          = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7235_abs__if__raw,axiom,
% 9.20/5.21      ( abs_abs_int
% 9.20/5.21      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7236_abs__if__raw,axiom,
% 9.20/5.21      ( abs_abs_real
% 9.20/5.21      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7237_abs__if__raw,axiom,
% 9.20/5.21      ( abs_abs_rat
% 9.20/5.21      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7238_abs__if__raw,axiom,
% 9.20/5.21      ( abs_abs_Code_integer
% 9.20/5.21      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7239_abs__of__neg,axiom,
% 9.20/5.21      ! [A: int] :
% 9.20/5.21        ( ( ord_less_int @ A @ zero_zero_int )
% 9.20/5.21       => ( ( abs_abs_int @ A )
% 9.20/5.21          = ( uminus_uminus_int @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7240_abs__of__neg,axiom,
% 9.20/5.21      ! [A: real] :
% 9.20/5.21        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.21       => ( ( abs_abs_real @ A )
% 9.20/5.21          = ( uminus_uminus_real @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7241_abs__of__neg,axiom,
% 9.20/5.21      ! [A: rat] :
% 9.20/5.21        ( ( ord_less_rat @ A @ zero_zero_rat )
% 9.20/5.21       => ( ( abs_abs_rat @ A )
% 9.20/5.21          = ( uminus_uminus_rat @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7242_abs__of__neg,axiom,
% 9.20/5.21      ! [A: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
% 9.20/5.21       => ( ( abs_abs_Code_integer @ A )
% 9.20/5.21          = ( uminus1351360451143612070nteger @ A ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7243_abs__if,axiom,
% 9.20/5.21      ( abs_abs_int
% 9.20/5.21      = ( ^ [A3: int] : ( if_int @ ( ord_less_int @ A3 @ zero_zero_int ) @ ( uminus_uminus_int @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7244_abs__if,axiom,
% 9.20/5.21      ( abs_abs_real
% 9.20/5.21      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7245_abs__if,axiom,
% 9.20/5.21      ( abs_abs_rat
% 9.20/5.21      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7246_abs__if,axiom,
% 9.20/5.21      ( abs_abs_Code_integer
% 9.20/5.21      = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7247_abs__diff__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7248_abs__diff__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7249_abs__diff__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7250_abs__diff__triangle__ineq,axiom,
% 9.20/5.21      ! [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 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7251_abs__triangle__ineq4,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7252_abs__triangle__ineq4,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7253_abs__triangle__ineq4,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7254_abs__triangle__ineq4,axiom,
% 9.20/5.21      ! [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 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7255_abs__diff__le__iff,axiom,
% 9.20/5.21      ! [X3: code_integer,A: code_integer,R2: code_integer] :
% 9.20/5.21        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7256_abs__diff__le__iff,axiom,
% 9.20/5.21      ! [X3: real,A: real,R2: real] :
% 9.20/5.21        ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7257_abs__diff__le__iff,axiom,
% 9.20/5.21      ! [X3: rat,A: rat,R2: rat] :
% 9.20/5.21        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7258_abs__diff__le__iff,axiom,
% 9.20/5.21      ! [X3: int,A: int,R2: int] :
% 9.20/5.21        ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7259_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( groups8097168146408367636l_real @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7260_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( groups8778361861064173332t_real @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7261_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( groups5808333547571424918x_real @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7262_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7263_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( ( member_nat @ X3 @ A2 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.21              = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.21           => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7264_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7265_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7266_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_real,X3: real,G: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.21           => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7267_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_int,X3: int,G: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.21           => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7268_sum_Oinsert__if,axiom,
% 9.20/5.21      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
% 9.20/5.21          & ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.21           => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.21              = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7269_abs__diff__less__iff,axiom,
% 9.20/5.21      ! [X3: code_integer,A: code_integer,R2: code_integer] :
% 9.20/5.21        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7270_abs__diff__less__iff,axiom,
% 9.20/5.21      ! [X3: real,A: real,R2: real] :
% 9.20/5.21        ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7271_abs__diff__less__iff,axiom,
% 9.20/5.21      ! [X3: rat,A: rat,R2: rat] :
% 9.20/5.21        ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7272_abs__diff__less__iff,axiom,
% 9.20/5.21      ! [X3: int,A: int,R2: int] :
% 9.20/5.21        ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
% 9.20/5.21        = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
% 9.20/5.21          & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7273_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ S6 )
% 9.20/5.21       => ( ( finite_finite_real @ T5 )
% 9.20/5.21         => ( ! [A4: real] :
% 9.20/5.21                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: real] :
% 9.20/5.21                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: real] :
% 9.20/5.21                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: real] :
% 9.20/5.21                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: real] :
% 9.20/5.21                        ( ( member_real @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_complex ) )
% 9.20/5.21                   => ( ! [B4: real] :
% 9.20/5.21                          ( ( member_real @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_complex ) )
% 9.20/5.21                     => ( ! [A4: real] :
% 9.20/5.21                            ( ( member_real @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups5754745047067104278omplex @ G @ S3 )
% 9.20/5.21                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7274_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
% 9.20/5.21        ( ( finite_finite_real @ S6 )
% 9.20/5.21       => ( ( finite_finite_int @ T5 )
% 9.20/5.21         => ( ! [A4: real] :
% 9.20/5.21                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: real] :
% 9.20/5.21                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: int] :
% 9.20/5.21                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: int] :
% 9.20/5.21                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: real] :
% 9.20/5.21                        ( ( member_real @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_complex ) )
% 9.20/5.21                   => ( ! [B4: int] :
% 9.20/5.21                          ( ( member_int @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_complex ) )
% 9.20/5.21                     => ( ! [A4: real] :
% 9.20/5.21                            ( ( member_real @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups5754745047067104278omplex @ G @ S3 )
% 9.20/5.21                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7275_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
% 9.20/5.21        ( ( finite_finite_int @ S6 )
% 9.20/5.21       => ( ( finite_finite_real @ T5 )
% 9.20/5.21         => ( ! [A4: int] :
% 9.20/5.21                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: int] :
% 9.20/5.21                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: real] :
% 9.20/5.21                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: real] :
% 9.20/5.21                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: int] :
% 9.20/5.21                        ( ( member_int @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_complex ) )
% 9.20/5.21                   => ( ! [B4: real] :
% 9.20/5.21                          ( ( member_real @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_complex ) )
% 9.20/5.21                     => ( ! [A4: int] :
% 9.20/5.21                            ( ( member_int @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups3049146728041665814omplex @ G @ S3 )
% 9.20/5.21                          = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7276_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
% 9.20/5.21        ( ( finite_finite_int @ S6 )
% 9.20/5.21       => ( ( finite_finite_int @ T5 )
% 9.20/5.21         => ( ! [A4: int] :
% 9.20/5.21                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: int] :
% 9.20/5.21                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: int] :
% 9.20/5.21                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: int] :
% 9.20/5.21                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: int] :
% 9.20/5.21                        ( ( member_int @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_complex ) )
% 9.20/5.21                   => ( ! [B4: int] :
% 9.20/5.21                          ( ( member_int @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_complex ) )
% 9.20/5.21                     => ( ! [A4: int] :
% 9.20/5.21                            ( ( member_int @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups3049146728041665814omplex @ G @ S3 )
% 9.20/5.21                          = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7277_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ S6 )
% 9.20/5.21       => ( ( finite_finite_real @ T5 )
% 9.20/5.21         => ( ! [A4: real] :
% 9.20/5.21                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: real] :
% 9.20/5.21                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: real] :
% 9.20/5.21                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: real] :
% 9.20/5.21                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: real] :
% 9.20/5.21                        ( ( member_real @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: real] :
% 9.20/5.21                          ( ( member_real @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: real] :
% 9.20/5.21                            ( ( member_real @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 9.20/5.21                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7278_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
% 9.20/5.21        ( ( finite_finite_real @ S6 )
% 9.20/5.21       => ( ( finite_finite_int @ T5 )
% 9.20/5.21         => ( ! [A4: real] :
% 9.20/5.21                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: real] :
% 9.20/5.21                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: int] :
% 9.20/5.21                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: int] :
% 9.20/5.21                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: real] :
% 9.20/5.21                        ( ( member_real @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: int] :
% 9.20/5.21                          ( ( member_int @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: real] :
% 9.20/5.21                            ( ( member_real @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 9.20/5.21                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7279_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_real,T5: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
% 9.20/5.21        ( ( finite_finite_real @ S6 )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T5 )
% 9.20/5.21         => ( ! [A4: real] :
% 9.20/5.21                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: real] :
% 9.20/5.21                  ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: complex] :
% 9.20/5.21                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: complex] :
% 9.20/5.21                      ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_real @ ( I @ B4 ) @ ( minus_minus_set_real @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: real] :
% 9.20/5.21                        ( ( member_real @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: complex] :
% 9.20/5.21                          ( ( member_complex @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: real] :
% 9.20/5.21                            ( ( member_real @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8097168146408367636l_real @ G @ S3 )
% 9.20/5.21                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7280_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
% 9.20/5.21        ( ( finite_finite_int @ S6 )
% 9.20/5.21       => ( ( finite_finite_real @ T5 )
% 9.20/5.21         => ( ! [A4: int] :
% 9.20/5.21                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: int] :
% 9.20/5.21                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_real @ ( J @ A4 ) @ ( minus_minus_set_real @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: real] :
% 9.20/5.21                    ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: real] :
% 9.20/5.21                      ( ( member_real @ B4 @ ( minus_minus_set_real @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: int] :
% 9.20/5.21                        ( ( member_int @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: real] :
% 9.20/5.21                          ( ( member_real @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: int] :
% 9.20/5.21                            ( ( member_int @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 9.20/5.21                          = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7281_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ S6 )
% 9.20/5.21       => ( ( finite_finite_int @ T5 )
% 9.20/5.21         => ( ! [A4: int] :
% 9.20/5.21                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: int] :
% 9.20/5.21                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_int @ ( J @ A4 ) @ ( minus_minus_set_int @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: int] :
% 9.20/5.21                    ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: int] :
% 9.20/5.21                      ( ( member_int @ B4 @ ( minus_minus_set_int @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: int] :
% 9.20/5.21                        ( ( member_int @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: int] :
% 9.20/5.21                          ( ( member_int @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: int] :
% 9.20/5.21                            ( ( member_int @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 9.20/5.21                          = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7282_sum_Oreindex__bij__witness__not__neutral,axiom,
% 9.20/5.21      ! [S6: set_int,T5: set_complex,S3: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
% 9.20/5.21        ( ( finite_finite_int @ S6 )
% 9.20/5.21       => ( ( finite3207457112153483333omplex @ T5 )
% 9.20/5.21         => ( ! [A4: int] :
% 9.20/5.21                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21               => ( ( I @ ( J @ A4 ) )
% 9.20/5.21                  = A4 ) )
% 9.20/5.21           => ( ! [A4: int] :
% 9.20/5.21                  ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
% 9.20/5.21                 => ( member_complex @ ( J @ A4 ) @ ( minus_811609699411566653omplex @ T3 @ T5 ) ) )
% 9.20/5.21             => ( ! [B4: complex] :
% 9.20/5.21                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 9.20/5.21                   => ( ( J @ ( I @ B4 ) )
% 9.20/5.21                      = B4 ) )
% 9.20/5.21               => ( ! [B4: complex] :
% 9.20/5.21                      ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ T3 @ T5 ) )
% 9.20/5.21                     => ( member_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
% 9.20/5.21                 => ( ! [A4: int] :
% 9.20/5.21                        ( ( member_int @ A4 @ S6 )
% 9.20/5.21                       => ( ( G @ A4 )
% 9.20/5.21                          = zero_zero_real ) )
% 9.20/5.21                   => ( ! [B4: complex] :
% 9.20/5.21                          ( ( member_complex @ B4 @ T5 )
% 9.20/5.21                         => ( ( H2 @ B4 )
% 9.20/5.21                            = zero_zero_real ) )
% 9.20/5.21                     => ( ! [A4: int] :
% 9.20/5.21                            ( ( member_int @ A4 @ S3 )
% 9.20/5.21                           => ( ( H2 @ ( J @ A4 ) )
% 9.20/5.21                              = ( G @ A4 ) ) )
% 9.20/5.21                       => ( ( groups8778361861064173332t_real @ G @ S3 )
% 9.20/5.21                          = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7283_sum__eq__Suc0__iff,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.20/5.21            = ( suc @ zero_zero_nat ) )
% 9.20/5.21          = ( ? [X: int] :
% 9.20/5.21                ( ( member_int @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = ( suc @ zero_zero_nat ) )
% 9.20/5.21                & ! [Y: int] :
% 9.20/5.21                    ( ( member_int @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7284_sum__eq__Suc0__iff,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.20/5.21            = ( suc @ zero_zero_nat ) )
% 9.20/5.21          = ( ? [X: complex] :
% 9.20/5.21                ( ( member_complex @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = ( suc @ zero_zero_nat ) )
% 9.20/5.21                & ! [Y: complex] :
% 9.20/5.21                    ( ( member_complex @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7285_sum__eq__Suc0__iff,axiom,
% 9.20/5.21      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.20/5.21        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.21       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 9.20/5.21            = ( suc @ zero_zero_nat ) )
% 9.20/5.21          = ( ? [X: product_prod_nat_nat] :
% 9.20/5.21                ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = ( suc @ zero_zero_nat ) )
% 9.20/5.21                & ! [Y: product_prod_nat_nat] :
% 9.20/5.21                    ( ( member8440522571783428010at_nat @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7286_sum__eq__Suc0__iff,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > nat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.20/5.21            = ( suc @ zero_zero_nat ) )
% 9.20/5.21          = ( ? [X: nat] :
% 9.20/5.21                ( ( member_nat @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = ( suc @ zero_zero_nat ) )
% 9.20/5.21                & ! [Y: nat] :
% 9.20/5.21                    ( ( member_nat @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7287_sum__SucD,axiom,
% 9.20/5.21      ! [F: nat > nat,A2: set_nat,N2: nat] :
% 9.20/5.21        ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.20/5.21          = ( suc @ N2 ) )
% 9.20/5.21       => ? [X4: nat] :
% 9.20/5.21            ( ( member_nat @ X4 @ A2 )
% 9.20/5.21            & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7288_sum__eq__1__iff,axiom,
% 9.20/5.21      ! [A2: set_int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
% 9.20/5.21            = one_one_nat )
% 9.20/5.21          = ( ? [X: int] :
% 9.20/5.21                ( ( member_int @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = one_one_nat )
% 9.20/5.21                & ! [Y: int] :
% 9.20/5.21                    ( ( member_int @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7289_sum__eq__1__iff,axiom,
% 9.20/5.21      ! [A2: set_complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
% 9.20/5.21            = one_one_nat )
% 9.20/5.21          = ( ? [X: complex] :
% 9.20/5.21                ( ( member_complex @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = one_one_nat )
% 9.20/5.21                & ! [Y: complex] :
% 9.20/5.21                    ( ( member_complex @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7290_sum__eq__1__iff,axiom,
% 9.20/5.21      ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.20/5.21        ( ( finite6177210948735845034at_nat @ A2 )
% 9.20/5.21       => ( ( ( groups977919841031483927at_nat @ F @ A2 )
% 9.20/5.21            = one_one_nat )
% 9.20/5.21          = ( ? [X: product_prod_nat_nat] :
% 9.20/5.21                ( ( member8440522571783428010at_nat @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = one_one_nat )
% 9.20/5.21                & ! [Y: product_prod_nat_nat] :
% 9.20/5.21                    ( ( member8440522571783428010at_nat @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7291_sum__eq__1__iff,axiom,
% 9.20/5.21      ! [A2: set_nat,F: nat > nat] :
% 9.20/5.21        ( ( finite_finite_nat @ A2 )
% 9.20/5.21       => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
% 9.20/5.21            = one_one_nat )
% 9.20/5.21          = ( ? [X: nat] :
% 9.20/5.21                ( ( member_nat @ X @ A2 )
% 9.20/5.21                & ( ( F @ X )
% 9.20/5.21                  = one_one_nat )
% 9.20/5.21                & ! [Y: nat] :
% 9.20/5.21                    ( ( member_nat @ Y @ A2 )
% 9.20/5.21                   => ( ( X != Y )
% 9.20/5.21                     => ( ( F @ Y )
% 9.20/5.21                        = zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7292_abs__real__def,axiom,
% 9.20/5.21      ( abs_abs_real
% 9.20/5.21      = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7293_zabs__def,axiom,
% 9.20/5.21      ( abs_abs_int
% 9.20/5.21      = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7294_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_real,F: real > real,B2: real,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7295_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_int,F: int > real,B2: real,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7296_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > real,B2: real,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7297_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_real,F: real > rat,B2: rat,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7298_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
% 9.20/5.21        ( ( finite_finite_nat @ S )
% 9.20/5.21       => ( ! [I3: nat] :
% 9.20/5.21              ( ( member_nat @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_nat @ I @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7299_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_int,F: int > rat,B2: rat,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7300_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7301_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_real,F: real > nat,B2: nat,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7302_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_int,F: int > nat,B2: nat,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7303_sum__nonneg__leq__bound,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > nat,B2: nat,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.20/5.21              = B2 )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7304_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_real,F: real > real,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups8097168146408367636l_real @ F @ S )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_real ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7305_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_int,F: int > real,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups8778361861064173332t_real @ F @ S )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_real ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7306_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > real,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5808333547571424918x_real @ F @ S )
% 9.20/5.21              = zero_zero_real )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_real ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7307_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_real,F: real > rat,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups1300246762558778688al_rat @ F @ S )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7308_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_nat,F: nat > rat,I: nat] :
% 9.20/5.21        ( ( finite_finite_nat @ S )
% 9.20/5.21       => ( ! [I3: nat] :
% 9.20/5.21              ( ( member_nat @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups2906978787729119204at_rat @ F @ S )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21           => ( ( member_nat @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7309_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_int,F: int > rat,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups3906332499630173760nt_rat @ F @ S )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7310_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > rat,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5058264527183730370ex_rat @ F @ S )
% 9.20/5.21              = zero_zero_rat )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7311_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_real,F: real > nat,I: real] :
% 9.20/5.21        ( ( finite_finite_real @ S )
% 9.20/5.21       => ( ! [I3: real] :
% 9.20/5.21              ( ( member_real @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups1935376822645274424al_nat @ F @ S )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21           => ( ( member_real @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7312_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_int,F: int > nat,I: int] :
% 9.20/5.21        ( ( finite_finite_int @ S )
% 9.20/5.21       => ( ! [I3: int] :
% 9.20/5.21              ( ( member_int @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups4541462559716669496nt_nat @ F @ S )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21           => ( ( member_int @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7313_sum__nonneg__0,axiom,
% 9.20/5.21      ! [S: set_complex,F: complex > nat,I: complex] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ S )
% 9.20/5.21       => ( ! [I3: complex] :
% 9.20/5.21              ( ( member_complex @ I3 @ S )
% 9.20/5.21             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21         => ( ( ( groups5693394587270226106ex_nat @ F @ S )
% 9.20/5.21              = zero_zero_nat )
% 9.20/5.21           => ( ( member_complex @ I @ S )
% 9.20/5.21             => ( ( F @ I )
% 9.20/5.21                = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7314_dvd__imp__le__int,axiom,
% 9.20/5.21      ! [I: int,D: int] :
% 9.20/5.21        ( ( I != zero_zero_int )
% 9.20/5.21       => ( ( dvd_dvd_int @ D @ I )
% 9.20/5.21         => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7315_abs__mod__less,axiom,
% 9.20/5.21      ! [L: int,K: int] :
% 9.20/5.21        ( ( L != zero_zero_int )
% 9.20/5.21       => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7316_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups5754745047067104278omplex @ G
% 9.20/5.21            @ ( minus_minus_set_real @ A2
% 9.20/5.21              @ ( collect_real
% 9.20/5.21                @ ^ [X: real] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_complex ) ) ) )
% 9.20/5.21          = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7317_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > complex] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3049146728041665814omplex @ G
% 9.20/5.21            @ ( minus_minus_set_int @ A2
% 9.20/5.21              @ ( collect_int
% 9.20/5.21                @ ^ [X: int] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_complex ) ) ) )
% 9.20/5.21          = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7318_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups8097168146408367636l_real @ G
% 9.20/5.21            @ ( minus_minus_set_real @ A2
% 9.20/5.21              @ ( collect_real
% 9.20/5.21                @ ^ [X: real] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) )
% 9.20/5.21          = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7319_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups8778361861064173332t_real @ G
% 9.20/5.21            @ ( minus_minus_set_int @ A2
% 9.20/5.21              @ ( collect_int
% 9.20/5.21                @ ^ [X: int] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) )
% 9.20/5.21          = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7320_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5808333547571424918x_real @ G
% 9.20/5.21            @ ( minus_811609699411566653omplex @ A2
% 9.20/5.21              @ ( collect_complex
% 9.20/5.21                @ ^ [X: complex] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_real ) ) ) )
% 9.20/5.21          = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7321_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups1300246762558778688al_rat @ G
% 9.20/5.21            @ ( minus_minus_set_real @ A2
% 9.20/5.21              @ ( collect_real
% 9.20/5.21                @ ^ [X: real] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) )
% 9.20/5.21          = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7322_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups3906332499630173760nt_rat @ G
% 9.20/5.21            @ ( minus_minus_set_int @ A2
% 9.20/5.21              @ ( collect_int
% 9.20/5.21                @ ^ [X: int] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) )
% 9.20/5.21          = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7323_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_complex,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.21       => ( ( groups5058264527183730370ex_rat @ G
% 9.20/5.21            @ ( minus_811609699411566653omplex @ A2
% 9.20/5.21              @ ( collect_complex
% 9.20/5.21                @ ^ [X: complex] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_rat ) ) ) )
% 9.20/5.21          = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7324_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_real,G: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ A2 )
% 9.20/5.21       => ( ( groups1935376822645274424al_nat @ G
% 9.20/5.21            @ ( minus_minus_set_real @ A2
% 9.20/5.21              @ ( collect_real
% 9.20/5.21                @ ^ [X: real] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_nat ) ) ) )
% 9.20/5.21          = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7325_sum_Osetdiff__irrelevant,axiom,
% 9.20/5.21      ! [A2: set_int,G: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ A2 )
% 9.20/5.21       => ( ( groups4541462559716669496nt_nat @ G
% 9.20/5.21            @ ( minus_minus_set_int @ A2
% 9.20/5.21              @ ( collect_int
% 9.20/5.21                @ ^ [X: int] :
% 9.20/5.21                    ( ( G @ X )
% 9.20/5.21                    = zero_zero_nat ) ) ) )
% 9.20/5.21          = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7326_sum__power__add,axiom,
% 9.20/5.21      ! [X3: complex,M: nat,I5: set_nat] :
% 9.20/5.21        ( ( groups2073611262835488442omplex
% 9.20/5.21          @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 9.20/5.21          @ I5 )
% 9.20/5.21        = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7327_sum__power__add,axiom,
% 9.20/5.21      ! [X3: rat,M: nat,I5: set_nat] :
% 9.20/5.21        ( ( groups2906978787729119204at_rat
% 9.20/5.21          @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 9.20/5.21          @ I5 )
% 9.20/5.21        = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7328_sum__power__add,axiom,
% 9.20/5.21      ! [X3: int,M: nat,I5: set_nat] :
% 9.20/5.21        ( ( groups3539618377306564664at_int
% 9.20/5.21          @ ^ [I2: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 9.20/5.21          @ I5 )
% 9.20/5.21        = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7329_sum__power__add,axiom,
% 9.20/5.21      ! [X3: real,M: nat,I5: set_nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I2 ) )
% 9.20/5.21          @ I5 )
% 9.20/5.21        = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7330_sum_OatLeastAtMost__rev,axiom,
% 9.20/5.21      ! [G: nat > nat,N2: nat,M: nat] :
% 9.20/5.21        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 9.20/5.21        = ( groups3542108847815614940at_nat
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7331_sum_OatLeastAtMost__rev,axiom,
% 9.20/5.21      ! [G: nat > real,N2: nat,M: nat] :
% 9.20/5.21        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
% 9.20/5.21        = ( groups6591440286371151544t_real
% 9.20/5.21          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
% 9.20/5.21          @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7332_tanh__real__gt__neg1,axiom,
% 9.20/5.21      ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7333_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_real,I: real,F: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( member_real @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: real] :
% 9.20/5.21                  ( ( member_real @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7334_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_int,I: int,F: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( member_int @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: int] :
% 9.20/5.21                  ( ( member_int @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7335_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_complex,I: complex,F: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( member_complex @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: complex] :
% 9.20/5.21                  ( ( member_complex @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7336_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_real,I: real,F: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( member_real @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: real] :
% 9.20/5.21                  ( ( member_real @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7337_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_nat,I: nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ I5 )
% 9.20/5.21       => ( ( member_nat @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: nat] :
% 9.20/5.21                  ( ( member_nat @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7338_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_int,I: int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( member_int @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: int] :
% 9.20/5.21                  ( ( member_int @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7339_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_complex,I: complex,F: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( member_complex @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: complex] :
% 9.20/5.21                  ( ( member_complex @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7340_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_real,I: real,F: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( member_real @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: real] :
% 9.20/5.21                  ( ( member_real @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7341_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_int,I: int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( member_int @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: int] :
% 9.20/5.21                  ( ( member_int @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7342_sum__pos2,axiom,
% 9.20/5.21      ! [I5: set_complex,I: complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( member_complex @ I @ I5 )
% 9.20/5.21         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.20/5.21           => ( ! [I3: complex] :
% 9.20/5.21                  ( ( member_complex @ I3 @ I5 )
% 9.20/5.21                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21             => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7343_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_complex,F: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [I3: complex] :
% 9.20/5.21                ( ( member_complex @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7344_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_real,F: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_real )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7345_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_int,F: int > real] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_int )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7346_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_complex,F: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [I3: complex] :
% 9.20/5.21                ( ( member_complex @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7347_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_real,F: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_real )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7348_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_nat,F: nat > rat] :
% 9.20/5.21        ( ( finite_finite_nat @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_nat )
% 9.20/5.21         => ( ! [I3: nat] :
% 9.20/5.21                ( ( member_nat @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7349_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_int,F: int > rat] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_int )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7350_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_complex,F: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_complex )
% 9.20/5.21         => ( ! [I3: complex] :
% 9.20/5.21                ( ( member_complex @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7351_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_real,F: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_real )
% 9.20/5.21         => ( ! [I3: real] :
% 9.20/5.21                ( ( member_real @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7352_sum__pos,axiom,
% 9.20/5.21      ! [I5: set_int,F: int > nat] :
% 9.20/5.21        ( ( finite_finite_int @ I5 )
% 9.20/5.21       => ( ( I5 != bot_bot_set_int )
% 9.20/5.21         => ( ! [I3: int] :
% 9.20/5.21                ( ( member_int @ I3 @ I5 )
% 9.20/5.21               => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
% 9.20/5.21           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7353_abs__add__one__gt__zero,axiom,
% 9.20/5.21      ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7354_abs__add__one__gt__zero,axiom,
% 9.20/5.21      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7355_abs__add__one__gt__zero,axiom,
% 9.20/5.21      ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7356_abs__add__one__gt__zero,axiom,
% 9.20/5.21      ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7357_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_complex ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5754745047067104278omplex @ G @ T3 )
% 9.20/5.21                = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7358_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_real ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups8097168146408367636l_real @ G @ T3 )
% 9.20/5.21                = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7359_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_real ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.20/5.21                = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7360_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_rat ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups1300246762558778688al_rat @ G @ T3 )
% 9.20/5.21                = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7361_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_rat ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.20/5.21                = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7362_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_nat ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups1935376822645274424al_nat @ G @ T3 )
% 9.20/5.21                = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7363_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_nat ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.20/5.21                = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7364_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_int ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups1932886352136224148al_int @ G @ T3 )
% 9.20/5.21                = ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7365_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_int ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 9.20/5.21                = ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7366_sum_Omono__neutral__cong__right,axiom,
% 9.20/5.21      ! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
% 9.20/5.21        ( ( finite_finite_nat @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: nat] :
% 9.20/5.21                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.21               => ( ( G @ X4 )
% 9.20/5.21                  = zero_zero_complex ) )
% 9.20/5.21           => ( ! [X4: nat] :
% 9.20/5.21                  ( ( member_nat @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.20/5.21                = ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7367_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_complex ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5754745047067104278omplex @ G @ S3 )
% 9.20/5.21                = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7368_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_real ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups8097168146408367636l_real @ G @ S3 )
% 9.20/5.21                = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7369_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_real ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5808333547571424918x_real @ G @ S3 )
% 9.20/5.21                = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7370_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_rat ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups1300246762558778688al_rat @ G @ S3 )
% 9.20/5.21                = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7371_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_rat ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 9.20/5.21                = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7372_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_nat ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups1935376822645274424al_nat @ G @ S3 )
% 9.20/5.21                = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7373_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
% 9.20/5.21        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.21       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: complex] :
% 9.20/5.21                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_nat ) )
% 9.20/5.21           => ( ! [X4: complex] :
% 9.20/5.21                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.21                    = ( H2 @ X4 ) ) )
% 9.20/5.21             => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 9.20/5.21                = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.21  
% 9.20/5.21  thf(fact_7374_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.21      ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
% 9.20/5.21        ( ( finite_finite_real @ T3 )
% 9.20/5.21       => ( ( ord_less_eq_set_real @ S3 @ T3 )
% 9.20/5.21         => ( ! [X4: real] :
% 9.20/5.21                ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
% 9.20/5.21               => ( ( H2 @ X4 )
% 9.20/5.21                  = zero_zero_int ) )
% 9.20/5.21           => ( ! [X4: real] :
% 9.20/5.21                  ( ( member_real @ X4 @ S3 )
% 9.20/5.21                 => ( ( G @ X4 )
% 9.20/5.22                    = ( H2 @ X4 ) ) )
% 9.20/5.22             => ( ( groups1932886352136224148al_int @ G @ S3 )
% 9.20/5.22                = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7375_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( H2 @ X4 )
% 9.20/5.22                  = zero_zero_int ) )
% 9.20/5.22           => ( ! [X4: complex] :
% 9.20/5.22                  ( ( member_complex @ X4 @ S3 )
% 9.20/5.22                 => ( ( G @ X4 )
% 9.20/5.22                    = ( H2 @ X4 ) ) )
% 9.20/5.22             => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 9.20/5.22                = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7376_sum_Omono__neutral__cong__left,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( H2 @ X4 )
% 9.20/5.22                  = zero_zero_complex ) )
% 9.20/5.22           => ( ! [X4: nat] :
% 9.20/5.22                  ( ( member_nat @ X4 @ S3 )
% 9.20/5.22                 => ( ( G @ X4 )
% 9.20/5.22                    = ( H2 @ X4 ) ) )
% 9.20/5.22             => ( ( groups2073611262835488442omplex @ G @ S3 )
% 9.20/5.22                = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7377_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_real ) )
% 9.20/5.22           => ( ( groups5808333547571424918x_real @ G @ T3 )
% 9.20/5.22              = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7378_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
% 9.20/5.22              = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7379_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_nat ) )
% 9.20/5.22           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
% 9.20/5.22              = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7380_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_int ) )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int @ G @ T3 )
% 9.20/5.22              = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7381_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_complex ) )
% 9.20/5.22           => ( ( groups2073611262835488442omplex @ G @ T3 )
% 9.20/5.22              = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7382_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups2906978787729119204at_rat @ G @ T3 )
% 9.20/5.22              = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7383_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_int ) )
% 9.20/5.22           => ( ( groups3539618377306564664at_int @ G @ T3 )
% 9.20/5.22              = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7384_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > complex] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_complex ) )
% 9.20/5.22           => ( ( groups3049146728041665814omplex @ G @ T3 )
% 9.20/5.22              = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7385_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_real ) )
% 9.20/5.22           => ( ( groups8778361861064173332t_real @ G @ T3 )
% 9.20/5.22              = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7386_sum_Omono__neutral__right,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat @ G @ T3 )
% 9.20/5.22              = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7387_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_real ) )
% 9.20/5.22           => ( ( groups5808333547571424918x_real @ G @ S3 )
% 9.20/5.22              = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7388_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
% 9.20/5.22              = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7389_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_nat ) )
% 9.20/5.22           => ( ( groups5693394587270226106ex_nat @ G @ S3 )
% 9.20/5.22              = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7390_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_complex,S3: set_complex,G: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ T3 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: complex] :
% 9.20/5.22                ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_int ) )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int @ G @ S3 )
% 9.20/5.22              = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7391_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_complex ) )
% 9.20/5.22           => ( ( groups2073611262835488442omplex @ G @ S3 )
% 9.20/5.22              = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7392_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups2906978787729119204at_rat @ G @ S3 )
% 9.20/5.22              = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7393_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_nat,S3: set_nat,G: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: nat] :
% 9.20/5.22                ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_int ) )
% 9.20/5.22           => ( ( groups3539618377306564664at_int @ G @ S3 )
% 9.20/5.22              = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7394_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > complex] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_complex ) )
% 9.20/5.22           => ( ( groups3049146728041665814omplex @ G @ S3 )
% 9.20/5.22              = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7395_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_real ) )
% 9.20/5.22           => ( ( groups8778361861064173332t_real @ G @ S3 )
% 9.20/5.22              = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7396_sum_Omono__neutral__left,axiom,
% 9.20/5.22      ! [T3: set_int,S3: set_int,G: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ T3 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ S3 @ T3 )
% 9.20/5.22         => ( ! [X4: int] :
% 9.20/5.22                ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
% 9.20/5.22               => ( ( G @ X4 )
% 9.20/5.22                  = zero_zero_rat ) )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat @ G @ S3 )
% 9.20/5.22              = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7397_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_complex ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_complex ) )
% 9.20/5.22               => ( ( ( groups5754745047067104278omplex @ G @ C4 )
% 9.20/5.22                    = ( groups5754745047067104278omplex @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.20/5.22                    = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7398_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_real ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_real ) )
% 9.20/5.22               => ( ( ( groups8097168146408367636l_real @ G @ C4 )
% 9.20/5.22                    = ( groups8097168146408367636l_real @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.20/5.22                    = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7399_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_real ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_real ) )
% 9.20/5.22               => ( ( ( groups5808333547571424918x_real @ G @ C4 )
% 9.20/5.22                    = ( groups5808333547571424918x_real @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.20/5.22                    = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7400_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_rat ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_rat ) )
% 9.20/5.22               => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
% 9.20/5.22                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.20/5.22                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7401_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_rat ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_rat ) )
% 9.20/5.22               => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 9.20/5.22                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.20/5.22                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7402_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_nat ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_nat ) )
% 9.20/5.22               => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
% 9.20/5.22                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.20/5.22                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7403_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_nat ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_nat ) )
% 9.20/5.22               => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 9.20/5.22                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.20/5.22                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7404_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_int ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_int ) )
% 9.20/5.22               => ( ( ( groups1932886352136224148al_int @ G @ C4 )
% 9.20/5.22                    = ( groups1932886352136224148al_int @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.20/5.22                    = ( groups1932886352136224148al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7405_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_int ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_int ) )
% 9.20/5.22               => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
% 9.20/5.22                    = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.20/5.22                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7406_sum_Osame__carrierI,axiom,
% 9.20/5.22      ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: nat] :
% 9.20/5.22                  ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_complex ) )
% 9.20/5.22             => ( ! [B4: nat] :
% 9.20/5.22                    ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_complex ) )
% 9.20/5.22               => ( ( ( groups2073611262835488442omplex @ G @ C4 )
% 9.20/5.22                    = ( groups2073611262835488442omplex @ H2 @ C4 ) )
% 9.20/5.22                 => ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.20/5.22                    = ( groups2073611262835488442omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7407_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_complex ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_complex ) )
% 9.20/5.22               => ( ( ( groups5754745047067104278omplex @ G @ A2 )
% 9.20/5.22                    = ( groups5754745047067104278omplex @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups5754745047067104278omplex @ G @ C4 )
% 9.20/5.22                    = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7408_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_real ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_real ) )
% 9.20/5.22               => ( ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.20/5.22                    = ( groups8097168146408367636l_real @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups8097168146408367636l_real @ G @ C4 )
% 9.20/5.22                    = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7409_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_real ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_real ) )
% 9.20/5.22               => ( ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.20/5.22                    = ( groups5808333547571424918x_real @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups5808333547571424918x_real @ G @ C4 )
% 9.20/5.22                    = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7410_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_rat ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_rat ) )
% 9.20/5.22               => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.20/5.22                    = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups1300246762558778688al_rat @ G @ C4 )
% 9.20/5.22                    = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7411_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_rat ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_rat ) )
% 9.20/5.22               => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.20/5.22                    = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups5058264527183730370ex_rat @ G @ C4 )
% 9.20/5.22                    = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7412_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_nat ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_nat ) )
% 9.20/5.22               => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.20/5.22                    = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups1935376822645274424al_nat @ G @ C4 )
% 9.20/5.22                    = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7413_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_nat ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_nat ) )
% 9.20/5.22               => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.20/5.22                    = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups5693394587270226106ex_nat @ G @ C4 )
% 9.20/5.22                    = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7414_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_real @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: real] :
% 9.20/5.22                  ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_int ) )
% 9.20/5.22             => ( ! [B4: real] :
% 9.20/5.22                    ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_int ) )
% 9.20/5.22               => ( ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.20/5.22                    = ( groups1932886352136224148al_int @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups1932886352136224148al_int @ G @ C4 )
% 9.20/5.22                    = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7415_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ C4 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: complex] :
% 9.20/5.22                  ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_int ) )
% 9.20/5.22             => ( ! [B4: complex] :
% 9.20/5.22                    ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_int ) )
% 9.20/5.22               => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.20/5.22                    = ( groups5690904116761175830ex_int @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups5690904116761175830ex_int @ G @ C4 )
% 9.20/5.22                    = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7416_sum_Osame__carrier,axiom,
% 9.20/5.22      ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ C4 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ C4 )
% 9.20/5.22         => ( ( ord_less_eq_set_nat @ B2 @ C4 )
% 9.20/5.22           => ( ! [A4: nat] :
% 9.20/5.22                  ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
% 9.20/5.22                 => ( ( G @ A4 )
% 9.20/5.22                    = zero_zero_complex ) )
% 9.20/5.22             => ( ! [B4: nat] :
% 9.20/5.22                    ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
% 9.20/5.22                   => ( ( H2 @ B4 )
% 9.20/5.22                      = zero_zero_complex ) )
% 9.20/5.22               => ( ( ( groups2073611262835488442omplex @ G @ A2 )
% 9.20/5.22                    = ( groups2073611262835488442omplex @ H2 @ B2 ) )
% 9.20/5.22                  = ( ( groups2073611262835488442omplex @ G @ C4 )
% 9.20/5.22                    = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7417_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,G: complex > real] :
% 9.20/5.22        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22       => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.20/5.22            = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7418_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,G: complex > rat] :
% 9.20/5.22        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22       => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7419_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,G: complex > nat] :
% 9.20/5.22        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22       => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7420_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,G: complex > int] :
% 9.20/5.22        ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22       => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.20/5.22            = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7421_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,G: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_nat @ A2 )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7422_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,G: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_nat @ A2 )
% 9.20/5.22         => ( ( groups3539618377306564664at_int @ G @ A2 )
% 9.20/5.22            = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7423_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_int,A2: set_int,G: int > real] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_int @ A2 )
% 9.20/5.22         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.20/5.22            = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7424_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_int,A2: set_int,G: int > rat] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_int @ A2 )
% 9.20/5.22         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7425_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_int,A2: set_int,G: int > nat] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_int @ A2 )
% 9.20/5.22         => ( ( groups4541462559716669496nt_nat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7426_sum_Osubset__diff,axiom,
% 9.20/5.22      ! [B2: set_int,A2: set_int,G: int > int] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22       => ( ( finite_finite_int @ A2 )
% 9.20/5.22         => ( ( groups4538972089207619220nt_int @ G @ A2 )
% 9.20/5.22            = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7427_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_complex,B2: set_complex,F: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22         => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7428_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_complex,B2: set_complex,F: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7429_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_complex,B2: set_complex,F: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22         => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7430_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_nat,B2: set_nat,F: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7431_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_nat,B2: set_nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22         => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7432_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_int,B2: set_int,F: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22         => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7433_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_int,B2: set_int,F: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7434_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_int,B2: set_int,F: int > int] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22         => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7435_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_complex,B2: set_complex,F: complex > complex] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22         => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7436_sum__diff,axiom,
% 9.20/5.22      ! [A2: set_nat,B2: set_nat,F: nat > real] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22         => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7437_of__int__leD,axiom,
% 9.20/5.22      ! [N2: int,X3: code_integer] :
% 9.20/5.22        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7438_of__int__leD,axiom,
% 9.20/5.22      ! [N2: int,X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7439_of__int__leD,axiom,
% 9.20/5.22      ! [N2: int,X3: rat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7440_of__int__leD,axiom,
% 9.20/5.22      ! [N2: int,X3: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_eq_int @ one_one_int @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7441_sum__diff__nat,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
% 9.20/5.22         => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7442_sum__diff__nat,axiom,
% 9.20/5.22      ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.20/5.22        ( ( finite6177210948735845034at_nat @ B2 )
% 9.20/5.22       => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
% 9.20/5.22         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7443_sum__diff__nat,axiom,
% 9.20/5.22      ! [B2: set_int,A2: set_int,F: int > nat] :
% 9.20/5.22        ( ( finite_finite_int @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_int @ B2 @ A2 )
% 9.20/5.22         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7444_sum__diff__nat,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,F: nat > nat] :
% 9.20/5.22        ( ( finite_finite_nat @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ B2 @ A2 )
% 9.20/5.22         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7445_of__int__lessD,axiom,
% 9.20/5.22      ! [N2: int,X3: code_integer] :
% 9.20/5.22        ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7446_of__int__lessD,axiom,
% 9.20/5.22      ! [N2: int,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7447_of__int__lessD,axiom,
% 9.20/5.22      ! [N2: int,X3: rat] :
% 9.20/5.22        ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7448_of__int__lessD,axiom,
% 9.20/5.22      ! [N2: int,X3: int] :
% 9.20/5.22        ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
% 9.20/5.22       => ( ( N2 = zero_zero_int )
% 9.20/5.22          | ( ord_less_int @ one_one_int @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7449_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
% 9.20/5.22        ( ( ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member8757157785044589968at_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
% 9.20/5.22            = ( groups3860910324918113789at_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7450_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: set_nat,A2: set_set_nat,F: set_nat > nat] :
% 9.20/5.22        ( ( ( member_set_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member_set_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups8294997508430121362at_nat @ F @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
% 9.20/5.22            = ( groups8294997508430121362at_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7451_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
% 9.20/5.22        ( ( ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member8440522571783428010at_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
% 9.20/5.22            = ( groups977919841031483927at_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7452_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: real,A2: set_real,F: real > nat] :
% 9.20/5.22        ( ( ( member_real @ A @ A2 )
% 9.20/5.22         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member_real @ A @ A2 )
% 9.20/5.22         => ( ( groups1935376822645274424al_nat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22            = ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7453_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: int,A2: set_int,F: int > nat] :
% 9.20/5.22        ( ( ( member_int @ A @ A2 )
% 9.20/5.22         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member_int @ A @ A2 )
% 9.20/5.22         => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22            = ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7454_sum__diff1__nat,axiom,
% 9.20/5.22      ! [A: nat,A2: set_nat,F: nat > nat] :
% 9.20/5.22        ( ( ( member_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.22            = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22        & ( ~ ( member_nat @ A @ A2 )
% 9.20/5.22         => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.22            = ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7455_sum__shift__lb__Suc0__0,axiom,
% 9.20/5.22      ! [F: nat > complex,K: nat] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_complex )
% 9.20/5.22       => ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.20/5.22          = ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7456_sum__shift__lb__Suc0__0,axiom,
% 9.20/5.22      ! [F: nat > rat,K: nat] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_rat )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.20/5.22          = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7457_sum__shift__lb__Suc0__0,axiom,
% 9.20/5.22      ! [F: nat > int,K: nat] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_int )
% 9.20/5.22       => ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.20/5.22          = ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7458_sum__shift__lb__Suc0__0,axiom,
% 9.20/5.22      ! [F: nat > nat,K: nat] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_nat )
% 9.20/5.22       => ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.20/5.22          = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7459_sum__shift__lb__Suc0__0,axiom,
% 9.20/5.22      ! [F: nat > real,K: nat] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_real )
% 9.20/5.22       => ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
% 9.20/5.22          = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7460_sum_OatLeast0__atMost__Suc,axiom,
% 9.20/5.22      ! [G: nat > rat,N2: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7461_sum_OatLeast0__atMost__Suc,axiom,
% 9.20/5.22      ! [G: nat > int,N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7462_sum_OatLeast0__atMost__Suc,axiom,
% 9.20/5.22      ! [G: nat > nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7463_sum_OatLeast0__atMost__Suc,axiom,
% 9.20/5.22      ! [G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7464_sum_OatLeast__Suc__atMost,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7465_sum_OatLeast__Suc__atMost,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7466_sum_OatLeast__Suc__atMost,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7467_sum_OatLeast__Suc__atMost,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7468_sum_Onat__ivl__Suc_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7469_sum_Onat__ivl__Suc_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7470_sum_Onat__ivl__Suc_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7471_sum_Onat__ivl__Suc_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7472_round__diff__minimal,axiom,
% 9.20/5.22      ! [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 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7473_round__diff__minimal,axiom,
% 9.20/5.22      ! [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 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7474_zdvd__mult__cancel1,axiom,
% 9.20/5.22      ! [M: int,N2: int] :
% 9.20/5.22        ( ( M != zero_zero_int )
% 9.20/5.22       => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
% 9.20/5.22          = ( ( abs_abs_int @ N2 )
% 9.20/5.22            = one_one_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7475_sum_OSuc__reindex__ivl,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ M )
% 9.20/5.22            @ ( groups2906978787729119204at_rat
% 9.20/5.22              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7476_sum_OSuc__reindex__ivl,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_int @ ( G @ M )
% 9.20/5.22            @ ( groups3539618377306564664at_int
% 9.20/5.22              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7477_sum_OSuc__reindex__ivl,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_nat @ ( G @ M )
% 9.20/5.22            @ ( groups3542108847815614940at_nat
% 9.20/5.22              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7478_sum_OSuc__reindex__ivl,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ M )
% 9.20/5.22            @ ( groups6591440286371151544t_real
% 9.20/5.22              @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7479_sum__Suc__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7480_sum__Suc__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7481_sum__Suc__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
% 9.20/5.22       => ( ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7482_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,F: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: real] :
% 9.20/5.22                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7483_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,F: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: complex] :
% 9.20/5.22                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7484_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,F: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: real] :
% 9.20/5.22                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7485_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,F: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: complex] :
% 9.20/5.22                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7486_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,F: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: nat] :
% 9.20/5.22                ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7487_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,F: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: real] :
% 9.20/5.22                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7488_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,F: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: complex] :
% 9.20/5.22                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7489_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,F: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: real] :
% 9.20/5.22                ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7490_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,F: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: complex] :
% 9.20/5.22                ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7491_sum__mono2,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.22         => ( ! [B4: nat] :
% 9.20/5.22                ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.20/5.22               => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
% 9.20/5.22           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7492_abs__le__square__iff,axiom,
% 9.20/5.22      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.22        ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y3 ) )
% 9.20/5.22        = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7493_abs__le__square__iff,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7494_abs__le__square__iff,axiom,
% 9.20/5.22      ! [X3: rat,Y3: rat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7495_abs__le__square__iff,axiom,
% 9.20/5.22      ! [X3: int,Y3: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7496_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_complex,X3: complex,G: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( member_complex @ X3 @ A2 )
% 9.20/5.22         => ( ( groups5808333547571424918x_real @ G @ A2 )
% 9.20/5.22            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7497_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_complex,X3: complex,G: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( member_complex @ X3 @ A2 )
% 9.20/5.22         => ( ( groups5058264527183730370ex_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7498_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_complex,X3: complex,G: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( member_complex @ X3 @ A2 )
% 9.20/5.22         => ( ( groups5693394587270226106ex_nat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7499_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_complex,X3: complex,G: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( member_complex @ X3 @ A2 )
% 9.20/5.22         => ( ( groups5690904116761175830ex_int @ G @ A2 )
% 9.20/5.22            = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7500_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_real,X3: real,G: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( member_real @ X3 @ A2 )
% 9.20/5.22         => ( ( groups8097168146408367636l_real @ G @ A2 )
% 9.20/5.22            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7501_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_real,X3: real,G: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( member_real @ X3 @ A2 )
% 9.20/5.22         => ( ( groups1300246762558778688al_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7502_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_real,X3: real,G: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( member_real @ X3 @ A2 )
% 9.20/5.22         => ( ( groups1935376822645274424al_nat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7503_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_real,X3: real,G: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( member_real @ X3 @ A2 )
% 9.20/5.22         => ( ( groups1932886352136224148al_int @ G @ A2 )
% 9.20/5.22            = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7504_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_int,X3: int,G: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( member_int @ X3 @ A2 )
% 9.20/5.22         => ( ( groups8778361861064173332t_real @ G @ A2 )
% 9.20/5.22            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7505_sum_Oremove,axiom,
% 9.20/5.22      ! [A2: set_int,X3: int,G: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( member_int @ X3 @ A2 )
% 9.20/5.22         => ( ( groups3906332499630173760nt_rat @ G @ A2 )
% 9.20/5.22            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7506_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_complex,G: complex > real,X3: complex] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7507_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_complex,G: complex > rat,X3: complex] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7508_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_complex,G: complex > nat,X3: complex] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7509_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_complex,G: complex > int,X3: complex] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7510_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_real,G: real > real,X3: real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7511_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_real,G: real > rat,X3: real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7512_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_real,G: real > nat,X3: real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7513_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_real,G: real > int,X3: real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7514_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_int,G: int > real,X3: int] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7515_sum_Oinsert__remove,axiom,
% 9.20/5.22      ! [A2: set_int,G: int > rat,X3: int] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.22          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7516_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_complex,A: complex,F: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7517_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_real,A: real,F: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( minus_minus_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups8097168146408367636l_real @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7518_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_int,A: int,F: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7519_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_complex,A: complex,F: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7520_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_real,A: real,F: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7521_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_int,A: int,F: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7522_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_complex,A: complex,F: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22       => ( ( ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ A2 )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
% 9.20/5.22              = ( groups5690904116761175830ex_int @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7523_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_real,A: real,F: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ A2 )
% 9.20/5.22       => ( ( ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( minus_minus_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ A2 )
% 9.20/5.22           => ( ( groups1932886352136224148al_int @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
% 9.20/5.22              = ( groups1932886352136224148al_int @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7524_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_int,A: int,F: int > int] :
% 9.20/5.22        ( ( finite_finite_int @ A2 )
% 9.20/5.22       => ( ( ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_int @ A @ A2 )
% 9.20/5.22           => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
% 9.20/5.22              = ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7525_sum__diff1,axiom,
% 9.20/5.22      ! [A2: set_nat,A: nat,F: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( ( ( member_nat @ A @ A2 )
% 9.20/5.22           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.22              = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
% 9.20/5.22          & ( ~ ( member_nat @ A @ A2 )
% 9.20/5.22           => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
% 9.20/5.22              = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7526_sum_Oub__add__nat,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 9.20/5.22          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7527_sum_Oub__add__nat,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > int,P4: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 9.20/5.22       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 9.20/5.22          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7528_sum_Oub__add__nat,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 9.20/5.22       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 9.20/5.22          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7529_sum_Oub__add__nat,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,G: nat > real,P4: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
% 9.20/5.22       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
% 9.20/5.22          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7530_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.22       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5808333547571424918x_real
% 9.20/5.22                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5808333547571424918x_real
% 9.20/5.22                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7531_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.22       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat
% 9.20/5.22                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5058264527183730370ex_rat
% 9.20/5.22                @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7532_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.22       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5693394587270226106ex_nat
% 9.20/5.22                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5693394587270226106ex_nat
% 9.20/5.22                @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7533_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.22       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int
% 9.20/5.22                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.22           => ( ( groups5690904116761175830ex_int
% 9.20/5.22                @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7534_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_real,A: real,B: real > real,C: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ S3 )
% 9.20/5.22       => ( ( ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups8097168146408367636l_real
% 9.20/5.22                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups8097168146408367636l_real
% 9.20/5.22                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7535_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ S3 )
% 9.20/5.22       => ( ( ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1300246762558778688al_rat
% 9.20/5.22                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1300246762558778688al_rat
% 9.20/5.22                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7536_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ S3 )
% 9.20/5.22       => ( ( ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1935376822645274424al_nat
% 9.20/5.22                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1935376822645274424al_nat
% 9.20/5.22                @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7537_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_real,A: real,B: real > int,C: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ S3 )
% 9.20/5.22       => ( ( ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1932886352136224148al_int
% 9.20/5.22                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.22           => ( ( groups1932886352136224148al_int
% 9.20/5.22                @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7538_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_int,A: int,B: int > real,C: int > real] :
% 9.20/5.22        ( ( finite_finite_int @ S3 )
% 9.20/5.22       => ( ( ( member_int @ A @ S3 )
% 9.20/5.22           => ( ( groups8778361861064173332t_real
% 9.20/5.22                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.22           => ( ( groups8778361861064173332t_real
% 9.20/5.22                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7539_sum_Odelta__remove,axiom,
% 9.20/5.22      ! [S3: set_int,A: int,B: int > rat,C: int > rat] :
% 9.20/5.22        ( ( finite_finite_int @ S3 )
% 9.20/5.22       => ( ( ( member_int @ A @ S3 )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat
% 9.20/5.22                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
% 9.20/5.22          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.22           => ( ( groups3906332499630173760nt_rat
% 9.20/5.22                @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
% 9.20/5.22                @ S3 )
% 9.20/5.22              = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7540_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_complex,X7: set_complex] :
% 9.20/5.22        ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite3207457112153483333omplex @ X7 )
% 9.20/5.22         => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
% 9.20/5.22            = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7541_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_P6011104703257516679at_nat,X7: set_Pr1261947904930325089at_nat] :
% 9.20/5.22        ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite6177210948735845034at_nat @ X7 )
% 9.20/5.22         => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X7 )
% 9.20/5.22            = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7542_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
% 9.20/5.22        ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite5795047828879050333T_VEBT @ X7 )
% 9.20/5.22         => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
% 9.20/5.22            = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7543_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_o,X7: set_o] :
% 9.20/5.22        ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite_finite_o @ X7 )
% 9.20/5.22         => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X7 )
% 9.20/5.22            = ( size_size_list_o @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7544_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_int,X7: set_int] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite_finite_int @ X7 )
% 9.20/5.22         => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
% 9.20/5.22            = ( size_size_list_int @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7545_sum__count__set,axiom,
% 9.20/5.22      ! [Xs: list_nat,X7: set_nat] :
% 9.20/5.22        ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
% 9.20/5.22       => ( ( finite_finite_nat @ X7 )
% 9.20/5.22         => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
% 9.20/5.22            = ( size_size_list_nat @ Xs ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7546_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,B: real,F: real > real] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: real] :
% 9.20/5.22                    ( ( member_real @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7547_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: complex] :
% 9.20/5.22                    ( ( member_complex @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7548_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: real] :
% 9.20/5.22                    ( ( member_real @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7549_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: complex] :
% 9.20/5.22                    ( ( member_complex @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7550_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > rat] :
% 9.20/5.22        ( ( finite_finite_nat @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.22         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: nat] :
% 9.20/5.22                    ( ( member_nat @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7551_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: real] :
% 9.20/5.22                    ( ( member_real @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7552_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: complex] :
% 9.20/5.22                    ( ( member_complex @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7553_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_real,A2: set_real,B: real,F: real > int] :
% 9.20/5.22        ( ( finite_finite_real @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_real @ A2 @ B2 )
% 9.20/5.22         => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: real] :
% 9.20/5.22                    ( ( member_real @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7554_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_complex,A2: set_complex,B: complex,F: complex > int] :
% 9.20/5.22        ( ( finite3207457112153483333omplex @ B2 )
% 9.20/5.22       => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
% 9.20/5.22         => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: complex] :
% 9.20/5.22                    ( ( member_complex @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7555_sum__strict__mono2,axiom,
% 9.20/5.22      ! [B2: set_nat,A2: set_nat,B: nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ B2 )
% 9.20/5.22       => ( ( ord_less_eq_set_nat @ A2 @ B2 )
% 9.20/5.22         => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
% 9.20/5.22           => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
% 9.20/5.22             => ( ! [X4: nat] :
% 9.20/5.22                    ( ( member_nat @ X4 @ B2 )
% 9.20/5.22                   => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
% 9.20/5.22               => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7556_member__le__sum,axiom,
% 9.20/5.22      ! [I: complex,A2: set_complex,F: complex > real] :
% 9.20/5.22        ( ( member_complex @ I @ A2 )
% 9.20/5.22       => ( ! [X4: complex] :
% 9.20/5.22              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.20/5.22             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22           => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7557_member__le__sum,axiom,
% 9.20/5.22      ! [I: real,A2: set_real,F: real > real] :
% 9.20/5.22        ( ( member_real @ I @ A2 )
% 9.20/5.22       => ( ! [X4: real] :
% 9.20/5.22              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.20/5.22             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_real @ A2 )
% 9.20/5.22           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7558_member__le__sum,axiom,
% 9.20/5.22      ! [I: int,A2: set_int,F: int > real] :
% 9.20/5.22        ( ( member_int @ I @ A2 )
% 9.20/5.22       => ( ! [X4: int] :
% 9.20/5.22              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.20/5.22             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_int @ A2 )
% 9.20/5.22           => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7559_member__le__sum,axiom,
% 9.20/5.22      ! [I: complex,A2: set_complex,F: complex > rat] :
% 9.20/5.22        ( ( member_complex @ I @ A2 )
% 9.20/5.22       => ( ! [X4: complex] :
% 9.20/5.22              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.20/5.22             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7560_member__le__sum,axiom,
% 9.20/5.22      ! [I: real,A2: set_real,F: real > rat] :
% 9.20/5.22        ( ( member_real @ I @ A2 )
% 9.20/5.22       => ( ! [X4: real] :
% 9.20/5.22              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.20/5.22             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_real @ A2 )
% 9.20/5.22           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7561_member__le__sum,axiom,
% 9.20/5.22      ! [I: int,A2: set_int,F: int > rat] :
% 9.20/5.22        ( ( member_int @ I @ A2 )
% 9.20/5.22       => ( ! [X4: int] :
% 9.20/5.22              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.20/5.22             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_int @ A2 )
% 9.20/5.22           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7562_member__le__sum,axiom,
% 9.20/5.22      ! [I: nat,A2: set_nat,F: nat > rat] :
% 9.20/5.22        ( ( member_nat @ I @ A2 )
% 9.20/5.22       => ( ! [X4: nat] :
% 9.20/5.22              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
% 9.20/5.22             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_nat @ A2 )
% 9.20/5.22           => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7563_member__le__sum,axiom,
% 9.20/5.22      ! [I: complex,A2: set_complex,F: complex > nat] :
% 9.20/5.22        ( ( member_complex @ I @ A2 )
% 9.20/5.22       => ( ! [X4: complex] :
% 9.20/5.22              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
% 9.20/5.22             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.22           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7564_member__le__sum,axiom,
% 9.20/5.22      ! [I: real,A2: set_real,F: real > nat] :
% 9.20/5.22        ( ( member_real @ I @ A2 )
% 9.20/5.22       => ( ! [X4: real] :
% 9.20/5.22              ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
% 9.20/5.22             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_real @ A2 )
% 9.20/5.22           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7565_member__le__sum,axiom,
% 9.20/5.22      ! [I: int,A2: set_int,F: int > nat] :
% 9.20/5.22        ( ( member_int @ I @ A2 )
% 9.20/5.22       => ( ! [X4: int] :
% 9.20/5.22              ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
% 9.20/5.22             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
% 9.20/5.22         => ( ( finite_finite_int @ A2 )
% 9.20/5.22           => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7566_power2__le__iff__abs__le,axiom,
% 9.20/5.22      ! [Y3: code_integer,X3: code_integer] :
% 9.20/5.22        ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
% 9.20/5.22       => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.22          = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7567_power2__le__iff__abs__le,axiom,
% 9.20/5.22      ! [Y3: real,X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.22          = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7568_power2__le__iff__abs__le,axiom,
% 9.20/5.22      ! [Y3: rat,X3: rat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
% 9.20/5.22       => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.22          = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7569_power2__le__iff__abs__le,axiom,
% 9.20/5.22      ! [Y3: int,X3: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.22       => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.22          = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7570_abs__sqrt__wlog,axiom,
% 9.20/5.22      ! [P: code_integer > code_integer > $o,X3: code_integer] :
% 9.20/5.22        ( ! [X4: code_integer] :
% 9.20/5.22            ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
% 9.20/5.22           => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.22       => ( P @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7571_abs__sqrt__wlog,axiom,
% 9.20/5.22      ! [P: real > real > $o,X3: real] :
% 9.20/5.22        ( ! [X4: real] :
% 9.20/5.22            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.20/5.22           => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.22       => ( P @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7572_abs__sqrt__wlog,axiom,
% 9.20/5.22      ! [P: rat > rat > $o,X3: rat] :
% 9.20/5.22        ( ! [X4: rat] :
% 9.20/5.22            ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
% 9.20/5.22           => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.22       => ( P @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7573_abs__sqrt__wlog,axiom,
% 9.20/5.22      ! [P: int > int > $o,X3: int] :
% 9.20/5.22        ( ! [X4: int] :
% 9.20/5.22            ( ( ord_less_eq_int @ zero_zero_int @ X4 )
% 9.20/5.22           => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.22       => ( P @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7574_abs__square__le__1,axiom,
% 9.20/5.22      ! [X3: code_integer] :
% 9.20/5.22        ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.20/5.22        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7575_abs__square__le__1,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.20/5.22        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7576_abs__square__le__1,axiom,
% 9.20/5.22      ! [X3: rat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.20/5.22        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7577_abs__square__le__1,axiom,
% 9.20/5.22      ! [X3: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.20/5.22        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7578_abs__square__less__1,axiom,
% 9.20/5.22      ! [X3: code_integer] :
% 9.20/5.22        ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
% 9.20/5.22        = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7579_abs__square__less__1,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
% 9.20/5.22        = ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7580_abs__square__less__1,axiom,
% 9.20/5.22      ! [X3: rat] :
% 9.20/5.22        ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
% 9.20/5.22        = ( ord_less_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7581_abs__square__less__1,axiom,
% 9.20/5.22      ! [X3: int] :
% 9.20/5.22        ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
% 9.20/5.22        = ( ord_less_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7582_power__mono__even,axiom,
% 9.20/5.22      ! [N2: nat,A: code_integer,B: code_integer] :
% 9.20/5.22        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.22       => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
% 9.20/5.22         => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7583_power__mono__even,axiom,
% 9.20/5.22      ! [N2: nat,A: real,B: real] :
% 9.20/5.22        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
% 9.20/5.22         => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7584_power__mono__even,axiom,
% 9.20/5.22      ! [N2: nat,A: rat,B: rat] :
% 9.20/5.22        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.22       => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
% 9.20/5.22         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7585_power__mono__even,axiom,
% 9.20/5.22      ! [N2: nat,A: int,B: int] :
% 9.20/5.22        ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.22       => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
% 9.20/5.22         => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7586_nat__intermed__int__val,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > int,K: int] :
% 9.20/5.22        ( ! [I3: nat] :
% 9.20/5.22            ( ( ( ord_less_eq_nat @ M @ I3 )
% 9.20/5.22              & ( ord_less_nat @ I3 @ N2 ) )
% 9.20/5.22           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.20/5.22       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( ord_less_eq_int @ ( F @ M ) @ K )
% 9.20/5.22           => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 9.20/5.22             => ? [I3: nat] :
% 9.20/5.22                  ( ( ord_less_eq_nat @ M @ I3 )
% 9.20/5.22                  & ( ord_less_eq_nat @ I3 @ N2 )
% 9.20/5.22                  & ( ( F @ I3 )
% 9.20/5.22                    = K ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7587_incr__lemma,axiom,
% 9.20/5.22      ! [D: int,Z2: int,X3: int] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.22       => ( ord_less_int @ Z2 @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7588_decr__lemma,axiom,
% 9.20/5.22      ! [D: int,X3: int,Z2: int] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ D )
% 9.20/5.22       => ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7589_sum__natinterval__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > complex] :
% 9.20/5.22        ( ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups2073611262835488442omplex
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 9.20/5.22        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups2073611262835488442omplex
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_complex ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7590_sum__natinterval__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > rat] :
% 9.20/5.22        ( ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 9.20/5.22        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_rat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7591_sum__natinterval__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > int] :
% 9.20/5.22        ( ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups3539618377306564664at_int
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 9.20/5.22        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups3539618377306564664at_int
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7592_sum__natinterval__diff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > real] :
% 9.20/5.22        ( ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups6591440286371151544t_real
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
% 9.20/5.22        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22         => ( ( groups6591440286371151544t_real
% 9.20/5.22              @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22              @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7593_sum__telescope_H_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 9.20/5.22          = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7594_sum__telescope_H_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 9.20/5.22          = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7595_sum__telescope_H_H,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,F: nat > real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
% 9.20/5.22          = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7596_mask__eq__sum__exp,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int )
% 9.20/5.22        = ( groups3539618377306564664at_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.22          @ ( collect_nat
% 9.20/5.22            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7597_mask__eq__sum__exp,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat )
% 9.20/5.22        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.22          @ ( collect_nat
% 9.20/5.22            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7598_nat__ivt__aux,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > int,K: int] :
% 9.20/5.22        ( ! [I3: nat] :
% 9.20/5.22            ( ( ord_less_nat @ I3 @ N2 )
% 9.20/5.22           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.20/5.22       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.20/5.22         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 9.20/5.22           => ? [I3: nat] :
% 9.20/5.22                ( ( ord_less_eq_nat @ I3 @ N2 )
% 9.20/5.22                & ( ( F @ I3 )
% 9.20/5.22                  = K ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7599_divmod__nat__def,axiom,
% 9.20/5.22      ( divmod_nat
% 9.20/5.22      = ( ^ [M2: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7600_sum__gp__multiplied,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,X3: complex] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 9.20/5.22          = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7601_sum__gp__multiplied,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,X3: rat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 9.20/5.22          = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7602_sum__gp__multiplied,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,X3: int] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 9.20/5.22          = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7603_sum__gp__multiplied,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,X3: real] :
% 9.20/5.22        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.22       => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
% 9.20/5.22          = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7604_sum_Oin__pairs,axiom,
% 9.20/5.22      ! [G: nat > rat,M: nat,N2: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.22        = ( groups2906978787729119204at_rat
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7605_sum_Oin__pairs,axiom,
% 9.20/5.22      ! [G: nat > int,M: nat,N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.22        = ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7606_sum_Oin__pairs,axiom,
% 9.20/5.22      ! [G: nat > nat,M: nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7607_sum_Oin__pairs,axiom,
% 9.20/5.22      ! [G: nat > real,M: nat,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
% 9.20/5.22        = ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7608_nat0__intermed__int__val,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > int,K: int] :
% 9.20/5.22        ( ! [I3: nat] :
% 9.20/5.22            ( ( ord_less_nat @ I3 @ N2 )
% 9.20/5.22           => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
% 9.20/5.22       => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
% 9.20/5.22         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
% 9.20/5.22           => ? [I3: nat] :
% 9.20/5.22                ( ( ord_less_eq_nat @ I3 @ N2 )
% 9.20/5.22                & ( ( F @ I3 )
% 9.20/5.22                  = K ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7609_mask__eq__sum__exp__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.22          @ ( collect_nat
% 9.20/5.22            @ ^ [Q5: nat] : ( ord_less_nat @ Q5 @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7610_gauss__sum__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [X: nat] : X
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7611_of__int__round__abs__le,axiom,
% 9.20/5.22      ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ X3 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7612_of__int__round__abs__le,axiom,
% 9.20/5.22      ! [X3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ X3 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7613_listrel1p__def,axiom,
% 9.20/5.22      ( listrel1p_int
% 9.20/5.22      = ( ^ [R5: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R5 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7614_lemma__interval,axiom,
% 9.20/5.22      ! [A: real,X3: real,B: real] :
% 9.20/5.22        ( ( ord_less_real @ A @ X3 )
% 9.20/5.22       => ( ( ord_less_real @ X3 @ B )
% 9.20/5.22         => ? [D4: real] :
% 9.20/5.22              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.22              & ! [Y6: real] :
% 9.20/5.22                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
% 9.20/5.22                 => ( ( ord_less_eq_real @ A @ Y6 )
% 9.20/5.22                    & ( ord_less_eq_real @ Y6 @ B ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7615_sum__gp,axiom,
% 9.20/5.22      ! [N2: nat,M: nat,X3: complex] :
% 9.20/5.22        ( ( ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_complex ) )
% 9.20/5.22        & ( ~ ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( ( X3 = one_one_complex )
% 9.20/5.22             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 9.20/5.22            & ( ( X3 != one_one_complex )
% 9.20/5.22             => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7616_sum__gp,axiom,
% 9.20/5.22      ! [N2: nat,M: nat,X3: rat] :
% 9.20/5.22        ( ( ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_rat ) )
% 9.20/5.22        & ( ~ ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( ( X3 = one_one_rat )
% 9.20/5.22             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 9.20/5.22            & ( ( X3 != one_one_rat )
% 9.20/5.22             => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7617_sum__gp,axiom,
% 9.20/5.22      ! [N2: nat,M: nat,X3: real] :
% 9.20/5.22        ( ( ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22            = zero_zero_real ) )
% 9.20/5.22        & ( ~ ( ord_less_nat @ N2 @ M )
% 9.20/5.22         => ( ( ( X3 = one_one_real )
% 9.20/5.22             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
% 9.20/5.22            & ( ( X3 != one_one_real )
% 9.20/5.22             => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
% 9.20/5.22                = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7618_gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.22        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7619_gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.22        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7620_gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.22        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7621_lemma__interval__lt,axiom,
% 9.20/5.22      ! [A: real,X3: real,B: real] :
% 9.20/5.22        ( ( ord_less_real @ A @ X3 )
% 9.20/5.22       => ( ( ord_less_real @ X3 @ B )
% 9.20/5.22         => ? [D4: real] :
% 9.20/5.22              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.22              & ! [Y6: real] :
% 9.20/5.22                  ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
% 9.20/5.22                 => ( ( ord_less_real @ A @ Y6 )
% 9.20/5.22                    & ( ord_less_real @ Y6 @ B ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7622_arctan__double,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.22       => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X3 ) )
% 9.20/5.22          = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7623_of__nat__eq__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ( semiri1314217659103216013at_int @ M )
% 9.20/5.22          = ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( M = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7624_of__nat__eq__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ( semiri5074537144036343181t_real @ M )
% 9.20/5.22          = ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( M = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7625_of__nat__eq__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.20/5.22          = ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( M = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7626_split__part,axiom,
% 9.20/5.22      ! [P: $o,Q: int > int > $o] :
% 9.20/5.22        ( ( produc4947309494688390418_int_o
% 9.20/5.22          @ ^ [A3: int,B3: int] :
% 9.20/5.22              ( P
% 9.20/5.22              & ( Q @ A3 @ B3 ) ) )
% 9.20/5.22        = ( ^ [Ab: product_prod_int_int] :
% 9.20/5.22              ( P
% 9.20/5.22              & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7627_int__eq__iff__numeral,axiom,
% 9.20/5.22      ! [M: nat,V: num] :
% 9.20/5.22        ( ( ( semiri1314217659103216013at_int @ M )
% 9.20/5.22          = ( numeral_numeral_int @ V ) )
% 9.20/5.22        = ( M
% 9.20/5.22          = ( numeral_numeral_nat @ V ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7628_abs__of__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7629_abs__of__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
% 9.20/5.22        = ( semiri4939895301339042750nteger @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7630_abs__of__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7631_abs__of__nat,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7632_negative__eq__positive,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22          = ( semiri1314217659103216013at_int @ M ) )
% 9.20/5.22        = ( ( N2 = zero_zero_nat )
% 9.20/5.22          & ( M = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7633_of__int__of__nat__eq,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( semiri681578069525770553at_rat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7634_of__int__of__nat__eq,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( semiri1314217659103216013at_int @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7635_of__int__of__nat__eq,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ring_1_of_int_real @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( semiri5074537144036343181t_real @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7636_negative__zle,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7637_int__dvd__int__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( dvd_dvd_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7638_of__nat__eq__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ( semiri8010041392384452111omplex @ M )
% 9.20/5.22          = zero_zero_complex )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7639_of__nat__eq__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ( semiri681578069525770553at_rat @ M )
% 9.20/5.22          = zero_zero_rat )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7640_of__nat__eq__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ( semiri1314217659103216013at_int @ M )
% 9.20/5.22          = zero_zero_int )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7641_of__nat__eq__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ( semiri5074537144036343181t_real @ M )
% 9.20/5.22          = zero_zero_real )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7642_of__nat__eq__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ( semiri1316708129612266289at_nat @ M )
% 9.20/5.22          = zero_zero_nat )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7643_of__nat__0__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( zero_zero_complex
% 9.20/5.22          = ( semiri8010041392384452111omplex @ N2 ) )
% 9.20/5.22        = ( zero_zero_nat = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7644_of__nat__0__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( zero_zero_rat
% 9.20/5.22          = ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( zero_zero_nat = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7645_of__nat__0__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( zero_zero_int
% 9.20/5.22          = ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( zero_zero_nat = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7646_of__nat__0__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( zero_zero_real
% 9.20/5.22          = ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( zero_zero_nat = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7647_of__nat__0__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( zero_zero_nat
% 9.20/5.22          = ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( zero_zero_nat = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7648_of__nat__0,axiom,
% 9.20/5.22      ( ( semiri8010041392384452111omplex @ zero_zero_nat )
% 9.20/5.22      = zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7649_of__nat__0,axiom,
% 9.20/5.22      ( ( semiri681578069525770553at_rat @ zero_zero_nat )
% 9.20/5.22      = zero_zero_rat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7650_of__nat__0,axiom,
% 9.20/5.22      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.20/5.22      = zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7651_of__nat__0,axiom,
% 9.20/5.22      ( ( semiri5074537144036343181t_real @ zero_zero_nat )
% 9.20/5.22      = zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7652_of__nat__0,axiom,
% 9.20/5.22      ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
% 9.20/5.22      = zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7653_of__nat__less__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7654_of__nat__less__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7655_of__nat__less__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7656_of__nat__less__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7657_of__nat__le__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7658_of__nat__le__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7659_of__nat__le__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7660_of__nat__le__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7661_of__nat__add,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.22        = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7662_of__nat__add,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7663_of__nat__add,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.22        = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7664_of__nat__add,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.22        = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7665_of__nat__mult,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.22        = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7666_of__nat__mult,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.22        = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7667_of__nat__mult,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.22        = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7668_of__nat__mult,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.22        = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7669_of__nat__mult,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
% 9.20/5.22        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7670_of__nat__1,axiom,
% 9.20/5.22      ( ( semiri8010041392384452111omplex @ one_one_nat )
% 9.20/5.22      = one_one_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7671_of__nat__1,axiom,
% 9.20/5.22      ( ( semiri681578069525770553at_rat @ one_one_nat )
% 9.20/5.22      = one_one_rat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7672_of__nat__1,axiom,
% 9.20/5.22      ( ( semiri1314217659103216013at_int @ one_one_nat )
% 9.20/5.22      = one_one_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7673_of__nat__1,axiom,
% 9.20/5.22      ( ( semiri5074537144036343181t_real @ one_one_nat )
% 9.20/5.22      = one_one_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7674_of__nat__1,axiom,
% 9.20/5.22      ( ( semiri1316708129612266289at_nat @ one_one_nat )
% 9.20/5.22      = one_one_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7675_of__nat__1__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( one_one_complex
% 9.20/5.22          = ( semiri8010041392384452111omplex @ N2 ) )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7676_of__nat__1__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( one_one_rat
% 9.20/5.22          = ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7677_of__nat__1__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( one_one_int
% 9.20/5.22          = ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7678_of__nat__1__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( one_one_real
% 9.20/5.22          = ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7679_of__nat__1__eq__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( one_one_nat
% 9.20/5.22          = ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7680_of__nat__eq__1__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( semiri8010041392384452111omplex @ N2 )
% 9.20/5.22          = one_one_complex )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7681_of__nat__eq__1__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( semiri681578069525770553at_rat @ N2 )
% 9.20/5.22          = one_one_rat )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7682_of__nat__eq__1__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( semiri1314217659103216013at_int @ N2 )
% 9.20/5.22          = one_one_int )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7683_of__nat__eq__1__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( semiri5074537144036343181t_real @ N2 )
% 9.20/5.22          = one_one_real )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7684_of__nat__eq__1__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( semiri1316708129612266289at_nat @ N2 )
% 9.20/5.22          = one_one_nat )
% 9.20/5.22        = ( N2 = one_one_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7685_negative__zless,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7686_arctan__less__zero__iff,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
% 9.20/5.22        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7687_zero__less__arctan__iff,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
% 9.20/5.22        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7688_of__nat__of__bool,axiom,
% 9.20/5.22      ! [P: $o] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.20/5.22        = ( zero_n3304061248610475627l_real @ P ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7689_of__nat__of__bool,axiom,
% 9.20/5.22      ! [P: $o] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.20/5.22        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7690_of__nat__of__bool,axiom,
% 9.20/5.22      ! [P: $o] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
% 9.20/5.22        = ( zero_n2684676970156552555ol_int @ P ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7691_of__nat__sum,axiom,
% 9.20/5.22      ! [F: complex > nat,A2: set_complex] :
% 9.20/5.22        ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
% 9.20/5.22        = ( groups7754918857620584856omplex
% 9.20/5.22          @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
% 9.20/5.22          @ A2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7692_of__nat__sum,axiom,
% 9.20/5.22      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 9.20/5.22        = ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 9.20/5.22          @ A2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7693_of__nat__sum,axiom,
% 9.20/5.22      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
% 9.20/5.22          @ A2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7694_of__nat__sum,axiom,
% 9.20/5.22      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 9.20/5.22        = ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
% 9.20/5.22          @ A2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7695_of__nat__le__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7696_of__nat__le__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7697_of__nat__le__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7698_of__nat__le__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
% 9.20/5.22        = ( M = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7699_of__nat__Suc,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
% 9.20/5.22        = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7700_of__nat__Suc,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
% 9.20/5.22        = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7701_of__nat__Suc,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
% 9.20/5.22        = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7702_of__nat__Suc,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
% 9.20/5.22        = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7703_of__nat__Suc,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
% 9.20/5.22        = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7704_real__of__nat__less__numeral__iff,axiom,
% 9.20/5.22      ! [N2: nat,W2: num] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W2 ) )
% 9.20/5.22        = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7705_numeral__less__real__of__nat__iff,axiom,
% 9.20/5.22      ! [W2: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ ( numeral_numeral_nat @ W2 ) @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7706_numeral__le__real__of__nat__iff,axiom,
% 9.20/5.22      ! [N2: num,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7707_of__nat__0__less__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7708_of__nat__0__less__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7709_of__nat__0__less__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7710_of__nat__0__less__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7711_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7712_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7713_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7714_of__nat__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7715_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7716_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7717_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7718_of__nat__less__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7719_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7720_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7721_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7722_of__nat__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,B: nat,W2: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7723_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7724_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7725_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7726_of__nat__le__of__nat__power__cancel__iff,axiom,
% 9.20/5.22      ! [B: nat,W2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7727_of__nat__zero__less__power__iff,axiom,
% 9.20/5.22      ! [X3: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N2 ) )
% 9.20/5.22        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7728_of__nat__zero__less__power__iff,axiom,
% 9.20/5.22      ! [X3: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N2 ) )
% 9.20/5.22        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7729_of__nat__zero__less__power__iff,axiom,
% 9.20/5.22      ! [X3: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N2 ) )
% 9.20/5.22        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7730_of__nat__zero__less__power__iff,axiom,
% 9.20/5.22      ! [X3: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N2 ) )
% 9.20/5.22        = ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7731_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7732_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7733_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7734_numeral__power__less__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.20/5.22        = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7735_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7736_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7737_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7738_of__nat__less__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7739_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7740_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7741_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7742_of__nat__le__numeral__power__cancel__iff,axiom,
% 9.20/5.22      ! [X3: nat,I: num,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7743_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7744_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7745_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7746_numeral__power__le__of__nat__cancel__iff,axiom,
% 9.20/5.22      ! [I: num,N2: nat,X3: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7747_prod_Odisc__eq__case,axiom,
% 9.20/5.22      ! [Prod: product_prod_int_int] :
% 9.20/5.22        ( produc4947309494688390418_int_o
% 9.20/5.22        @ ^ [Uu3: int,Uv3: int] : $true
% 9.20/5.22        @ Prod ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7748_int__sum,axiom,
% 9.20/5.22      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
% 9.20/5.22        = ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 9.20/5.22          @ A2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7749_real__arch__simple,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22      ? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7750_real__arch__simple,axiom,
% 9.20/5.22      ! [X3: rat] :
% 9.20/5.22      ? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7751_reals__Archimedean2,axiom,
% 9.20/5.22      ! [X3: rat] :
% 9.20/5.22      ? [N3: nat] : ( ord_less_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7752_reals__Archimedean2,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22      ? [N3: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7753_mult__of__nat__commute,axiom,
% 9.20/5.22      ! [X3: nat,Y3: complex] :
% 9.20/5.22        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X3 ) @ Y3 )
% 9.20/5.22        = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7754_mult__of__nat__commute,axiom,
% 9.20/5.22      ! [X3: nat,Y3: rat] :
% 9.20/5.22        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y3 )
% 9.20/5.22        = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7755_mult__of__nat__commute,axiom,
% 9.20/5.22      ! [X3: nat,Y3: int] :
% 9.20/5.22        ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
% 9.20/5.22        = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7756_mult__of__nat__commute,axiom,
% 9.20/5.22      ! [X3: nat,Y3: real] :
% 9.20/5.22        ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y3 )
% 9.20/5.22        = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7757_mult__of__nat__commute,axiom,
% 9.20/5.22      ! [X3: nat,Y3: nat] :
% 9.20/5.22        ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
% 9.20/5.22        = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7758_int__cases2,axiom,
% 9.20/5.22      ! [Z2: int] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( Z2
% 9.20/5.22           != ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( Z2
% 9.20/5.22             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7759_arctan__less__iff,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 9.20/5.22        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7760_arctan__monotone,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.22       => ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7761_int__diff__cases,axiom,
% 9.20/5.22      ! [Z2: int] :
% 9.20/5.22        ~ ! [M3: nat,N3: nat] :
% 9.20/5.22            ( Z2
% 9.20/5.22           != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7762_of__nat__less__of__int__iff,axiom,
% 9.20/5.22      ! [N2: nat,X3: int] :
% 9.20/5.22        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X3 ) )
% 9.20/5.22        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7763_of__nat__less__of__int__iff,axiom,
% 9.20/5.22      ! [N2: nat,X3: int] :
% 9.20/5.22        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X3 ) )
% 9.20/5.22        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7764_of__nat__less__of__int__iff,axiom,
% 9.20/5.22      ! [N2: nat,X3: int] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X3 ) )
% 9.20/5.22        = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7765_of__nat__0__le__iff,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri5074537144036343181t_real @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7766_of__nat__0__le__iff,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7767_of__nat__0__le__iff,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7768_of__nat__0__le__iff,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7769_of__nat__less__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7770_of__nat__less__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7771_of__nat__less__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7772_of__nat__less__0__iff,axiom,
% 9.20/5.22      ! [M: nat] :
% 9.20/5.22        ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7773_of__nat__neq__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
% 9.20/5.22       != zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7774_of__nat__neq__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
% 9.20/5.22       != zero_zero_rat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7775_of__nat__neq__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 9.20/5.22       != zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7776_of__nat__neq__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
% 9.20/5.22       != zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7777_of__nat__neq__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
% 9.20/5.22       != zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7778_of__nat__less__imp__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
% 9.20/5.22       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7779_of__nat__less__imp__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7780_of__nat__less__imp__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.22       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7781_of__nat__less__imp__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
% 9.20/5.22       => ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7782_less__imp__of__nat__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.22       => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7783_less__imp__of__nat__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.22       => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7784_less__imp__of__nat__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.22       => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7785_less__imp__of__nat__less,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ M @ N2 )
% 9.20/5.22       => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7786_of__nat__mono,axiom,
% 9.20/5.22      ! [I: nat,J: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.22       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7787_of__nat__mono,axiom,
% 9.20/5.22      ! [I: nat,J: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.22       => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7788_of__nat__mono,axiom,
% 9.20/5.22      ! [I: nat,J: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.22       => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7789_of__nat__mono,axiom,
% 9.20/5.22      ! [I: nat,J: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.22       => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7790_int__ops_I1_J,axiom,
% 9.20/5.22      ( ( semiri1314217659103216013at_int @ zero_zero_nat )
% 9.20/5.22      = zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7791_nat__int__comparison_I2_J,axiom,
% 9.20/5.22      ( ord_less_nat
% 9.20/5.22      = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7792_int__of__nat__induct,axiom,
% 9.20/5.22      ! [P: int > $o,Z2: int] :
% 9.20/5.22        ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22       => ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
% 9.20/5.22         => ( P @ Z2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7793_int__cases,axiom,
% 9.20/5.22      ! [Z2: int] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( Z2
% 9.20/5.22           != ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( Z2
% 9.20/5.22             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7794_zle__int,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7795_nat__int__comparison_I3_J,axiom,
% 9.20/5.22      ( ord_less_eq_nat
% 9.20/5.22      = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7796_zero__le__imp__eq__int,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.22       => ? [N3: nat] :
% 9.20/5.22            ( K
% 9.20/5.22            = ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7797_nonneg__int__cases,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( K
% 9.20/5.22             != ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7798_zadd__int__left,axiom,
% 9.20/5.22      ! [M: nat,N2: nat,Z2: int] :
% 9.20/5.22        ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z2 ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7799_int__ops_I5_J,axiom,
% 9.20/5.22      ! [A: nat,B: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7800_int__plus,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7801_zle__iff__zadd,axiom,
% 9.20/5.22      ( ord_less_eq_int
% 9.20/5.22      = ( ^ [W3: int,Z6: int] :
% 9.20/5.22          ? [N: nat] :
% 9.20/5.22            ( Z6
% 9.20/5.22            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7802_not__int__zless__negative,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7803_nat__less__as__int,axiom,
% 9.20/5.22      ( ord_less_nat
% 9.20/5.22      = ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7804_nat__leq__as__int,axiom,
% 9.20/5.22      ( ord_less_eq_nat
% 9.20/5.22      = ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7805_ex__less__of__nat__mult,axiom,
% 9.20/5.22      ! [X3: rat,Y3: rat] :
% 9.20/5.22        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.20/5.22       => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7806_ex__less__of__nat__mult,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22       => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7807_of__nat__diff,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7808_of__nat__diff,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7809_of__nat__diff,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7810_of__nat__diff,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
% 9.20/5.22          = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7811_reals__Archimedean3,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22       => ! [Y6: real] :
% 9.20/5.22          ? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7812_int__cases4,axiom,
% 9.20/5.22      ! [M: int] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( M
% 9.20/5.22           != ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.22             => ( M
% 9.20/5.22               != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7813_int__zle__neg,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.20/5.22        = ( ( N2 = zero_zero_nat )
% 9.20/5.22          & ( M = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7814_int__ops_I4_J,axiom,
% 9.20/5.22      ! [A: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( suc @ A ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7815_int__Suc,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
% 9.20/5.22        = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7816_zless__iff__Suc__zadd,axiom,
% 9.20/5.22      ( ord_less_int
% 9.20/5.22      = ( ^ [W3: int,Z6: int] :
% 9.20/5.22          ? [N: nat] :
% 9.20/5.22            ( Z6
% 9.20/5.22            = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7817_negative__zle__0,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7818_nonpos__int__cases,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_eq_int @ K @ zero_zero_int )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( K
% 9.20/5.22             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7819_sum__roots__unity,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ord_less_nat @ one_one_nat @ N2 )
% 9.20/5.22       => ( ( groups7754918857620584856omplex
% 9.20/5.22            @ ^ [X: complex] : X
% 9.20/5.22            @ ( collect_complex
% 9.20/5.22              @ ^ [Z6: complex] :
% 9.20/5.22                  ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.22                  = one_one_complex ) ) )
% 9.20/5.22          = zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7820_sum__nth__roots,axiom,
% 9.20/5.22      ! [N2: nat,C: complex] :
% 9.20/5.22        ( ( ord_less_nat @ one_one_nat @ N2 )
% 9.20/5.22       => ( ( groups7754918857620584856omplex
% 9.20/5.22            @ ^ [X: complex] : X
% 9.20/5.22            @ ( collect_complex
% 9.20/5.22              @ ^ [Z6: complex] :
% 9.20/5.22                  ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.22                  = C ) ) )
% 9.20/5.22          = zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7821_mod__mult2__eq_H,axiom,
% 9.20/5.22      ! [A: code_integer,M: nat,N2: nat] :
% 9.20/5.22        ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
% 9.20/5.22        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7822_mod__mult2__eq_H,axiom,
% 9.20/5.22      ! [A: int,M: nat,N2: nat] :
% 9.20/5.22        ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7823_mod__mult2__eq_H,axiom,
% 9.20/5.22      ! [A: nat,M: nat,N2: nat] :
% 9.20/5.22        ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7824_pos__int__cases,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( ( K
% 9.20/5.22                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7825_zero__less__imp__eq__int,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_int @ zero_zero_int @ K )
% 9.20/5.22       => ? [N3: nat] :
% 9.20/5.22            ( ( ord_less_nat @ zero_zero_nat @ N3 )
% 9.20/5.22            & ( K
% 9.20/5.22              = ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7826_int__cases3,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( K != zero_zero_int )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ( K
% 9.20/5.22                = ( semiri1314217659103216013at_int @ N3 ) )
% 9.20/5.22             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
% 9.20/5.22         => ~ ! [N3: nat] :
% 9.20/5.22                ( ( K
% 9.20/5.22                  = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.20/5.22               => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7827_nat__less__real__le,axiom,
% 9.20/5.22      ( ord_less_nat
% 9.20/5.22      = ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7828_nat__le__real__less,axiom,
% 9.20/5.22      ( ord_less_eq_nat
% 9.20/5.22      = ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7829_zmult__zless__mono2__lemma,axiom,
% 9.20/5.22      ! [I: int,J: int,K: nat] :
% 9.20/5.22        ( ( ord_less_int @ I @ J )
% 9.20/5.22       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.22         => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7830_not__zle__0__negative,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7831_negative__zless__0,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7832_negD,axiom,
% 9.20/5.22      ! [X3: int] :
% 9.20/5.22        ( ( ord_less_int @ X3 @ zero_zero_int )
% 9.20/5.22       => ? [N3: nat] :
% 9.20/5.22            ( X3
% 9.20/5.22            = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7833_int__ops_I6_J,axiom,
% 9.20/5.22      ! [A: nat,B: nat] :
% 9.20/5.22        ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.20/5.22         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.22            = zero_zero_int ) )
% 9.20/5.22        & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
% 9.20/5.22         => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
% 9.20/5.22            = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7834_nat__approx__posE,axiom,
% 9.20/5.22      ! [E2: rat] :
% 9.20/5.22        ( ( ord_less_rat @ zero_zero_rat @ E2 )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7835_nat__approx__posE,axiom,
% 9.20/5.22      ! [E2: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ~ ( ord_less_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7836_of__nat__less__two__power,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7837_of__nat__less__two__power,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7838_of__nat__less__two__power,axiom,
% 9.20/5.22      ! [N2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7839_inverse__of__nat__le,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( N2 != zero_zero_nat )
% 9.20/5.22         => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7840_inverse__of__nat__le,axiom,
% 9.20/5.22      ! [N2: nat,M: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.22       => ( ( N2 != zero_zero_nat )
% 9.20/5.22         => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7841_real__archimedian__rdiv__eq__0,axiom,
% 9.20/5.22      ! [X3: real,C: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.22       => ( ( ord_less_eq_real @ zero_zero_real @ C )
% 9.20/5.22         => ( ! [M3: nat] :
% 9.20/5.22                ( ( ord_less_nat @ zero_zero_nat @ M3 )
% 9.20/5.22               => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M3 ) @ X3 ) @ C ) )
% 9.20/5.22           => ( X3 = zero_zero_real ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7842_neg__int__cases,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.22       => ~ ! [N3: nat] :
% 9.20/5.22              ( ( K
% 9.20/5.22                = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
% 9.20/5.22             => ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7843_zdiff__int__split,axiom,
% 9.20/5.22      ! [P: int > $o,X3: nat,Y3: nat] :
% 9.20/5.22        ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y3 ) ) )
% 9.20/5.22        = ( ( ( ord_less_eq_nat @ Y3 @ X3 )
% 9.20/5.22           => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
% 9.20/5.22          & ( ( ord_less_nat @ X3 @ Y3 )
% 9.20/5.22           => ( P @ zero_zero_int ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7844_ln__realpow,axiom,
% 9.20/5.22      ! [X3: real,N2: nat] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22       => ( ( ln_ln_real @ ( power_power_real @ X3 @ N2 ) )
% 9.20/5.22          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7845_arctan__add,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.22       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.20/5.22         => ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
% 9.20/5.22            = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7846_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7847_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7848_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7849_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7850_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7851_double__gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7852_double__arith__series,axiom,
% 9.20/5.22      ! [A: complex,D: complex,N2: nat] :
% 9.20/5.22        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups2073611262835488442omplex
% 9.20/5.22            @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7853_double__arith__series,axiom,
% 9.20/5.22      ! [A: extended_enat,D: extended_enat,N2: nat] :
% 9.20/5.22        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups7108830773950497114d_enat
% 9.20/5.22            @ ^ [I2: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7854_double__arith__series,axiom,
% 9.20/5.22      ! [A: rat,D: rat,N2: nat] :
% 9.20/5.22        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7855_double__arith__series,axiom,
% 9.20/5.22      ! [A: int,D: int,N2: nat] :
% 9.20/5.22        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7856_double__arith__series,axiom,
% 9.20/5.22      ! [A: nat,D: nat,N2: nat] :
% 9.20/5.22        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups3542108847815614940at_nat
% 9.20/5.22            @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7857_double__arith__series,axiom,
% 9.20/5.22      ! [A: real,D: real,N2: nat] :
% 9.20/5.22        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I2 ) @ D ) )
% 9.20/5.22            @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.22        = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7858_gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7859_gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide_divide_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7860_gauss__sum,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide_divide_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7861_arith__series,axiom,
% 9.20/5.22      ! [A: code_integer,D: code_integer,N2: nat] :
% 9.20/5.22        ( ( groups7501900531339628137nteger
% 9.20/5.22          @ ^ [I2: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I2 ) @ D ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7862_arith__series,axiom,
% 9.20/5.22      ! [A: int,D: int,N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ D ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7863_arith__series,axiom,
% 9.20/5.22      ! [A: nat,D: nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ D ) )
% 9.20/5.22          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.22        = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7864_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( groups2073611262835488442omplex @ semiri8010041392384452111omplex @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7865_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( groups7108830773950497114d_enat @ semiri4216267220026989637d_enat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7866_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7867_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( groups3539618377306564664at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7868_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( groups3542108847815614940at_nat @ semiri1316708129612266289at_nat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7869_double__gauss__sum__from__Suc__0,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( groups6591440286371151544t_real @ semiri5074537144036343181t_real @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
% 9.20/5.22        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7870_sum__gp__offset,axiom,
% 9.20/5.22      ! [X3: complex,M: nat,N2: nat] :
% 9.20/5.22        ( ( ( X3 = one_one_complex )
% 9.20/5.22         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
% 9.20/5.22        & ( ( X3 != one_one_complex )
% 9.20/5.22         => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7871_sum__gp__offset,axiom,
% 9.20/5.22      ! [X3: rat,M: nat,N2: nat] :
% 9.20/5.22        ( ( ( X3 = one_one_rat )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
% 9.20/5.22        & ( ( X3 != one_one_rat )
% 9.20/5.22         => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7872_sum__gp__offset,axiom,
% 9.20/5.22      ! [X3: real,M: nat,N2: nat] :
% 9.20/5.22        ( ( ( X3 = one_one_real )
% 9.20/5.22         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
% 9.20/5.22        & ( ( X3 != one_one_real )
% 9.20/5.22         => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
% 9.20/5.22            = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7873_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri8010041392384452111omplex
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
% 9.20/5.22            @ ( produc1917071388513777916omplex
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7874_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri4216267220026989637d_enat
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_Extended_enat @ ( N = zero_zero_nat ) @ zero_z5237406670263579293d_enat
% 9.20/5.22            @ ( produc2676513652042109336d_enat
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7875_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri681578069525770553at_rat
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
% 9.20/5.22            @ ( produc6207742614233964070at_rat
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7876_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri1314217659103216013at_int
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
% 9.20/5.22            @ ( produc6840382203811409530at_int
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7877_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri5074537144036343181t_real
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 9.20/5.22            @ ( produc1703576794950452218t_real
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7878_of__nat__code__if,axiom,
% 9.20/5.22      ( semiri1316708129612266289at_nat
% 9.20/5.22      = ( ^ [N: nat] :
% 9.20/5.22            ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 9.20/5.22            @ ( produc6842872674320459806at_nat
% 9.20/5.22              @ ^ [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 ) )
% 9.20/5.22              @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7879_monoseq__arctan__series,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.22       => ( topolo6980174941875973593q_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7880_lemma__termdiff3,axiom,
% 9.20/5.22      ! [H2: real,Z2: real,K5: real,N2: nat] :
% 9.20/5.22        ( ( H2 != zero_zero_real )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
% 9.20/5.22         => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
% 9.20/5.22           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V7735802525324610683m_real @ H2 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7881_lemma__termdiff3,axiom,
% 9.20/5.22      ! [H2: complex,Z2: complex,K5: real,N2: nat] :
% 9.20/5.22        ( ( H2 != zero_zero_complex )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
% 9.20/5.22         => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
% 9.20/5.22           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) @ ( power_power_real @ K5 @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( real_V1022390504157884413omplex @ H2 ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7882_ln__series,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.20/5.22         => ( ( ln_ln_real @ X3 )
% 9.20/5.22            = ( suminf_real
% 9.20/5.22              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N @ one_one_nat ) ) ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7883_lemma__termdiff2,axiom,
% 9.20/5.22      ! [H2: complex,Z2: complex,N2: nat] :
% 9.20/5.22        ( ( H2 != zero_zero_complex )
% 9.20/5.22       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.20/5.22          = ( times_times_complex @ H2
% 9.20/5.22            @ ( groups2073611262835488442omplex
% 9.20/5.22              @ ^ [P6: nat] :
% 9.20/5.22                  ( groups2073611262835488442omplex
% 9.20/5.22                  @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.20/5.22                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.20/5.22              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7884_lemma__termdiff2,axiom,
% 9.20/5.22      ! [H2: rat,Z2: rat,N2: nat] :
% 9.20/5.22        ( ( H2 != zero_zero_rat )
% 9.20/5.22       => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N2 ) @ ( power_power_rat @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.20/5.22          = ( times_times_rat @ H2
% 9.20/5.22            @ ( groups2906978787729119204at_rat
% 9.20/5.22              @ ^ [P6: nat] :
% 9.20/5.22                  ( groups2906978787729119204at_rat
% 9.20/5.22                  @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q5 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.20/5.22                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.20/5.22              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7885_lemma__termdiff2,axiom,
% 9.20/5.22      ! [H2: real,Z2: real,N2: nat] :
% 9.20/5.22        ( ( H2 != zero_zero_real )
% 9.20/5.22       => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
% 9.20/5.22          = ( times_times_real @ H2
% 9.20/5.22            @ ( groups6591440286371151544t_real
% 9.20/5.22              @ ^ [P6: nat] :
% 9.20/5.22                  ( groups6591440286371151544t_real
% 9.20/5.22                  @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ Q5 ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Q5 ) ) )
% 9.20/5.22                  @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) @ P6 ) ) )
% 9.20/5.22              @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7886_arctan__series,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.22       => ( ( arctan @ X3 )
% 9.20/5.22          = ( suminf_real
% 9.20/5.22            @ ^ [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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7887_lessThan__iff,axiom,
% 9.20/5.22      ! [I: set_nat,K: set_nat] :
% 9.20/5.22        ( ( member_set_nat @ I @ ( set_or890127255671739683et_nat @ K ) )
% 9.20/5.22        = ( ord_less_set_nat @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7888_lessThan__iff,axiom,
% 9.20/5.22      ! [I: rat,K: rat] :
% 9.20/5.22        ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
% 9.20/5.22        = ( ord_less_rat @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7889_lessThan__iff,axiom,
% 9.20/5.22      ! [I: num,K: num] :
% 9.20/5.22        ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
% 9.20/5.22        = ( ord_less_num @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7890_lessThan__iff,axiom,
% 9.20/5.22      ! [I: nat,K: nat] :
% 9.20/5.22        ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
% 9.20/5.22        = ( ord_less_nat @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7891_lessThan__iff,axiom,
% 9.20/5.22      ! [I: int,K: int] :
% 9.20/5.22        ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
% 9.20/5.22        = ( ord_less_int @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7892_lessThan__iff,axiom,
% 9.20/5.22      ! [I: real,K: real] :
% 9.20/5.22        ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
% 9.20/5.22        = ( ord_less_real @ I @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7893_finite__lessThan,axiom,
% 9.20/5.22      ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7894_lessThan__subset__iff,axiom,
% 9.20/5.22      ! [X3: rat,Y3: rat] :
% 9.20/5.22        ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7895_lessThan__subset__iff,axiom,
% 9.20/5.22      ! [X3: num,Y3: num] :
% 9.20/5.22        ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7896_lessThan__subset__iff,axiom,
% 9.20/5.22      ! [X3: nat,Y3: nat] :
% 9.20/5.22        ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7897_lessThan__subset__iff,axiom,
% 9.20/5.22      ! [X3: int,Y3: int] :
% 9.20/5.22        ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7898_lessThan__subset__iff,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
% 9.20/5.22        = ( ord_less_eq_real @ X3 @ Y3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7899_lessThan__0,axiom,
% 9.20/5.22      ( ( set_ord_lessThan_nat @ zero_zero_nat )
% 9.20/5.22      = bot_bot_set_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7900_single__Diff__lessThan,axiom,
% 9.20/5.22      ! [K: nat] :
% 9.20/5.22        ( ( minus_minus_set_nat @ ( insert_nat @ K @ bot_bot_set_nat ) @ ( set_ord_lessThan_nat @ K ) )
% 9.20/5.22        = ( insert_nat @ K @ bot_bot_set_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7901_single__Diff__lessThan,axiom,
% 9.20/5.22      ! [K: int] :
% 9.20/5.22        ( ( minus_minus_set_int @ ( insert_int @ K @ bot_bot_set_int ) @ ( set_ord_lessThan_int @ K ) )
% 9.20/5.22        = ( insert_int @ K @ bot_bot_set_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7902_single__Diff__lessThan,axiom,
% 9.20/5.22      ! [K: real] :
% 9.20/5.22        ( ( minus_minus_set_real @ ( insert_real @ K @ bot_bot_set_real ) @ ( set_or5984915006950818249n_real @ K ) )
% 9.20/5.22        = ( insert_real @ K @ bot_bot_set_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7903_sum_OlessThan__Suc,axiom,
% 9.20/5.22      ! [G: nat > rat,N2: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7904_sum_OlessThan__Suc,axiom,
% 9.20/5.22      ! [G: nat > int,N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7905_sum_OlessThan__Suc,axiom,
% 9.20/5.22      ! [G: nat > nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7906_sum_OlessThan__Suc,axiom,
% 9.20/5.22      ! [G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7907_powser__zero,axiom,
% 9.20/5.22      ! [F: nat > complex] :
% 9.20/5.22        ( ( suminf_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) )
% 9.20/5.22        = ( F @ zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7908_powser__zero,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ( suminf_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) )
% 9.20/5.22        = ( F @ zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7909_int__int__eq,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ( semiri1314217659103216013at_int @ M )
% 9.20/5.22          = ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.22        = ( M = N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7910_lessThan__non__empty,axiom,
% 9.20/5.22      ! [X3: int] :
% 9.20/5.22        ( ( set_ord_lessThan_int @ X3 )
% 9.20/5.22       != bot_bot_set_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7911_lessThan__non__empty,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( set_or5984915006950818249n_real @ X3 )
% 9.20/5.22       != bot_bot_set_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7912_infinite__Iio,axiom,
% 9.20/5.22      ! [A: int] :
% 9.20/5.22        ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7913_infinite__Iio,axiom,
% 9.20/5.22      ! [A: real] :
% 9.20/5.22        ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7914_lessThan__def,axiom,
% 9.20/5.22      ( set_or890127255671739683et_nat
% 9.20/5.22      = ( ^ [U2: set_nat] :
% 9.20/5.22            ( collect_set_nat
% 9.20/5.22            @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7915_lessThan__def,axiom,
% 9.20/5.22      ( set_ord_lessThan_rat
% 9.20/5.22      = ( ^ [U2: rat] :
% 9.20/5.22            ( collect_rat
% 9.20/5.22            @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7916_lessThan__def,axiom,
% 9.20/5.22      ( set_ord_lessThan_num
% 9.20/5.22      = ( ^ [U2: num] :
% 9.20/5.22            ( collect_num
% 9.20/5.22            @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7917_lessThan__def,axiom,
% 9.20/5.22      ( set_ord_lessThan_nat
% 9.20/5.22      = ( ^ [U2: nat] :
% 9.20/5.22            ( collect_nat
% 9.20/5.22            @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7918_lessThan__def,axiom,
% 9.20/5.22      ( set_ord_lessThan_int
% 9.20/5.22      = ( ^ [U2: int] :
% 9.20/5.22            ( collect_int
% 9.20/5.22            @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7919_lessThan__def,axiom,
% 9.20/5.22      ( set_or5984915006950818249n_real
% 9.20/5.22      = ( ^ [U2: real] :
% 9.20/5.22            ( collect_real
% 9.20/5.22            @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7920_Iio__eq__empty__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( set_ord_lessThan_nat @ N2 )
% 9.20/5.22          = bot_bot_set_nat )
% 9.20/5.22        = ( N2 = bot_bot_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7921_lessThan__strict__subset__iff,axiom,
% 9.20/5.22      ! [M: rat,N2: rat] :
% 9.20/5.22        ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
% 9.20/5.22        = ( ord_less_rat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7922_lessThan__strict__subset__iff,axiom,
% 9.20/5.22      ! [M: num,N2: num] :
% 9.20/5.22        ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
% 9.20/5.22        = ( ord_less_num @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7923_lessThan__strict__subset__iff,axiom,
% 9.20/5.22      ! [M: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.22        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7924_lessThan__strict__subset__iff,axiom,
% 9.20/5.22      ! [M: int,N2: int] :
% 9.20/5.22        ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
% 9.20/5.22        = ( ord_less_int @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7925_lessThan__strict__subset__iff,axiom,
% 9.20/5.22      ! [M: real,N2: real] :
% 9.20/5.22        ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
% 9.20/5.22        = ( ord_less_real @ M @ N2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7926_lessThan__Suc,axiom,
% 9.20/5.22      ! [K: nat] :
% 9.20/5.22        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.20/5.22        = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7927_lessThan__empty__iff,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( ( set_ord_lessThan_nat @ N2 )
% 9.20/5.22          = bot_bot_set_nat )
% 9.20/5.22        = ( N2 = zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7928_finite__nat__iff__bounded,axiom,
% 9.20/5.22      ( finite_finite_nat
% 9.20/5.22      = ( ^ [S7: set_nat] :
% 9.20/5.22          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7929_finite__nat__bounded,axiom,
% 9.20/5.22      ! [S3: set_nat] :
% 9.20/5.22        ( ( finite_finite_nat @ S3 )
% 9.20/5.22       => ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7930_sum_Onat__diff__reindex,axiom,
% 9.20/5.22      ! [G: nat > nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7931_sum_Onat__diff__reindex,axiom,
% 9.20/5.22      ! [G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.22        = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7932_sum__diff__distrib,axiom,
% 9.20/5.22      ! [Q: int > nat,P: int > nat,N2: int] :
% 9.20/5.22        ( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.20/5.22       => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
% 9.20/5.22          = ( groups4541462559716669496nt_nat
% 9.20/5.22            @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.20/5.22            @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7933_sum__diff__distrib,axiom,
% 9.20/5.22      ! [Q: real > nat,P: real > nat,N2: real] :
% 9.20/5.22        ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.20/5.22       => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
% 9.20/5.22          = ( groups1935376822645274424al_nat
% 9.20/5.22            @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.20/5.22            @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7934_sum__diff__distrib,axiom,
% 9.20/5.22      ! [Q: nat > nat,P: nat > nat,N2: nat] :
% 9.20/5.22        ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
% 9.20/5.22       => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
% 9.20/5.22          = ( groups3542108847815614940at_nat
% 9.20/5.22            @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7935_sum_OlessThan__Suc__shift,axiom,
% 9.20/5.22      ! [G: nat > rat,N2: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_rat @ ( G @ zero_zero_nat )
% 9.20/5.22          @ ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7936_sum_OlessThan__Suc__shift,axiom,
% 9.20/5.22      ! [G: nat > int,N2: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_int @ ( G @ zero_zero_nat )
% 9.20/5.22          @ ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7937_sum_OlessThan__Suc__shift,axiom,
% 9.20/5.22      ! [G: nat > nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_nat @ ( G @ zero_zero_nat )
% 9.20/5.22          @ ( groups3542108847815614940at_nat
% 9.20/5.22            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7938_sum_OlessThan__Suc__shift,axiom,
% 9.20/5.22      ! [G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_real @ ( G @ zero_zero_nat )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7939_sum__lessThan__telescope,axiom,
% 9.20/5.22      ! [F: nat > rat,M: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7940_sum__lessThan__telescope,axiom,
% 9.20/5.22      ! [F: nat > int,M: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7941_sum__lessThan__telescope,axiom,
% 9.20/5.22      ! [F: nat > real,M: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7942_sum__lessThan__telescope_H,axiom,
% 9.20/5.22      ! [F: nat > rat,M: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7943_sum__lessThan__telescope_H,axiom,
% 9.20/5.22      ! [F: nat > int,M: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7944_sum__lessThan__telescope_H,axiom,
% 9.20/5.22      ! [F: nat > real,M: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7945_sum_OatLeast1__atMost__eq,axiom,
% 9.20/5.22      ! [G: nat > nat,N2: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7946_sum_OatLeast1__atMost__eq,axiom,
% 9.20/5.22      ! [G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.22        = ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ N2 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7947_lemma__termdiff1,axiom,
% 9.20/5.22      ! [Z2: complex,H2: complex,M: nat] :
% 9.20/5.22        ( ( groups2073611262835488442omplex
% 9.20/5.22          @ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ P6 ) ) @ ( power_power_complex @ Z2 @ M ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( groups2073611262835488442omplex
% 9.20/5.22          @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7948_lemma__termdiff1,axiom,
% 9.20/5.22      ! [Z2: rat,H2: rat,M: nat] :
% 9.20/5.22        ( ( groups2906978787729119204at_rat
% 9.20/5.22          @ ^ [P6: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ P6 ) ) @ ( power_power_rat @ Z2 @ M ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( groups2906978787729119204at_rat
% 9.20/5.22          @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P6 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7949_lemma__termdiff1,axiom,
% 9.20/5.22      ! [Z2: int,H2: int,M: nat] :
% 9.20/5.22        ( ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ P6 ) ) @ ( power_power_int @ Z2 @ M ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( groups3539618377306564664at_int
% 9.20/5.22          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7950_lemma__termdiff1,axiom,
% 9.20/5.22      ! [Z2: real,H2: real,M: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ P6 ) ) @ ( power_power_real @ Z2 @ M ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) )
% 9.20/5.22        = ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ M ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7951_diff__power__eq__sum,axiom,
% 9.20/5.22      ! [X3: complex,N2: nat,Y3: complex] :
% 9.20/5.22        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) @ ( power_power_complex @ Y3 @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.20/5.22          @ ( groups2073611262835488442omplex
% 9.20/5.22            @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P6 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7952_diff__power__eq__sum,axiom,
% 9.20/5.22      ! [X3: rat,N2: nat,Y3: rat] :
% 9.20/5.22        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) @ ( power_power_rat @ Y3 @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.20/5.22          @ ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P6 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7953_diff__power__eq__sum,axiom,
% 9.20/5.22      ! [X3: int,N2: nat,Y3: int] :
% 9.20/5.22        ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N2 ) ) @ ( power_power_int @ Y3 @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.20/5.22          @ ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X3 @ P6 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7954_diff__power__eq__sum,axiom,
% 9.20/5.22      ! [X3: real,N2: nat,Y3: real] :
% 9.20/5.22        ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) @ ( power_power_real @ Y3 @ ( suc @ N2 ) ) )
% 9.20/5.22        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X3 @ P6 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7955_power__diff__sumr2,axiom,
% 9.20/5.22      ! [X3: complex,N2: nat,Y3: complex] :
% 9.20/5.22        ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ N2 ) )
% 9.20/5.22        = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
% 9.20/5.22          @ ( groups2073611262835488442omplex
% 9.20/5.22            @ ^ [I2: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_complex @ X3 @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7956_power__diff__sumr2,axiom,
% 9.20/5.22      ! [X3: rat,N2: nat,Y3: rat] :
% 9.20/5.22        ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N2 ) @ ( power_power_rat @ Y3 @ N2 ) )
% 9.20/5.22        = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
% 9.20/5.22          @ ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [I2: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_rat @ X3 @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7957_power__diff__sumr2,axiom,
% 9.20/5.22      ! [X3: int,N2: nat,Y3: int] :
% 9.20/5.22        ( ( minus_minus_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ N2 ) )
% 9.20/5.22        = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
% 9.20/5.22          @ ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [I2: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_int @ X3 @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7958_power__diff__sumr2,axiom,
% 9.20/5.22      ! [X3: real,N2: nat,Y3: real] :
% 9.20/5.22        ( ( minus_minus_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ N2 ) )
% 9.20/5.22        = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_real @ X3 @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7959_real__sum__nat__ivl__bounded2,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
% 9.20/5.22        ( ! [P7: nat] :
% 9.20/5.22            ( ( ord_less_nat @ P7 @ N2 )
% 9.20/5.22           => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
% 9.20/5.22       => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
% 9.20/5.22         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7960_real__sum__nat__ivl__bounded2,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > int,K5: int,K: nat] :
% 9.20/5.22        ( ! [P7: nat] :
% 9.20/5.22            ( ( ord_less_nat @ P7 @ N2 )
% 9.20/5.22           => ( ord_less_eq_int @ ( F @ P7 ) @ K5 ) )
% 9.20/5.22       => ( ( ord_less_eq_int @ zero_zero_int @ K5 )
% 9.20/5.22         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ K5 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7961_real__sum__nat__ivl__bounded2,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > nat,K5: nat,K: nat] :
% 9.20/5.22        ( ! [P7: nat] :
% 9.20/5.22            ( ( ord_less_nat @ P7 @ N2 )
% 9.20/5.22           => ( ord_less_eq_nat @ ( F @ P7 ) @ K5 ) )
% 9.20/5.22       => ( ( ord_less_eq_nat @ zero_zero_nat @ K5 )
% 9.20/5.22         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ K5 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7962_real__sum__nat__ivl__bounded2,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > real,K5: real,K: nat] :
% 9.20/5.22        ( ! [P7: nat] :
% 9.20/5.22            ( ( ord_less_nat @ P7 @ N2 )
% 9.20/5.22           => ( ord_less_eq_real @ ( F @ P7 ) @ K5 ) )
% 9.20/5.22       => ( ( ord_less_eq_real @ zero_zero_real @ K5 )
% 9.20/5.22         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ K5 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7963_one__diff__power__eq_H,axiom,
% 9.20/5.22      ! [X3: complex,N2: nat] :
% 9.20/5.22        ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N2 ) )
% 9.20/5.22        = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
% 9.20/5.22          @ ( groups2073611262835488442omplex
% 9.20/5.22            @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7964_one__diff__power__eq_H,axiom,
% 9.20/5.22      ! [X3: rat,N2: nat] :
% 9.20/5.22        ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N2 ) )
% 9.20/5.22        = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
% 9.20/5.22          @ ( groups2906978787729119204at_rat
% 9.20/5.22            @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7965_one__diff__power__eq_H,axiom,
% 9.20/5.22      ! [X3: int,N2: nat] :
% 9.20/5.22        ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N2 ) )
% 9.20/5.22        = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
% 9.20/5.22          @ ( groups3539618377306564664at_int
% 9.20/5.22            @ ^ [I2: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7966_one__diff__power__eq_H,axiom,
% 9.20/5.22      ! [X3: real,N2: nat] :
% 9.20/5.22        ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N2 ) )
% 9.20/5.22        = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7967_sum__split__even__odd,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real,N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.22        = ( plus_plus_real
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.22          @ ( groups6591440286371151544t_real
% 9.20/5.22            @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
% 9.20/5.22            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7968_norm__le__zero__iff,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
% 9.20/5.22        = ( X3 = zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7969_norm__le__zero__iff,axiom,
% 9.20/5.22      ! [X3: complex] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
% 9.20/5.22        = ( X3 = zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7970_zero__less__norm__iff,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.20/5.22        = ( X3 != zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7971_zero__less__norm__iff,axiom,
% 9.20/5.22      ! [X3: complex] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.20/5.22        = ( X3 != zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7972_suminf__geometric,axiom,
% 9.20/5.22      ! [C: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.20/5.22       => ( ( suminf_real @ ( power_power_real @ C ) )
% 9.20/5.22          = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7973_suminf__geometric,axiom,
% 9.20/5.22      ! [C: complex] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.20/5.22       => ( ( suminf_complex @ ( power_power_complex @ C ) )
% 9.20/5.22          = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7974_norm__zero,axiom,
% 9.20/5.22      ( ( real_V7735802525324610683m_real @ zero_zero_real )
% 9.20/5.22      = zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7975_norm__zero,axiom,
% 9.20/5.22      ( ( real_V1022390504157884413omplex @ zero_zero_complex )
% 9.20/5.22      = zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7976_norm__eq__zero,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ( real_V7735802525324610683m_real @ X3 )
% 9.20/5.22          = zero_zero_real )
% 9.20/5.22        = ( X3 = zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7977_norm__eq__zero,axiom,
% 9.20/5.22      ! [X3: complex] :
% 9.20/5.22        ( ( ( real_V1022390504157884413omplex @ X3 )
% 9.20/5.22          = zero_zero_real )
% 9.20/5.22        = ( X3 = zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7978_suminf__zero,axiom,
% 9.20/5.22      ( ( suminf_complex
% 9.20/5.22        @ ^ [N: nat] : zero_zero_complex )
% 9.20/5.22      = zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7979_suminf__zero,axiom,
% 9.20/5.22      ( ( suminf_real
% 9.20/5.22        @ ^ [N: nat] : zero_zero_real )
% 9.20/5.22      = zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7980_suminf__zero,axiom,
% 9.20/5.22      ( ( suminf_nat
% 9.20/5.22        @ ^ [N: nat] : zero_zero_nat )
% 9.20/5.22      = zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7981_suminf__zero,axiom,
% 9.20/5.22      ( ( suminf_int
% 9.20/5.22        @ ^ [N: nat] : zero_zero_int )
% 9.20/5.22      = zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7982_norm__not__less__zero,axiom,
% 9.20/5.22      ! [X3: complex] :
% 9.20/5.22        ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7983_norm__uminus__minus,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] :
% 9.20/5.22        ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) )
% 9.20/5.22        = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7984_norm__uminus__minus,axiom,
% 9.20/5.22      ! [X3: complex,Y3: complex] :
% 9.20/5.22        ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 ) )
% 9.20/5.22        = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7985_nonzero__norm__divide,axiom,
% 9.20/5.22      ! [B: real,A: real] :
% 9.20/5.22        ( ( B != zero_zero_real )
% 9.20/5.22       => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
% 9.20/5.22          = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7986_nonzero__norm__divide,axiom,
% 9.20/5.22      ! [B: complex,A: complex] :
% 9.20/5.22        ( ( B != zero_zero_complex )
% 9.20/5.22       => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
% 9.20/5.22          = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7987_power__eq__imp__eq__norm,axiom,
% 9.20/5.22      ! [W2: real,N2: nat,Z2: real] :
% 9.20/5.22        ( ( ( power_power_real @ W2 @ N2 )
% 9.20/5.22          = ( power_power_real @ Z2 @ N2 ) )
% 9.20/5.22       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.22         => ( ( real_V7735802525324610683m_real @ W2 )
% 9.20/5.22            = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7988_power__eq__imp__eq__norm,axiom,
% 9.20/5.22      ! [W2: complex,N2: nat,Z2: complex] :
% 9.20/5.22        ( ( ( power_power_complex @ W2 @ N2 )
% 9.20/5.22          = ( power_power_complex @ Z2 @ N2 ) )
% 9.20/5.22       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.22         => ( ( real_V1022390504157884413omplex @ W2 )
% 9.20/5.22            = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7989_norm__mult__less,axiom,
% 9.20/5.22      ! [X3: real,R2: real,Y3: real,S: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 9.20/5.22         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7990_norm__mult__less,axiom,
% 9.20/5.22      ! [X3: complex,R2: real,Y3: complex,S: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 9.20/5.22         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7991_norm__triangle__lt,axiom,
% 9.20/5.22      ! [X3: real,Y3: real,E2: real] :
% 9.20/5.22        ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 9.20/5.22       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7992_norm__triangle__lt,axiom,
% 9.20/5.22      ! [X3: complex,Y3: complex,E2: real] :
% 9.20/5.22        ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 9.20/5.22       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7993_norm__add__less,axiom,
% 9.20/5.22      ! [X3: real,R2: real,Y3: real,S: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
% 9.20/5.22         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7994_norm__add__less,axiom,
% 9.20/5.22      ! [X3: complex,R2: real,Y3: complex,S: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
% 9.20/5.22         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7995_norm__triangle__mono,axiom,
% 9.20/5.22      ! [A: real,R2: real,B: real,S: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
% 9.20/5.22         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7996_norm__triangle__mono,axiom,
% 9.20/5.22      ! [A: complex,R2: real,B: complex,S: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
% 9.20/5.22       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
% 9.20/5.22         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7997_norm__triangle__ineq,axiom,
% 9.20/5.22      ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7998_norm__triangle__ineq,axiom,
% 9.20/5.22      ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_7999_norm__triangle__le,axiom,
% 9.20/5.22      ! [X3: real,Y3: real,E2: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
% 9.20/5.22       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8000_norm__triangle__le,axiom,
% 9.20/5.22      ! [X3: complex,Y3: complex,E2: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
% 9.20/5.22       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8001_norm__add__leD,axiom,
% 9.20/5.22      ! [A: real,B: real,C: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
% 9.20/5.22       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8002_norm__add__leD,axiom,
% 9.20/5.22      ! [A: complex,B: complex,C: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
% 9.20/5.22       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8003_norm__diff__triangle__less,axiom,
% 9.20/5.22      ! [X3: real,Y3: real,E1: real,Z2: real,E22: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z2 ) ) @ E22 )
% 9.20/5.22         => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8004_norm__diff__triangle__less,axiom,
% 9.20/5.22      ! [X3: complex,Y3: complex,E1: real,Z2: complex,E22: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z2 ) ) @ E22 )
% 9.20/5.22         => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8005_norm__diff__ineq,axiom,
% 9.20/5.22      ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8006_norm__diff__ineq,axiom,
% 9.20/5.22      ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8007_suminf__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_complex ) )
% 9.20/5.22         => ( ( suminf_complex @ F )
% 9.20/5.22            = ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8008_suminf__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_int ) )
% 9.20/5.22         => ( ( suminf_int @ F )
% 9.20/5.22            = ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8009_suminf__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > nat] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_nat ) )
% 9.20/5.22         => ( ( suminf_nat @ F )
% 9.20/5.22            = ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8010_suminf__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > real] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_real ) )
% 9.20/5.22         => ( ( suminf_real @ F )
% 9.20/5.22            = ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8011_power__eq__1__iff,axiom,
% 9.20/5.22      ! [W2: real,N2: nat] :
% 9.20/5.22        ( ( ( power_power_real @ W2 @ N2 )
% 9.20/5.22          = one_one_real )
% 9.20/5.22       => ( ( ( real_V7735802525324610683m_real @ W2 )
% 9.20/5.22            = one_one_real )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8012_power__eq__1__iff,axiom,
% 9.20/5.22      ! [W2: complex,N2: nat] :
% 9.20/5.22        ( ( ( power_power_complex @ W2 @ N2 )
% 9.20/5.22          = one_one_complex )
% 9.20/5.22       => ( ( ( real_V1022390504157884413omplex @ W2 )
% 9.20/5.22            = one_one_real )
% 9.20/5.22          | ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8013_norm__diff__triangle__ineq,axiom,
% 9.20/5.22      ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8014_norm__diff__triangle__ineq,axiom,
% 9.20/5.22      ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8015_sum__bounds__lt__plus1,axiom,
% 9.20/5.22      ! [F: nat > nat,Mm: nat] :
% 9.20/5.22        ( ( groups3542108847815614940at_nat
% 9.20/5.22          @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ Mm ) )
% 9.20/5.22        = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8016_sum__bounds__lt__plus1,axiom,
% 9.20/5.22      ! [F: nat > real,Mm: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ Mm ) )
% 9.20/5.22        = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8017_sumr__cos__zero__one,axiom,
% 9.20/5.22      ! [N2: nat] :
% 9.20/5.22        ( ( groups6591440286371151544t_real
% 9.20/5.22          @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
% 9.20/5.22          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.22        = one_one_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8018_pi__series,axiom,
% 9.20/5.22      ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
% 9.20/5.22      = ( suminf_real
% 9.20/5.22        @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8019_summable__arctan__series,axiom,
% 9.20/5.22      ! [X3: real] :
% 9.20/5.22        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.22       => ( summable_real
% 9.20/5.22          @ ^ [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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8020_ceiling__log__nat__eq__powr__iff,axiom,
% 9.20/5.22      ! [B: nat,K: nat,N2: nat] :
% 9.20/5.22        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.22       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.22         => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.20/5.22              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
% 9.20/5.22            = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.20/5.22              & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8021_geometric__deriv__sums,axiom,
% 9.20/5.22      ! [Z2: real] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
% 9.20/5.22       => ( sums_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) )
% 9.20/5.22          @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8022_geometric__deriv__sums,axiom,
% 9.20/5.22      ! [Z2: complex] :
% 9.20/5.22        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
% 9.20/5.22       => ( sums_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) )
% 9.20/5.22          @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8023_summable__zero,axiom,
% 9.20/5.22      ( summable_complex
% 9.20/5.22      @ ^ [N: nat] : zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8024_summable__zero,axiom,
% 9.20/5.22      ( summable_real
% 9.20/5.22      @ ^ [N: nat] : zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8025_summable__zero,axiom,
% 9.20/5.22      ( summable_nat
% 9.20/5.22      @ ^ [N: nat] : zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8026_summable__zero,axiom,
% 9.20/5.22      ( summable_int
% 9.20/5.22      @ ^ [N: nat] : zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8027_summable__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > complex] :
% 9.20/5.22        ( summable_complex
% 9.20/5.22        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8028_summable__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > real] :
% 9.20/5.22        ( summable_real
% 9.20/5.22        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8029_summable__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > nat] :
% 9.20/5.22        ( summable_nat
% 9.20/5.22        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8030_summable__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > int] :
% 9.20/5.22        ( summable_int
% 9.20/5.22        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8031_sums__zero,axiom,
% 9.20/5.22      ( sums_complex
% 9.20/5.22      @ ^ [N: nat] : zero_zero_complex
% 9.20/5.22      @ zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8032_sums__zero,axiom,
% 9.20/5.22      ( sums_real
% 9.20/5.22      @ ^ [N: nat] : zero_zero_real
% 9.20/5.22      @ zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8033_sums__zero,axiom,
% 9.20/5.22      ( sums_nat
% 9.20/5.22      @ ^ [N: nat] : zero_zero_nat
% 9.20/5.22      @ zero_zero_nat ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8034_sums__zero,axiom,
% 9.20/5.22      ( sums_int
% 9.20/5.22      @ ^ [N: nat] : zero_zero_int
% 9.20/5.22      @ zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8035_summable__iff__shift,axiom,
% 9.20/5.22      ! [F: nat > real,K: nat] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 9.20/5.22        = ( summable_real @ F ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8036_cos__coeff__0,axiom,
% 9.20/5.22      ( ( cos_coeff @ zero_zero_nat )
% 9.20/5.22      = one_one_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8037_summable__cmult__iff,axiom,
% 9.20/5.22      ! [C: complex,F: nat > complex] :
% 9.20/5.22        ( ( summable_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 9.20/5.22        = ( ( C = zero_zero_complex )
% 9.20/5.22          | ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8038_summable__cmult__iff,axiom,
% 9.20/5.22      ! [C: real,F: nat > real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 9.20/5.22        = ( ( C = zero_zero_real )
% 9.20/5.22          | ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8039_summable__divide__iff,axiom,
% 9.20/5.22      ! [F: nat > complex,C: complex] :
% 9.20/5.22        ( ( summable_complex
% 9.20/5.22          @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
% 9.20/5.22        = ( ( C = zero_zero_complex )
% 9.20/5.22          | ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8040_summable__divide__iff,axiom,
% 9.20/5.22      ! [F: nat > real,C: real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
% 9.20/5.22        = ( ( C = zero_zero_real )
% 9.20/5.22          | ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8041_summable__If__finite,axiom,
% 9.20/5.22      ! [P: nat > $o,F: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.22       => ( summable_complex
% 9.20/5.22          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8042_summable__If__finite,axiom,
% 9.20/5.22      ! [P: nat > $o,F: nat > real] :
% 9.20/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.22       => ( summable_real
% 9.20/5.22          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8043_summable__If__finite,axiom,
% 9.20/5.22      ! [P: nat > $o,F: nat > nat] :
% 9.20/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.22       => ( summable_nat
% 9.20/5.22          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8044_summable__If__finite,axiom,
% 9.20/5.22      ! [P: nat > $o,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.22       => ( summable_int
% 9.20/5.22          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8045_summable__If__finite__set,axiom,
% 9.20/5.22      ! [A2: set_nat,F: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( summable_complex
% 9.20/5.22          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8046_summable__If__finite__set,axiom,
% 9.20/5.22      ! [A2: set_nat,F: nat > real] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( summable_real
% 9.20/5.22          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8047_summable__If__finite__set,axiom,
% 9.20/5.22      ! [A2: set_nat,F: nat > nat] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( summable_nat
% 9.20/5.22          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8048_summable__If__finite__set,axiom,
% 9.20/5.22      ! [A2: set_nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ A2 )
% 9.20/5.22       => ( summable_int
% 9.20/5.22          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8049_log__eq__one,axiom,
% 9.20/5.22      ! [A: real] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.22       => ( ( A != one_one_real )
% 9.20/5.22         => ( ( log @ A @ A )
% 9.20/5.22            = one_one_real ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8050_log__less__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.22           => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 9.20/5.22              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8051_log__less__one__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
% 9.20/5.22            = ( ord_less_real @ X3 @ A ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8052_one__less__log__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
% 9.20/5.22            = ( ord_less_real @ A @ X3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8053_log__less__zero__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 9.20/5.22            = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8054_zero__less__log__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 9.20/5.22            = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8055_log__le__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real,Y3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.22           => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
% 9.20/5.22              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8056_log__le__one__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
% 9.20/5.22            = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8057_one__le__log__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
% 9.20/5.22            = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8058_log__le__zero__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
% 9.20/5.22            = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8059_zero__le__log__cancel__iff,axiom,
% 9.20/5.22      ! [A: real,X3: real] :
% 9.20/5.22        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.22       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.22         => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
% 9.20/5.22            = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8060_powser__sums__zero__iff,axiom,
% 9.20/5.22      ! [A: nat > complex,X3: complex] :
% 9.20/5.22        ( ( sums_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 9.20/5.22          @ X3 )
% 9.20/5.22        = ( ( A @ zero_zero_nat )
% 9.20/5.22          = X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8061_powser__sums__zero__iff,axiom,
% 9.20/5.22      ! [A: nat > real,X3: real] :
% 9.20/5.22        ( ( sums_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 9.20/5.22          @ X3 )
% 9.20/5.22        = ( ( A @ zero_zero_nat )
% 9.20/5.22          = X3 ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8062_summable__geometric__iff,axiom,
% 9.20/5.22      ! [C: real] :
% 9.20/5.22        ( ( summable_real @ ( power_power_real @ C ) )
% 9.20/5.22        = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8063_summable__geometric__iff,axiom,
% 9.20/5.22      ! [C: complex] :
% 9.20/5.22        ( ( summable_complex @ ( power_power_complex @ C ) )
% 9.20/5.22        = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8064_log__pow__cancel,axiom,
% 9.20/5.22      ! [A: real,B: nat] :
% 9.20/5.22        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.22       => ( ( A != one_one_real )
% 9.20/5.22         => ( ( log @ A @ ( power_power_real @ A @ B ) )
% 9.20/5.22            = ( semiri5074537144036343181t_real @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8065_sums__0,axiom,
% 9.20/5.22      ! [F: nat > complex] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( ( F @ N3 )
% 9.20/5.22            = zero_zero_complex )
% 9.20/5.22       => ( sums_complex @ F @ zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8066_sums__0,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( ( F @ N3 )
% 9.20/5.22            = zero_zero_real )
% 9.20/5.22       => ( sums_real @ F @ zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8067_sums__0,axiom,
% 9.20/5.22      ! [F: nat > nat] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( ( F @ N3 )
% 9.20/5.22            = zero_zero_nat )
% 9.20/5.22       => ( sums_nat @ F @ zero_zero_nat ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8068_sums__0,axiom,
% 9.20/5.22      ! [F: nat > int] :
% 9.20/5.22        ( ! [N3: nat] :
% 9.20/5.22            ( ( F @ N3 )
% 9.20/5.22            = zero_zero_int )
% 9.20/5.22       => ( sums_int @ F @ zero_zero_int ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8069_sums__le,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real,S: real,T: real] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( sums_real @ F @ S )
% 9.20/5.22         => ( ( sums_real @ G @ T )
% 9.20/5.22           => ( ord_less_eq_real @ S @ T ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8070_sums__le,axiom,
% 9.20/5.22      ! [F: nat > nat,G: nat > nat,S: nat,T: nat] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( sums_nat @ F @ S )
% 9.20/5.22         => ( ( sums_nat @ G @ T )
% 9.20/5.22           => ( ord_less_eq_nat @ S @ T ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8071_sums__le,axiom,
% 9.20/5.22      ! [F: nat > int,G: nat > int,S: int,T: int] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( sums_int @ F @ S )
% 9.20/5.22         => ( ( sums_int @ G @ T )
% 9.20/5.22           => ( ord_less_eq_int @ S @ T ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8072_sums__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > complex] :
% 9.20/5.22        ( sums_complex
% 9.20/5.22        @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.20/5.22        @ ( F @ I ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8073_sums__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > real] :
% 9.20/5.22        ( sums_real
% 9.20/5.22        @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
% 9.20/5.22        @ ( F @ I ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8074_sums__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > nat] :
% 9.20/5.22        ( sums_nat
% 9.20/5.22        @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.20/5.22        @ ( F @ I ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8075_sums__single,axiom,
% 9.20/5.22      ! [I: nat,F: nat > int] :
% 9.20/5.22        ( sums_int
% 9.20/5.22        @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
% 9.20/5.22        @ ( F @ I ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8076_sums__add,axiom,
% 9.20/5.22      ! [F: nat > real,A: real,G: nat > real,B: real] :
% 9.20/5.22        ( ( sums_real @ F @ A )
% 9.20/5.22       => ( ( sums_real @ G @ B )
% 9.20/5.22         => ( sums_real
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.22            @ ( plus_plus_real @ A @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8077_sums__add,axiom,
% 9.20/5.22      ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
% 9.20/5.22        ( ( sums_nat @ F @ A )
% 9.20/5.22       => ( ( sums_nat @ G @ B )
% 9.20/5.22         => ( sums_nat
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.22            @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8078_sums__add,axiom,
% 9.20/5.22      ! [F: nat > int,A: int,G: nat > int,B: int] :
% 9.20/5.22        ( ( sums_int @ F @ A )
% 9.20/5.22       => ( ( sums_int @ G @ B )
% 9.20/5.22         => ( sums_int
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.22            @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8079_summable__comparison__test,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real] :
% 9.20/5.22        ( ? [N8: nat] :
% 9.20/5.22          ! [N3: nat] :
% 9.20/5.22            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.20/5.22           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.22       => ( ( summable_real @ G )
% 9.20/5.22         => ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8080_summable__comparison__test,axiom,
% 9.20/5.22      ! [F: nat > complex,G: nat > real] :
% 9.20/5.22        ( ? [N8: nat] :
% 9.20/5.22          ! [N3: nat] :
% 9.20/5.22            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.20/5.22           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.22       => ( ( summable_real @ G )
% 9.20/5.22         => ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8081_summable__comparison__test_H,axiom,
% 9.20/5.22      ! [G: nat > real,N7: nat,F: nat > real] :
% 9.20/5.22        ( ( summable_real @ G )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.20/5.22             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.22         => ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8082_summable__comparison__test_H,axiom,
% 9.20/5.22      ! [G: nat > real,N7: nat,F: nat > complex] :
% 9.20/5.22        ( ( summable_real @ G )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.20/5.22             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.22         => ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8083_summable__const__iff,axiom,
% 9.20/5.22      ! [C: complex] :
% 9.20/5.22        ( ( summable_complex
% 9.20/5.22          @ ^ [Uu3: nat] : C )
% 9.20/5.22        = ( C = zero_zero_complex ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8084_summable__const__iff,axiom,
% 9.20/5.22      ! [C: real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [Uu3: nat] : C )
% 9.20/5.22        = ( C = zero_zero_real ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8085_summable__add,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( ( summable_real @ G )
% 9.20/5.22         => ( summable_real
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8086_summable__add,axiom,
% 9.20/5.22      ! [F: nat > nat,G: nat > nat] :
% 9.20/5.22        ( ( summable_nat @ F )
% 9.20/5.22       => ( ( summable_nat @ G )
% 9.20/5.22         => ( summable_nat
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8087_summable__add,axiom,
% 9.20/5.22      ! [F: nat > int,G: nat > int] :
% 9.20/5.22        ( ( summable_int @ F )
% 9.20/5.22       => ( ( summable_int @ G )
% 9.20/5.22         => ( summable_int
% 9.20/5.22            @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8088_summable__Suc__iff,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 9.20/5.22        = ( summable_real @ F ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8089_summable__ignore__initial__segment,axiom,
% 9.20/5.22      ! [F: nat > real,K: nat] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8090_powser__insidea,axiom,
% 9.20/5.22      ! [F: nat > real,X3: real,Z2: real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.20/5.22         => ( summable_real
% 9.20/5.22            @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8091_powser__insidea,axiom,
% 9.20/5.22      ! [F: nat > complex,X3: complex,Z2: complex] :
% 9.20/5.22        ( ( summable_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 9.20/5.22       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.20/5.22         => ( summable_real
% 9.20/5.22            @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8092_suminf__le,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( summable_real @ F )
% 9.20/5.22         => ( ( summable_real @ G )
% 9.20/5.22           => ( ord_less_eq_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8093_suminf__le,axiom,
% 9.20/5.22      ! [F: nat > nat,G: nat > nat] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( summable_nat @ F )
% 9.20/5.22         => ( ( summable_nat @ G )
% 9.20/5.22           => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8094_suminf__le,axiom,
% 9.20/5.22      ! [F: nat > int,G: nat > int] :
% 9.20/5.22        ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.22       => ( ( summable_int @ F )
% 9.20/5.22         => ( ( summable_int @ G )
% 9.20/5.22           => ( ord_less_eq_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8095_summable__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > complex] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_complex ) )
% 9.20/5.22         => ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8096_summable__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > real] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_real ) )
% 9.20/5.22         => ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8097_summable__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > nat] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_nat ) )
% 9.20/5.22         => ( summable_nat @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8098_summable__finite,axiom,
% 9.20/5.22      ! [N7: set_nat,F: nat > int] :
% 9.20/5.22        ( ( finite_finite_nat @ N7 )
% 9.20/5.22       => ( ! [N3: nat] :
% 9.20/5.22              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.22             => ( ( F @ N3 )
% 9.20/5.22                = zero_zero_int ) )
% 9.20/5.22         => ( summable_int @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8099_sums__mult__iff,axiom,
% 9.20/5.22      ! [C: complex,F: nat > complex,D: complex] :
% 9.20/5.22        ( ( C != zero_zero_complex )
% 9.20/5.22       => ( ( sums_complex
% 9.20/5.22            @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 9.20/5.22            @ ( times_times_complex @ C @ D ) )
% 9.20/5.22          = ( sums_complex @ F @ D ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8100_sums__mult__iff,axiom,
% 9.20/5.22      ! [C: real,F: nat > real,D: real] :
% 9.20/5.22        ( ( C != zero_zero_real )
% 9.20/5.22       => ( ( sums_real
% 9.20/5.22            @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 9.20/5.22            @ ( times_times_real @ C @ D ) )
% 9.20/5.22          = ( sums_real @ F @ D ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8101_sums__mult2__iff,axiom,
% 9.20/5.22      ! [C: complex,F: nat > complex,D: complex] :
% 9.20/5.22        ( ( C != zero_zero_complex )
% 9.20/5.22       => ( ( sums_complex
% 9.20/5.22            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ C )
% 9.20/5.22            @ ( times_times_complex @ D @ C ) )
% 9.20/5.22          = ( sums_complex @ F @ D ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8102_sums__mult2__iff,axiom,
% 9.20/5.22      ! [C: real,F: nat > real,D: real] :
% 9.20/5.22        ( ( C != zero_zero_real )
% 9.20/5.22       => ( ( sums_real
% 9.20/5.22            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ C )
% 9.20/5.22            @ ( times_times_real @ D @ C ) )
% 9.20/5.22          = ( sums_real @ F @ D ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8103_summable__mult__D,axiom,
% 9.20/5.22      ! [C: complex,F: nat > complex] :
% 9.20/5.22        ( ( summable_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) ) )
% 9.20/5.22       => ( ( C != zero_zero_complex )
% 9.20/5.22         => ( summable_complex @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8104_summable__mult__D,axiom,
% 9.20/5.22      ! [C: real,F: nat > real] :
% 9.20/5.22        ( ( summable_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) ) )
% 9.20/5.22       => ( ( C != zero_zero_real )
% 9.20/5.22         => ( summable_real @ F ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8105_summable__zero__power,axiom,
% 9.20/5.22      summable_real @ ( power_power_real @ zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8106_summable__zero__power,axiom,
% 9.20/5.22      summable_int @ ( power_power_int @ zero_zero_int ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8107_summable__zero__power,axiom,
% 9.20/5.22      summable_complex @ ( power_power_complex @ zero_zero_complex ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8108_pi__not__less__zero,axiom,
% 9.20/5.22      ~ ( ord_less_real @ pi @ zero_zero_real ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8109_pi__gt__zero,axiom,
% 9.20/5.22      ord_less_real @ zero_zero_real @ pi ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8110_suminf__add,axiom,
% 9.20/5.22      ! [F: nat > real,G: nat > real] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( ( summable_real @ G )
% 9.20/5.22         => ( ( plus_plus_real @ ( suminf_real @ F ) @ ( suminf_real @ G ) )
% 9.20/5.22            = ( suminf_real
% 9.20/5.22              @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8111_suminf__add,axiom,
% 9.20/5.22      ! [F: nat > nat,G: nat > nat] :
% 9.20/5.22        ( ( summable_nat @ F )
% 9.20/5.22       => ( ( summable_nat @ G )
% 9.20/5.22         => ( ( plus_plus_nat @ ( suminf_nat @ F ) @ ( suminf_nat @ G ) )
% 9.20/5.22            = ( suminf_nat
% 9.20/5.22              @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8112_suminf__add,axiom,
% 9.20/5.22      ! [F: nat > int,G: nat > int] :
% 9.20/5.22        ( ( summable_int @ F )
% 9.20/5.22       => ( ( summable_int @ G )
% 9.20/5.22         => ( ( plus_plus_int @ ( suminf_int @ F ) @ ( suminf_int @ G ) )
% 9.20/5.22            = ( suminf_int
% 9.20/5.22              @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8113_suminf__eq__zero__iff,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.22         => ( ( ( suminf_real @ F )
% 9.20/5.22              = zero_zero_real )
% 9.20/5.22            = ( ! [N: nat] :
% 9.20/5.22                  ( ( F @ N )
% 9.20/5.22                  = zero_zero_real ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8114_suminf__eq__zero__iff,axiom,
% 9.20/5.22      ! [F: nat > nat] :
% 9.20/5.22        ( ( summable_nat @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.22         => ( ( ( suminf_nat @ F )
% 9.20/5.22              = zero_zero_nat )
% 9.20/5.22            = ( ! [N: nat] :
% 9.20/5.22                  ( ( F @ N )
% 9.20/5.22                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8115_suminf__eq__zero__iff,axiom,
% 9.20/5.22      ! [F: nat > int] :
% 9.20/5.22        ( ( summable_int @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.22         => ( ( ( suminf_int @ F )
% 9.20/5.22              = zero_zero_int )
% 9.20/5.22            = ( ! [N: nat] :
% 9.20/5.22                  ( ( F @ N )
% 9.20/5.22                  = zero_zero_int ) ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8116_suminf__nonneg,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_eq_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8117_suminf__nonneg,axiom,
% 9.20/5.22      ! [F: nat > nat] :
% 9.20/5.22        ( ( summable_nat @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_eq_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8118_suminf__nonneg,axiom,
% 9.20/5.22      ! [F: nat > int] :
% 9.20/5.22        ( ( summable_int @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_eq_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8119_suminf__pos,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( ( summable_real @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8120_suminf__pos,axiom,
% 9.20/5.22      ! [F: nat > nat] :
% 9.20/5.22        ( ( summable_nat @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8121_suminf__pos,axiom,
% 9.20/5.22      ! [F: nat > int] :
% 9.20/5.22        ( ( summable_int @ F )
% 9.20/5.22       => ( ! [N3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.22         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8122_sums__mult__D,axiom,
% 9.20/5.22      ! [C: complex,F: nat > complex,A: complex] :
% 9.20/5.22        ( ( sums_complex
% 9.20/5.22          @ ^ [N: nat] : ( times_times_complex @ C @ ( F @ N ) )
% 9.20/5.22          @ A )
% 9.20/5.22       => ( ( C != zero_zero_complex )
% 9.20/5.22         => ( sums_complex @ F @ ( divide1717551699836669952omplex @ A @ C ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8123_sums__mult__D,axiom,
% 9.20/5.22      ! [C: real,F: nat > real,A: real] :
% 9.20/5.22        ( ( sums_real
% 9.20/5.22          @ ^ [N: nat] : ( times_times_real @ C @ ( F @ N ) )
% 9.20/5.22          @ A )
% 9.20/5.22       => ( ( C != zero_zero_real )
% 9.20/5.22         => ( sums_real @ F @ ( divide_divide_real @ A @ C ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8124_sums__Suc__imp,axiom,
% 9.20/5.22      ! [F: nat > complex,S: complex] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_complex )
% 9.20/5.22       => ( ( sums_complex
% 9.20/5.22            @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22            @ S )
% 9.20/5.22         => ( sums_complex @ F @ S ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8125_sums__Suc__imp,axiom,
% 9.20/5.22      ! [F: nat > real,S: real] :
% 9.20/5.22        ( ( ( F @ zero_zero_nat )
% 9.20/5.22          = zero_zero_real )
% 9.20/5.22       => ( ( sums_real
% 9.20/5.22            @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22            @ S )
% 9.20/5.22         => ( sums_real @ F @ S ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8126_sums__Suc,axiom,
% 9.20/5.22      ! [F: nat > real,L: real] :
% 9.20/5.22        ( ( sums_real
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22          @ L )
% 9.20/5.22       => ( sums_real @ F @ ( plus_plus_real @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8127_sums__Suc,axiom,
% 9.20/5.22      ! [F: nat > nat,L: nat] :
% 9.20/5.22        ( ( sums_nat
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22          @ L )
% 9.20/5.22       => ( sums_nat @ F @ ( plus_plus_nat @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8128_sums__Suc,axiom,
% 9.20/5.22      ! [F: nat > int,L: int] :
% 9.20/5.22        ( ( sums_int
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22          @ L )
% 9.20/5.22       => ( sums_int @ F @ ( plus_plus_int @ L @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8129_sums__Suc__iff,axiom,
% 9.20/5.22      ! [F: nat > real,S: real] :
% 9.20/5.22        ( ( sums_real
% 9.20/5.22          @ ^ [N: nat] : ( F @ ( suc @ N ) )
% 9.20/5.22          @ S )
% 9.20/5.22        = ( sums_real @ F @ ( plus_plus_real @ S @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8130_sums__zero__iff__shift,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > complex,S: complex] :
% 9.20/5.22        ( ! [I3: nat] :
% 9.20/5.22            ( ( ord_less_nat @ I3 @ N2 )
% 9.20/5.22           => ( ( F @ I3 )
% 9.20/5.22              = zero_zero_complex ) )
% 9.20/5.22       => ( ( sums_complex
% 9.20/5.22            @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 9.20/5.22            @ S )
% 9.20/5.22          = ( sums_complex @ F @ S ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8131_sums__zero__iff__shift,axiom,
% 9.20/5.22      ! [N2: nat,F: nat > real,S: real] :
% 9.20/5.22        ( ! [I3: nat] :
% 9.20/5.22            ( ( ord_less_nat @ I3 @ N2 )
% 9.20/5.22           => ( ( F @ I3 )
% 9.20/5.22              = zero_zero_real ) )
% 9.20/5.22       => ( ( sums_real
% 9.20/5.22            @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 9.20/5.22            @ S )
% 9.20/5.22          = ( sums_real @ F @ S ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8132_summable__0__powser,axiom,
% 9.20/5.22      ! [F: nat > complex] :
% 9.20/5.22        ( summable_complex
% 9.20/5.22        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8133_summable__0__powser,axiom,
% 9.20/5.22      ! [F: nat > real] :
% 9.20/5.22        ( summable_real
% 9.20/5.22        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 9.20/5.22  
% 9.20/5.22  thf(fact_8134_summable__zero__power_H,axiom,
% 9.20/5.23      ! [F: nat > complex] :
% 9.20/5.23        ( summable_complex
% 9.20/5.23        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8135_summable__zero__power_H,axiom,
% 9.20/5.23      ! [F: nat > real] :
% 9.20/5.23        ( summable_real
% 9.20/5.23        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ zero_zero_real @ N ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8136_summable__zero__power_H,axiom,
% 9.20/5.23      ! [F: nat > int] :
% 9.20/5.23        ( summable_int
% 9.20/5.23        @ ^ [N: nat] : ( times_times_int @ ( F @ N ) @ ( power_power_int @ zero_zero_int @ N ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8137_summable__powser__split__head,axiom,
% 9.20/5.23      ! [F: nat > complex,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23        = ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8138_summable__powser__split__head,axiom,
% 9.20/5.23      ! [F: nat > real,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23        = ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8139_powser__split__head_I3_J,axiom,
% 9.20/5.23      ! [F: nat > complex,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23       => ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8140_powser__split__head_I3_J,axiom,
% 9.20/5.23      ! [F: nat > real,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23       => ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8141_summable__powser__ignore__initial__segment,axiom,
% 9.20/5.23      ! [F: nat > complex,M: nat,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23        = ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8142_summable__powser__ignore__initial__segment,axiom,
% 9.20/5.23      ! [F: nat > real,M: nat,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23        = ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8143_sums__finite,axiom,
% 9.20/5.23      ! [N7: set_nat,F: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ N7 )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.23             => ( ( F @ N3 )
% 9.20/5.23                = zero_zero_complex ) )
% 9.20/5.23         => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8144_sums__finite,axiom,
% 9.20/5.23      ! [N7: set_nat,F: nat > int] :
% 9.20/5.23        ( ( finite_finite_nat @ N7 )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.23             => ( ( F @ N3 )
% 9.20/5.23                = zero_zero_int ) )
% 9.20/5.23         => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8145_sums__finite,axiom,
% 9.20/5.23      ! [N7: set_nat,F: nat > nat] :
% 9.20/5.23        ( ( finite_finite_nat @ N7 )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.23             => ( ( F @ N3 )
% 9.20/5.23                = zero_zero_nat ) )
% 9.20/5.23         => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8146_sums__finite,axiom,
% 9.20/5.23      ! [N7: set_nat,F: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ N7 )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ~ ( member_nat @ N3 @ N7 )
% 9.20/5.23             => ( ( F @ N3 )
% 9.20/5.23                = zero_zero_real ) )
% 9.20/5.23         => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8147_sums__If__finite,axiom,
% 9.20/5.23      ! [P: nat > $o,F: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.23       => ( sums_complex
% 9.20/5.23          @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.20/5.23          @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8148_sums__If__finite,axiom,
% 9.20/5.23      ! [P: nat > $o,F: nat > int] :
% 9.20/5.23        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.23       => ( sums_int
% 9.20/5.23          @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.20/5.23          @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8149_sums__If__finite,axiom,
% 9.20/5.23      ! [P: nat > $o,F: nat > nat] :
% 9.20/5.23        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.23       => ( sums_nat
% 9.20/5.23          @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.20/5.23          @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8150_sums__If__finite,axiom,
% 9.20/5.23      ! [P: nat > $o,F: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ ( collect_nat @ P ) )
% 9.20/5.23       => ( sums_real
% 9.20/5.23          @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.20/5.23          @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8151_sums__If__finite__set,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( sums_complex
% 9.20/5.23          @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
% 9.20/5.23          @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8152_sums__If__finite__set,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > int] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( sums_int
% 9.20/5.23          @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
% 9.20/5.23          @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8153_sums__If__finite__set,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > nat] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( sums_nat
% 9.20/5.23          @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
% 9.20/5.23          @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8154_sums__If__finite__set,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( sums_real
% 9.20/5.23          @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
% 9.20/5.23          @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8155_summable__norm__comparison__test,axiom,
% 9.20/5.23      ! [F: nat > complex,G: nat > real] :
% 9.20/5.23        ( ? [N8: nat] :
% 9.20/5.23          ! [N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.20/5.23           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.23       => ( ( summable_real @ G )
% 9.20/5.23         => ( summable_real
% 9.20/5.23            @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8156_summable__rabs__comparison__test,axiom,
% 9.20/5.23      ! [F: nat > real,G: nat > real] :
% 9.20/5.23        ( ? [N8: nat] :
% 9.20/5.23          ! [N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.20/5.23           => ( ord_less_eq_real @ ( abs_abs_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
% 9.20/5.23       => ( ( summable_real @ G )
% 9.20/5.23         => ( summable_real
% 9.20/5.23            @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8157_suminf__pos__iff,axiom,
% 9.20/5.23      ! [F: nat > real] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
% 9.20/5.23            = ( ? [I2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8158_suminf__pos__iff,axiom,
% 9.20/5.23      ! [F: nat > nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
% 9.20/5.23            = ( ? [I2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8159_suminf__pos__iff,axiom,
% 9.20/5.23      ! [F: nat > int] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
% 9.20/5.23            = ( ? [I2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8160_suminf__pos2,axiom,
% 9.20/5.23      ! [F: nat > real,I: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.20/5.23           => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8161_suminf__pos2,axiom,
% 9.20/5.23      ! [F: nat > nat,I: nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.20/5.23           => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8162_suminf__pos2,axiom,
% 9.20/5.23      ! [F: nat > int,I: nat] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.23         => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.20/5.23           => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8163_suminf__le__const,axiom,
% 9.20/5.23      ! [F: nat > int,X3: int] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8164_suminf__le__const,axiom,
% 9.20/5.23      ! [F: nat > nat,X3: nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8165_suminf__le__const,axiom,
% 9.20/5.23      ! [F: nat > real,X3: real] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8166_powser__sums__if,axiom,
% 9.20/5.23      ! [M: nat,Z2: complex] :
% 9.20/5.23        ( sums_complex
% 9.20/5.23        @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N ) )
% 9.20/5.23        @ ( power_power_complex @ Z2 @ M ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8167_powser__sums__if,axiom,
% 9.20/5.23      ! [M: nat,Z2: real] :
% 9.20/5.23        ( sums_real
% 9.20/5.23        @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N ) )
% 9.20/5.23        @ ( power_power_real @ Z2 @ M ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8168_powser__sums__if,axiom,
% 9.20/5.23      ! [M: nat,Z2: int] :
% 9.20/5.23        ( sums_int
% 9.20/5.23        @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N ) )
% 9.20/5.23        @ ( power_power_int @ Z2 @ M ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8169_powser__sums__zero,axiom,
% 9.20/5.23      ! [A: nat > complex] :
% 9.20/5.23        ( sums_complex
% 9.20/5.23        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
% 9.20/5.23        @ ( A @ zero_zero_nat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8170_powser__sums__zero,axiom,
% 9.20/5.23      ! [A: nat > real] :
% 9.20/5.23        ( sums_real
% 9.20/5.23        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
% 9.20/5.23        @ ( A @ zero_zero_nat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8171_log__base__change,axiom,
% 9.20/5.23      ! [A: real,B: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( log @ B @ X3 )
% 9.20/5.23            = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8172_log__of__power__eq,axiom,
% 9.20/5.23      ! [M: nat,B: real,N2: nat] :
% 9.20/5.23        ( ( ( semiri5074537144036343181t_real @ M )
% 9.20/5.23          = ( power_power_real @ B @ N2 ) )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ( semiri5074537144036343181t_real @ N2 )
% 9.20/5.23            = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8173_less__log__of__power,axiom,
% 9.20/5.23      ! [B: real,N2: nat,M: real] :
% 9.20/5.23        ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8174_powser__inside,axiom,
% 9.20/5.23      ! [F: nat > real,X3: real,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 9.20/5.23       => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
% 9.20/5.23         => ( summable_real
% 9.20/5.23            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8175_powser__inside,axiom,
% 9.20/5.23      ! [F: nat > complex,X3: complex,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 9.20/5.23       => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
% 9.20/5.23         => ( summable_complex
% 9.20/5.23            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8176_sums__iff__shift,axiom,
% 9.20/5.23      ! [F: nat > real,N2: nat,S: real] :
% 9.20/5.23        ( ( sums_real
% 9.20/5.23          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 9.20/5.23          @ S )
% 9.20/5.23        = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8177_summableI__nonneg__bounded,axiom,
% 9.20/5.23      ! [F: nat > int,X3: int] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( summable_int @ F ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8178_summableI__nonneg__bounded,axiom,
% 9.20/5.23      ! [F: nat > nat,X3: nat] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( summable_nat @ F ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8179_summableI__nonneg__bounded,axiom,
% 9.20/5.23      ! [F: nat > real,X3: real] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
% 9.20/5.23       => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
% 9.20/5.23         => ( summable_real @ F ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8180_sums__split__initial__segment,axiom,
% 9.20/5.23      ! [F: nat > real,S: real,N2: nat] :
% 9.20/5.23        ( ( sums_real @ F @ S )
% 9.20/5.23       => ( sums_real
% 9.20/5.23          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 9.20/5.23          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8181_sums__iff__shift_H,axiom,
% 9.20/5.23      ! [F: nat > real,N2: nat,S: real] :
% 9.20/5.23        ( ( sums_real
% 9.20/5.23          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
% 9.20/5.23          @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
% 9.20/5.23        = ( sums_real @ F @ S ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8182_complete__algebra__summable__geometric,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ one_one_real )
% 9.20/5.23       => ( summable_real @ ( power_power_real @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8183_complete__algebra__summable__geometric,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ one_one_real )
% 9.20/5.23       => ( summable_complex @ ( power_power_complex @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8184_summable__geometric,axiom,
% 9.20/5.23      ! [C: real] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.20/5.23       => ( summable_real @ ( power_power_real @ C ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8185_summable__geometric,axiom,
% 9.20/5.23      ! [C: complex] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.20/5.23       => ( summable_complex @ ( power_power_complex @ C ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8186_sums__If__finite__set_H,axiom,
% 9.20/5.23      ! [G: nat > real,S3: real,A2: set_nat,S6: real,F: nat > real] :
% 9.20/5.23        ( ( sums_real @ G @ S3 )
% 9.20/5.23       => ( ( finite_finite_nat @ A2 )
% 9.20/5.23         => ( ( S6
% 9.20/5.23              = ( plus_plus_real @ S3
% 9.20/5.23                @ ( groups6591440286371151544t_real
% 9.20/5.23                  @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.23                  @ A2 ) ) )
% 9.20/5.23           => ( sums_real
% 9.20/5.23              @ ^ [N: nat] : ( if_real @ ( member_nat @ N @ A2 ) @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.23              @ S6 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8187_suminf__split__head,axiom,
% 9.20/5.23      ! [F: nat > real] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ( suminf_real
% 9.20/5.23            @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
% 9.20/5.23          = ( minus_minus_real @ ( suminf_real @ F ) @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8188_pi__less__4,axiom,
% 9.20/5.23      ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8189_sum__le__suminf,axiom,
% 9.20/5.23      ! [F: nat > int,I5: set_nat] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ( finite_finite_nat @ I5 )
% 9.20/5.23         => ( ! [N3: nat] :
% 9.20/5.23                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.20/5.23               => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
% 9.20/5.23           => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8190_sum__le__suminf,axiom,
% 9.20/5.23      ! [F: nat > nat,I5: set_nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ( finite_finite_nat @ I5 )
% 9.20/5.23         => ( ! [N3: nat] :
% 9.20/5.23                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.20/5.23               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
% 9.20/5.23           => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8191_sum__le__suminf,axiom,
% 9.20/5.23      ! [F: nat > real,I5: set_nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ( finite_finite_nat @ I5 )
% 9.20/5.23         => ( ! [N3: nat] :
% 9.20/5.23                ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
% 9.20/5.23               => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
% 9.20/5.23           => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8192_log__mult,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.23             => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
% 9.20/5.23                = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8193_log__divide,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.23             => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
% 9.20/5.23                = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8194_le__log__of__power,axiom,
% 9.20/5.23      ! [B: real,N2: nat,M: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8195_log__base__pow,axiom,
% 9.20/5.23      ! [A: real,N2: nat,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( log @ ( power_power_real @ A @ N2 ) @ X3 )
% 9.20/5.23          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8196_log__nat__power,axiom,
% 9.20/5.23      ! [X3: real,B: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( log @ B @ ( power_power_real @ X3 @ N2 ) )
% 9.20/5.23          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8197_suminf__split__initial__segment,axiom,
% 9.20/5.23      ! [F: nat > real,K: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ( suminf_real @ F )
% 9.20/5.23          = ( plus_plus_real
% 9.20/5.23            @ ( suminf_real
% 9.20/5.23              @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 9.20/5.23            @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8198_suminf__minus__initial__segment,axiom,
% 9.20/5.23      ! [F: nat > real,K: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ( suminf_real
% 9.20/5.23            @ ^ [N: nat] : ( F @ ( plus_plus_nat @ N @ K ) ) )
% 9.20/5.23          = ( minus_minus_real @ ( suminf_real @ F ) @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8199_pi__half__less__two,axiom,
% 9.20/5.23      ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8200_sum__less__suminf,axiom,
% 9.20/5.23      ! [F: nat > int,N2: nat] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_int @ zero_zero_int @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8201_sum__less__suminf,axiom,
% 9.20/5.23      ! [F: nat > nat,N2: nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_nat @ zero_zero_nat @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8202_sum__less__suminf,axiom,
% 9.20/5.23      ! [F: nat > real,N2: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_real @ zero_zero_real @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8203_powser__split__head_I1_J,axiom,
% 9.20/5.23      ! [F: nat > complex,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23       => ( ( suminf_complex
% 9.20/5.23            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23          = ( plus_plus_complex @ ( F @ zero_zero_nat )
% 9.20/5.23            @ ( times_times_complex
% 9.20/5.23              @ ( suminf_complex
% 9.20/5.23                @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23              @ Z2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8204_powser__split__head_I1_J,axiom,
% 9.20/5.23      ! [F: nat > real,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23       => ( ( suminf_real
% 9.20/5.23            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23          = ( plus_plus_real @ ( F @ zero_zero_nat )
% 9.20/5.23            @ ( times_times_real
% 9.20/5.23              @ ( suminf_real
% 9.20/5.23                @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23              @ Z2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8205_powser__split__head_I2_J,axiom,
% 9.20/5.23      ! [F: nat > complex,Z2: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23       => ( ( times_times_complex
% 9.20/5.23            @ ( suminf_complex
% 9.20/5.23              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23            @ Z2 )
% 9.20/5.23          = ( minus_minus_complex
% 9.20/5.23            @ ( suminf_complex
% 9.20/5.23              @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
% 9.20/5.23            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8206_powser__split__head_I2_J,axiom,
% 9.20/5.23      ! [F: nat > real,Z2: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23       => ( ( times_times_real
% 9.20/5.23            @ ( suminf_real
% 9.20/5.23              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23            @ Z2 )
% 9.20/5.23          = ( minus_minus_real
% 9.20/5.23            @ ( suminf_real
% 9.20/5.23              @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
% 9.20/5.23            @ ( F @ zero_zero_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8207_log__of__power__less,axiom,
% 9.20/5.23      ! [M: nat,B: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23           => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8208_log__eq__div__ln__mult__log,axiom,
% 9.20/5.23      ! [A: real,B: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.23           => ( ( B != one_one_real )
% 9.20/5.23             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23               => ( ( log @ A @ X3 )
% 9.20/5.23                  = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8209_summable__partial__sum__bound,axiom,
% 9.20/5.23      ! [F: nat > complex,E2: real] :
% 9.20/5.23        ( ( summable_complex @ F )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.23         => ~ ! [N9: nat] :
% 9.20/5.23                ~ ! [M6: nat] :
% 9.20/5.23                    ( ( ord_less_eq_nat @ N9 @ M6 )
% 9.20/5.23                   => ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8210_summable__partial__sum__bound,axiom,
% 9.20/5.23      ! [F: nat > real,E2: real] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.23         => ~ ! [N9: nat] :
% 9.20/5.23                ~ ! [M6: nat] :
% 9.20/5.23                    ( ( ord_less_eq_nat @ N9 @ M6 )
% 9.20/5.23                   => ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8211_suminf__exist__split,axiom,
% 9.20/5.23      ! [R2: real,F: nat > real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.20/5.23       => ( ( summable_real @ F )
% 9.20/5.23         => ? [N9: nat] :
% 9.20/5.23            ! [N6: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N9 @ N6 )
% 9.20/5.23             => ( ord_less_real
% 9.20/5.23                @ ( real_V7735802525324610683m_real
% 9.20/5.23                  @ ( suminf_real
% 9.20/5.23                    @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
% 9.20/5.23                @ R2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8212_suminf__exist__split,axiom,
% 9.20/5.23      ! [R2: real,F: nat > complex] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ R2 )
% 9.20/5.23       => ( ( summable_complex @ F )
% 9.20/5.23         => ? [N9: nat] :
% 9.20/5.23            ! [N6: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N9 @ N6 )
% 9.20/5.23             => ( ord_less_real
% 9.20/5.23                @ ( real_V1022390504157884413omplex
% 9.20/5.23                  @ ( suminf_complex
% 9.20/5.23                    @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
% 9.20/5.23                @ R2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8213_summable__power__series,axiom,
% 9.20/5.23      ! [F: nat > real,Z2: real] :
% 9.20/5.23        ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
% 9.20/5.23       => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
% 9.20/5.23         => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
% 9.20/5.23           => ( ( ord_less_real @ Z2 @ one_one_real )
% 9.20/5.23             => ( summable_real
% 9.20/5.23                @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z2 @ I2 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8214_Abel__lemma,axiom,
% 9.20/5.23      ! [R2: real,R0: real,A: nat > complex,M7: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ R2 )
% 9.20/5.23       => ( ( ord_less_real @ R2 @ R0 )
% 9.20/5.23         => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
% 9.20/5.23           => ( summable_real
% 9.20/5.23              @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R2 @ N ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8215_pi__half__gt__zero,axiom,
% 9.20/5.23      ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8216_m2pi__less__pi,axiom,
% 9.20/5.23      ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8217_summable__ratio__test,axiom,
% 9.20/5.23      ! [C: real,N7: nat,F: nat > real] :
% 9.20/5.23        ( ( ord_less_real @ C @ one_one_real )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.20/5.23             => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) ) ) )
% 9.20/5.23         => ( summable_real @ F ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8218_summable__ratio__test,axiom,
% 9.20/5.23      ! [C: real,N7: nat,F: nat > complex] :
% 9.20/5.23        ( ( ord_less_real @ C @ one_one_real )
% 9.20/5.23       => ( ! [N3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N7 @ N3 )
% 9.20/5.23             => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ ( suc @ N3 ) ) ) @ ( times_times_real @ C @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) ) ) )
% 9.20/5.23         => ( summable_complex @ F ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8219_sum__less__suminf2,axiom,
% 9.20/5.23      ! [F: nat > int,N2: nat,I: nat] :
% 9.20/5.23        ( ( summable_int @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_eq_int @ zero_zero_int @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ N2 @ I )
% 9.20/5.23           => ( ( ord_less_int @ zero_zero_int @ ( F @ I ) )
% 9.20/5.23             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8220_sum__less__suminf2,axiom,
% 9.20/5.23      ! [F: nat > nat,N2: nat,I: nat] :
% 9.20/5.23        ( ( summable_nat @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ N2 @ I )
% 9.20/5.23           => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
% 9.20/5.23             => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8221_sum__less__suminf2,axiom,
% 9.20/5.23      ! [F: nat > real,N2: nat,I: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [M3: nat] :
% 9.20/5.23              ( ( ord_less_eq_nat @ N2 @ M3 )
% 9.20/5.23             => ( ord_less_eq_real @ zero_zero_real @ ( F @ M3 ) ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ N2 @ I )
% 9.20/5.23           => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
% 9.20/5.23             => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8222_arctan__ubound,axiom,
% 9.20/5.23      ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8223_geometric__sums,axiom,
% 9.20/5.23      ! [C: real] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
% 9.20/5.23       => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8224_geometric__sums,axiom,
% 9.20/5.23      ! [C: complex] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
% 9.20/5.23       => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8225_power__half__series,axiom,
% 9.20/5.23      ( sums_real
% 9.20/5.23      @ ^ [N: nat] : ( power_power_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( suc @ N ) )
% 9.20/5.23      @ one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8226_log__of__power__le,axiom,
% 9.20/5.23      ! [M: nat,B: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23           => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8227_minus__pi__half__less__zero,axiom,
% 9.20/5.23      ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8228_arctan__lbound,axiom,
% 9.20/5.23      ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8229_arctan__bounded,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.20/5.23        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8230_less__log2__of__power,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 9.20/5.23       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8231_le__log2__of__power,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
% 9.20/5.23       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8232_log2__of__power__less,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23         => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8233_log2__of__power__le,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23         => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8234_sum__pos__lt__pair,axiom,
% 9.20/5.23      ! [F: nat > real,K: nat] :
% 9.20/5.23        ( ( summable_real @ F )
% 9.20/5.23       => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
% 9.20/5.23         => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8235_ceiling__log__nat__eq__if,axiom,
% 9.20/5.23      ! [B: nat,N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.20/5.23       => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.23           => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.20/5.23              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8236_ceiling__log2__div2,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.23       => ( ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.20/5.23          = ( plus_plus_int @ ( archim7802044766580827645g_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( divide_divide_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) @ one_one_int ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8237_sin__cos__npi,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( sin_real @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8238_diffs__equiv,axiom,
% 9.20/5.23      ! [C: nat > complex,X3: complex] :
% 9.20/5.23        ( ( summable_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) )
% 9.20/5.23       => ( sums_complex
% 9.20/5.23          @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.20/5.23          @ ( suminf_complex
% 9.20/5.23            @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8239_diffs__equiv,axiom,
% 9.20/5.23      ! [C: nat > real,X3: real] :
% 9.20/5.23        ( ( summable_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) )
% 9.20/5.23       => ( sums_real
% 9.20/5.23          @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
% 9.20/5.23          @ ( suminf_real
% 9.20/5.23            @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8240_cos__pi__eq__zero,axiom,
% 9.20/5.23      ! [M: nat] :
% 9.20/5.23        ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8241_ceiling__log__eq__powr__iff,axiom,
% 9.20/5.23      ! [X3: real,B: real,K: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X3 ) )
% 9.20/5.23              = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
% 9.20/5.23            = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
% 9.20/5.23              & ( ord_less_eq_real @ X3 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8242_floor__log__nat__eq__powr__iff,axiom,
% 9.20/5.23      ! [B: nat,K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23         => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.20/5.23              = ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.23            = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.20/5.23              & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8243_sin__zero,axiom,
% 9.20/5.23      ( ( sin_complex @ zero_zero_complex )
% 9.20/5.23      = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8244_sin__zero,axiom,
% 9.20/5.23      ( ( sin_real @ zero_zero_real )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8245_powr__eq__0__iff,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( ( powr_real @ W2 @ Z2 )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23        = ( W2 = zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8246_powr__0,axiom,
% 9.20/5.23      ! [Z2: real] :
% 9.20/5.23        ( ( powr_real @ zero_zero_real @ Z2 )
% 9.20/5.23        = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8247_cos__zero,axiom,
% 9.20/5.23      ( ( cos_complex @ zero_zero_complex )
% 9.20/5.23      = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8248_cos__zero,axiom,
% 9.20/5.23      ( ( cos_real @ zero_zero_real )
% 9.20/5.23      = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8249_powr__zero__eq__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( X3 = zero_zero_real )
% 9.20/5.23         => ( ( powr_real @ X3 @ zero_zero_real )
% 9.20/5.23            = zero_zero_real ) )
% 9.20/5.23        & ( ( X3 != zero_zero_real )
% 9.20/5.23         => ( ( powr_real @ X3 @ zero_zero_real )
% 9.20/5.23            = one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8250_floor__zero,axiom,
% 9.20/5.23      ( ( archim6058952711729229775r_real @ zero_zero_real )
% 9.20/5.23      = zero_zero_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8251_floor__zero,axiom,
% 9.20/5.23      ( ( archim3151403230148437115or_rat @ zero_zero_rat )
% 9.20/5.23      = zero_zero_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8252_powr__gt__zero,axiom,
% 9.20/5.23      ! [X3: real,A: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
% 9.20/5.23        = ( X3 != zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8253_powr__less__cancel__iff,axiom,
% 9.20/5.23      ! [X3: real,A: real,B: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 9.20/5.23          = ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8254_powr__eq__one__iff,axiom,
% 9.20/5.23      ! [A: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ A )
% 9.20/5.23       => ( ( ( powr_real @ A @ X3 )
% 9.20/5.23            = one_one_real )
% 9.20/5.23          = ( X3 = zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8255_powr__le__cancel__iff,axiom,
% 9.20/5.23      ! [X3: real,A: real,B: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 9.20/5.23          = ( ord_less_eq_real @ A @ B ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8256_zero__le__floor,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8257_zero__le__floor,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ zero_zero_rat @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8258_sin__cos__squared__add3,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8259_sin__cos__squared__add3,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8260_floor__less__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
% 9.20/5.23        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8261_floor__less__zero,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
% 9.20/5.23        = ( ord_less_rat @ X3 @ zero_zero_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8262_numeral__le__floor,axiom,
% 9.20/5.23      ! [V: num,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8263_numeral__le__floor,axiom,
% 9.20/5.23      ! [V: num,X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8264_zero__less__floor,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8265_zero__less__floor,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8266_floor__le__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
% 9.20/5.23        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8267_floor__le__zero,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
% 9.20/5.23        = ( ord_less_rat @ X3 @ one_one_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8268_floor__less__numeral,axiom,
% 9.20/5.23      ! [X3: real,V: num] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8269_floor__less__numeral,axiom,
% 9.20/5.23      ! [X3: rat,V: num] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8270_one__le__floor,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8271_one__le__floor,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8272_floor__less__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 9.20/5.23        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8273_floor__less__one,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 9.20/5.23        = ( ord_less_rat @ X3 @ one_one_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8274_powr__log__cancel,axiom,
% 9.20/5.23      ! [A: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( powr_real @ A @ ( log @ A @ X3 ) )
% 9.20/5.23              = X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8275_log__powr__cancel,axiom,
% 9.20/5.23      ! [A: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
% 9.20/5.23            = Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8276_numeral__less__floor,axiom,
% 9.20/5.23      ! [V: num,X3: real] :
% 9.20/5.23        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8277_numeral__less__floor,axiom,
% 9.20/5.23      ! [V: num,X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8278_floor__le__numeral,axiom,
% 9.20/5.23      ! [X3: real,V: num] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8279_floor__le__numeral,axiom,
% 9.20/5.23      ! [X3: rat,V: num] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8280_one__less__floor,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8281_one__less__floor,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8282_floor__le__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8283_floor__le__one,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8284_neg__numeral__le__floor,axiom,
% 9.20/5.23      ! [V: num,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8285_neg__numeral__le__floor,axiom,
% 9.20/5.23      ! [V: num,X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8286_floor__less__neg__numeral,axiom,
% 9.20/5.23      ! [X3: real,V: num] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8287_floor__less__neg__numeral,axiom,
% 9.20/5.23      ! [X3: rat,V: num] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8288_sin__cos__squared__add2,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8289_sin__cos__squared__add2,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8290_sin__cos__squared__add,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8291_sin__cos__squared__add,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8292_neg__numeral__less__floor,axiom,
% 9.20/5.23      ! [V: num,X3: real] :
% 9.20/5.23        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8293_neg__numeral__less__floor,axiom,
% 9.20/5.23      ! [V: num,X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8294_floor__le__neg__numeral,axiom,
% 9.20/5.23      ! [X3: real,V: num] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8295_floor__le__neg__numeral,axiom,
% 9.20/5.23      ! [X3: rat,V: num] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8296_cos__one__sin__zero,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23          = one_one_complex )
% 9.20/5.23       => ( ( sin_complex @ X3 )
% 9.20/5.23          = zero_zero_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8297_cos__one__sin__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23          = one_one_real )
% 9.20/5.23       => ( ( sin_real @ X3 )
% 9.20/5.23          = zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8298_sin__add,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8299_sin__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8300_cos__diff,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8301_cos__diff,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8302_cos__add,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.20/5.23        = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8303_cos__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23        = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8304_sin__zero__norm__cos__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( sin_real @ X3 )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23       => ( ( real_V7735802525324610683m_real @ ( cos_real @ X3 ) )
% 9.20/5.23          = one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8305_sin__zero__norm__cos__one,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( ( sin_complex @ X3 )
% 9.20/5.23          = zero_zero_complex )
% 9.20/5.23       => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X3 ) )
% 9.20/5.23          = one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8306_sincos__principal__value,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23      ? [Y5: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y5 )
% 9.20/5.23        & ( ord_less_eq_real @ Y5 @ pi )
% 9.20/5.23        & ( ( sin_real @ Y5 )
% 9.20/5.23          = ( sin_real @ X3 ) )
% 9.20/5.23        & ( ( cos_real @ Y5 )
% 9.20/5.23          = ( cos_real @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8307_floor__mono,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.20/5.23       => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8308_floor__mono,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] :
% 9.20/5.23        ( ( ord_less_eq_rat @ X3 @ Y3 )
% 9.20/5.23       => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8309_of__int__floor__le,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8310_of__int__floor__le,axiom,
% 9.20/5.23      ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8311_floor__less__cancel,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) )
% 9.20/5.23       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8312_floor__less__cancel,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
% 9.20/5.23       => ( ord_less_rat @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8313_powr__non__neg,axiom,
% 9.20/5.23      ! [A: real,X3: real] :
% 9.20/5.23        ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8314_powr__less__mono2__neg,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ A @ zero_zero_real )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.23           => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8315_powr__less__cancel,axiom,
% 9.20/5.23      ! [X3: real,A: real,B: real] :
% 9.20/5.23        ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23         => ( ord_less_real @ A @ B ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8316_powr__less__mono,axiom,
% 9.20/5.23      ! [A: real,B: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ A @ B )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8317_le__floor__iff,axiom,
% 9.20/5.23      ! [Z2: int,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8318_le__floor__iff,axiom,
% 9.20/5.23      ! [Z2: int,X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8319_floor__less__iff,axiom,
% 9.20/5.23      ! [X3: real,Z2: int] :
% 9.20/5.23        ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8320_floor__less__iff,axiom,
% 9.20/5.23      ! [X3: rat,Z2: int] :
% 9.20/5.23        ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8321_powr__mono2_H,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ A @ zero_zero_real )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.20/5.23           => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8322_powr__less__mono2,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.23           => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8323_le__floor__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8324_le__floor__add,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8325_gr__one__powr,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.23         => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8326_powr__inj,axiom,
% 9.20/5.23      ! [A: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( A != one_one_real )
% 9.20/5.23         => ( ( ( powr_real @ A @ X3 )
% 9.20/5.23              = ( powr_real @ A @ Y3 ) )
% 9.20/5.23            = ( X3 = Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8327_int__add__floor,axiom,
% 9.20/5.23      ! [Z2: int,X3: real] :
% 9.20/5.23        ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8328_int__add__floor,axiom,
% 9.20/5.23      ! [Z2: int,X3: rat] :
% 9.20/5.23        ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8329_floor__add__int,axiom,
% 9.20/5.23      ! [X3: real,Z2: int] :
% 9.20/5.23        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 9.20/5.23        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8330_floor__add__int,axiom,
% 9.20/5.23      ! [X3: rat,Z2: int] :
% 9.20/5.23        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 9.20/5.23        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8331_sin__gt__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ pi )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8332_sin__times__sin,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 9.20/5.23        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8333_sin__times__sin,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8334_sin__times__cos,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 9.20/5.23        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8335_sin__times__cos,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8336_cos__times__sin,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 9.20/5.23        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8337_cos__times__sin,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8338_sin__plus__sin,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 9.20/5.23        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8339_sin__plus__sin,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 9.20/5.23        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8340_sin__diff__sin,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
% 9.20/5.23        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8341_sin__diff__sin,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
% 9.20/5.23        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8342_cos__diff__cos,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 9.20/5.23        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8343_cos__diff__cos,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 9.20/5.23        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8344_powr__add,axiom,
% 9.20/5.23      ! [X3: real,A: real,B: real] :
% 9.20/5.23        ( ( powr_real @ X3 @ ( plus_plus_real @ A @ B ) )
% 9.20/5.23        = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8345_one__add__floor,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
% 9.20/5.23        = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8346_one__add__floor,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
% 9.20/5.23        = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8347_floor__log__eq__powr__iff,axiom,
% 9.20/5.23      ! [X3: real,B: real,K: int] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23         => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X3 ) )
% 9.20/5.23              = K )
% 9.20/5.23            = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X3 )
% 9.20/5.23              & ( ord_less_real @ X3 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8348_powr__realpow,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.23          = ( power_power_real @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8349_less__log__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ Y3 @ ( log @ B @ X3 ) )
% 9.20/5.23            = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8350_log__less__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ ( log @ B @ X3 ) @ Y3 )
% 9.20/5.23            = ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8351_less__powr__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) )
% 9.20/5.23            = ( ord_less_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8352_powr__less__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 )
% 9.20/5.23            = ( ord_less_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8353_cos__mono__less__eq,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ X3 @ pi )
% 9.20/5.23         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.23           => ( ( ord_less_eq_real @ Y3 @ pi )
% 9.20/5.23             => ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
% 9.20/5.23                = ( ord_less_real @ Y3 @ X3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8354_cos__monotone__0__pi,axiom,
% 9.20/5.23      ! [Y3: real,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ X3 @ pi )
% 9.20/5.23           => ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8355_sin__eq__0__pi,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ pi )
% 9.20/5.23         => ( ( ( sin_real @ X3 )
% 9.20/5.23              = zero_zero_real )
% 9.20/5.23           => ( X3 = zero_zero_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8356_real__of__int__floor__add__one__gt,axiom,
% 9.20/5.23      ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8357_floor__eq,axiom,
% 9.20/5.23      ! [N2: int,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 9.20/5.23         => ( ( archim6058952711729229775r_real @ X3 )
% 9.20/5.23            = N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8358_sin__zero__pi__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
% 9.20/5.23       => ( ( ( sin_real @ X3 )
% 9.20/5.23            = zero_zero_real )
% 9.20/5.23          = ( X3 = zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8359_real__of__int__floor__gt__diff__one,axiom,
% 9.20/5.23      ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8360_diffs__def,axiom,
% 9.20/5.23      ( diffs_complex
% 9.20/5.23      = ( ^ [C3: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8361_diffs__def,axiom,
% 9.20/5.23      ( diffs_rat
% 9.20/5.23      = ( ^ [C3: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8362_diffs__def,axiom,
% 9.20/5.23      ( diffs_int
% 9.20/5.23      = ( ^ [C3: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8363_diffs__def,axiom,
% 9.20/5.23      ( diffs_real
% 9.20/5.23      = ( ^ [C3: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8364_sin__expansion__lemma,axiom,
% 9.20/5.23      ! [X3: real,M: nat] :
% 9.20/5.23        ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23        = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8365_floor__unique,axiom,
% 9.20/5.23      ! [Z2: int,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
% 9.20/5.23         => ( ( archim6058952711729229775r_real @ X3 )
% 9.20/5.23            = Z2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8366_floor__unique,axiom,
% 9.20/5.23      ! [Z2: int,X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
% 9.20/5.23       => ( ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
% 9.20/5.23         => ( ( archim3151403230148437115or_rat @ X3 )
% 9.20/5.23            = Z2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8367_floor__eq__iff,axiom,
% 9.20/5.23      ! [X3: real,A: int] :
% 9.20/5.23        ( ( ( archim6058952711729229775r_real @ X3 )
% 9.20/5.23          = A )
% 9.20/5.23        = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X3 )
% 9.20/5.23          & ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8368_floor__eq__iff,axiom,
% 9.20/5.23      ! [X3: rat,A: int] :
% 9.20/5.23        ( ( ( archim3151403230148437115or_rat @ X3 )
% 9.20/5.23          = A )
% 9.20/5.23        = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X3 )
% 9.20/5.23          & ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8369_floor__split,axiom,
% 9.20/5.23      ! [P: int > $o,T: real] :
% 9.20/5.23        ( ( P @ ( archim6058952711729229775r_real @ T ) )
% 9.20/5.23        = ( ! [I2: int] :
% 9.20/5.23              ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
% 9.20/5.23                & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
% 9.20/5.23             => ( P @ I2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8370_floor__split,axiom,
% 9.20/5.23      ! [P: int > $o,T: rat] :
% 9.20/5.23        ( ( P @ ( archim3151403230148437115or_rat @ T ) )
% 9.20/5.23        = ( ! [I2: int] :
% 9.20/5.23              ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
% 9.20/5.23                & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
% 9.20/5.23             => ( P @ I2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8371_le__mult__floor,axiom,
% 9.20/5.23      ! [A: real,B: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ A )
% 9.20/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ B )
% 9.20/5.23         => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8372_le__mult__floor,axiom,
% 9.20/5.23      ! [A: rat,B: rat] :
% 9.20/5.23        ( ( ord_less_eq_rat @ zero_zero_rat @ A )
% 9.20/5.23       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
% 9.20/5.23         => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8373_less__floor__iff,axiom,
% 9.20/5.23      ! [Z2: int,X3: real] :
% 9.20/5.23        ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8374_less__floor__iff,axiom,
% 9.20/5.23      ! [Z2: int,X3: rat] :
% 9.20/5.23        ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8375_floor__le__iff,axiom,
% 9.20/5.23      ! [X3: real,Z2: int] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
% 9.20/5.23        = ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8376_floor__le__iff,axiom,
% 9.20/5.23      ! [X3: rat,Z2: int] :
% 9.20/5.23        ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
% 9.20/5.23        = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8377_floor__correct,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 )
% 9.20/5.23        & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8378_floor__correct,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 )
% 9.20/5.23        & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8379_cos__expansion__lemma,axiom,
% 9.20/5.23      ! [X3: real,M: nat] :
% 9.20/5.23        ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23        = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8380_powr__neg__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
% 9.20/5.23          = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8381_sin__gt__zero__02,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8382_le__log__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) )
% 9.20/5.23            = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8383_log__le__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 )
% 9.20/5.23            = ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8384_le__powr__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) )
% 9.20/5.23            = ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8385_powr__le__iff,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 )
% 9.20/5.23            = ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8386_cos__two__less__zero,axiom,
% 9.20/5.23      ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8387_floor__eq2,axiom,
% 9.20/5.23      ! [N2: int,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
% 9.20/5.23         => ( ( archim6058952711729229775r_real @ X3 )
% 9.20/5.23            = N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8388_cos__monotone__minus__pi__0,axiom,
% 9.20/5.23      ! [Y3: real,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.20/5.23           => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8389_floor__divide__lower,axiom,
% 9.20/5.23      ! [Q4: real,P4: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.20/5.23       => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8390_floor__divide__lower,axiom,
% 9.20/5.23      ! [Q4: rat,P4: rat] :
% 9.20/5.23        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.20/5.23       => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8391_sincos__total__2pi,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23          = one_one_real )
% 9.20/5.23       => ~ ! [T6: real] :
% 9.20/5.23              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.20/5.23             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.20/5.23               => ( ( X3
% 9.20/5.23                    = ( cos_real @ T6 ) )
% 9.20/5.23                 => ( Y3
% 9.20/5.23                   != ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8392_ln__powr__bound,axiom,
% 9.20/5.23      ! [X3: real,A: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23         => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8393_ln__powr__bound2,axiom,
% 9.20/5.23      ! [X3: real,A: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.23         => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8394_log__add__eq__powr,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.23       => ( ( B != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( plus_plus_real @ ( log @ B @ X3 ) @ Y3 )
% 9.20/5.23              = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8395_add__log__eq__powr,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.23       => ( ( B != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( plus_plus_real @ Y3 @ ( log @ B @ X3 ) )
% 9.20/5.23              = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8396_minus__log__eq__powr,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.23       => ( ( B != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( minus_minus_real @ Y3 @ ( log @ B @ X3 ) )
% 9.20/5.23              = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8397_sin__pi__divide__n__ge__0,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( N2 != zero_zero_nat )
% 9.20/5.23       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8398_cos__times__cos,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 9.20/5.23        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8399_cos__times__cos,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8400_cos__plus__cos,axiom,
% 9.20/5.23      ! [W2: complex,Z2: complex] :
% 9.20/5.23        ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
% 9.20/5.23        = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8401_cos__plus__cos,axiom,
% 9.20/5.23      ! [W2: real,Z2: real] :
% 9.20/5.23        ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
% 9.20/5.23        = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8402_floor__divide__upper,axiom,
% 9.20/5.23      ! [Q4: real,P4: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ Q4 )
% 9.20/5.23       => ( 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 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8403_floor__divide__upper,axiom,
% 9.20/5.23      ! [Q4: rat,P4: rat] :
% 9.20/5.23        ( ( ord_less_rat @ zero_zero_rat @ Q4 )
% 9.20/5.23       => ( 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 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8404_round__def,axiom,
% 9.20/5.23      ( archim8280529875227126926d_real
% 9.20/5.23      = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8405_round__def,axiom,
% 9.20/5.23      ( archim7778729529865785530nd_rat
% 9.20/5.23      = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8406_sin__gt__zero2,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8407_sin__lt__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ pi @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.20/5.23         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8408_cos__double__less__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
% 9.20/5.23         => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8409_cos__gt__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8410_log__minus__eq__powr,axiom,
% 9.20/5.23      ! [B: real,X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.23       => ( ( B != one_one_real )
% 9.20/5.23         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( minus_minus_real @ ( log @ B @ X3 ) @ Y3 )
% 9.20/5.23              = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8411_termdiff__converges,axiom,
% 9.20/5.23      ! [X3: real,K5: real,C: nat > real] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
% 9.20/5.23       => ( ! [X4: real] :
% 9.20/5.23              ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
% 9.20/5.23             => ( summable_real
% 9.20/5.23                @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) ) )
% 9.20/5.23         => ( summable_real
% 9.20/5.23            @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8412_termdiff__converges,axiom,
% 9.20/5.23      ! [X3: complex,K5: real,C: nat > complex] :
% 9.20/5.23        ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
% 9.20/5.23       => ( ! [X4: complex] :
% 9.20/5.23              ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
% 9.20/5.23             => ( summable_complex
% 9.20/5.23                @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) )
% 9.20/5.23         => ( summable_complex
% 9.20/5.23            @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8413_powr__neg__numeral,axiom,
% 9.20/5.23      ! [X3: real,N2: num] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
% 9.20/5.23          = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8414_sin__le__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ pi @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.20/5.23         => ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8415_sin__less__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.23         => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8416_sin__mono__less__eq,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23           => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23             => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
% 9.20/5.23                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8417_sin__monotone__2pi,axiom,
% 9.20/5.23      ! [Y3: real,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ X3 )
% 9.20/5.23         => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23           => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8418_cos__gt__zero__pi,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8419_sin__pi__divide__n__gt__0,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.23       => ( ord_less_real @ zero_zero_real @ ( sin_real @ ( divide_divide_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8420_floor__log2__div2,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.23       => ( ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.20/5.23          = ( plus_plus_int @ ( archim6058952711729229775r_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ one_one_int ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8421_floor__log__nat__eq__if,axiom,
% 9.20/5.23      ! [B: nat,N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
% 9.20/5.23       => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
% 9.20/5.23           => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
% 9.20/5.23              = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8422_Maclaurin__minus__cos__expansion,axiom,
% 9.20/5.23      ! [N2: nat,X3: real] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.23         => ? [T6: real] :
% 9.20/5.23              ( ( ord_less_real @ X3 @ T6 )
% 9.20/5.23              & ( ord_less_real @ T6 @ zero_zero_real )
% 9.20/5.23              & ( ( cos_real @ X3 )
% 9.20/5.23                = ( plus_plus_real
% 9.20/5.23                  @ ( groups6591440286371151544t_real
% 9.20/5.23                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8423_Maclaurin__cos__expansion2,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23         => ? [T6: real] :
% 9.20/5.23              ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.20/5.23              & ( ord_less_real @ T6 @ X3 )
% 9.20/5.23              & ( ( cos_real @ X3 )
% 9.20/5.23                = ( plus_plus_real
% 9.20/5.23                  @ ( groups6591440286371151544t_real
% 9.20/5.23                    @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8424_sin__paired,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( sums_real
% 9.20/5.23        @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 9.20/5.23        @ ( sin_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8425_tan__double,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.20/5.23           != zero_zero_complex )
% 9.20/5.23         => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
% 9.20/5.23            = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8426_tan__double,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.20/5.23           != zero_zero_real )
% 9.20/5.23         => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
% 9.20/5.23            = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X3 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8427_arcosh__def,axiom,
% 9.20/5.23      ( arcosh_real
% 9.20/5.23      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8428_tan__zero,axiom,
% 9.20/5.23      ( ( tan_complex @ zero_zero_complex )
% 9.20/5.23      = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8429_tan__zero,axiom,
% 9.20/5.23      ( ( tan_real @ zero_zero_real )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8430_of__real__eq__0__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( real_V1803761363581548252l_real @ X3 )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23        = ( X3 = zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8431_of__real__eq__0__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( real_V4546457046886955230omplex @ X3 )
% 9.20/5.23          = zero_zero_complex )
% 9.20/5.23        = ( X3 = zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8432_of__real__0,axiom,
% 9.20/5.23      ( ( real_V1803761363581548252l_real @ zero_zero_real )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8433_of__real__0,axiom,
% 9.20/5.23      ( ( real_V4546457046886955230omplex @ zero_zero_real )
% 9.20/5.23      = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8434_fact__0,axiom,
% 9.20/5.23      ( ( semiri5044797733671781792omplex @ zero_zero_nat )
% 9.20/5.23      = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8435_fact__0,axiom,
% 9.20/5.23      ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
% 9.20/5.23      = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8436_fact__0,axiom,
% 9.20/5.23      ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
% 9.20/5.23      = one_one_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8437_fact__0,axiom,
% 9.20/5.23      ( ( semiri2265585572941072030t_real @ zero_zero_nat )
% 9.20/5.23      = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8438_fact__0,axiom,
% 9.20/5.23      ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
% 9.20/5.23      = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8439_of__real__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8440_of__real__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23        = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8441_fact__Suc__0,axiom,
% 9.20/5.23      ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
% 9.20/5.23      = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8442_fact__Suc__0,axiom,
% 9.20/5.23      ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
% 9.20/5.23      = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8443_fact__Suc__0,axiom,
% 9.20/5.23      ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
% 9.20/5.23      = one_one_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8444_fact__Suc__0,axiom,
% 9.20/5.23      ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
% 9.20/5.23      = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8445_fact__Suc__0,axiom,
% 9.20/5.23      ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
% 9.20/5.23      = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8446_fact__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8447_fact__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8448_fact__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8449_fact__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8450_fact__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8451_sin__of__real__pi,axiom,
% 9.20/5.23      ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8452_sin__of__real__pi,axiom,
% 9.20/5.23      ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
% 9.20/5.23      = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8453_norm__of__real__add1,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ one_one_real ) )
% 9.20/5.23        = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8454_norm__of__real__add1,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ one_one_complex ) )
% 9.20/5.23        = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8455_norm__of__real__addn,axiom,
% 9.20/5.23      ! [X3: real,B: num] :
% 9.20/5.23        ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( numeral_numeral_real @ B ) ) )
% 9.20/5.23        = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8456_norm__of__real__addn,axiom,
% 9.20/5.23      ! [X3: real,B: num] :
% 9.20/5.23        ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( numera6690914467698888265omplex @ B ) ) )
% 9.20/5.23        = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8457_cos__of__real__pi__half,axiom,
% 9.20/5.23      ( ( cos_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8458_cos__of__real__pi__half,axiom,
% 9.20/5.23      ( ( cos_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.20/5.23      = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8459_fact__nonzero,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri5044797733671781792omplex @ N2 )
% 9.20/5.23       != zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8460_fact__nonzero,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri773545260158071498ct_rat @ N2 )
% 9.20/5.23       != zero_zero_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8461_fact__nonzero,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri1406184849735516958ct_int @ N2 )
% 9.20/5.23       != zero_zero_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8462_fact__nonzero,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri2265585572941072030t_real @ N2 )
% 9.20/5.23       != zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8463_fact__nonzero,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( semiri1408675320244567234ct_nat @ N2 )
% 9.20/5.23       != zero_zero_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8464_fact__ge__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8465_fact__ge__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8466_fact__ge__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8467_fact__ge__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8468_fact__not__neg,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8469_fact__not__neg,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8470_fact__not__neg,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8471_fact__not__neg,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8472_fact__gt__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8473_fact__gt__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8474_fact__gt__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8475_fact__gt__zero,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8476_fact__ge__1,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8477_fact__ge__1,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8478_fact__ge__1,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8479_fact__ge__1,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8480_fact__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8481_fact__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8482_fact__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8483_fact__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8484_fact__dvd,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23       => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8485_fact__dvd,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23       => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8486_fact__dvd,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23       => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8487_fact__dvd,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23       => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8488_fact__less__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.23         => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8489_fact__less__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.23         => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8490_fact__less__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.23         => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8491_fact__less__mono,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.23         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8492_fact__fact__dvd__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8493_fact__fact__dvd__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8494_fact__fact__dvd__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] : ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ N2 ) ) @ ( semiri1406184849735516958ct_int @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8495_fact__fact__dvd__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] : ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8496_fact__fact__dvd__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] : ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ ( plus_plus_nat @ K @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8497_fact__mod,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
% 9.20/5.23          = zero_zero_int ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8498_fact__mod,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
% 9.20/5.23          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8499_fact__mod,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
% 9.20/5.23          = zero_zero_nat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8500_fact__le__power,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8501_fact__le__power,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1314217659103216013at_int @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8502_fact__le__power,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri5074537144036343181t_real @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8503_fact__le__power,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1316708129612266289at_nat @ ( power_power_nat @ N2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8504_norm__less__p1,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X3 ) ) @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8505_norm__less__p1,axiom,
% 9.20/5.23      ! [X3: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X3 ) ) @ one_one_complex ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8506_choose__dvd,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8507_choose__dvd,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8508_choose__dvd,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( dvd_dvd_int @ ( times_times_int @ ( semiri1406184849735516958ct_int @ K ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8509_choose__dvd,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( dvd_dvd_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8510_choose__dvd,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( dvd_dvd_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8511_fact__num__eq__if,axiom,
% 9.20/5.23      ( semiri5044797733671781792omplex
% 9.20/5.23      = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8512_fact__num__eq__if,axiom,
% 9.20/5.23      ( semiri773545260158071498ct_rat
% 9.20/5.23      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8513_fact__num__eq__if,axiom,
% 9.20/5.23      ( semiri1406184849735516958ct_int
% 9.20/5.23      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8514_fact__num__eq__if,axiom,
% 9.20/5.23      ( semiri2265585572941072030t_real
% 9.20/5.23      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8515_fact__num__eq__if,axiom,
% 9.20/5.23      ( semiri1408675320244567234ct_nat
% 9.20/5.23      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8516_fact__reduce,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( semiri5044797733671781792omplex @ N2 )
% 9.20/5.23          = ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8517_fact__reduce,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( semiri773545260158071498ct_rat @ N2 )
% 9.20/5.23          = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8518_fact__reduce,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( semiri1406184849735516958ct_int @ N2 )
% 9.20/5.23          = ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8519_fact__reduce,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( semiri2265585572941072030t_real @ N2 )
% 9.20/5.23          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8520_fact__reduce,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( semiri1408675320244567234ct_nat @ N2 )
% 9.20/5.23          = ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8521_tan__gt__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8522_lemma__tan__total,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.23       => ? [X4: real] :
% 9.20/5.23            ( ( ord_less_real @ zero_zero_real @ X4 )
% 9.20/5.23            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23            & ( ord_less_real @ Y3 @ ( tan_real @ X4 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8523_lemma__tan__total1,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23      ? [X4: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.20/5.23        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23        & ( ( tan_real @ X4 )
% 9.20/5.23          = Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8524_tan__mono__lt__eq,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23             => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.20/5.23                = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8525_tan__monotone_H,axiom,
% 9.20/5.23      ! [Y3: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23           => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23             => ( ( ord_less_real @ Y3 @ X3 )
% 9.20/5.23                = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8526_tan__monotone,axiom,
% 9.20/5.23      ! [Y3: real,X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ X3 )
% 9.20/5.23         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23           => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8527_tan__total,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23      ? [X4: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
% 9.20/5.23        & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23        & ( ( tan_real @ X4 )
% 9.20/5.23          = Y3 )
% 9.20/5.23        & ! [Y6: real] :
% 9.20/5.23            ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
% 9.20/5.23              & ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23              & ( ( tan_real @ Y6 )
% 9.20/5.23                = Y3 ) )
% 9.20/5.23           => ( Y6 = X4 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8528_add__tan__eq,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ( cos_complex @ Y3 )
% 9.20/5.23           != zero_zero_complex )
% 9.20/5.23         => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) )
% 9.20/5.23            = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8529_add__tan__eq,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ( cos_real @ Y3 )
% 9.20/5.23           != zero_zero_real )
% 9.20/5.23         => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.20/5.23            = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8530_tan__total__pos,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.23       => ? [X4: real] :
% 9.20/5.23            ( ( ord_less_eq_real @ zero_zero_real @ X4 )
% 9.20/5.23            & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23            & ( ( tan_real @ X4 )
% 9.20/5.23              = Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8531_tan__pos__pi2__le,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8532_tan__less__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.23         => ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8533_tan__mono__le__eq,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
% 9.20/5.23           => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23             => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
% 9.20/5.23                = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8534_tan__mono__le,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.20/5.23         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23           => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8535_tan__bound__pi2,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23       => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X3 ) ) @ one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8536_arctan,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
% 9.20/5.23        & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23        & ( ( tan_real @ ( arctan @ Y3 ) )
% 9.20/5.23          = Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8537_arctan__tan,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( arctan @ ( tan_real @ X3 ) )
% 9.20/5.23            = X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8538_arctan__unique,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ( tan_real @ X3 )
% 9.20/5.23              = Y3 )
% 9.20/5.23           => ( ( arctan @ Y3 )
% 9.20/5.23              = X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8539_Maclaurin__zero,axiom,
% 9.20/5.23      ! [X3: real,N2: nat,Diff: nat > complex > real] :
% 9.20/5.23        ( ( X3 = zero_zero_real )
% 9.20/5.23       => ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( groups6591440286371151544t_real
% 9.20/5.23              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23              @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23            = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8540_Maclaurin__zero,axiom,
% 9.20/5.23      ! [X3: real,N2: nat,Diff: nat > real > real] :
% 9.20/5.23        ( ( X3 = zero_zero_real )
% 9.20/5.23       => ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( groups6591440286371151544t_real
% 9.20/5.23              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23              @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23            = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8541_Maclaurin__zero,axiom,
% 9.20/5.23      ! [X3: real,N2: nat,Diff: nat > rat > real] :
% 9.20/5.23        ( ( X3 = zero_zero_real )
% 9.20/5.23       => ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( groups6591440286371151544t_real
% 9.20/5.23              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23              @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23            = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8542_Maclaurin__zero,axiom,
% 9.20/5.23      ! [X3: real,N2: nat,Diff: nat > nat > real] :
% 9.20/5.23        ( ( X3 = zero_zero_real )
% 9.20/5.23       => ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( groups6591440286371151544t_real
% 9.20/5.23              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23              @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23            = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8543_Maclaurin__zero,axiom,
% 9.20/5.23      ! [X3: real,N2: nat,Diff: nat > int > real] :
% 9.20/5.23        ( ( X3 = zero_zero_real )
% 9.20/5.23       => ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( groups6591440286371151544t_real
% 9.20/5.23              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23              @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23            = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8544_Maclaurin__lemma,axiom,
% 9.20/5.23      ! [H2: real,F: real > real,J: nat > real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.20/5.23       => ? [B9: real] :
% 9.20/5.23            ( ( F @ H2 )
% 9.20/5.23            = ( plus_plus_real
% 9.20/5.23              @ ( groups6591440286371151544t_real
% 9.20/5.23                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.20/5.23                @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23              @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8545_tan__add,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ( cos_complex @ Y3 )
% 9.20/5.23           != zero_zero_complex )
% 9.20/5.23         => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.20/5.23             != zero_zero_complex )
% 9.20/5.23           => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.20/5.23              = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8546_tan__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ( cos_real @ Y3 )
% 9.20/5.23           != zero_zero_real )
% 9.20/5.23         => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23             != zero_zero_real )
% 9.20/5.23           => ( ( tan_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23              = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8547_tan__diff,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ( cos_complex @ Y3 )
% 9.20/5.23           != zero_zero_complex )
% 9.20/5.23         => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.20/5.23             != zero_zero_complex )
% 9.20/5.23           => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
% 9.20/5.23              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8548_tan__diff,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ( cos_real @ Y3 )
% 9.20/5.23           != zero_zero_real )
% 9.20/5.23         => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.20/5.23             != zero_zero_real )
% 9.20/5.23           => ( ( tan_real @ ( minus_minus_real @ X3 @ Y3 ) )
% 9.20/5.23              = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8549_lemma__tan__add1,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( ( cos_complex @ X3 )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ( cos_complex @ Y3 )
% 9.20/5.23           != zero_zero_complex )
% 9.20/5.23         => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) )
% 9.20/5.23            = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8550_lemma__tan__add1,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( cos_real @ X3 )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ( cos_real @ Y3 )
% 9.20/5.23           != zero_zero_real )
% 9.20/5.23         => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) )
% 9.20/5.23            = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8551_minus__sin__cos__eq,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( uminus_uminus_real @ ( sin_real @ X3 ) )
% 9.20/5.23        = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8552_minus__sin__cos__eq,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( uminus1482373934393186551omplex @ ( sin_complex @ X3 ) )
% 9.20/5.23        = ( cos_complex @ ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8553_tan__total__pi4,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.23       => ? [Z3: real] :
% 9.20/5.23            ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
% 9.20/5.23            & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23            & ( ( tan_real @ Z3 )
% 9.20/5.23              = X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8554_tan__half,axiom,
% 9.20/5.23      ( tan_complex
% 9.20/5.23      = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8555_tan__half,axiom,
% 9.20/5.23      ( tan_real
% 9.20/5.23      = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8556_arsinh__def,axiom,
% 9.20/5.23      ( arsinh_real
% 9.20/5.23      = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8557_Maclaurin__sin__expansion3,axiom,
% 9.20/5.23      ! [N2: nat,X3: real] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23         => ? [T6: real] :
% 9.20/5.23              ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.20/5.23              & ( ord_less_real @ T6 @ X3 )
% 9.20/5.23              & ( ( sin_real @ X3 )
% 9.20/5.23                = ( plus_plus_real
% 9.20/5.23                  @ ( groups6591440286371151544t_real
% 9.20/5.23                    @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23                  @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8558_Maclaurin__sin__expansion4,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ? [T6: real] :
% 9.20/5.23            ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.20/5.23            & ( ord_less_eq_real @ T6 @ X3 )
% 9.20/5.23            & ( ( sin_real @ X3 )
% 9.20/5.23              = ( plus_plus_real
% 9.20/5.23                @ ( groups6591440286371151544t_real
% 9.20/5.23                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.23                  @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23                @ ( 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 @ N2 ) ) @ pi ) ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8559_sin__coeff__def,axiom,
% 9.20/5.23      ( sin_coeff
% 9.20/5.23      = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8560_Maclaurin__exp__lt,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( X3 != zero_zero_real )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23         => ? [T6: real] :
% 9.20/5.23              ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.20/5.23              & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 9.20/5.23              & ( ( exp_real @ X3 )
% 9.20/5.23                = ( plus_plus_real
% 9.20/5.23                  @ ( groups6591440286371151544t_real
% 9.20/5.23                    @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
% 9.20/5.23                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.23                  @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8561_round__altdef,axiom,
% 9.20/5.23      ( archim8280529875227126926d_real
% 9.20/5.23      = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8562_round__altdef,axiom,
% 9.20/5.23      ( archim7778729529865785530nd_rat
% 9.20/5.23      = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8563_exp__less__mono,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.23       => ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8564_exp__less__cancel__iff,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8565_exp__zero,axiom,
% 9.20/5.23      ( ( exp_complex @ zero_zero_complex )
% 9.20/5.23      = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8566_exp__zero,axiom,
% 9.20/5.23      ( ( exp_real @ zero_zero_real )
% 9.20/5.23      = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8567_frac__of__int,axiom,
% 9.20/5.23      ! [Z2: int] :
% 9.20/5.23        ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
% 9.20/5.23        = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8568_frac__of__int,axiom,
% 9.20/5.23      ! [Z2: int] :
% 9.20/5.23        ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
% 9.20/5.23        = zero_zero_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8569_sin__coeff__0,axiom,
% 9.20/5.23      ( ( sin_coeff @ zero_zero_nat )
% 9.20/5.23      = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8570_one__less__exp__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
% 9.20/5.23        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8571_exp__less__one__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
% 9.20/5.23        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8572_exp__ln__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.23          = X3 )
% 9.20/5.23        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8573_exp__ln,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( exp_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.23          = X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8574_fact__mono__nat,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8575_fact__ge__self,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8576_exp__less__cancel,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.20/5.23       => ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8577_exp__not__eq__zero,axiom,
% 9.20/5.23      ! [X3: complex] :
% 9.20/5.23        ( ( exp_complex @ X3 )
% 9.20/5.23       != zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8578_exp__not__eq__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( exp_real @ X3 )
% 9.20/5.23       != zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8579_fact__less__mono__nat,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.23       => ( ( ord_less_nat @ M @ N2 )
% 9.20/5.23         => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8580_exp__total,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ Y3 )
% 9.20/5.23       => ? [X4: real] :
% 9.20/5.23            ( ( exp_real @ X4 )
% 9.20/5.23            = Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8581_exp__gt__zero,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8582_not__exp__less__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8583_mult__exp__exp,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) )
% 9.20/5.23        = ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8584_mult__exp__exp,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
% 9.20/5.23        = ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8585_exp__add__commuting,axiom,
% 9.20/5.23      ! [X3: complex,Y3: complex] :
% 9.20/5.23        ( ( ( times_times_complex @ X3 @ Y3 )
% 9.20/5.23          = ( times_times_complex @ Y3 @ X3 ) )
% 9.20/5.23       => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
% 9.20/5.23          = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8586_exp__add__commuting,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( times_times_real @ X3 @ Y3 )
% 9.20/5.23          = ( times_times_real @ Y3 @ X3 ) )
% 9.20/5.23       => ( ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23          = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8587_frac__ge__0,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8588_frac__ge__0,axiom,
% 9.20/5.23      ! [X3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8589_frac__lt__1,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X3 ) @ one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8590_frac__lt__1,axiom,
% 9.20/5.23      ! [X3: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X3 ) @ one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8591_frac__1__eq,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ one_one_real ) )
% 9.20/5.23        = ( archim2898591450579166408c_real @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8592_frac__1__eq,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
% 9.20/5.23        = ( archimedean_frac_rat @ X3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8593_fact__ge__Suc__0__nat,axiom,
% 9.20/5.23      ! [N2: nat] : ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8594_exp__gt__one,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8595_dvd__fact,axiom,
% 9.20/5.23      ! [M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.20/5.23       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23         => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8596_fact__diff__Suc,axiom,
% 9.20/5.23      ! [N2: nat,M: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ ( suc @ M ) )
% 9.20/5.23       => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
% 9.20/5.23          = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8597_ln__ge__iff,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X3 ) )
% 9.20/5.23          = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8598_fact__div__fact__le__pow,axiom,
% 9.20/5.23      ! [R2: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ R2 @ N2 )
% 9.20/5.23       => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R2 ) ) ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8599_powr__def,axiom,
% 9.20/5.23      ( powr_real
% 9.20/5.23      = ( ^ [X: real,A3: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8600_exp__divide__power__eq,axiom,
% 9.20/5.23      ! [N2: nat,X3: complex] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
% 9.20/5.23          = ( exp_complex @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8601_exp__divide__power__eq,axiom,
% 9.20/5.23      ! [N2: nat,X3: real] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
% 9.20/5.23          = ( exp_real @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8602_frac__eq,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ( archim2898591450579166408c_real @ X3 )
% 9.20/5.23          = X3 )
% 9.20/5.23        = ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23          & ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8603_frac__eq,axiom,
% 9.20/5.23      ! [X3: rat] :
% 9.20/5.23        ( ( ( archimedean_frac_rat @ X3 )
% 9.20/5.23          = X3 )
% 9.20/5.23        = ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
% 9.20/5.23          & ( ord_less_rat @ X3 @ one_one_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8604_frac__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.20/5.23         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
% 9.20/5.23        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.20/5.23         => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23            = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8605_frac__add,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] :
% 9.20/5.23        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.20/5.23         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
% 9.20/5.23        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.20/5.23         => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 9.20/5.23            = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8606_tanh__altdef,axiom,
% 9.20/5.23      ( tanh_real
% 9.20/5.23      = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8607_sin__coeff__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( sin_coeff @ ( suc @ N2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( cos_coeff @ N2 ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8608_floor__add,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.20/5.23         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
% 9.20/5.23        & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
% 9.20/5.23         => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8609_floor__add,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] :
% 9.20/5.23        ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.20/5.23         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
% 9.20/5.23        & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
% 9.20/5.23         => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
% 9.20/5.23            = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8610_cos__coeff__Suc,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( cos_coeff @ ( suc @ N2 ) )
% 9.20/5.23        = ( divide_divide_real @ ( uminus_uminus_real @ ( sin_coeff @ N2 ) ) @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8611_exp__ge__one__plus__x__over__n__power__n,axiom,
% 9.20/5.23      ! [N2: nat,X3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X3 )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23         => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8612_exp__ge__one__minus__x__over__n__power__n,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23         => ( ord_less_eq_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8613_log__base__10__eq2,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23          = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8614_log__base__10__eq1,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
% 9.20/5.23          = ( 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 @ X3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8615_sin__tan,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23       => ( ( sin_real @ X3 )
% 9.20/5.23          = ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8616_cos__tan,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23       => ( ( cos_real @ X3 )
% 9.20/5.23          = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8617_monoseq__def,axiom,
% 9.20/5.23      ( topolo6980174941875973593q_real
% 9.20/5.23      = ( ^ [X5: nat > real] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8618_monoseq__def,axiom,
% 9.20/5.23      ( topolo3100542954746470799et_int
% 9.20/5.23      = ( ^ [X5: nat > set_int] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_set_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8619_monoseq__def,axiom,
% 9.20/5.23      ( topolo4267028734544971653eq_rat
% 9.20/5.23      = ( ^ [X5: nat > rat] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_rat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8620_monoseq__def,axiom,
% 9.20/5.23      ( topolo1459490580787246023eq_num
% 9.20/5.23      = ( ^ [X5: nat > num] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_num @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8621_monoseq__def,axiom,
% 9.20/5.23      ( topolo4902158794631467389eq_nat
% 9.20/5.23      = ( ^ [X5: nat > nat] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_nat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8622_monoseq__def,axiom,
% 9.20/5.23      ( topolo4899668324122417113eq_int
% 9.20/5.23      = ( ^ [X5: nat > int] :
% 9.20/5.23            ( ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
% 9.20/5.23            | ! [M2: nat,N: nat] :
% 9.20/5.23                ( ( ord_less_eq_nat @ M2 @ N )
% 9.20/5.23               => ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8623_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > real] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8624_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > set_int] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8625_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > rat] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8626_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > num] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8627_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > nat] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8628_monoI2,axiom,
% 9.20/5.23      ! [X7: nat > int] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
% 9.20/5.23       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8629_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > real] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8630_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > set_int] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8631_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > rat] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8632_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > num] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8633_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > nat] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8634_monoI1,axiom,
% 9.20/5.23      ! [X7: nat > int] :
% 9.20/5.23        ( ! [M3: nat,N3: nat] :
% 9.20/5.23            ( ( ord_less_eq_nat @ M3 @ N3 )
% 9.20/5.23           => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
% 9.20/5.23       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8635_real__sqrt__less__iff,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
% 9.20/5.23        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8636_real__sqrt__lt__0__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
% 9.20/5.23        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8637_real__sqrt__gt__0__iff,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
% 9.20/5.23        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8638_real__sqrt__lt__1__iff,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( sqrt @ X3 ) @ one_one_real )
% 9.20/5.23        = ( ord_less_real @ X3 @ one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8639_real__sqrt__gt__1__iff,axiom,
% 9.20/5.23      ! [Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
% 9.20/5.23        = ( ord_less_real @ one_one_real @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8640_real__sqrt__less__mono,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.23       => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8641_real__sqrt__gt__zero,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8642_sqrt2__less__2,axiom,
% 9.20/5.23      ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8643_real__less__rsqrt,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
% 9.20/5.23       => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8644_lemma__real__divide__sqrt__less,axiom,
% 9.20/5.23      ! [U: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ U )
% 9.20/5.23       => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8645_real__less__lsqrt,axiom,
% 9.20/5.23      ! [X3: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.23         => ( ( ord_less_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23           => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8646_ln__sqrt,axiom,
% 9.20/5.23      ! [X3: real] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ( ln_ln_real @ ( sqrt @ X3 ) )
% 9.20/5.23          = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8647_arsinh__real__aux,axiom,
% 9.20/5.23      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8648_real__sqrt__sum__squares__less,axiom,
% 9.20/5.23      ! [X3: real,U: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23       => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23         => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8649_sqrt__sum__squares__half__less,axiom,
% 9.20/5.23      ! [X3: real,U: real,Y3: real] :
% 9.20/5.23        ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.23           => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.23             => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8650_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo6980174941875973593q_real
% 9.20/5.23      = ( ^ [X5: nat > real] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_real @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8651_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo3100542954746470799et_int
% 9.20/5.23      = ( ^ [X5: nat > set_int] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8652_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo4267028734544971653eq_rat
% 9.20/5.23      = ( ^ [X5: nat > rat] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8653_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo1459490580787246023eq_num
% 9.20/5.23      = ( ^ [X5: nat > num] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_num @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8654_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo4902158794631467389eq_nat
% 9.20/5.23      = ( ^ [X5: nat > nat] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8655_monoseq__Suc,axiom,
% 9.20/5.23      ( topolo4899668324122417113eq_int
% 9.20/5.23      = ( ^ [X5: nat > int] :
% 9.20/5.23            ( ! [N: nat] : ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
% 9.20/5.23            | ! [N: nat] : ( ord_less_eq_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8656_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > real] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8657_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > set_int] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8658_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > rat] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8659_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > num] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8660_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > nat] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8661_mono__SucI2,axiom,
% 9.20/5.23      ! [X7: nat > int] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
% 9.20/5.23       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8662_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > real] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo6980174941875973593q_real @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8663_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > set_int] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo3100542954746470799et_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8664_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > rat] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo4267028734544971653eq_rat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8665_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > num] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo1459490580787246023eq_num @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8666_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > nat] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo4902158794631467389eq_nat @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8667_mono__SucI1,axiom,
% 9.20/5.23      ! [X7: nat > int] :
% 9.20/5.23        ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
% 9.20/5.23       => ( topolo4899668324122417113eq_int @ X7 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8668_pochhammer__double,axiom,
% 9.20/5.23      ! [Z2: complex,N2: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.23        = ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8669_pochhammer__double,axiom,
% 9.20/5.23      ! [Z2: rat,N2: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.23        = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8670_pochhammer__double,axiom,
% 9.20/5.23      ! [Z2: real,N2: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.23        = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8671_of__nat__code,axiom,
% 9.20/5.23      ( semiri8010041392384452111omplex
% 9.20/5.23      = ( ^ [N: nat] :
% 9.20/5.23            ( semiri2816024913162550771omplex
% 9.20/5.23            @ ^ [I2: complex] : ( plus_plus_complex @ I2 @ one_one_complex )
% 9.20/5.23            @ N
% 9.20/5.23            @ zero_zero_complex ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8672_of__nat__code,axiom,
% 9.20/5.23      ( semiri681578069525770553at_rat
% 9.20/5.23      = ( ^ [N: nat] :
% 9.20/5.23            ( semiri7787848453975740701ux_rat
% 9.20/5.23            @ ^ [I2: rat] : ( plus_plus_rat @ I2 @ one_one_rat )
% 9.20/5.23            @ N
% 9.20/5.23            @ zero_zero_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8673_of__nat__code,axiom,
% 9.20/5.23      ( semiri1314217659103216013at_int
% 9.20/5.23      = ( ^ [N: nat] :
% 9.20/5.23            ( semiri8420488043553186161ux_int
% 9.20/5.23            @ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
% 9.20/5.23            @ N
% 9.20/5.23            @ zero_zero_int ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8674_of__nat__code,axiom,
% 9.20/5.23      ( semiri5074537144036343181t_real
% 9.20/5.23      = ( ^ [N: nat] :
% 9.20/5.23            ( semiri7260567687927622513x_real
% 9.20/5.23            @ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
% 9.20/5.23            @ N
% 9.20/5.23            @ zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8675_of__nat__code,axiom,
% 9.20/5.23      ( semiri1316708129612266289at_nat
% 9.20/5.23      = ( ^ [N: nat] :
% 9.20/5.23            ( semiri8422978514062236437ux_nat
% 9.20/5.23            @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
% 9.20/5.23            @ N
% 9.20/5.23            @ zero_zero_nat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8676_gchoose__row__sum__weighted,axiom,
% 9.20/5.23      ! [R2: complex,M: nat] :
% 9.20/5.23        ( ( groups2073611262835488442omplex
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.20/5.23        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8677_gchoose__row__sum__weighted,axiom,
% 9.20/5.23      ! [R2: rat,M: nat] :
% 9.20/5.23        ( ( groups2906978787729119204at_rat
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.20/5.23        = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8678_gchoose__row__sum__weighted,axiom,
% 9.20/5.23      ! [R2: real,M: nat] :
% 9.20/5.23        ( ( groups6591440286371151544t_real
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
% 9.20/5.23        = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8679_central__binomial__lower__bound,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ord_less_eq_real @ ( divide_divide_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ N2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8680_complex__unimodular__polar,axiom,
% 9.20/5.23      ! [Z2: complex] :
% 9.20/5.23        ( ( ( real_V1022390504157884413omplex @ Z2 )
% 9.20/5.23          = one_one_real )
% 9.20/5.23       => ~ ! [T6: real] :
% 9.20/5.23              ( ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.20/5.23             => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
% 9.20/5.23               => ( Z2
% 9.20/5.23                 != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8681_binomial__Suc__n,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( binomial @ ( suc @ N2 ) @ N2 )
% 9.20/5.23        = ( suc @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8682_gbinomial__0_I2_J,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8683_gbinomial__0_I2_J,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8684_gbinomial__0_I2_J,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8685_gbinomial__0_I2_J,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8686_gbinomial__0_I2_J,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8687_binomial__1,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.23        = N2 ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8688_binomial__0__Suc,axiom,
% 9.20/5.23      ! [K: nat] :
% 9.20/5.23        ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
% 9.20/5.23        = zero_zero_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8689_binomial__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( binomial @ N2 @ K )
% 9.20/5.23          = zero_zero_nat )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8690_gbinomial__0_I1_J,axiom,
% 9.20/5.23      ! [A: complex] :
% 9.20/5.23        ( ( gbinomial_complex @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8691_gbinomial__0_I1_J,axiom,
% 9.20/5.23      ! [A: real] :
% 9.20/5.23        ( ( gbinomial_real @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8692_gbinomial__0_I1_J,axiom,
% 9.20/5.23      ! [A: rat] :
% 9.20/5.23        ( ( gbinomial_rat @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8693_gbinomial__0_I1_J,axiom,
% 9.20/5.23      ! [A: nat] :
% 9.20/5.23        ( ( gbinomial_nat @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8694_gbinomial__0_I1_J,axiom,
% 9.20/5.23      ! [A: int] :
% 9.20/5.23        ( ( gbinomial_int @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8695_binomial__Suc__Suc,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8696_pochhammer__0,axiom,
% 9.20/5.23      ! [A: complex] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8697_pochhammer__0,axiom,
% 9.20/5.23      ! [A: real] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8698_pochhammer__0,axiom,
% 9.20/5.23      ! [A: rat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8699_pochhammer__0,axiom,
% 9.20/5.23      ! [A: nat] :
% 9.20/5.23        ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8700_pochhammer__0,axiom,
% 9.20/5.23      ! [A: int] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
% 9.20/5.23        = one_one_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8701_binomial__n__0,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( binomial @ N2 @ zero_zero_nat )
% 9.20/5.23        = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8702_zero__less__binomial__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) )
% 9.20/5.23        = ( ord_less_eq_nat @ K @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8703_binomial__eq__0,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( binomial @ N2 @ K )
% 9.20/5.23          = zero_zero_nat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8704_Suc__times__binomial,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8705_Suc__times__binomial__eq,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) )
% 9.20/5.23        = ( times_times_nat @ ( binomial @ ( suc @ N2 ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8706_binomial__symmetric,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( binomial @ N2 @ K )
% 9.20/5.23          = ( binomial @ N2 @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8707_choose__mult__lemma,axiom,
% 9.20/5.23      ! [M: nat,R2: nat,K: nat] :
% 9.20/5.23        ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
% 9.20/5.23        = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8708_binomial__le__pow,axiom,
% 9.20/5.23      ! [R2: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ R2 @ N2 )
% 9.20/5.23       => ( ord_less_eq_nat @ ( binomial @ N2 @ R2 ) @ ( power_power_nat @ N2 @ R2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8709_pochhammer__pos,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8710_pochhammer__pos,axiom,
% 9.20/5.23      ! [X3: rat,N2: nat] :
% 9.20/5.23        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.20/5.23       => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8711_pochhammer__pos,axiom,
% 9.20/5.23      ! [X3: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.23       => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8712_pochhammer__pos,axiom,
% 9.20/5.23      ! [X3: int,N2: nat] :
% 9.20/5.23        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.20/5.23       => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8713_pochhammer__neq__0__mono,axiom,
% 9.20/5.23      ! [A: complex,M: nat,N2: nat] :
% 9.20/5.23        ( ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.20/5.23         != zero_zero_complex )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 9.20/5.23           != zero_zero_complex ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8714_pochhammer__neq__0__mono,axiom,
% 9.20/5.23      ! [A: real,M: nat,N2: nat] :
% 9.20/5.23        ( ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.20/5.23         != zero_zero_real )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 9.20/5.23           != zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8715_pochhammer__neq__0__mono,axiom,
% 9.20/5.23      ! [A: rat,M: nat,N2: nat] :
% 9.20/5.23        ( ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.20/5.23         != zero_zero_rat )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 9.20/5.23           != zero_zero_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8716_pochhammer__eq__0__mono,axiom,
% 9.20/5.23      ! [A: complex,N2: nat,M: nat] :
% 9.20/5.23        ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 9.20/5.23          = zero_zero_complex )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s2602460028002588243omplex @ A @ M )
% 9.20/5.23            = zero_zero_complex ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8717_pochhammer__eq__0__mono,axiom,
% 9.20/5.23      ! [A: real,N2: nat,M: nat] :
% 9.20/5.23        ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s7457072308508201937r_real @ A @ M )
% 9.20/5.23            = zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8718_pochhammer__eq__0__mono,axiom,
% 9.20/5.23      ! [A: rat,N2: nat,M: nat] :
% 9.20/5.23        ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 9.20/5.23          = zero_zero_rat )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.23         => ( ( comm_s4028243227959126397er_rat @ A @ M )
% 9.20/5.23            = zero_zero_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8719_zero__less__binomial,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ord_less_nat @ zero_zero_nat @ ( binomial @ N2 @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8720_Suc__times__binomial__add,axiom,
% 9.20/5.23      ! [A: nat,B: nat] :
% 9.20/5.23        ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
% 9.20/5.23        = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8721_choose__mult,axiom,
% 9.20/5.23      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.23       => ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23         => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
% 9.20/5.23            = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8722_binomial__Suc__Suc__eq__times,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( binomial @ ( suc @ N2 ) @ ( suc @ K ) )
% 9.20/5.23        = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( binomial @ N2 @ K ) ) @ ( suc @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8723_gbinomial__pochhammer_H,axiom,
% 9.20/5.23      ( gbinomial_complex
% 9.20/5.23      = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8724_gbinomial__pochhammer_H,axiom,
% 9.20/5.23      ( gbinomial_rat
% 9.20/5.23      = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8725_gbinomial__pochhammer_H,axiom,
% 9.20/5.23      ( gbinomial_real
% 9.20/5.23      = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8726_gbinomial__Suc__Suc,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8727_gbinomial__Suc__Suc,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8728_gbinomial__Suc__Suc,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8729_gbinomial__of__nat__symmetric,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ K )
% 9.20/5.23          = ( gbinomial_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8730_pochhammer__nonneg,axiom,
% 9.20/5.23      ! [X3: real,N2: nat] :
% 9.20/5.23        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.23       => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8731_pochhammer__nonneg,axiom,
% 9.20/5.23      ! [X3: rat,N2: nat] :
% 9.20/5.23        ( ( ord_less_rat @ zero_zero_rat @ X3 )
% 9.20/5.23       => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8732_pochhammer__nonneg,axiom,
% 9.20/5.23      ! [X3: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ X3 )
% 9.20/5.23       => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8733_pochhammer__nonneg,axiom,
% 9.20/5.23      ! [X3: int,N2: nat] :
% 9.20/5.23        ( ( ord_less_int @ zero_zero_int @ X3 )
% 9.20/5.23       => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8734_pochhammer__0__left,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ( N2 = zero_zero_nat )
% 9.20/5.23         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 9.20/5.23            = one_one_complex ) )
% 9.20/5.23        & ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( comm_s2602460028002588243omplex @ zero_zero_complex @ N2 )
% 9.20/5.23            = zero_zero_complex ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8735_pochhammer__0__left,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ( N2 = zero_zero_nat )
% 9.20/5.23         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 9.20/5.23            = one_one_real ) )
% 9.20/5.23        & ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( comm_s7457072308508201937r_real @ zero_zero_real @ N2 )
% 9.20/5.23            = zero_zero_real ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8736_pochhammer__0__left,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ( N2 = zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 9.20/5.23            = one_one_rat ) )
% 9.20/5.23        & ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
% 9.20/5.23            = zero_zero_rat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8737_pochhammer__0__left,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ( N2 = zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 9.20/5.23            = one_one_nat ) )
% 9.20/5.23        & ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4663373288045622133er_nat @ zero_zero_nat @ N2 )
% 9.20/5.23            = zero_zero_nat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8738_pochhammer__0__left,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ( ( N2 = zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 9.20/5.23            = one_one_int ) )
% 9.20/5.23        & ( ( N2 != zero_zero_nat )
% 9.20/5.23         => ( ( comm_s4660882817536571857er_int @ zero_zero_int @ N2 )
% 9.20/5.23            = zero_zero_int ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8739_binomial__absorption,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8740_gbinomial__addition__formula,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ A @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_complex @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8741_gbinomial__addition__formula,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ A @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_real @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ ( suc @ K ) ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8742_gbinomial__addition__formula,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ A @ ( suc @ K ) )
% 9.20/5.23        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8743_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.20/5.23      ! [K: nat,A: real] :
% 9.20/5.23        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ K ) @ A )
% 9.20/5.23       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8744_gbinomial__ge__n__over__k__pow__k,axiom,
% 9.20/5.23      ! [K: nat,A: rat] :
% 9.20/5.23        ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
% 9.20/5.23       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8745_gbinomial__mult__1_H,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ A )
% 9.20/5.23        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8746_gbinomial__mult__1_H,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
% 9.20/5.23        = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8747_gbinomial__mult__1_H,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( times_times_real @ ( gbinomial_real @ A @ K ) @ A )
% 9.20/5.23        = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8748_gbinomial__mult__1,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( times_times_complex @ A @ ( gbinomial_complex @ A @ K ) )
% 9.20/5.23        = ( plus_plus_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ K ) @ ( gbinomial_complex @ A @ K ) ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8749_gbinomial__mult__1,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
% 9.20/5.23        = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8750_gbinomial__mult__1,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( times_times_real @ A @ ( gbinomial_real @ A @ K ) )
% 9.20/5.23        = ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ K ) @ ( gbinomial_real @ A @ K ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8751_binomial__fact__lemma,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( times_times_nat @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( binomial @ N2 @ K ) )
% 9.20/5.23          = ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8752_pochhammer__rec,axiom,
% 9.20/5.23      ! [A: complex,N2: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_complex @ A @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8753_pochhammer__rec,axiom,
% 9.20/5.23      ! [A: real,N2: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_real @ A @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ one_one_real ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8754_pochhammer__rec,axiom,
% 9.20/5.23      ! [A: rat,N2: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8755_pochhammer__rec,axiom,
% 9.20/5.23      ! [A: nat,N2: nat] :
% 9.20/5.23        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_nat @ A @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8756_pochhammer__rec,axiom,
% 9.20/5.23      ! [A: int,N2: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_int @ A @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ one_one_int ) @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8757_pochhammer__rec_H,axiom,
% 9.20/5.23      ! [Z2: complex,N2: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8758_pochhammer__rec_H,axiom,
% 9.20/5.23      ! [Z2: rat,N2: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8759_pochhammer__rec_H,axiom,
% 9.20/5.23      ! [Z2: int,N2: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8760_pochhammer__rec_H,axiom,
% 9.20/5.23      ! [Z2: real,N2: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8761_pochhammer__rec_H,axiom,
% 9.20/5.23      ! [Z2: nat,N2: nat] :
% 9.20/5.23        ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8762_pochhammer__Suc,axiom,
% 9.20/5.23      ! [A: complex,N2: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ A @ N2 ) @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8763_pochhammer__Suc,axiom,
% 9.20/5.23      ! [A: rat,N2: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8764_pochhammer__Suc,axiom,
% 9.20/5.23      ! [A: int,N2: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_int @ ( comm_s4660882817536571857er_int @ A @ N2 ) @ ( plus_plus_int @ A @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8765_pochhammer__Suc,axiom,
% 9.20/5.23      ! [A: real,N2: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_real @ ( comm_s7457072308508201937r_real @ A @ N2 ) @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8766_pochhammer__Suc,axiom,
% 9.20/5.23      ! [A: nat,N2: nat] :
% 9.20/5.23        ( ( comm_s4663373288045622133er_nat @ A @ ( suc @ N2 ) )
% 9.20/5.23        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ A @ N2 ) @ ( plus_plus_nat @ A @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8767_pochhammer__eq__0__iff,axiom,
% 9.20/5.23      ! [A: complex,N2: nat] :
% 9.20/5.23        ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
% 9.20/5.23          = zero_zero_complex )
% 9.20/5.23        = ( ? [K3: nat] :
% 9.20/5.23              ( ( ord_less_nat @ K3 @ N2 )
% 9.20/5.23              & ( A
% 9.20/5.23                = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8768_pochhammer__eq__0__iff,axiom,
% 9.20/5.23      ! [A: rat,N2: nat] :
% 9.20/5.23        ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
% 9.20/5.23          = zero_zero_rat )
% 9.20/5.23        = ( ? [K3: nat] :
% 9.20/5.23              ( ( ord_less_nat @ K3 @ N2 )
% 9.20/5.23              & ( A
% 9.20/5.23                = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8769_pochhammer__eq__0__iff,axiom,
% 9.20/5.23      ! [A: real,N2: nat] :
% 9.20/5.23        ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23        = ( ? [K3: nat] :
% 9.20/5.23              ( ( ord_less_nat @ K3 @ N2 )
% 9.20/5.23              & ( A
% 9.20/5.23                = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8770_pochhammer__of__nat__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_complex )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8771_pochhammer__of__nat__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_rat )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8772_pochhammer__of__nat__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 9.20/5.23          = zero_z3403309356797280102nteger )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8773_pochhammer__of__nat__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_int )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8774_pochhammer__of__nat__eq__0__iff,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_real )
% 9.20/5.23        = ( ord_less_nat @ N2 @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8775_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8776_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8777_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 9.20/5.23          = zero_z3403309356797280102nteger ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8778_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_int ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8779_pochhammer__of__nat__eq__0__lemma,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ N2 @ K )
% 9.20/5.23       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 9.20/5.23          = zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8780_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ K )
% 9.20/5.23         != zero_zero_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8781_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
% 9.20/5.23         != zero_zero_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8782_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
% 9.20/5.23         != zero_z3403309356797280102nteger ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8783_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ K )
% 9.20/5.23         != zero_zero_int ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8784_pochhammer__of__nat__eq__0__lemma_H,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ K )
% 9.20/5.23         != zero_zero_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8785_pochhammer__product_H,axiom,
% 9.20/5.23      ! [Z2: complex,N2: nat,M: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.23        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8786_pochhammer__product_H,axiom,
% 9.20/5.23      ! [Z2: rat,N2: nat,M: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.23        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8787_pochhammer__product_H,axiom,
% 9.20/5.23      ! [Z2: int,N2: nat,M: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.23        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8788_pochhammer__product_H,axiom,
% 9.20/5.23      ! [Z2: real,N2: nat,M: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.23        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8789_pochhammer__product_H,axiom,
% 9.20/5.23      ! [Z2: nat,N2: nat,M: nat] :
% 9.20/5.23        ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
% 9.20/5.23        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8790_binomial__ge__n__over__k__pow__k,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ord_less_eq_real @ ( power_power_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ K ) ) @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8791_binomial__ge__n__over__k__pow__k,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8792_binomial__maximum_H,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ K ) @ ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8793_binomial__mono,axiom,
% 9.20/5.23      ! [K: nat,K6: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ K6 )
% 9.20/5.23       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 9.20/5.23         => ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8794_binomial__antimono,axiom,
% 9.20/5.23      ! [K: nat,K6: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ K6 )
% 9.20/5.23       => ( ( ord_less_eq_nat @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ K )
% 9.20/5.23         => ( ( ord_less_eq_nat @ K6 @ N2 )
% 9.20/5.23           => ( ord_less_eq_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8795_binomial__maximum,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8796_binomial__le__pow2,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] : ( ord_less_eq_nat @ ( binomial @ N2 @ K ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8797_choose__reduce__nat,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23         => ( ( binomial @ N2 @ K )
% 9.20/5.23            = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8798_times__binomial__minus1__eq,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( times_times_nat @ K @ ( binomial @ N2 @ K ) )
% 9.20/5.23          = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8799_Suc__times__gbinomial,axiom,
% 9.20/5.23      ! [K: nat,A: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8800_Suc__times__gbinomial,axiom,
% 9.20/5.23      ! [K: nat,A: rat] :
% 9.20/5.23        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8801_Suc__times__gbinomial,axiom,
% 9.20/5.23      ! [K: nat,A: real] :
% 9.20/5.23        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_real @ ( plus_plus_real @ A @ one_one_real ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8802_gbinomial__absorption,axiom,
% 9.20/5.23      ! [K: nat,A: complex] :
% 9.20/5.23        ( ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) @ ( gbinomial_complex @ A @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8803_gbinomial__absorption,axiom,
% 9.20/5.23      ! [K: nat,A: rat] :
% 9.20/5.23        ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8804_gbinomial__absorption,axiom,
% 9.20/5.23      ! [K: nat,A: real] :
% 9.20/5.23        ( ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) @ ( gbinomial_real @ A @ ( suc @ K ) ) )
% 9.20/5.23        = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8805_binomial__altdef__nat,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( binomial @ N2 @ K )
% 9.20/5.23          = ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( times_times_nat @ ( semiri1408675320244567234ct_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8806_gbinomial__trinomial__revision,axiom,
% 9.20/5.23      ! [K: nat,M: nat,A: complex] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.23       => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
% 9.20/5.23          = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8807_gbinomial__trinomial__revision,axiom,
% 9.20/5.23      ! [K: nat,M: nat,A: rat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.23       => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
% 9.20/5.23          = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8808_gbinomial__trinomial__revision,axiom,
% 9.20/5.23      ! [K: nat,M: nat,A: real] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ M )
% 9.20/5.23       => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
% 9.20/5.23          = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8809_pochhammer__product,axiom,
% 9.20/5.23      ! [M: nat,N2: nat,Z2: complex] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( comm_s2602460028002588243omplex @ Z2 @ N2 )
% 9.20/5.23          = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8810_pochhammer__product,axiom,
% 9.20/5.23      ! [M: nat,N2: nat,Z2: rat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( comm_s4028243227959126397er_rat @ Z2 @ N2 )
% 9.20/5.23          = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8811_pochhammer__product,axiom,
% 9.20/5.23      ! [M: nat,N2: nat,Z2: int] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( comm_s4660882817536571857er_int @ Z2 @ N2 )
% 9.20/5.23          = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8812_pochhammer__product,axiom,
% 9.20/5.23      ! [M: nat,N2: nat,Z2: real] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( comm_s7457072308508201937r_real @ Z2 @ N2 )
% 9.20/5.23          = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8813_pochhammer__product,axiom,
% 9.20/5.23      ! [M: nat,N2: nat,Z2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.23       => ( ( comm_s4663373288045622133er_nat @ Z2 @ N2 )
% 9.20/5.23          = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8814_binomial__strict__antimono,axiom,
% 9.20/5.23      ! [K: nat,K6: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ K @ K6 )
% 9.20/5.23       => ( ( ord_less_eq_nat @ N2 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) )
% 9.20/5.23         => ( ( ord_less_eq_nat @ K6 @ N2 )
% 9.20/5.23           => ( ord_less_nat @ ( binomial @ N2 @ K6 ) @ ( binomial @ N2 @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8815_binomial__strict__mono,axiom,
% 9.20/5.23      ! [K: nat,K6: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ K @ K6 )
% 9.20/5.23       => ( ( ord_less_eq_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K6 ) @ N2 )
% 9.20/5.23         => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ K6 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8816_binomial__less__binomial__Suc,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_nat @ K @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.23       => ( ord_less_nat @ ( binomial @ N2 @ K ) @ ( binomial @ N2 @ ( suc @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8817_central__binomial__odd,axiom,
% 9.20/5.23      ! [N2: nat] :
% 9.20/5.23        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.23       => ( ( binomial @ N2 @ ( suc @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.23          = ( binomial @ N2 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8818_binomial__addition__formula,axiom,
% 9.20/5.23      ! [N2: nat,K: nat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.23       => ( ( binomial @ N2 @ ( suc @ K ) )
% 9.20/5.23          = ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8819_binomial__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) )
% 9.20/5.23          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8820_binomial__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
% 9.20/5.23          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8821_binomial__fact,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) )
% 9.20/5.23          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8822_fact__binomial,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( times_times_complex @ ( semiri5044797733671781792omplex @ K ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K ) ) )
% 9.20/5.23          = ( divide1717551699836669952omplex @ ( semiri5044797733671781792omplex @ N2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8823_fact__binomial,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
% 9.20/5.23          = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8824_fact__binomial,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( ord_less_eq_nat @ K @ N2 )
% 9.20/5.23       => ( ( times_times_real @ ( semiri2265585572941072030t_real @ K ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K ) ) )
% 9.20/5.23          = ( divide_divide_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8825_gbinomial__rec,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8826_gbinomial__rec,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8827_gbinomial__rec,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8828_gbinomial__factors,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ one_one_complex ) @ ( semiri8010041392384452111omplex @ ( suc @ K ) ) ) @ ( gbinomial_complex @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8829_gbinomial__factors,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8830_gbinomial__factors,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ ( plus_plus_real @ A @ one_one_real ) @ ( suc @ K ) )
% 9.20/5.23        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ A @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( suc @ K ) ) ) @ ( gbinomial_real @ A @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8831_pochhammer__absorb__comp,axiom,
% 9.20/5.23      ! [R2: complex,K: nat] :
% 9.20/5.23        ( ( times_times_complex @ ( minus_minus_complex @ R2 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R2 ) @ K ) )
% 9.20/5.23        = ( times_times_complex @ R2 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R2 ) @ one_one_complex ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8832_pochhammer__absorb__comp,axiom,
% 9.20/5.23      ! [R2: rat,K: nat] :
% 9.20/5.23        ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
% 9.20/5.23        = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8833_pochhammer__absorb__comp,axiom,
% 9.20/5.23      ! [R2: code_integer,K: nat] :
% 9.20/5.23        ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
% 9.20/5.23        = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8834_pochhammer__absorb__comp,axiom,
% 9.20/5.23      ! [R2: int,K: nat] :
% 9.20/5.23        ( ( times_times_int @ ( minus_minus_int @ R2 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R2 ) @ K ) )
% 9.20/5.23        = ( times_times_int @ R2 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R2 ) @ one_one_int ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8835_pochhammer__absorb__comp,axiom,
% 9.20/5.23      ! [R2: real,K: nat] :
% 9.20/5.23        ( ( times_times_real @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R2 ) @ K ) )
% 9.20/5.23        = ( times_times_real @ R2 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R2 ) @ one_one_real ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8836_gbinomial__minus,axiom,
% 9.20/5.23      ! [A: complex,K: nat] :
% 9.20/5.23        ( ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K )
% 9.20/5.23        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8837_gbinomial__minus,axiom,
% 9.20/5.23      ! [A: rat,K: nat] :
% 9.20/5.23        ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
% 9.20/5.23        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8838_gbinomial__minus,axiom,
% 9.20/5.23      ! [A: real,K: nat] :
% 9.20/5.23        ( ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K )
% 9.20/5.23        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8839_gbinomial__reduce__nat,axiom,
% 9.20/5.23      ! [K: nat,A: complex] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_complex @ A @ K )
% 9.20/5.23          = ( 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 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8840_gbinomial__reduce__nat,axiom,
% 9.20/5.23      ! [K: nat,A: real] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_real @ A @ K )
% 9.20/5.23          = ( 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 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8841_gbinomial__reduce__nat,axiom,
% 9.20/5.23      ! [K: nat,A: rat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_rat @ A @ K )
% 9.20/5.23          = ( 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 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8842_pochhammer__minus,axiom,
% 9.20/5.23      ! [B: complex,K: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
% 9.20/5.23        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8843_pochhammer__minus,axiom,
% 9.20/5.23      ! [B: rat,K: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
% 9.20/5.23        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8844_pochhammer__minus,axiom,
% 9.20/5.23      ! [B: code_integer,K: nat] :
% 9.20/5.23        ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
% 9.20/5.23        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8845_pochhammer__minus,axiom,
% 9.20/5.23      ! [B: int,K: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
% 9.20/5.23        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8846_pochhammer__minus,axiom,
% 9.20/5.23      ! [B: real,K: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
% 9.20/5.23        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8847_pochhammer__minus_H,axiom,
% 9.20/5.23      ! [B: complex,K: nat] :
% 9.20/5.23        ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
% 9.20/5.23        = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8848_pochhammer__minus_H,axiom,
% 9.20/5.23      ! [B: rat,K: nat] :
% 9.20/5.23        ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
% 9.20/5.23        = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8849_pochhammer__minus_H,axiom,
% 9.20/5.23      ! [B: code_integer,K: nat] :
% 9.20/5.23        ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
% 9.20/5.23        = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8850_pochhammer__minus_H,axiom,
% 9.20/5.23      ! [B: int,K: nat] :
% 9.20/5.23        ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
% 9.20/5.23        = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8851_pochhammer__minus_H,axiom,
% 9.20/5.23      ! [B: real,K: nat] :
% 9.20/5.23        ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
% 9.20/5.23        = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8852_gbinomial__sum__up__index,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( groups2073611262835488442omplex
% 9.20/5.23          @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ K )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.23        = ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8853_gbinomial__sum__up__index,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( groups2906978787729119204at_rat
% 9.20/5.23          @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.23        = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8854_gbinomial__sum__up__index,axiom,
% 9.20/5.23      ! [K: nat,N2: nat] :
% 9.20/5.23        ( ( groups6591440286371151544t_real
% 9.20/5.23          @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ K )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.23        = ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8855_gbinomial__absorption_H,axiom,
% 9.20/5.23      ! [K: nat,A: complex] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_complex @ A @ K )
% 9.20/5.23          = ( times_times_complex @ ( divide1717551699836669952omplex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8856_gbinomial__absorption_H,axiom,
% 9.20/5.23      ! [K: nat,A: rat] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_rat @ A @ K )
% 9.20/5.23          = ( 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 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8857_gbinomial__absorption_H,axiom,
% 9.20/5.23      ! [K: nat,A: real] :
% 9.20/5.23        ( ( ord_less_nat @ zero_zero_nat @ K )
% 9.20/5.23       => ( ( gbinomial_real @ A @ K )
% 9.20/5.23          = ( 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 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8858_binomial__code,axiom,
% 9.20/5.23      ( binomial
% 9.20/5.23      = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8859_gbinomial__code,axiom,
% 9.20/5.23      ( gbinomial_complex
% 9.20/5.23      = ( ^ [A3: complex,K3: nat] :
% 9.20/5.23            ( if_complex @ ( K3 = zero_zero_nat ) @ one_one_complex
% 9.20/5.23            @ ( divide1717551699836669952omplex
% 9.20/5.23              @ ( set_fo1517530859248394432omplex
% 9.20/5.23                @ ^ [L2: nat] : ( times_times_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ L2 ) ) )
% 9.20/5.23                @ zero_zero_nat
% 9.20/5.23                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.20/5.23                @ one_one_complex )
% 9.20/5.23              @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8860_gbinomial__code,axiom,
% 9.20/5.23      ( gbinomial_rat
% 9.20/5.23      = ( ^ [A3: rat,K3: nat] :
% 9.20/5.23            ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
% 9.20/5.23            @ ( divide_divide_rat
% 9.20/5.23              @ ( set_fo1949268297981939178at_rat
% 9.20/5.23                @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
% 9.20/5.23                @ zero_zero_nat
% 9.20/5.23                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.20/5.23                @ one_one_rat )
% 9.20/5.23              @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8861_gbinomial__code,axiom,
% 9.20/5.23      ( gbinomial_real
% 9.20/5.23      = ( ^ [A3: real,K3: nat] :
% 9.20/5.23            ( if_real @ ( K3 = zero_zero_nat ) @ one_one_real
% 9.20/5.23            @ ( divide_divide_real
% 9.20/5.23              @ ( set_fo3111899725591712190t_real
% 9.20/5.23                @ ^ [L2: nat] : ( times_times_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ L2 ) ) )
% 9.20/5.23                @ zero_zero_nat
% 9.20/5.23                @ ( minus_minus_nat @ K3 @ one_one_nat )
% 9.20/5.23                @ one_one_real )
% 9.20/5.23              @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8862_pochhammer__times__pochhammer__half,axiom,
% 9.20/5.23      ! [Z2: complex,N2: nat] :
% 9.20/5.23        ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 9.20/5.23        = ( groups6464643781859351333omplex
% 9.20/5.23          @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8863_pochhammer__times__pochhammer__half,axiom,
% 9.20/5.23      ! [Z2: rat,N2: nat] :
% 9.20/5.23        ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 9.20/5.23        = ( groups73079841787564623at_rat
% 9.20/5.23          @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8864_pochhammer__times__pochhammer__half,axiom,
% 9.20/5.23      ! [Z2: real,N2: nat] :
% 9.20/5.23        ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
% 9.20/5.23        = ( groups129246275422532515t_real
% 9.20/5.23          @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.23          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8865_pochhammer__code,axiom,
% 9.20/5.23      ( comm_s2602460028002588243omplex
% 9.20/5.23      = ( ^ [A3: complex,N: nat] :
% 9.20/5.23            ( if_complex @ ( N = zero_zero_nat ) @ one_one_complex
% 9.20/5.23            @ ( set_fo1517530859248394432omplex
% 9.20/5.23              @ ^ [O: nat] : ( times_times_complex @ ( plus_plus_complex @ A3 @ ( semiri8010041392384452111omplex @ O ) ) )
% 9.20/5.23              @ zero_zero_nat
% 9.20/5.23              @ ( minus_minus_nat @ N @ one_one_nat )
% 9.20/5.23              @ one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8866_pochhammer__code,axiom,
% 9.20/5.23      ( comm_s4028243227959126397er_rat
% 9.20/5.23      = ( ^ [A3: rat,N: nat] :
% 9.20/5.23            ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
% 9.20/5.23            @ ( set_fo1949268297981939178at_rat
% 9.20/5.23              @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
% 9.20/5.23              @ zero_zero_nat
% 9.20/5.23              @ ( minus_minus_nat @ N @ one_one_nat )
% 9.20/5.23              @ one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8867_pochhammer__code,axiom,
% 9.20/5.23      ( comm_s4660882817536571857er_int
% 9.20/5.23      = ( ^ [A3: int,N: nat] :
% 9.20/5.23            ( if_int @ ( N = zero_zero_nat ) @ one_one_int
% 9.20/5.23            @ ( set_fo2581907887559384638at_int
% 9.20/5.23              @ ^ [O: nat] : ( times_times_int @ ( plus_plus_int @ A3 @ ( semiri1314217659103216013at_int @ O ) ) )
% 9.20/5.23              @ zero_zero_nat
% 9.20/5.23              @ ( minus_minus_nat @ N @ one_one_nat )
% 9.20/5.23              @ one_one_int ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8868_pochhammer__code,axiom,
% 9.20/5.23      ( comm_s7457072308508201937r_real
% 9.20/5.23      = ( ^ [A3: real,N: nat] :
% 9.20/5.23            ( if_real @ ( N = zero_zero_nat ) @ one_one_real
% 9.20/5.23            @ ( set_fo3111899725591712190t_real
% 9.20/5.23              @ ^ [O: nat] : ( times_times_real @ ( plus_plus_real @ A3 @ ( semiri5074537144036343181t_real @ O ) ) )
% 9.20/5.23              @ zero_zero_nat
% 9.20/5.23              @ ( minus_minus_nat @ N @ one_one_nat )
% 9.20/5.23              @ one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8869_pochhammer__code,axiom,
% 9.20/5.23      ( comm_s4663373288045622133er_nat
% 9.20/5.23      = ( ^ [A3: nat,N: nat] :
% 9.20/5.23            ( if_nat @ ( N = zero_zero_nat ) @ one_one_nat
% 9.20/5.23            @ ( set_fo2584398358068434914at_nat
% 9.20/5.23              @ ^ [O: nat] : ( times_times_nat @ ( plus_plus_nat @ A3 @ ( semiri1316708129612266289at_nat @ O ) ) )
% 9.20/5.23              @ zero_zero_nat
% 9.20/5.23              @ ( minus_minus_nat @ N @ one_one_nat )
% 9.20/5.23              @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8870_gbinomial__partial__row__sum,axiom,
% 9.20/5.23      ! [A: complex,M: nat] :
% 9.20/5.23        ( ( groups2073611262835488442omplex
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
% 9.20/5.23          @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.23        = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8871_gbinomial__partial__row__sum,axiom,
% 9.20/5.23      ! [A: rat,M: nat] :
% 9.20/5.23        ( ( groups2906978787729119204at_rat
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
% 9.20/5.23          @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.23        = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8872_gbinomial__partial__row__sum,axiom,
% 9.20/5.23      ! [A: real,M: nat] :
% 9.20/5.23        ( ( groups6591440286371151544t_real
% 9.20/5.23          @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
% 9.20/5.23          @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.23        = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8873_atMost__iff,axiom,
% 9.20/5.23      ! [I: real,K: real] :
% 9.20/5.23        ( ( member_real @ I @ ( set_ord_atMost_real @ K ) )
% 9.20/5.23        = ( ord_less_eq_real @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8874_atMost__iff,axiom,
% 9.20/5.23      ! [I: set_nat,K: set_nat] :
% 9.20/5.23        ( ( member_set_nat @ I @ ( set_or4236626031148496127et_nat @ K ) )
% 9.20/5.23        = ( ord_less_eq_set_nat @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8875_atMost__iff,axiom,
% 9.20/5.23      ! [I: set_int,K: set_int] :
% 9.20/5.23        ( ( member_set_int @ I @ ( set_or58775011639299419et_int @ K ) )
% 9.20/5.23        = ( ord_less_eq_set_int @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8876_atMost__iff,axiom,
% 9.20/5.23      ! [I: rat,K: rat] :
% 9.20/5.23        ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
% 9.20/5.23        = ( ord_less_eq_rat @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8877_atMost__iff,axiom,
% 9.20/5.23      ! [I: num,K: num] :
% 9.20/5.23        ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
% 9.20/5.23        = ( ord_less_eq_num @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8878_atMost__iff,axiom,
% 9.20/5.23      ! [I: nat,K: nat] :
% 9.20/5.23        ( ( member_nat @ I @ ( set_ord_atMost_nat @ K ) )
% 9.20/5.23        = ( ord_less_eq_nat @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8879_atMost__iff,axiom,
% 9.20/5.23      ! [I: int,K: int] :
% 9.20/5.23        ( ( member_int @ I @ ( set_ord_atMost_int @ K ) )
% 9.20/5.23        = ( ord_less_eq_int @ I @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8880_finite__atMost,axiom,
% 9.20/5.23      ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8881_of__nat__prod,axiom,
% 9.20/5.23      ! [F: int > nat,A2: set_int] :
% 9.20/5.23        ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
% 9.20/5.23        = ( groups1705073143266064639nt_int
% 9.20/5.23          @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8882_of__nat__prod,axiom,
% 9.20/5.23      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.23        ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 9.20/5.23        = ( groups129246275422532515t_real
% 9.20/5.23          @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8883_of__nat__prod,axiom,
% 9.20/5.23      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.23        ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 9.20/5.23        = ( groups708209901874060359at_nat
% 9.20/5.23          @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8884_of__nat__prod,axiom,
% 9.20/5.23      ! [F: nat > nat,A2: set_nat] :
% 9.20/5.23        ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
% 9.20/5.23        = ( groups705719431365010083at_int
% 9.20/5.23          @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8885_of__int__prod,axiom,
% 9.20/5.23      ! [F: nat > int,A2: set_nat] :
% 9.20/5.23        ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
% 9.20/5.23        = ( groups129246275422532515t_real
% 9.20/5.23          @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8886_of__int__prod,axiom,
% 9.20/5.23      ! [F: nat > int,A2: set_nat] :
% 9.20/5.23        ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
% 9.20/5.23        = ( groups73079841787564623at_rat
% 9.20/5.23          @ ^ [X: nat] : ( ring_1_of_int_rat @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8887_of__int__prod,axiom,
% 9.20/5.23      ! [F: nat > int,A2: set_nat] :
% 9.20/5.23        ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
% 9.20/5.23        = ( groups705719431365010083at_int
% 9.20/5.23          @ ^ [X: nat] : ( ring_1_of_int_int @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8888_of__int__prod,axiom,
% 9.20/5.23      ! [F: int > int,A2: set_int] :
% 9.20/5.23        ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 9.20/5.23        = ( groups2316167850115554303t_real
% 9.20/5.23          @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8889_of__int__prod,axiom,
% 9.20/5.23      ! [F: int > int,A2: set_int] :
% 9.20/5.23        ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 9.20/5.23        = ( groups1072433553688619179nt_rat
% 9.20/5.23          @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8890_of__int__prod,axiom,
% 9.20/5.23      ! [F: int > int,A2: set_int] :
% 9.20/5.23        ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
% 9.20/5.23        = ( groups1705073143266064639nt_int
% 9.20/5.23          @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
% 9.20/5.23          @ A2 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8891_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( ( groups6464643781859351333omplex @ F @ A2 )
% 9.20/5.23            = zero_zero_complex )
% 9.20/5.23          = ( ? [X: nat] :
% 9.20/5.23                ( ( member_nat @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_complex ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8892_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_int,F: int > complex] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( ( groups7440179247065528705omplex @ F @ A2 )
% 9.20/5.23            = zero_zero_complex )
% 9.20/5.23          = ( ? [X: int] :
% 9.20/5.23                ( ( member_int @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_complex ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8893_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_complex,F: complex > complex] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( ( groups3708469109370488835omplex @ F @ A2 )
% 9.20/5.23            = zero_zero_complex )
% 9.20/5.23          = ( ? [X: complex] :
% 9.20/5.23                ( ( member_complex @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_complex ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8894_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( ( groups129246275422532515t_real @ F @ A2 )
% 9.20/5.23            = zero_zero_real )
% 9.20/5.23          = ( ? [X: nat] :
% 9.20/5.23                ( ( member_nat @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_real ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8895_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_int,F: int > real] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( ( groups2316167850115554303t_real @ F @ A2 )
% 9.20/5.23            = zero_zero_real )
% 9.20/5.23          = ( ? [X: int] :
% 9.20/5.23                ( ( member_int @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_real ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8896_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_complex,F: complex > real] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( ( groups766887009212190081x_real @ F @ A2 )
% 9.20/5.23            = zero_zero_real )
% 9.20/5.23          = ( ? [X: complex] :
% 9.20/5.23                ( ( member_complex @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_real ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8897_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_nat,F: nat > rat] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( ( groups73079841787564623at_rat @ F @ A2 )
% 9.20/5.23            = zero_zero_rat )
% 9.20/5.23          = ( ? [X: nat] :
% 9.20/5.23                ( ( member_nat @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8898_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_int,F: int > rat] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
% 9.20/5.23            = zero_zero_rat )
% 9.20/5.23          = ( ? [X: int] :
% 9.20/5.23                ( ( member_int @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8899_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_complex,F: complex > rat] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
% 9.20/5.23            = zero_zero_rat )
% 9.20/5.23          = ( ? [X: complex] :
% 9.20/5.23                ( ( member_complex @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_rat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8900_prod__zero__iff,axiom,
% 9.20/5.23      ! [A2: set_int,F: int > nat] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
% 9.20/5.23            = zero_zero_nat )
% 9.20/5.23          = ( ? [X: int] :
% 9.20/5.23                ( ( member_int @ X @ A2 )
% 9.20/5.23                & ( ( F @ X )
% 9.20/5.23                  = zero_zero_nat ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8901_prod_Oempty,axiom,
% 9.20/5.23      ! [G: real > complex] :
% 9.20/5.23        ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8902_prod_Oempty,axiom,
% 9.20/5.23      ! [G: real > real] :
% 9.20/5.23        ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8903_prod_Oempty,axiom,
% 9.20/5.23      ! [G: real > rat] :
% 9.20/5.23        ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
% 9.20/5.23        = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8904_prod_Oempty,axiom,
% 9.20/5.23      ! [G: real > nat] :
% 9.20/5.23        ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
% 9.20/5.23        = one_one_nat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8905_prod_Oempty,axiom,
% 9.20/5.23      ! [G: real > int] :
% 9.20/5.23        ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
% 9.20/5.23        = one_one_int ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8906_prod_Oempty,axiom,
% 9.20/5.23      ! [G: nat > complex] :
% 9.20/5.23        ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8907_prod_Oempty,axiom,
% 9.20/5.23      ! [G: nat > real] :
% 9.20/5.23        ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8908_prod_Oempty,axiom,
% 9.20/5.23      ! [G: nat > rat] :
% 9.20/5.23        ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
% 9.20/5.23        = one_one_rat ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8909_prod_Oempty,axiom,
% 9.20/5.23      ! [G: int > complex] :
% 9.20/5.23        ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
% 9.20/5.23        = one_one_complex ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8910_prod_Oempty,axiom,
% 9.20/5.23      ! [G: int > real] :
% 9.20/5.23        ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
% 9.20/5.23        = one_one_real ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8911_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_nat,G: nat > complex] :
% 9.20/5.23        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( groups6464643781859351333omplex @ G @ A2 )
% 9.20/5.23          = one_one_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8912_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_int,G: int > complex] :
% 9.20/5.23        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( groups7440179247065528705omplex @ G @ A2 )
% 9.20/5.23          = one_one_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8913_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_complex,G: complex > complex] :
% 9.20/5.23        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( groups3708469109370488835omplex @ G @ A2 )
% 9.20/5.23          = one_one_complex ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8914_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_nat,G: nat > real] :
% 9.20/5.23        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( groups129246275422532515t_real @ G @ A2 )
% 9.20/5.23          = one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8915_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_int,G: int > real] :
% 9.20/5.23        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( groups2316167850115554303t_real @ G @ A2 )
% 9.20/5.23          = one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8916_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_complex,G: complex > real] :
% 9.20/5.23        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( groups766887009212190081x_real @ G @ A2 )
% 9.20/5.23          = one_one_real ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8917_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_nat,G: nat > rat] :
% 9.20/5.23        ( ~ ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( groups73079841787564623at_rat @ G @ A2 )
% 9.20/5.23          = one_one_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8918_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_int,G: int > rat] :
% 9.20/5.23        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( groups1072433553688619179nt_rat @ G @ A2 )
% 9.20/5.23          = one_one_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8919_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_complex,G: complex > rat] :
% 9.20/5.23        ( ~ ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( groups225925009352817453ex_rat @ G @ A2 )
% 9.20/5.23          = one_one_rat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8920_prod_Oinfinite,axiom,
% 9.20/5.23      ! [A2: set_int,G: int > nat] :
% 9.20/5.23        ( ~ ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( groups1707563613775114915nt_nat @ G @ A2 )
% 9.20/5.23          = one_one_nat ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8921_atMost__subset__iff,axiom,
% 9.20/5.23      ! [X3: set_int,Y3: set_int] :
% 9.20/5.23        ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y3 ) )
% 9.20/5.23        = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8922_atMost__subset__iff,axiom,
% 9.20/5.23      ! [X3: rat,Y3: rat] :
% 9.20/5.23        ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y3 ) )
% 9.20/5.23        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8923_atMost__subset__iff,axiom,
% 9.20/5.23      ! [X3: num,Y3: num] :
% 9.20/5.23        ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y3 ) )
% 9.20/5.23        = ( ord_less_eq_num @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8924_atMost__subset__iff,axiom,
% 9.20/5.23      ! [X3: nat,Y3: nat] :
% 9.20/5.23        ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y3 ) )
% 9.20/5.23        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8925_atMost__subset__iff,axiom,
% 9.20/5.23      ! [X3: int,Y3: int] :
% 9.20/5.23        ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y3 ) )
% 9.20/5.23        = ( ord_less_eq_int @ X3 @ Y3 ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8926_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,B: nat,F: real > nat] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8927_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_int,A: int,B: nat,F: int > nat] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( member_int @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8928_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,B: nat,F: complex > nat] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8929_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,B: int,F: real > int] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8930_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,B: int,F: complex > int] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8931_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,B: code_integer,F: real > code_integer] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8932_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( member_nat @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8933_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_int,A: int,B: code_integer,F: int > code_integer] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( member_int @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8934_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,B: code_integer,F: complex > code_integer] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_Code_integer @ B @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8935_dvd__prod__eqI,axiom,
% 9.20/5.23      ! [A2: set_nat,A: nat,B: nat,F: nat > nat] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( member_nat @ A @ A2 )
% 9.20/5.23         => ( ( B
% 9.20/5.23              = ( F @ A ) )
% 9.20/5.23           => ( dvd_dvd_nat @ B @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8936_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,F: real > nat] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8937_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_int,A: int,F: int > nat] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( member_int @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8938_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,F: complex > nat] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8939_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,F: real > int] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_int @ ( F @ A ) @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8940_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,F: complex > int] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_int @ ( F @ A ) @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8941_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_real,A: real,F: real > code_integer] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ( member_real @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8942_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_nat,A: nat,F: nat > code_integer] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( member_nat @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8943_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_int,A: int,F: int > code_integer] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ( member_int @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8944_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_complex,A: complex,F: complex > code_integer] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ( member_complex @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8945_dvd__prodI,axiom,
% 9.20/5.23      ! [A2: set_nat,A: nat,F: nat > nat] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ( member_nat @ A @ A2 )
% 9.20/5.23         => ( dvd_dvd_nat @ ( F @ A ) @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8946_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > complex] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups713298508707869441omplex
% 9.20/5.23                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups713298508707869441omplex
% 9.20/5.23                @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8947_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups6464643781859351333omplex
% 9.20/5.23                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups6464643781859351333omplex
% 9.20/5.23                @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8948_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_int,A: int,B: int > complex] :
% 9.20/5.23        ( ( finite_finite_int @ S3 )
% 9.20/5.23       => ( ( ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups7440179247065528705omplex
% 9.20/5.23                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups7440179247065528705omplex
% 9.20/5.23                @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8949_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_complex,A: complex,B: complex > complex] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.23       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups3708469109370488835omplex
% 9.20/5.23                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups3708469109370488835omplex
% 9.20/5.23                @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8950_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > real] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups1681761925125756287l_real
% 9.20/5.23                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups1681761925125756287l_real
% 9.20/5.23                @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8951_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups129246275422532515t_real
% 9.20/5.23                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups129246275422532515t_real
% 9.20/5.23                @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8952_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_int,A: int,B: int > real] :
% 9.20/5.23        ( ( finite_finite_int @ S3 )
% 9.20/5.23       => ( ( ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups2316167850115554303t_real
% 9.20/5.23                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups2316167850115554303t_real
% 9.20/5.23                @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8953_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_complex,A: complex,B: complex > real] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.23       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups766887009212190081x_real
% 9.20/5.23                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups766887009212190081x_real
% 9.20/5.23                @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8954_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > rat] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups4061424788464935467al_rat
% 9.20/5.23                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups4061424788464935467al_rat
% 9.20/5.23                @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8955_prod_Odelta,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > rat] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups73079841787564623at_rat
% 9.20/5.23                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups73079841787564623at_rat
% 9.20/5.23                @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8956_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > complex] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups713298508707869441omplex
% 9.20/5.23                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups713298508707869441omplex
% 9.20/5.23                @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8957_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups6464643781859351333omplex
% 9.20/5.23                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups6464643781859351333omplex
% 9.20/5.23                @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8958_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_int,A: int,B: int > complex] :
% 9.20/5.23        ( ( finite_finite_int @ S3 )
% 9.20/5.23       => ( ( ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups7440179247065528705omplex
% 9.20/5.23                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups7440179247065528705omplex
% 9.20/5.23                @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8959_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_complex,A: complex,B: complex > complex] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.23       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups3708469109370488835omplex
% 9.20/5.23                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups3708469109370488835omplex
% 9.20/5.23                @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_complex ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8960_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > real] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups1681761925125756287l_real
% 9.20/5.23                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups1681761925125756287l_real
% 9.20/5.23                @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8961_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups129246275422532515t_real
% 9.20/5.23                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups129246275422532515t_real
% 9.20/5.23                @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8962_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_int,A: int,B: int > real] :
% 9.20/5.23        ( ( finite_finite_int @ S3 )
% 9.20/5.23       => ( ( ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups2316167850115554303t_real
% 9.20/5.23                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_int @ A @ S3 )
% 9.20/5.23           => ( ( groups2316167850115554303t_real
% 9.20/5.23                @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8963_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_complex,A: complex,B: complex > real] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ S3 )
% 9.20/5.23       => ( ( ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups766887009212190081x_real
% 9.20/5.23                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_complex @ A @ S3 )
% 9.20/5.23           => ( ( groups766887009212190081x_real
% 9.20/5.23                @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_real ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8964_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_real,A: real,B: real > rat] :
% 9.20/5.23        ( ( finite_finite_real @ S3 )
% 9.20/5.23       => ( ( ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups4061424788464935467al_rat
% 9.20/5.23                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_real @ A @ S3 )
% 9.20/5.23           => ( ( groups4061424788464935467al_rat
% 9.20/5.23                @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8965_prod_Odelta_H,axiom,
% 9.20/5.23      ! [S3: set_nat,A: nat,B: nat > rat] :
% 9.20/5.23        ( ( finite_finite_nat @ S3 )
% 9.20/5.23       => ( ( ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups73079841787564623at_rat
% 9.20/5.23                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = ( B @ A ) ) )
% 9.20/5.23          & ( ~ ( member_nat @ A @ S3 )
% 9.20/5.23           => ( ( groups73079841787564623at_rat
% 9.20/5.23                @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
% 9.20/5.23                @ S3 )
% 9.20/5.23              = one_one_rat ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8966_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_real,X3: real,G: real > complex] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.23         => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_complex @ ( G @ X3 ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8967_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_nat,X3: nat,G: nat > complex] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.23         => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_complex @ ( G @ X3 ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8968_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_int,X3: int,G: int > complex] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.23         => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_complex @ ( G @ X3 ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8969_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_complex,X3: complex,G: complex > complex] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.23         => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_complex @ ( G @ X3 ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8970_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_real,X3: real,G: real > real] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.23         => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8971_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_nat,X3: nat,G: nat > real] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.23         => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8972_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_int,X3: int,G: int > real] :
% 9.20/5.23        ( ( finite_finite_int @ A2 )
% 9.20/5.23       => ( ~ ( member_int @ X3 @ A2 )
% 9.20/5.23         => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8973_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_complex,X3: complex,G: complex > real] :
% 9.20/5.23        ( ( finite3207457112153483333omplex @ A2 )
% 9.20/5.23       => ( ~ ( member_complex @ X3 @ A2 )
% 9.20/5.23         => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8974_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_real,X3: real,G: real > rat] :
% 9.20/5.23        ( ( finite_finite_real @ A2 )
% 9.20/5.23       => ( ~ ( member_real @ X3 @ A2 )
% 9.20/5.23         => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8975_prod_Oinsert,axiom,
% 9.20/5.23      ! [A2: set_nat,X3: nat,G: nat > rat] :
% 9.20/5.23        ( ( finite_finite_nat @ A2 )
% 9.20/5.23       => ( ~ ( member_nat @ X3 @ A2 )
% 9.20/5.23         => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
% 9.20/5.23            = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8976_Icc__subset__Iic__iff,axiom,
% 9.20/5.23      ! [L: set_int,H2: set_int,H3: set_int] :
% 9.20/5.23        ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H2 ) @ ( set_or58775011639299419et_int @ H3 ) )
% 9.20/5.23        = ( ~ ( ord_less_eq_set_int @ L @ H2 )
% 9.20/5.23          | ( ord_less_eq_set_int @ H2 @ H3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8977_Icc__subset__Iic__iff,axiom,
% 9.20/5.23      ! [L: rat,H2: rat,H3: rat] :
% 9.20/5.23        ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
% 9.20/5.23        = ( ~ ( ord_less_eq_rat @ L @ H2 )
% 9.20/5.23          | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8978_Icc__subset__Iic__iff,axiom,
% 9.20/5.23      ! [L: num,H2: num,H3: num] :
% 9.20/5.23        ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
% 9.20/5.23        = ( ~ ( ord_less_eq_num @ L @ H2 )
% 9.20/5.23          | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).
% 9.20/5.23  
% 9.20/5.23  thf(fact_8979_Icc__subset__Iic__iff,axiom,
% 9.20/5.23      ! [L: nat,H2: nat,H3: nat] :
% 9.20/5.23        ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H2 ) @ ( set_ord_atMost_nat @ H3 ) )
% 9.20/5.24        = ( ~ ( ord_less_eq_nat @ L @ H2 )
% 9.20/5.24          | ( ord_less_eq_nat @ H2 @ H3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8980_Icc__subset__Iic__iff,axiom,
% 9.20/5.24      ! [L: int,H2: int,H3: int] :
% 9.20/5.24        ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H2 ) @ ( set_ord_atMost_int @ H3 ) )
% 9.20/5.24        = ( ~ ( ord_less_eq_int @ L @ H2 )
% 9.20/5.24          | ( ord_less_eq_int @ H2 @ H3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8981_Icc__subset__Iic__iff,axiom,
% 9.20/5.24      ! [L: real,H2: real,H3: real] :
% 9.20/5.24        ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H2 ) @ ( set_ord_atMost_real @ H3 ) )
% 9.20/5.24        = ( ~ ( ord_less_eq_real @ L @ H2 )
% 9.20/5.24          | ( ord_less_eq_real @ H2 @ H3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8982_sum_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > rat,N2: nat] :
% 9.20/5.24        ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8983_sum_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > int,N2: nat] :
% 9.20/5.24        ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8984_sum_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > nat,N2: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8985_sum_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > real,N2: nat] :
% 9.20/5.24        ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8986_prod_OlessThan__Suc,axiom,
% 9.20/5.24      ! [G: nat > complex,N2: nat] :
% 9.20/5.24        ( ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8987_prod_OlessThan__Suc,axiom,
% 9.20/5.24      ! [G: nat > real,N2: nat] :
% 9.20/5.24        ( ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8988_prod_OlessThan__Suc,axiom,
% 9.20/5.24      ! [G: nat > rat,N2: nat] :
% 9.20/5.24        ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8989_prod_OlessThan__Suc,axiom,
% 9.20/5.24      ! [G: nat > nat,N2: nat] :
% 9.20/5.24        ( ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8990_prod_OlessThan__Suc,axiom,
% 9.20/5.24      ! [G: nat > int,N2: nat] :
% 9.20/5.24        ( ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8991_prod_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > complex,N2: nat] :
% 9.20/5.24        ( ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8992_prod_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > real,N2: nat] :
% 9.20/5.24        ( ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8993_prod_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > rat,N2: nat] :
% 9.20/5.24        ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8994_prod_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > nat,N2: nat] :
% 9.20/5.24        ( ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8995_prod_OatMost__Suc,axiom,
% 9.20/5.24      ! [G: nat > int,N2: nat] :
% 9.20/5.24        ( ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
% 9.20/5.24        = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8996_atMost__0,axiom,
% 9.20/5.24      ( ( set_ord_atMost_nat @ zero_zero_nat )
% 9.20/5.24      = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8997_prod_Ocl__ivl__Suc,axiom,
% 9.20/5.24      ! [N2: nat,M: nat,G: nat > complex] :
% 9.20/5.24        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = one_one_complex ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8998_prod_Ocl__ivl__Suc,axiom,
% 9.20/5.24      ! [N2: nat,M: nat,G: nat > real] :
% 9.20/5.24        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = one_one_real ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_8999_prod_Ocl__ivl__Suc,axiom,
% 9.20/5.24      ! [N2: nat,M: nat,G: nat > rat] :
% 9.20/5.24        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = one_one_rat ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9000_prod_Ocl__ivl__Suc,axiom,
% 9.20/5.24      ! [N2: nat,M: nat,G: nat > nat] :
% 9.20/5.24        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = one_one_nat ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9001_prod_Ocl__ivl__Suc,axiom,
% 9.20/5.24      ! [N2: nat,M: nat,G: nat > int] :
% 9.20/5.24        ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = one_one_int ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24         => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
% 9.20/5.24            = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9002_atMost__atLeast0,axiom,
% 9.20/5.24      ( set_ord_atMost_nat
% 9.20/5.24      = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9003_lessThan__Suc__atMost,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( set_ord_lessThan_nat @ ( suc @ K ) )
% 9.20/5.24        = ( set_ord_atMost_nat @ K ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9004_atMost__Suc,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( set_ord_atMost_nat @ ( suc @ K ) )
% 9.20/5.24        = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9005_finite__nat__iff__bounded__le,axiom,
% 9.20/5.24      ( finite_finite_nat
% 9.20/5.24      = ( ^ [S7: set_nat] :
% 9.20/5.24          ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9006_sum__choose__upper,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [K3: nat] : ( binomial @ K3 @ M )
% 9.20/5.24          @ ( set_ord_atMost_nat @ N2 ) )
% 9.20/5.24        = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9007_sum__choose__lower,axiom,
% 9.20/5.24      ! [R2: nat,N2: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
% 9.20/5.24          @ ( set_ord_atMost_nat @ N2 ) )
% 9.20/5.24        = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9008_choose__rising__sum_I1_J,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 9.20/5.24          @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.24        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9009_choose__rising__sum_I2_J,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
% 9.20/5.24          @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.24        = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9010_atLeast1__atMost__eq__remove0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.24        = ( minus_minus_set_nat @ ( set_ord_atMost_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9011_fact__eq__fact__times,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.24       => ( ( semiri1408675320244567234ct_nat @ M )
% 9.20/5.24          = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
% 9.20/5.24            @ ( groups708209901874060359at_nat
% 9.20/5.24              @ ^ [X: nat] : X
% 9.20/5.24              @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9012_sum__choose__diagonal,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24       => ( ( groups3542108847815614940at_nat
% 9.20/5.24            @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
% 9.20/5.24            @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.24          = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9013_vandermonde,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,R2: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K3 ) ) )
% 9.20/5.24          @ ( set_ord_atMost_nat @ R2 ) )
% 9.20/5.24        = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9014_fact__div__fact,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.24       => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
% 9.20/5.24          = ( groups708209901874060359at_nat
% 9.20/5.24            @ ^ [X: nat] : X
% 9.20/5.24            @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9015_binomial,axiom,
% 9.20/5.24      ! [A: nat,B: nat,N2: nat] :
% 9.20/5.24        ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
% 9.20/5.24        = ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
% 9.20/5.24          @ ( set_ord_atMost_nat @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9016_polynomial__product__nat,axiom,
% 9.20/5.24      ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X3: nat] :
% 9.20/5.24        ( ! [I3: nat] :
% 9.20/5.24            ( ( ord_less_nat @ M @ I3 )
% 9.20/5.24           => ( ( A @ I3 )
% 9.20/5.24              = zero_zero_nat ) )
% 9.20/5.24       => ( ! [J2: nat] :
% 9.20/5.24              ( ( ord_less_nat @ N2 @ J2 )
% 9.20/5.24             => ( ( B @ J2 )
% 9.20/5.24                = zero_zero_nat ) )
% 9.20/5.24         => ( ( times_times_nat
% 9.20/5.24              @ ( groups3542108847815614940at_nat
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X3 @ I2 ) )
% 9.20/5.24                @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.24              @ ( groups3542108847815614940at_nat
% 9.20/5.24                @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
% 9.20/5.24                @ ( set_ord_atMost_nat @ N2 ) ) )
% 9.20/5.24            = ( groups3542108847815614940at_nat
% 9.20/5.24              @ ^ [R5: nat] :
% 9.20/5.24                  ( times_times_nat
% 9.20/5.24                  @ ( groups3542108847815614940at_nat
% 9.20/5.24                    @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
% 9.20/5.24                    @ ( set_ord_atMost_nat @ R5 ) )
% 9.20/5.24                  @ ( power_power_nat @ X3 @ R5 ) )
% 9.20/5.24              @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9017_binomial__r__part__sum,axiom,
% 9.20/5.24      ! [M: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
% 9.20/5.24        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9018_cot__less__zero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.24         => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9019_int__ge__less__than2__def,axiom,
% 9.20/5.24      ( int_ge_less_than2
% 9.20/5.24      = ( ^ [D5: int] :
% 9.20/5.24            ( collec213857154873943460nt_int
% 9.20/5.24            @ ( produc4947309494688390418_int_o
% 9.20/5.24              @ ^ [Z7: int,Z6: int] :
% 9.20/5.24                  ( ( ord_less_eq_int @ D5 @ Z6 )
% 9.20/5.24                  & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9020_of__nat__id,axiom,
% 9.20/5.24      ( semiri1316708129612266289at_nat
% 9.20/5.24      = ( ^ [N: nat] : N ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9021_prod__int__plus__eq,axiom,
% 9.20/5.24      ! [I: nat,J: nat] :
% 9.20/5.24        ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
% 9.20/5.24        = ( groups1705073143266064639nt_int
% 9.20/5.24          @ ^ [X: int] : X
% 9.20/5.24          @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9022_cot__gt__zero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.24         => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9023_int__ge__less__than__def,axiom,
% 9.20/5.24      ( int_ge_less_than
% 9.20/5.24      = ( ^ [D5: int] :
% 9.20/5.24            ( collec213857154873943460nt_int
% 9.20/5.24            @ ( produc4947309494688390418_int_o
% 9.20/5.24              @ ^ [Z7: int,Z6: int] :
% 9.20/5.24                  ( ( ord_less_eq_int @ D5 @ Z7 )
% 9.20/5.24                  & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9024_arctan__def,axiom,
% 9.20/5.24      ( arctan
% 9.20/5.24      = ( ^ [Y: real] :
% 9.20/5.24            ( the_real
% 9.20/5.24            @ ^ [X: real] :
% 9.20/5.24                ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
% 9.20/5.24                & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
% 9.20/5.24                & ( ( tan_real @ X )
% 9.20/5.24                  = Y ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9025_modulo__int__unfold,axiom,
% 9.20/5.24      ! [L: int,K: int,N2: nat,M: nat] :
% 9.20/5.24        ( ( ( ( ( sgn_sgn_int @ L )
% 9.20/5.24              = zero_zero_int )
% 9.20/5.24            | ( ( sgn_sgn_int @ K )
% 9.20/5.24              = zero_zero_int )
% 9.20/5.24            | ( N2 = zero_zero_nat ) )
% 9.20/5.24         => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24            = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
% 9.20/5.24        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.20/5.24                = zero_zero_int )
% 9.20/5.24              | ( ( sgn_sgn_int @ K )
% 9.20/5.24                = zero_zero_int )
% 9.20/5.24              | ( N2 = zero_zero_nat ) )
% 9.20/5.24         => ( ( ( ( sgn_sgn_int @ K )
% 9.20/5.24                = ( sgn_sgn_int @ L ) )
% 9.20/5.24             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24                = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
% 9.20/5.24            & ( ( ( sgn_sgn_int @ K )
% 9.20/5.24               != ( sgn_sgn_int @ L ) )
% 9.20/5.24             => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24                = ( times_times_int @ ( sgn_sgn_int @ L )
% 9.20/5.24                  @ ( minus_minus_int
% 9.20/5.24                    @ ( semiri1314217659103216013at_int
% 9.20/5.24                      @ ( times_times_nat @ N2
% 9.20/5.24                        @ ( zero_n2687167440665602831ol_nat
% 9.20/5.24                          @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
% 9.20/5.24                    @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9026_powr__int,axiom,
% 9.20/5.24      ! [X3: real,I: int] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
% 9.20/5.24           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 9.20/5.24              = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
% 9.20/5.24           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
% 9.20/5.24              = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9027_nat__int,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.24        = N2 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9028_nat__numeral,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( nat2 @ ( numeral_numeral_int @ K ) )
% 9.20/5.24        = ( numeral_numeral_nat @ K ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9029_nat__of__bool,axiom,
% 9.20/5.24      ! [P: $o] :
% 9.20/5.24        ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
% 9.20/5.24        = ( zero_n2687167440665602831ol_nat @ P ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9030_nat__1,axiom,
% 9.20/5.24      ( ( nat2 @ one_one_int )
% 9.20/5.24      = ( suc @ zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9031_nat__0__iff,axiom,
% 9.20/5.24      ! [I: int] :
% 9.20/5.24        ( ( ( nat2 @ I )
% 9.20/5.24          = zero_zero_nat )
% 9.20/5.24        = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9032_nat__le__0,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 9.20/5.24       => ( ( nat2 @ Z2 )
% 9.20/5.24          = zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9033_zless__nat__conj,axiom,
% 9.20/5.24      ! [W2: int,Z2: int] :
% 9.20/5.24        ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.20/5.24        = ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.24          & ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9034_nat__neg__numeral,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.24        = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9035_nat__zminus__int,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24        = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9036_int__nat__eq,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 9.20/5.24            = Z2 ) )
% 9.20/5.24        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24         => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 9.20/5.24            = zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9037_sgn__mult__dvd__iff,axiom,
% 9.20/5.24      ! [R2: int,L: int,K: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ L ) @ K )
% 9.20/5.24        = ( ( dvd_dvd_int @ L @ K )
% 9.20/5.24          & ( ( R2 = zero_zero_int )
% 9.20/5.24           => ( K = zero_zero_int ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9038_mult__sgn__dvd__iff,axiom,
% 9.20/5.24      ! [L: int,R2: int,K: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ ( times_times_int @ L @ ( sgn_sgn_int @ R2 ) ) @ K )
% 9.20/5.24        = ( ( dvd_dvd_int @ L @ K )
% 9.20/5.24          & ( ( R2 = zero_zero_int )
% 9.20/5.24           => ( K = zero_zero_int ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9039_dvd__sgn__mult__iff,axiom,
% 9.20/5.24      ! [L: int,R2: int,K: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ L @ ( times_times_int @ ( sgn_sgn_int @ R2 ) @ K ) )
% 9.20/5.24        = ( ( dvd_dvd_int @ L @ K )
% 9.20/5.24          | ( R2 = zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9040_dvd__mult__sgn__iff,axiom,
% 9.20/5.24      ! [L: int,K: int,R2: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ L @ ( times_times_int @ K @ ( sgn_sgn_int @ R2 ) ) )
% 9.20/5.24        = ( ( dvd_dvd_int @ L @ K )
% 9.20/5.24          | ( R2 = zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9041_zero__less__nat__eq,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
% 9.20/5.24        = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9042_diff__nat__numeral,axiom,
% 9.20/5.24      ! [V: num,V3: num] :
% 9.20/5.24        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ ( numeral_numeral_nat @ V3 ) )
% 9.20/5.24        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ V3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9043_numeral__power__eq__nat__cancel__iff,axiom,
% 9.20/5.24      ! [X3: num,N2: nat,Y3: int] :
% 9.20/5.24        ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 )
% 9.20/5.24          = ( nat2 @ Y3 ) )
% 9.20/5.24        = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
% 9.20/5.24          = Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9044_nat__eq__numeral__power__cancel__iff,axiom,
% 9.20/5.24      ! [Y3: int,X3: num,N2: nat] :
% 9.20/5.24        ( ( ( nat2 @ Y3 )
% 9.20/5.24          = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 9.20/5.24        = ( Y3
% 9.20/5.24          = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9045_nat__abs__dvd__iff,axiom,
% 9.20/5.24      ! [K: int,N2: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
% 9.20/5.24        = ( dvd_dvd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9046_dvd__nat__abs__iff,axiom,
% 9.20/5.24      ! [N2: nat,K: int] :
% 9.20/5.24        ( ( dvd_dvd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 9.20/5.24        = ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9047_nat__ceiling__le__eq,axiom,
% 9.20/5.24      ! [X3: real,A: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
% 9.20/5.24        = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9048_one__less__nat__eq,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
% 9.20/5.24        = ( ord_less_int @ one_one_int @ Z2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9049_nat__numeral__diff__1,axiom,
% 9.20/5.24      ! [V: num] :
% 9.20/5.24        ( ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat )
% 9.20/5.24        = ( nat2 @ ( minus_minus_int @ ( numeral_numeral_int @ V ) @ one_one_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9050_nat__less__numeral__power__cancel__iff,axiom,
% 9.20/5.24      ! [A: int,X3: num,N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 9.20/5.24        = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9051_numeral__power__less__nat__cancel__iff,axiom,
% 9.20/5.24      ! [X3: num,N2: nat,A: int] :
% 9.20/5.24        ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
% 9.20/5.24        = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9052_nat__le__numeral__power__cancel__iff,axiom,
% 9.20/5.24      ! [A: int,X3: num,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
% 9.20/5.24        = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9053_numeral__power__le__nat__cancel__iff,axiom,
% 9.20/5.24      ! [X3: num,N2: nat,A: int] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
% 9.20/5.24        = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9054_nat__zero__as__int,axiom,
% 9.20/5.24      ( zero_zero_nat
% 9.20/5.24      = ( nat2 @ zero_zero_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9055_nat__mono,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ X3 @ Y3 )
% 9.20/5.24       => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9056_int__sgnE,axiom,
% 9.20/5.24      ! [K: int] :
% 9.20/5.24        ~ ! [N3: nat,L4: int] :
% 9.20/5.24            ( K
% 9.20/5.24           != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9057_ex__nat,axiom,
% 9.20/5.24      ( ( ^ [P2: nat > $o] :
% 9.20/5.24          ? [X6: nat] : ( P2 @ X6 ) )
% 9.20/5.24      = ( ^ [P3: nat > $o] :
% 9.20/5.24          ? [X: int] :
% 9.20/5.24            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.20/5.24            & ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9058_all__nat,axiom,
% 9.20/5.24      ( ( ^ [P2: nat > $o] :
% 9.20/5.24          ! [X6: nat] : ( P2 @ X6 ) )
% 9.20/5.24      = ( ^ [P3: nat > $o] :
% 9.20/5.24          ! [X: int] :
% 9.20/5.24            ( ( ord_less_eq_int @ zero_zero_int @ X )
% 9.20/5.24           => ( P3 @ ( nat2 @ X ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9059_eq__nat__nat__iff,axiom,
% 9.20/5.24      ! [Z2: int,Z8: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 9.20/5.24         => ( ( ( nat2 @ Z2 )
% 9.20/5.24              = ( nat2 @ Z8 ) )
% 9.20/5.24            = ( Z2 = Z8 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9060_nat__mono__iff,axiom,
% 9.20/5.24      ! [Z2: int,W2: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.20/5.24          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9061_zless__nat__eq__int__zless,axiom,
% 9.20/5.24      ! [M: nat,Z2: int] :
% 9.20/5.24        ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
% 9.20/5.24        = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9062_nat__le__iff,axiom,
% 9.20/5.24      ! [X3: int,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N2 )
% 9.20/5.24        = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9063_nat__0__le,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
% 9.20/5.24          = Z2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9064_int__eq__iff,axiom,
% 9.20/5.24      ! [M: nat,Z2: int] :
% 9.20/5.24        ( ( ( semiri1314217659103216013at_int @ M )
% 9.20/5.24          = Z2 )
% 9.20/5.24        = ( ( M
% 9.20/5.24            = ( nat2 @ Z2 ) )
% 9.20/5.24          & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9065_nat__int__add,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
% 9.20/5.24        = ( plus_plus_nat @ A @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9066_nat__abs__mult__distrib,axiom,
% 9.20/5.24      ! [W2: int,Z2: int] :
% 9.20/5.24        ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
% 9.20/5.24        = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9067_nat__plus__as__int,axiom,
% 9.20/5.24      ( plus_plus_nat
% 9.20/5.24      = ( ^ [A3: nat,B3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9068_zsgn__def,axiom,
% 9.20/5.24      ( sgn_sgn_int
% 9.20/5.24      = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9069_powr__real__of__int,axiom,
% 9.20/5.24      ! [X3: real,N2: int] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
% 9.20/5.24           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 9.20/5.24              = ( power_power_real @ X3 @ ( nat2 @ N2 ) ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
% 9.20/5.24           => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 9.20/5.24              = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9070_nat__less__eq__zless,axiom,
% 9.20/5.24      ! [W2: int,Z2: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.20/5.24          = ( ord_less_int @ W2 @ Z2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9071_nat__le__eq__zle,axiom,
% 9.20/5.24      ! [W2: int,Z2: int] :
% 9.20/5.24        ( ( ( ord_less_int @ zero_zero_int @ W2 )
% 9.20/5.24          | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
% 9.20/5.24       => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
% 9.20/5.24          = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9072_nat__eq__iff,axiom,
% 9.20/5.24      ! [W2: int,M: nat] :
% 9.20/5.24        ( ( ( nat2 @ W2 )
% 9.20/5.24          = M )
% 9.20/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24           => ( W2
% 9.20/5.24              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24           => ( M = zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9073_nat__eq__iff2,axiom,
% 9.20/5.24      ! [M: nat,W2: int] :
% 9.20/5.24        ( ( M
% 9.20/5.24          = ( nat2 @ W2 ) )
% 9.20/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24           => ( W2
% 9.20/5.24              = ( semiri1314217659103216013at_int @ M ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24           => ( M = zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9074_split__nat,axiom,
% 9.20/5.24      ! [P: nat > $o,I: int] :
% 9.20/5.24        ( ( P @ ( nat2 @ I ) )
% 9.20/5.24        = ( ! [N: nat] :
% 9.20/5.24              ( ( I
% 9.20/5.24                = ( semiri1314217659103216013at_int @ N ) )
% 9.20/5.24             => ( P @ N ) )
% 9.20/5.24          & ( ( ord_less_int @ I @ zero_zero_int )
% 9.20/5.24           => ( P @ zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9075_le__nat__iff,axiom,
% 9.20/5.24      ! [K: int,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ K )
% 9.20/5.24       => ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
% 9.20/5.24          = ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9076_nat__add__distrib,axiom,
% 9.20/5.24      ! [Z2: int,Z8: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 9.20/5.24         => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z8 ) )
% 9.20/5.24            = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9077_nat__mult__distrib,axiom,
% 9.20/5.24      ! [Z2: int,Z8: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
% 9.20/5.24          = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9078_Suc__as__int,axiom,
% 9.20/5.24      ( suc
% 9.20/5.24      = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9079_forall__pos__mono__1,axiom,
% 9.20/5.24      ! [P: real > $o,E2: real] :
% 9.20/5.24        ( ! [D4: real,E: real] :
% 9.20/5.24            ( ( ord_less_real @ D4 @ E )
% 9.20/5.24           => ( ( P @ D4 )
% 9.20/5.24             => ( P @ E ) ) )
% 9.20/5.24       => ( ! [N3: nat] : ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.24           => ( P @ E2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9080_forall__pos__mono,axiom,
% 9.20/5.24      ! [P: real > $o,E2: real] :
% 9.20/5.24        ( ! [D4: real,E: real] :
% 9.20/5.24            ( ( ord_less_real @ D4 @ E )
% 9.20/5.24           => ( ( P @ D4 )
% 9.20/5.24             => ( P @ E ) ) )
% 9.20/5.24       => ( ! [N3: nat] :
% 9.20/5.24              ( ( N3 != zero_zero_nat )
% 9.20/5.24             => ( P @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) ) )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.24           => ( P @ E2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9081_real__arch__inverse,axiom,
% 9.20/5.24      ! [E2: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ E2 )
% 9.20/5.24        = ( ? [N: nat] :
% 9.20/5.24              ( ( N != zero_zero_nat )
% 9.20/5.24              & ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.20/5.24              & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N ) ) @ E2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9082_nat__diff__distrib,axiom,
% 9.20/5.24      ! [Z8: int,Z2: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
% 9.20/5.24       => ( ( ord_less_eq_int @ Z8 @ Z2 )
% 9.20/5.24         => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) )
% 9.20/5.24            = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9083_nat__diff__distrib_H,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.24       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.24         => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
% 9.20/5.24            = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9084_nat__abs__triangle__ineq,axiom,
% 9.20/5.24      ! [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 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9085_nat__floor__neg,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.20/5.24       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.24          = zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9086_nat__power__eq,axiom,
% 9.20/5.24      ! [Z2: int,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( nat2 @ ( power_power_int @ Z2 @ N2 ) )
% 9.20/5.24          = ( power_power_nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9087_ln__inverse,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
% 9.20/5.24          = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9088_floor__eq3,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 9.20/5.24         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.24            = N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9089_le__nat__floor,axiom,
% 9.20/5.24      ! [X3: nat,A: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
% 9.20/5.24       => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9090_divide__int__def,axiom,
% 9.20/5.24      ( divide_divide_int
% 9.20/5.24      = ( ^ [K3: int,L2: int] :
% 9.20/5.24            ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
% 9.20/5.24            @ ( if_int
% 9.20/5.24              @ ( ( sgn_sgn_int @ K3 )
% 9.20/5.24                = ( sgn_sgn_int @ L2 ) )
% 9.20/5.24              @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
% 9.20/5.24              @ ( uminus_uminus_int
% 9.20/5.24                @ ( semiri1314217659103216013at_int
% 9.20/5.24                  @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
% 9.20/5.24                    @ ( zero_n2687167440665602831ol_nat
% 9.20/5.24                      @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9091_nat__2,axiom,
% 9.20/5.24      ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.24      = ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9092_Suc__nat__eq__nat__zadd1,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24       => ( ( suc @ ( nat2 @ Z2 ) )
% 9.20/5.24          = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9093_nat__less__iff,axiom,
% 9.20/5.24      ! [W2: int,M: nat] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ W2 )
% 9.20/5.24       => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
% 9.20/5.24          = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9094_nat__mult__distrib__neg,axiom,
% 9.20/5.24      ! [Z2: int,Z8: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
% 9.20/5.24       => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
% 9.20/5.24          = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9095_nat__abs__int__diff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.24         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.20/5.24            = ( minus_minus_nat @ B @ A ) ) )
% 9.20/5.24        & ( ~ ( ord_less_eq_nat @ A @ B )
% 9.20/5.24         => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
% 9.20/5.24            = ( minus_minus_nat @ A @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9096_log__inverse,axiom,
% 9.20/5.24      ! [A: real,X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.24       => ( ( A != one_one_real )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24           => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
% 9.20/5.24              = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9097_floor__eq4,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
% 9.20/5.24         => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
% 9.20/5.24            = N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9098_diff__nat__eq__if,axiom,
% 9.20/5.24      ! [Z8: int,Z2: int] :
% 9.20/5.24        ( ( ( ord_less_int @ Z8 @ zero_zero_int )
% 9.20/5.24         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 9.20/5.24            = ( nat2 @ Z2 ) ) )
% 9.20/5.24        & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
% 9.20/5.24         => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
% 9.20/5.24            = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9099_nat__dvd__iff,axiom,
% 9.20/5.24      ! [Z2: int,M: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
% 9.20/5.24        = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24           => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
% 9.20/5.24           => ( M = zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9100_eucl__rel__int__remainderI,axiom,
% 9.20/5.24      ! [R2: int,L: int,K: int,Q4: int] :
% 9.20/5.24        ( ( ( sgn_sgn_int @ R2 )
% 9.20/5.24          = ( sgn_sgn_int @ L ) )
% 9.20/5.24       => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
% 9.20/5.24         => ( ( K
% 9.20/5.24              = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
% 9.20/5.24           => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9101_plus__inverse__ge__2,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9102_real__inv__sqrt__pow2,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
% 9.20/5.24          = ( inverse_inverse_real @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9103_eucl__rel__int_Ocases,axiom,
% 9.20/5.24      ! [A12: int,A23: int,A32: product_prod_int_int] :
% 9.20/5.24        ( ( eucl_rel_int @ A12 @ A23 @ A32 )
% 9.20/5.24       => ( ( ( A23 = zero_zero_int )
% 9.20/5.24           => ( A32
% 9.20/5.24             != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
% 9.20/5.24         => ( ! [Q2: int] :
% 9.20/5.24                ( ( A32
% 9.20/5.24                  = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
% 9.20/5.24               => ( ( A23 != zero_zero_int )
% 9.20/5.24                 => ( A12
% 9.20/5.24                   != ( times_times_int @ Q2 @ A23 ) ) ) )
% 9.20/5.24           => ~ ! [R4: int,Q2: int] :
% 9.20/5.24                  ( ( A32
% 9.20/5.24                    = ( product_Pair_int_int @ Q2 @ R4 ) )
% 9.20/5.24                 => ( ( ( sgn_sgn_int @ R4 )
% 9.20/5.24                      = ( sgn_sgn_int @ A23 ) )
% 9.20/5.24                   => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
% 9.20/5.24                     => ( A12
% 9.20/5.24                       != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9104_eucl__rel__int_Osimps,axiom,
% 9.20/5.24      ( eucl_rel_int
% 9.20/5.24      = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
% 9.20/5.24            ( ? [K3: int] :
% 9.20/5.24                ( ( A1 = K3 )
% 9.20/5.24                & ( A22 = zero_zero_int )
% 9.20/5.24                & ( A33
% 9.20/5.24                  = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
% 9.20/5.24            | ? [L2: int,K3: int,Q5: int] :
% 9.20/5.24                ( ( A1 = K3 )
% 9.20/5.24                & ( A22 = L2 )
% 9.20/5.24                & ( A33
% 9.20/5.24                  = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
% 9.20/5.24                & ( L2 != zero_zero_int )
% 9.20/5.24                & ( K3
% 9.20/5.24                  = ( times_times_int @ Q5 @ L2 ) ) )
% 9.20/5.24            | ? [R5: int,L2: int,K3: int,Q5: int] :
% 9.20/5.24                ( ( A1 = K3 )
% 9.20/5.24                & ( A22 = L2 )
% 9.20/5.24                & ( A33
% 9.20/5.24                  = ( product_Pair_int_int @ Q5 @ R5 ) )
% 9.20/5.24                & ( ( sgn_sgn_int @ R5 )
% 9.20/5.24                  = ( sgn_sgn_int @ L2 ) )
% 9.20/5.24                & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
% 9.20/5.24                & ( K3
% 9.20/5.24                  = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9105_divide__int__unfold,axiom,
% 9.20/5.24      ! [L: int,K: int,N2: nat,M: nat] :
% 9.20/5.24        ( ( ( ( ( sgn_sgn_int @ L )
% 9.20/5.24              = zero_zero_int )
% 9.20/5.24            | ( ( sgn_sgn_int @ K )
% 9.20/5.24              = zero_zero_int )
% 9.20/5.24            | ( N2 = zero_zero_nat ) )
% 9.20/5.24         => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24            = zero_zero_int ) )
% 9.20/5.24        & ( ~ ( ( ( sgn_sgn_int @ L )
% 9.20/5.24                = zero_zero_int )
% 9.20/5.24              | ( ( sgn_sgn_int @ K )
% 9.20/5.24                = zero_zero_int )
% 9.20/5.24              | ( N2 = zero_zero_nat ) )
% 9.20/5.24         => ( ( ( ( sgn_sgn_int @ K )
% 9.20/5.24                = ( sgn_sgn_int @ L ) )
% 9.20/5.24             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24                = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
% 9.20/5.24            & ( ( ( sgn_sgn_int @ K )
% 9.20/5.24               != ( sgn_sgn_int @ L ) )
% 9.20/5.24             => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
% 9.20/5.24                = ( uminus_uminus_int
% 9.20/5.24                  @ ( semiri1314217659103216013at_int
% 9.20/5.24                    @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
% 9.20/5.24                      @ ( zero_n2687167440665602831ol_nat
% 9.20/5.24                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9106_sinh__ln__real,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.24          = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9107_arcsin__lt__bounded,axiom,
% 9.20/5.24      ! [Y3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.20/5.24       => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.20/5.24         => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
% 9.20/5.24            & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9108_sinh__real__less__iff,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
% 9.20/5.24        = ( ord_less_real @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9109_sinh__real__neg__iff,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
% 9.20/5.24        = ( ord_less_real @ X3 @ zero_zero_real ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9110_sinh__real__pos__iff,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
% 9.20/5.24        = ( ord_less_real @ zero_zero_real @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9111_sinh__less__cosh__real,axiom,
% 9.20/5.24      ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9112_cosh__real__pos,axiom,
% 9.20/5.24      ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9113_cosh__real__strict__mono,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.24         => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9114_cosh__real__nonneg__less__iff,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.24         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.20/5.24            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9115_cosh__real__nonpos__less__iff,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ X3 @ zero_zero_real )
% 9.20/5.24       => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
% 9.20/5.24         => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
% 9.20/5.24            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9116_sgn__real__def,axiom,
% 9.20/5.24      ( sgn_sgn_real
% 9.20/5.24      = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9117_arcsin__less__arcsin,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.24         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.20/5.24           => ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9118_arcsin__less__mono,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.24       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.20/5.24         => ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
% 9.20/5.24            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9119_sgn__power__injE,axiom,
% 9.20/5.24      ! [A: real,N2: nat,X3: real,B: real] :
% 9.20/5.24        ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
% 9.20/5.24          = X3 )
% 9.20/5.24       => ( ( X3
% 9.20/5.24            = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
% 9.20/5.24         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24           => ( A = B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9120_cos__arcsin__nonzero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( ( cos_real @ ( arcsin @ X3 ) )
% 9.20/5.24           != zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9121_floor__real__def,axiom,
% 9.20/5.24      ( archim6058952711729229775r_real
% 9.20/5.24      = ( ^ [X: real] :
% 9.20/5.24            ( the_int
% 9.20/5.24            @ ^ [Z6: int] :
% 9.20/5.24                ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z6 ) @ X )
% 9.20/5.24                & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9122_cosh__ln__real,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
% 9.20/5.24          = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9123_floor__rat__def,axiom,
% 9.20/5.24      ( archim3151403230148437115or_rat
% 9.20/5.24      = ( ^ [X: rat] :
% 9.20/5.24            ( the_int
% 9.20/5.24            @ ^ [Z6: int] :
% 9.20/5.24                ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X )
% 9.20/5.24                & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9124_VEBT_Osize_I3_J,axiom,
% 9.20/5.24      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.20/5.24        ( ( size_size_VEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.20/5.24        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ size_size_VEBT_VEBT @ X13 ) @ ( size_size_VEBT_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9125_sum__power2,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ K ) )
% 9.20/5.24        = ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K ) @ one_one_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9126_finite__atLeastLessThan,axiom,
% 9.20/5.24      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9127_atLeastLessThan__singleton,axiom,
% 9.20/5.24      ! [M: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
% 9.20/5.24        = ( insert_nat @ M @ bot_bot_set_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9128_sgn__rat__def,axiom,
% 9.20/5.24      ( sgn_sgn_rat
% 9.20/5.24      = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9129_abs__rat__def,axiom,
% 9.20/5.24      ( abs_abs_rat
% 9.20/5.24      = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9130_less__eq__rat__def,axiom,
% 9.20/5.24      ( ord_less_eq_rat
% 9.20/5.24      = ( ^ [X: rat,Y: rat] :
% 9.20/5.24            ( ( ord_less_rat @ X @ Y )
% 9.20/5.24            | ( X = Y ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9131_obtain__pos__sum,axiom,
% 9.20/5.24      ! [R2: rat] :
% 9.20/5.24        ( ( ord_less_rat @ zero_zero_rat @ R2 )
% 9.20/5.24       => ~ ! [S2: rat] :
% 9.20/5.24              ( ( ord_less_rat @ zero_zero_rat @ S2 )
% 9.20/5.24             => ! [T6: rat] :
% 9.20/5.24                  ( ( ord_less_rat @ zero_zero_rat @ T6 )
% 9.20/5.24                 => ( R2
% 9.20/5.24                   != ( plus_plus_rat @ S2 @ T6 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9132_ex__nat__less__eq,axiom,
% 9.20/5.24      ! [N2: nat,P: nat > $o] :
% 9.20/5.24        ( ( ? [M2: nat] :
% 9.20/5.24              ( ( ord_less_nat @ M2 @ N2 )
% 9.20/5.24              & ( P @ M2 ) ) )
% 9.20/5.24        = ( ? [X: nat] :
% 9.20/5.24              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.24              & ( P @ X ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9133_all__nat__less__eq,axiom,
% 9.20/5.24      ! [N2: nat,P: nat > $o] :
% 9.20/5.24        ( ( ! [M2: nat] :
% 9.20/5.24              ( ( ord_less_nat @ M2 @ N2 )
% 9.20/5.24             => ( P @ M2 ) ) )
% 9.20/5.24        = ( ! [X: nat] :
% 9.20/5.24              ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.24             => ( P @ X ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9134_atLeastLessThanSuc__atLeastAtMost,axiom,
% 9.20/5.24      ! [L: nat,U: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
% 9.20/5.24        = ( set_or1269000886237332187st_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9135_lessThan__atLeast0,axiom,
% 9.20/5.24      ( set_ord_lessThan_nat
% 9.20/5.24      = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9136_atLeastLessThan0,axiom,
% 9.20/5.24      ! [M: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
% 9.20/5.24        = bot_bot_set_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9137_atLeast0__lessThan__Suc,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 9.20/5.24        = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9138_subset__eq__atLeast0__lessThan__finite,axiom,
% 9.20/5.24      ! [N7: set_nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.24       => ( finite_finite_nat @ N7 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9139_atLeastLessThanSuc,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 9.20/5.24            = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
% 9.20/5.24        & ( ~ ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24         => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
% 9.20/5.24            = bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9140_prod__Suc__Suc__fact,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
% 9.20/5.24        = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9141_prod__Suc__fact,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.24        = ( semiri1408675320244567234ct_nat @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9142_atLeastLessThan__nat__numeral,axiom,
% 9.20/5.24      ! [M: nat,K: num] :
% 9.20/5.24        ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.20/5.24         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24            = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
% 9.20/5.24        & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
% 9.20/5.24         => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24            = bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9143_atLeast1__lessThan__eq__remove0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.24        = ( minus_minus_set_nat @ ( set_ord_lessThan_nat @ N2 ) @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9144_Chebyshev__sum__upper__nat,axiom,
% 9.20/5.24      ! [N2: nat,A: nat > nat,B: nat > nat] :
% 9.20/5.24        ( ! [I3: nat,J2: nat] :
% 9.20/5.24            ( ( ord_less_eq_nat @ I3 @ J2 )
% 9.20/5.24           => ( ( ord_less_nat @ J2 @ N2 )
% 9.20/5.24             => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
% 9.20/5.24       => ( ! [I3: nat,J2: nat] :
% 9.20/5.24              ( ( ord_less_eq_nat @ I3 @ J2 )
% 9.20/5.24             => ( ( ord_less_nat @ J2 @ N2 )
% 9.20/5.24               => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
% 9.20/5.24         => ( ord_less_eq_nat
% 9.20/5.24            @ ( times_times_nat @ N2
% 9.20/5.24              @ ( groups3542108847815614940at_nat
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
% 9.20/5.24                @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
% 9.20/5.24            @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9145_VEBT_Osize__gen_I1_J,axiom,
% 9.20/5.24      ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT] :
% 9.20/5.24        ( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
% 9.20/5.24        = ( plus_plus_nat @ ( plus_plus_nat @ ( size_list_VEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9146_finite__atLeastLessThan__int,axiom,
% 9.20/5.24      ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9147_finite__atLeastZeroLessThan__int,axiom,
% 9.20/5.24      ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9148_VEBT_Osize__gen_I2_J,axiom,
% 9.20/5.24      ! [X21: $o,X222: $o] :
% 9.20/5.24        ( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
% 9.20/5.24        = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9149_rat__inverse__code,axiom,
% 9.20/5.24      ! [P4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9150_normalize__negative,axiom,
% 9.20/5.24      ! [Q4: int,P4: int] :
% 9.20/5.24        ( ( ord_less_int @ Q4 @ zero_zero_int )
% 9.20/5.24       => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24          = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9151_Cauchy__iff2,axiom,
% 9.20/5.24      ( topolo4055970368930404560y_real
% 9.20/5.24      = ( ^ [X5: nat > real] :
% 9.20/5.24          ! [J3: nat] :
% 9.20/5.24          ? [M8: nat] :
% 9.20/5.24          ! [M2: nat] :
% 9.20/5.24            ( ( ord_less_eq_nat @ M8 @ M2 )
% 9.20/5.24           => ! [N: nat] :
% 9.20/5.24                ( ( ord_less_eq_nat @ M8 @ N )
% 9.20/5.24               => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9152_vebt__mint_Opelims,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.20/5.24        ( ( ( vEBT_vebt_mint @ X3 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
% 9.20/5.24         => ( ! [A4: $o,B4: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.24               => ( ( ( A4
% 9.20/5.24                     => ( Y3
% 9.20/5.24                        = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.24                    & ( ~ A4
% 9.20/5.24                     => ( ( B4
% 9.20/5.24                         => ( Y3
% 9.20/5.24                            = ( some_nat @ one_one_nat ) ) )
% 9.20/5.24                        & ( ~ B4
% 9.20/5.24                         => ( Y3 = none_nat ) ) ) ) )
% 9.20/5.24                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 9.20/5.24           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.24                 => ( ( Y3 = none_nat )
% 9.20/5.24                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.20/5.24             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.24                    ( ( X3
% 9.20/5.24                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.20/5.24                   => ( ( Y3
% 9.20/5.24                        = ( some_nat @ Mi2 ) )
% 9.20/5.24                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9153_quotient__of__number_I3_J,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
% 9.20/5.24        = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9154_normalize__denom__zero,axiom,
% 9.20/5.24      ! [P4: int] :
% 9.20/5.24        ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
% 9.20/5.24        = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9155_rat__one__code,axiom,
% 9.20/5.24      ( ( quotient_of @ one_one_rat )
% 9.20/5.24      = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9156_rat__zero__code,axiom,
% 9.20/5.24      ( ( quotient_of @ zero_zero_rat )
% 9.20/5.24      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9157_quotient__of__number_I5_J,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
% 9.20/5.24        = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9158_quotient__of__number_I4_J,axiom,
% 9.20/5.24      ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
% 9.20/5.24      = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9159_rat__divide__code,axiom,
% 9.20/5.24      ! [P4: rat,Q4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int,C3: int] :
% 9.20/5.24              ( produc4245557441103728435nt_int
% 9.20/5.24              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) ) )
% 9.20/5.24              @ ( quotient_of @ Q4 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9160_rat__times__code,axiom,
% 9.20/5.24      ! [P4: rat,Q4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int,C3: int] :
% 9.20/5.24              ( produc4245557441103728435nt_int
% 9.20/5.24              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C3 @ D5 ) ) )
% 9.20/5.24              @ ( quotient_of @ Q4 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9161_quotient__of__div,axiom,
% 9.20/5.24      ! [R2: rat,N2: int,D: int] :
% 9.20/5.24        ( ( ( quotient_of @ R2 )
% 9.20/5.24          = ( product_Pair_int_int @ N2 @ D ) )
% 9.20/5.24       => ( R2
% 9.20/5.24          = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9162_rat__minus__code,axiom,
% 9.20/5.24      ! [P4: rat,Q4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int,C3: int] :
% 9.20/5.24              ( produc4245557441103728435nt_int
% 9.20/5.24              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
% 9.20/5.24              @ ( quotient_of @ Q4 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9163_quotient__of__denom__pos,axiom,
% 9.20/5.24      ! [R2: rat,P4: int,Q4: int] :
% 9.20/5.24        ( ( ( quotient_of @ R2 )
% 9.20/5.24          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9164_rat__plus__code,axiom,
% 9.20/5.24      ! [P4: rat,Q4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int,C3: int] :
% 9.20/5.24              ( produc4245557441103728435nt_int
% 9.20/5.24              @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
% 9.20/5.24              @ ( quotient_of @ Q4 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9165_normalize__denom__pos,axiom,
% 9.20/5.24      ! [R2: product_prod_int_int,P4: int,Q4: int] :
% 9.20/5.24        ( ( ( normalize @ R2 )
% 9.20/5.24          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24       => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9166_normalize__crossproduct,axiom,
% 9.20/5.24      ! [Q4: int,S: int,P4: int,R2: int] :
% 9.20/5.24        ( ( Q4 != zero_zero_int )
% 9.20/5.24       => ( ( S != zero_zero_int )
% 9.20/5.24         => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24              = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
% 9.20/5.24           => ( ( times_times_int @ P4 @ S )
% 9.20/5.24              = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9167_rat__uminus__code,axiom,
% 9.20/5.24      ! [P4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9168_rat__less__code,axiom,
% 9.20/5.24      ( ord_less_rat
% 9.20/5.24      = ( ^ [P6: rat,Q5: rat] :
% 9.20/5.24            ( produc4947309494688390418_int_o
% 9.20/5.24            @ ^ [A3: int,C3: int] :
% 9.20/5.24                ( produc4947309494688390418_int_o
% 9.20/5.24                @ ^ [B3: int,D5: int] : ( ord_less_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) )
% 9.20/5.24                @ ( quotient_of @ Q5 ) )
% 9.20/5.24            @ ( quotient_of @ P6 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9169_rat__abs__code,axiom,
% 9.20/5.24      ! [P4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
% 9.20/5.24        = ( produc4245557441103728435nt_int
% 9.20/5.24          @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
% 9.20/5.24          @ ( quotient_of @ P4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9170_vebt__maxt_Opelims,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Y3: option_nat] :
% 9.20/5.24        ( ( ( vEBT_vebt_maxt @ X3 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
% 9.20/5.24         => ( ! [A4: $o,B4: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ A4 @ B4 ) )
% 9.20/5.24               => ( ( ( B4
% 9.20/5.24                     => ( Y3
% 9.20/5.24                        = ( some_nat @ one_one_nat ) ) )
% 9.20/5.24                    & ( ~ B4
% 9.20/5.24                     => ( ( A4
% 9.20/5.24                         => ( Y3
% 9.20/5.24                            = ( some_nat @ zero_zero_nat ) ) )
% 9.20/5.24                        & ( ~ A4
% 9.20/5.24                         => ( Y3 = none_nat ) ) ) ) )
% 9.20/5.24                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
% 9.20/5.24           => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
% 9.20/5.24                 => ( ( Y3 = none_nat )
% 9.20/5.24                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
% 9.20/5.24             => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
% 9.20/5.24                    ( ( X3
% 9.20/5.24                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
% 9.20/5.24                   => ( ( Y3
% 9.20/5.24                        = ( some_nat @ Ma2 ) )
% 9.20/5.24                     => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9171_quotient__of__int,axiom,
% 9.20/5.24      ! [A: int] :
% 9.20/5.24        ( ( quotient_of @ ( of_int @ A ) )
% 9.20/5.24        = ( product_Pair_int_int @ A @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9172_VEBT__internal_OminNull_Opelims_I1_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Y3: $o] :
% 9.20/5.24        ( ( ( vEBT_VEBT_minNull @ X3 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.20/5.24         => ( ( ( X3
% 9.20/5.24                = ( vEBT_Leaf @ $false @ $false ) )
% 9.20/5.24             => ( Y3
% 9.20/5.24               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
% 9.20/5.24           => ( ! [Uv2: $o] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.20/5.24                 => ( ~ Y3
% 9.20/5.24                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
% 9.20/5.24             => ( ! [Uu2: $o] :
% 9.20/5.24                    ( ( X3
% 9.20/5.24                      = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.20/5.24                   => ( ~ Y3
% 9.20/5.24                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
% 9.20/5.24               => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.24                      ( ( X3
% 9.20/5.24                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.20/5.24                     => ( Y3
% 9.20/5.24                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
% 9.20/5.24                 => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.24                        ( ( X3
% 9.20/5.24                          = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.20/5.24                       => ( ~ Y3
% 9.20/5.24                         => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9173_VEBT__internal_OminNull_Opelims_I2_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT] :
% 9.20/5.24        ( ( vEBT_VEBT_minNull @ X3 )
% 9.20/5.24       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.20/5.24         => ( ( ( X3
% 9.20/5.24                = ( vEBT_Leaf @ $false @ $false ) )
% 9.20/5.24             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
% 9.20/5.24           => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
% 9.20/5.24                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9174_VEBT__internal_OminNull_Opelims_I3_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT] :
% 9.20/5.24        ( ~ ( vEBT_VEBT_minNull @ X3 )
% 9.20/5.24       => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
% 9.20/5.24         => ( ! [Uv2: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ $true @ Uv2 ) )
% 9.20/5.24               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
% 9.20/5.24           => ( ! [Uu2: $o] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Leaf @ Uu2 @ $true ) )
% 9.20/5.24                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
% 9.20/5.24             => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
% 9.20/5.24                    ( ( X3
% 9.20/5.24                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
% 9.20/5.24                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9175_Frct__code__post_I5_J,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
% 9.20/5.24        = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9176_divmod__step__integer__def,axiom,
% 9.20/5.24      ( unique4921790084139445826nteger
% 9.20/5.24      = ( ^ [L2: num] :
% 9.20/5.24            ( produc6916734918728496179nteger
% 9.20/5.24            @ ^ [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 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9177_divmod__integer_H__def,axiom,
% 9.20/5.24      ( unique3479559517661332726nteger
% 9.20/5.24      = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9178_sgn__integer__code,axiom,
% 9.20/5.24      ( sgn_sgn_Code_integer
% 9.20/5.24      = ( ^ [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 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9179_exhaustive__integer_H_Ocases,axiom,
% 9.20/5.24      ! [X3: produc8763457246119570046nteger] :
% 9.20/5.24        ~ ! [F2: code_integer > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
% 9.20/5.24            ( X3
% 9.20/5.24           != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9180_full__exhaustive__integer_H_Ocases,axiom,
% 9.20/5.24      ! [X3: produc1908205239877642774nteger] :
% 9.20/5.24        ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
% 9.20/5.24            ( X3
% 9.20/5.24           != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9181_zero__natural_Orsp,axiom,
% 9.20/5.24      zero_zero_nat = zero_zero_nat ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9182_Frct__code__post_I1_J,axiom,
% 9.20/5.24      ! [A: int] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
% 9.20/5.24        = zero_zero_rat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9183_Frct__code__post_I2_J,axiom,
% 9.20/5.24      ! [A: int] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
% 9.20/5.24        = zero_zero_rat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9184_Frct__code__post_I8_J,axiom,
% 9.20/5.24      ! [A: int,B: int] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
% 9.20/5.24        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9185_Frct__code__post_I7_J,axiom,
% 9.20/5.24      ! [A: int,B: int] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
% 9.20/5.24        = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9186_Frct__code__post_I3_J,axiom,
% 9.20/5.24      ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
% 9.20/5.24      = one_one_rat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9187_Frct__code__post_I4_J,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
% 9.20/5.24        = ( numeral_numeral_rat @ K ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9188_Frct__code__post_I6_J,axiom,
% 9.20/5.24      ! [K: num,L: num] :
% 9.20/5.24        ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
% 9.20/5.24        = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9189_integer__of__int__code,axiom,
% 9.20/5.24      ( code_integer_of_int
% 9.20/5.24      = ( ^ [K3: int] :
% 9.20/5.24            ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
% 9.20/5.24            @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
% 9.20/5.24              @ ( if_Code_integer
% 9.20/5.24                @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
% 9.20/5.24                  = zero_zero_int )
% 9.20/5.24                @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9190_pair__leqI2,axiom,
% 9.20/5.24      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ A @ B )
% 9.20/5.24       => ( ( ord_less_eq_nat @ S @ T )
% 9.20/5.24         => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9191_card__Collect__less__nat,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( finite_card_nat
% 9.20/5.24          @ ( collect_nat
% 9.20/5.24            @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
% 9.20/5.24        = N2 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9192_card__atMost,axiom,
% 9.20/5.24      ! [U: nat] :
% 9.20/5.24        ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
% 9.20/5.24        = ( suc @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9193_card__Collect__le__nat,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( finite_card_nat
% 9.20/5.24          @ ( collect_nat
% 9.20/5.24            @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N2 ) ) )
% 9.20/5.24        = ( suc @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9194_card__atLeastAtMost,axiom,
% 9.20/5.24      ! [L: nat,U: nat] :
% 9.20/5.24        ( ( finite_card_nat @ ( set_or1269000886237332187st_nat @ L @ U ) )
% 9.20/5.24        = ( minus_minus_nat @ ( suc @ U ) @ L ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9195_abs__integer__code,axiom,
% 9.20/5.24      ( abs_abs_Code_integer
% 9.20/5.24      = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9196_less__integer__code_I1_J,axiom,
% 9.20/5.24      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9197_less__integer_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: int,X3: int] :
% 9.20/5.24        ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 9.20/5.24        = ( ord_less_int @ Xa2 @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9198_card__less,axiom,
% 9.20/5.24      ! [M7: set_nat,I: nat] :
% 9.20/5.24        ( ( member_nat @ zero_zero_nat @ M7 )
% 9.20/5.24       => ( ( finite_card_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [K3: nat] :
% 9.20/5.24                  ( ( member_nat @ K3 @ M7 )
% 9.20/5.24                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
% 9.20/5.24         != zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9199_card__less__Suc,axiom,
% 9.20/5.24      ! [M7: set_nat,I: nat] :
% 9.20/5.24        ( ( member_nat @ zero_zero_nat @ M7 )
% 9.20/5.24       => ( ( suc
% 9.20/5.24            @ ( finite_card_nat
% 9.20/5.24              @ ( collect_nat
% 9.20/5.24                @ ^ [K3: nat] :
% 9.20/5.24                    ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 9.20/5.24                    & ( ord_less_nat @ K3 @ I ) ) ) ) )
% 9.20/5.24          = ( finite_card_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [K3: nat] :
% 9.20/5.24                  ( ( member_nat @ K3 @ M7 )
% 9.20/5.24                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9200_card__less__Suc2,axiom,
% 9.20/5.24      ! [M7: set_nat,I: nat] :
% 9.20/5.24        ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 9.20/5.24       => ( ( finite_card_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [K3: nat] :
% 9.20/5.24                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
% 9.20/5.24                  & ( ord_less_nat @ K3 @ I ) ) ) )
% 9.20/5.24          = ( finite_card_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [K3: nat] :
% 9.20/5.24                  ( ( member_nat @ K3 @ M7 )
% 9.20/5.24                  & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9201_subset__card__intvl__is__intvl,axiom,
% 9.20/5.24      ! [A2: set_nat,K: nat] :
% 9.20/5.24        ( ( ord_less_eq_set_nat @ A2 @ ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) )
% 9.20/5.24       => ( A2
% 9.20/5.24          = ( set_or4665077453230672383an_nat @ K @ ( plus_plus_nat @ K @ ( finite_card_nat @ A2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9202_subset__eq__atLeast0__lessThan__card,axiom,
% 9.20/5.24      ! [N7: set_nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
% 9.20/5.24       => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9203_card__sum__le__nat__sum,axiom,
% 9.20/5.24      ! [S3: set_nat] :
% 9.20/5.24        ( ord_less_eq_nat
% 9.20/5.24        @ ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [X: nat] : X
% 9.20/5.24          @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
% 9.20/5.24        @ ( groups3542108847815614940at_nat
% 9.20/5.24          @ ^ [X: nat] : X
% 9.20/5.24          @ S3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9204_card__nth__roots,axiom,
% 9.20/5.24      ! [C: complex,N2: nat] :
% 9.20/5.24        ( ( C != zero_zero_complex )
% 9.20/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( finite_card_complex
% 9.20/5.24              @ ( collect_complex
% 9.20/5.24                @ ^ [Z6: complex] :
% 9.20/5.24                    ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.24                    = C ) ) )
% 9.20/5.24            = N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9205_card__roots__unity__eq,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( finite_card_complex
% 9.20/5.24            @ ( collect_complex
% 9.20/5.24              @ ^ [Z6: complex] :
% 9.20/5.24                  ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.24                  = one_one_complex ) ) )
% 9.20/5.24          = N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9206_pair__leqI1,axiom,
% 9.20/5.24      ! [A: nat,B: nat,S: nat,T: nat] :
% 9.20/5.24        ( ( ord_less_nat @ A @ B )
% 9.20/5.24       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9207_int__of__integer__code,axiom,
% 9.20/5.24      ( code_int_of_integer
% 9.20/5.24      = ( ^ [K3: code_integer] :
% 9.20/5.24            ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
% 9.20/5.24            @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
% 9.20/5.24              @ ( produc1553301316500091796er_int
% 9.20/5.24                @ ^ [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 ) )
% 9.20/5.24                @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9208_less__integer_Orep__eq,axiom,
% 9.20/5.24      ( ord_le6747313008572928689nteger
% 9.20/5.24      = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9209_integer__less__iff,axiom,
% 9.20/5.24      ( ord_le6747313008572928689nteger
% 9.20/5.24      = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9210_divmod__integer__def,axiom,
% 9.20/5.24      ( code_divmod_integer
% 9.20/5.24      = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9211_nat__of__integer__code,axiom,
% 9.20/5.24      ( code_nat_of_integer
% 9.20/5.24      = ( ^ [K3: code_integer] :
% 9.20/5.24            ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
% 9.20/5.24            @ ( produc1555791787009142072er_nat
% 9.20/5.24              @ ^ [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 ) )
% 9.20/5.24              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9212_bit__cut__integer__def,axiom,
% 9.20/5.24      ( code_bit_cut_integer
% 9.20/5.24      = ( ^ [K3: code_integer] :
% 9.20/5.24            ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
% 9.20/5.24            @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9213_divmod__abs__def,axiom,
% 9.20/5.24      ( code_divmod_abs
% 9.20/5.24      = ( ^ [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 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9214_nat__of__integer__non__positive,axiom,
% 9.20/5.24      ! [K: code_integer] :
% 9.20/5.24        ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
% 9.20/5.24       => ( ( code_nat_of_integer @ K )
% 9.20/5.24          = zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9215_nat__of__integer__code__post_I1_J,axiom,
% 9.20/5.24      ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
% 9.20/5.24      = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9216_divmod__abs__code_I6_J,axiom,
% 9.20/5.24      ! [J: code_integer] :
% 9.20/5.24        ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
% 9.20/5.24        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9217_divmod__abs__code_I5_J,axiom,
% 9.20/5.24      ! [J: code_integer] :
% 9.20/5.24        ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
% 9.20/5.24        = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9218_bit__cut__integer__code,axiom,
% 9.20/5.24      ( code_bit_cut_integer
% 9.20/5.24      = ( ^ [K3: code_integer] :
% 9.20/5.24            ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
% 9.20/5.24            @ ( produc9125791028180074456eger_o
% 9.20/5.24              @ ^ [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 ) )
% 9.20/5.24              @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9219_divmod__integer__code,axiom,
% 9.20/5.24      ( code_divmod_integer
% 9.20/5.24      = ( ^ [K3: code_integer,L2: code_integer] :
% 9.20/5.24            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.20/5.24            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
% 9.20/5.24              @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.20/5.24                @ ( produc6916734918728496179nteger
% 9.20/5.24                  @ ^ [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 ) ) )
% 9.20/5.24                  @ ( code_divmod_abs @ K3 @ L2 ) ) )
% 9.20/5.24              @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 9.20/5.24                @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
% 9.20/5.24                  @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
% 9.20/5.24                    @ ( produc6916734918728496179nteger
% 9.20/5.24                      @ ^ [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 ) ) )
% 9.20/5.24                      @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9220_nat_Odisc__eq__case_I2_J,axiom,
% 9.20/5.24      ! [Nat: nat] :
% 9.20/5.24        ( ( Nat != zero_zero_nat )
% 9.20/5.24        = ( case_nat_o @ $false
% 9.20/5.24          @ ^ [Uu3: nat] : $true
% 9.20/5.24          @ Nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9221_nat_Odisc__eq__case_I1_J,axiom,
% 9.20/5.24      ! [Nat: nat] :
% 9.20/5.24        ( ( Nat = zero_zero_nat )
% 9.20/5.24        = ( case_nat_o @ $true
% 9.20/5.24          @ ^ [Uu3: nat] : $false
% 9.20/5.24          @ Nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9222_less__eq__nat_Osimps_I2_J,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
% 9.20/5.24        = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9223_diff__Suc,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
% 9.20/5.24        = ( case_nat_nat @ zero_zero_nat
% 9.20/5.24          @ ^ [K3: nat] : K3
% 9.20/5.24          @ ( minus_minus_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9224_binomial__def,axiom,
% 9.20/5.24      ( binomial
% 9.20/5.24      = ( ^ [N: nat,K3: nat] :
% 9.20/5.24            ( finite_card_set_nat
% 9.20/5.24            @ ( collect_set_nat
% 9.20/5.24              @ ^ [K7: set_nat] :
% 9.20/5.24                  ( ( member_set_nat @ K7 @ ( pow_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) ) )
% 9.20/5.24                  & ( ( finite_card_nat @ K7 )
% 9.20/5.24                    = K3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9225_pred__def,axiom,
% 9.20/5.24      ( pred
% 9.20/5.24      = ( case_nat_nat @ zero_zero_nat
% 9.20/5.24        @ ^ [X24: nat] : X24 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9226_wmin__insertI,axiom,
% 9.20/5.24      ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( member8440522571783428010at_nat @ X3 @ XS )
% 9.20/5.24       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
% 9.20/5.24         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
% 9.20/5.24           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9227_wmax__insertI,axiom,
% 9.20/5.24      ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( member8440522571783428010at_nat @ Y3 @ YS )
% 9.20/5.24       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
% 9.20/5.24         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
% 9.20/5.24           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9228_bezw__0,axiom,
% 9.20/5.24      ! [X3: nat] :
% 9.20/5.24        ( ( bezw @ X3 @ zero_zero_nat )
% 9.20/5.24        = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9229_push__bit__negative__int__iff,axiom,
% 9.20/5.24      ! [N2: nat,K: int] :
% 9.20/5.24        ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.24        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9230_push__bit__of__Suc__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_se547839408752420682it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9231_wmin__emptyI,axiom,
% 9.20/5.24      ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9232_wmax__emptyI,axiom,
% 9.20/5.24      ! [X7: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( finite6177210948735845034at_nat @ X7 )
% 9.20/5.24       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9233_bit__push__bit__iff__int,axiom,
% 9.20/5.24      ! [M: nat,K: int,N2: nat] :
% 9.20/5.24        ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
% 9.20/5.24        = ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24          & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9234_bit__push__bit__iff__nat,axiom,
% 9.20/5.24      ! [M: nat,Q4: nat,N2: nat] :
% 9.20/5.24        ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N2 )
% 9.20/5.24        = ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24          & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9235_smax__insertI,axiom,
% 9.20/5.24      ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
% 9.20/5.24       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
% 9.20/5.24         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
% 9.20/5.24           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9236_smin__insertI,axiom,
% 9.20/5.24      ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( member8440522571783428010at_nat @ X3 @ XS )
% 9.20/5.24       => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
% 9.20/5.24         => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
% 9.20/5.24           => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9237_prod__decode__aux_Osimps,axiom,
% 9.20/5.24      ( nat_prod_decode_aux
% 9.20/5.24      = ( ^ [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 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9238_prod__decode__aux_Oelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.20/5.24        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 9.20/5.24          & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9239_smax__emptyI,axiom,
% 9.20/5.24      ! [Y7: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( finite6177210948735845034at_nat @ Y7 )
% 9.20/5.24       => ( ( Y7 != bot_bo2099793752762293965at_nat )
% 9.20/5.24         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9240_smin__emptyI,axiom,
% 9.20/5.24      ! [X7: set_Pr1261947904930325089at_nat] :
% 9.20/5.24        ( ( X7 != bot_bo2099793752762293965at_nat )
% 9.20/5.24       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9241_Suc__0__div__numeral,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( divide_divide_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24        = ( product_fst_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9242_Suc__0__mod__numeral,axiom,
% 9.20/5.24      ! [K: num] :
% 9.20/5.24        ( ( modulo_modulo_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24        = ( product_snd_nat_nat @ ( unique5055182867167087721od_nat @ one @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9243_finite__enumerate,axiom,
% 9.20/5.24      ! [S3: set_nat] :
% 9.20/5.24        ( ( finite_finite_nat @ S3 )
% 9.20/5.24       => ? [R4: nat > nat] :
% 9.20/5.24            ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
% 9.20/5.24            & ! [N6: nat] :
% 9.20/5.24                ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S3 ) )
% 9.20/5.24               => ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9244_Sup__nat__empty,axiom,
% 9.20/5.24      ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
% 9.20/5.24      = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9245_Inf__nat__def1,axiom,
% 9.20/5.24      ! [K5: set_nat] :
% 9.20/5.24        ( ( K5 != bot_bot_set_nat )
% 9.20/5.24       => ( member_nat @ ( complete_Inf_Inf_nat @ K5 ) @ K5 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9246_quotient__of__denom__pos_H,axiom,
% 9.20/5.24      ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9247_bezw__non__0,axiom,
% 9.20/5.24      ! [Y3: nat,X3: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ Y3 )
% 9.20/5.24       => ( ( bezw @ X3 @ Y3 )
% 9.20/5.24          = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9248_bezw_Oelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.20/5.24        ( ( ( bezw @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.20/5.24          & ( ( Xa2 != zero_zero_nat )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9249_bezw_Osimps,axiom,
% 9.20/5.24      ( bezw
% 9.20/5.24      = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( 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 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9250_rat__sgn__code,axiom,
% 9.20/5.24      ! [P4: rat] :
% 9.20/5.24        ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
% 9.20/5.24        = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9251_bezw_Opelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
% 9.20/5.24        ( ( ( bezw @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.24                 => ( Y3
% 9.20/5.24                    = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
% 9.20/5.24                & ( ( Xa2 != zero_zero_nat )
% 9.20/5.24                 => ( Y3
% 9.20/5.24                    = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
% 9.20/5.24             => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9252_prod__decode__aux_Opelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
% 9.20/5.24        ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.20/5.24                 => ( Y3
% 9.20/5.24                    = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
% 9.20/5.24                & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
% 9.20/5.24                 => ( Y3
% 9.20/5.24                    = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
% 9.20/5.24             => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9253_normalize__def,axiom,
% 9.20/5.24      ( normalize
% 9.20/5.24      = ( ^ [P6: product_prod_int_int] :
% 9.20/5.24            ( 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 ) ) ) )
% 9.20/5.24            @ ( if_Pro3027730157355071871nt_int
% 9.20/5.24              @ ( ( product_snd_int_int @ P6 )
% 9.20/5.24                = zero_zero_int )
% 9.20/5.24              @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
% 9.20/5.24              @ ( 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 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9254_gcd__1__int,axiom,
% 9.20/5.24      ! [M: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ M @ one_one_int )
% 9.20/5.24        = one_one_int ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9255_gcd__neg2__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9256_gcd__neg1__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9257_abs__gcd__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( abs_abs_int @ ( gcd_gcd_int @ X3 @ Y3 ) )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9258_gcd__abs1__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ ( abs_abs_int @ X3 ) @ Y3 )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9259_gcd__abs2__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ X3 @ ( abs_abs_int @ Y3 ) )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9260_gcd__pos__int,axiom,
% 9.20/5.24      ! [M: int,N2: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
% 9.20/5.24        = ( ( M != zero_zero_int )
% 9.20/5.24          | ( N2 != zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9261_gcd__neg__numeral__1__int,axiom,
% 9.20/5.24      ! [N2: num,X3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X3 )
% 9.20/5.24        = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9262_gcd__neg__numeral__2__int,axiom,
% 9.20/5.24      ! [X3: int,N2: num] :
% 9.20/5.24        ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9263_gcd__0__int,axiom,
% 9.20/5.24      ! [X3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ X3 @ zero_zero_int )
% 9.20/5.24        = ( abs_abs_int @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9264_gcd__0__left__int,axiom,
% 9.20/5.24      ! [X3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ zero_zero_int @ X3 )
% 9.20/5.24        = ( abs_abs_int @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9265_gcd__proj1__if__dvd__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ X3 @ Y3 )
% 9.20/5.24       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 9.20/5.24          = ( abs_abs_int @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9266_gcd__proj2__if__dvd__int,axiom,
% 9.20/5.24      ! [Y3: int,X3: int] :
% 9.20/5.24        ( ( dvd_dvd_int @ Y3 @ X3 )
% 9.20/5.24       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 9.20/5.24          = ( abs_abs_int @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9267_gcd__idem__int,axiom,
% 9.20/5.24      ! [X3: int] :
% 9.20/5.24        ( ( gcd_gcd_int @ X3 @ X3 )
% 9.20/5.24        = ( abs_abs_int @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9268_gcd__red__int,axiom,
% 9.20/5.24      ( gcd_gcd_int
% 9.20/5.24      = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9269_gcd__ge__0__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9270_bezout__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int] :
% 9.20/5.24      ? [U3: int,V2: int] :
% 9.20/5.24        ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
% 9.20/5.24        = ( gcd_gcd_int @ X3 @ Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9271_gcd__mult__distrib__int,axiom,
% 9.20/5.24      ! [K: int,M: int,N2: int] :
% 9.20/5.24        ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
% 9.20/5.24        = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9272_gcd__le2__int,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9273_gcd__le1__int,axiom,
% 9.20/5.24      ! [A: int,B: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ A )
% 9.20/5.24       => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9274_gcd__cases__int,axiom,
% 9.20/5.24      ! [X3: int,Y3: int,P: int > $o] :
% 9.20/5.24        ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.24         => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.24           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
% 9.20/5.24       => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.24           => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.20/5.24             => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.20/5.24         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.20/5.24             => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
% 9.20/5.24               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
% 9.20/5.24           => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
% 9.20/5.24               => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
% 9.20/5.24                 => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
% 9.20/5.24             => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9275_gcd__unique__int,axiom,
% 9.20/5.24      ! [D: int,A: int,B: int] :
% 9.20/5.24        ( ( ( ord_less_eq_int @ zero_zero_int @ D )
% 9.20/5.24          & ( dvd_dvd_int @ D @ A )
% 9.20/5.24          & ( dvd_dvd_int @ D @ B )
% 9.20/5.24          & ! [E3: int] :
% 9.20/5.24              ( ( ( dvd_dvd_int @ E3 @ A )
% 9.20/5.24                & ( dvd_dvd_int @ E3 @ B ) )
% 9.20/5.24             => ( dvd_dvd_int @ E3 @ D ) ) )
% 9.20/5.24        = ( D
% 9.20/5.24          = ( gcd_gcd_int @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9276_gcd__non__0__int,axiom,
% 9.20/5.24      ! [Y3: int,X3: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ Y3 )
% 9.20/5.24       => ( ( gcd_gcd_int @ X3 @ Y3 )
% 9.20/5.24          = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9277_gcd__code__int,axiom,
% 9.20/5.24      ( gcd_gcd_int
% 9.20/5.24      = ( ^ [K3: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9278_gcd__nat_Oeq__neutr__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24          = zero_zero_nat )
% 9.20/5.24        = ( ( A = zero_zero_nat )
% 9.20/5.24          & ( B = zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9279_gcd__nat_Oleft__neutral,axiom,
% 9.20/5.24      ! [A: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ zero_zero_nat @ A )
% 9.20/5.24        = A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9280_gcd__nat_Oneutr__eq__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( zero_zero_nat
% 9.20/5.24          = ( gcd_gcd_nat @ A @ B ) )
% 9.20/5.24        = ( ( A = zero_zero_nat )
% 9.20/5.24          & ( B = zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9281_gcd__nat_Oright__neutral,axiom,
% 9.20/5.24      ! [A: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ A @ zero_zero_nat )
% 9.20/5.24        = A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9282_gcd__0__nat,axiom,
% 9.20/5.24      ! [X3: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ X3 @ zero_zero_nat )
% 9.20/5.24        = X3 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9283_gcd__0__left__nat,axiom,
% 9.20/5.24      ! [X3: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ zero_zero_nat @ X3 )
% 9.20/5.24        = X3 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9284_gcd__1__nat,axiom,
% 9.20/5.24      ! [M: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ M @ one_one_nat )
% 9.20/5.24        = one_one_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9285_gcd__proj2__if__dvd__nat,axiom,
% 9.20/5.24      ! [Y3: nat,X3: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ Y3 @ X3 )
% 9.20/5.24       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 9.20/5.24          = Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9286_gcd__proj1__if__dvd__nat,axiom,
% 9.20/5.24      ! [X3: nat,Y3: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ X3 @ Y3 )
% 9.20/5.24       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 9.20/5.24          = X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9287_gcd__nat_Obounded__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat,C: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 9.20/5.24        = ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24          & ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9288_gcd__nat_Oabsorb2,axiom,
% 9.20/5.24      ! [B: nat,A: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.24       => ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24          = B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9289_gcd__nat_Oabsorb1,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24       => ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24          = A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9290_gcd__Suc__0,axiom,
% 9.20/5.24      ! [M: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( suc @ zero_zero_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9291_gcd__pos__nat,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
% 9.20/5.24        = ( ( M != zero_zero_nat )
% 9.20/5.24          | ( N2 != zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9292_or__negative__int__iff,axiom,
% 9.20/5.24      ! [K: int,L: int] :
% 9.20/5.24        ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
% 9.20/5.24        = ( ( ord_less_int @ K @ zero_zero_int )
% 9.20/5.24          | ( ord_less_int @ L @ zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9293_gcd__int__int__eq,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
% 9.20/5.24        = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9294_or__nat__numerals_I4_J,axiom,
% 9.20/5.24      ! [X3: num] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9295_or__nat__numerals_I2_J,axiom,
% 9.20/5.24      ! [Y3: num] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
% 9.20/5.24        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9296_gcd__nat__abs__right__eq,axiom,
% 9.20/5.24      ! [N2: nat,K: int] :
% 9.20/5.24        ( ( gcd_gcd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
% 9.20/5.24        = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9297_gcd__nat__abs__left__eq,axiom,
% 9.20/5.24      ! [K: int,N2: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
% 9.20/5.24        = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9298_or__nat__numerals_I3_J,axiom,
% 9.20/5.24      ! [X3: num] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9299_or__nat__numerals_I1_J,axiom,
% 9.20/5.24      ! [Y3: num] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
% 9.20/5.24        = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9300_gcd__mult__distrib__nat,axiom,
% 9.20/5.24      ! [K: nat,M: nat,N2: nat] :
% 9.20/5.24        ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
% 9.20/5.24        = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9301_gcd__red__nat,axiom,
% 9.20/5.24      ( gcd_gcd_nat
% 9.20/5.24      = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9302_gcd__unique__nat,axiom,
% 9.20/5.24      ! [D: nat,A: nat,B: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ D @ A )
% 9.20/5.24          & ( dvd_dvd_nat @ D @ B )
% 9.20/5.24          & ! [E3: nat] :
% 9.20/5.24              ( ( ( dvd_dvd_nat @ E3 @ A )
% 9.20/5.24                & ( dvd_dvd_nat @ E3 @ B ) )
% 9.20/5.24             => ( dvd_dvd_nat @ E3 @ D ) ) )
% 9.20/5.24        = ( D
% 9.20/5.24          = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9303_gcd__nat_Ostrict__coboundedI2,axiom,
% 9.20/5.24      ! [B: nat,C: nat,A: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.24          & ( B != C ) )
% 9.20/5.24       => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 9.20/5.24          & ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24           != C ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9304_gcd__nat_Ostrict__coboundedI1,axiom,
% 9.20/5.24      ! [A: nat,C: nat,B: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.24          & ( A != C ) )
% 9.20/5.24       => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
% 9.20/5.24          & ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24           != C ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9305_gcd__nat_Ostrict__order__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24          & ( A != B ) )
% 9.20/5.24        = ( ( A
% 9.20/5.24            = ( gcd_gcd_nat @ A @ B ) )
% 9.20/5.24          & ( A != B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9306_gcd__nat_Ostrict__boundedE,axiom,
% 9.20/5.24      ! [A: nat,B: nat,C: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 9.20/5.24          & ( A
% 9.20/5.24           != ( gcd_gcd_nat @ B @ C ) ) )
% 9.20/5.24       => ~ ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24              & ( A != B ) )
% 9.20/5.24           => ~ ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.24                & ( A != C ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9307_gcd__nat_OcoboundedI2,axiom,
% 9.20/5.24      ! [B: nat,C: nat,A: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ B @ C )
% 9.20/5.24       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9308_gcd__nat_OcoboundedI1,axiom,
% 9.20/5.24      ! [A: nat,C: nat,B: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.24       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9309_gcd__nat_Oabsorb__iff2,axiom,
% 9.20/5.24      ( dvd_dvd_nat
% 9.20/5.24      = ( ^ [B3: nat,A3: nat] :
% 9.20/5.24            ( ( gcd_gcd_nat @ A3 @ B3 )
% 9.20/5.24            = B3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9310_gcd__nat_Oabsorb__iff1,axiom,
% 9.20/5.24      ( dvd_dvd_nat
% 9.20/5.24      = ( ^ [A3: nat,B3: nat] :
% 9.20/5.24            ( ( gcd_gcd_nat @ A3 @ B3 )
% 9.20/5.24            = A3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9311_gcd__nat_Ocobounded2,axiom,
% 9.20/5.24      ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9312_gcd__nat_Ocobounded1,axiom,
% 9.20/5.24      ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9313_gcd__nat_Oorder__iff,axiom,
% 9.20/5.24      ( dvd_dvd_nat
% 9.20/5.24      = ( ^ [A3: nat,B3: nat] :
% 9.20/5.24            ( A3
% 9.20/5.24            = ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9314_gcd__nat_OboundedI,axiom,
% 9.20/5.24      ! [A: nat,B: nat,C: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24       => ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.24         => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9315_gcd__nat_OboundedE,axiom,
% 9.20/5.24      ! [A: nat,B: nat,C: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
% 9.20/5.24       => ~ ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24           => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9316_gcd__nat_Oabsorb4,axiom,
% 9.20/5.24      ! [B: nat,A: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ B @ A )
% 9.20/5.24          & ( B != A ) )
% 9.20/5.24       => ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24          = B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9317_gcd__nat_Oabsorb3,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24          & ( A != B ) )
% 9.20/5.24       => ( ( gcd_gcd_nat @ A @ B )
% 9.20/5.24          = A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9318_gcd__nat_OorderI,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( A
% 9.20/5.24          = ( gcd_gcd_nat @ A @ B ) )
% 9.20/5.24       => ( dvd_dvd_nat @ A @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9319_gcd__nat_OorderE,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ B )
% 9.20/5.24       => ( A
% 9.20/5.24          = ( gcd_gcd_nat @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9320_gcd__nat_Omono,axiom,
% 9.20/5.24      ! [A: nat,C: nat,B: nat,D: nat] :
% 9.20/5.24        ( ( dvd_dvd_nat @ A @ C )
% 9.20/5.24       => ( ( dvd_dvd_nat @ B @ D )
% 9.20/5.24         => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9321_gcd__le1__nat,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( A != zero_zero_nat )
% 9.20/5.24       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9322_gcd__le2__nat,axiom,
% 9.20/5.24      ! [B: nat,A: nat] :
% 9.20/5.24        ( ( B != zero_zero_nat )
% 9.20/5.24       => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9323_gcd__diff1__nat,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ N2 @ M )
% 9.20/5.24       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
% 9.20/5.24          = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9324_gcd__diff2__nat,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.24       => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
% 9.20/5.24          = ( gcd_gcd_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9325_gcd__nat_Oelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: nat] :
% 9.20/5.24        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.24           => ( Y3 = X3 ) )
% 9.20/5.24          & ( ( Xa2 != zero_zero_nat )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9326_gcd__nat_Osimps,axiom,
% 9.20/5.24      ( gcd_gcd_nat
% 9.20/5.24      = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9327_gcd__non__0__nat,axiom,
% 9.20/5.24      ! [Y3: nat,X3: nat] :
% 9.20/5.24        ( ( Y3 != zero_zero_nat )
% 9.20/5.24       => ( ( gcd_gcd_nat @ X3 @ Y3 )
% 9.20/5.24          = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9328_gcd__integer_Orep__eq,axiom,
% 9.20/5.24      ! [X3: code_integer,Xa2: code_integer] :
% 9.20/5.24        ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X3 @ Xa2 ) )
% 9.20/5.24        = ( gcd_gcd_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9329_gcd__integer_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: int,X3: int] :
% 9.20/5.24        ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
% 9.20/5.24        = ( code_integer_of_int @ ( gcd_gcd_int @ Xa2 @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9330_bezout__nat,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( A != zero_zero_nat )
% 9.20/5.24       => ? [X4: nat,Y5: nat] :
% 9.20/5.24            ( ( times_times_nat @ A @ X4 )
% 9.20/5.24            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9331_bezout__gcd__nat_H,axiom,
% 9.20/5.24      ! [B: nat,A: nat] :
% 9.20/5.24      ? [X4: nat,Y5: nat] :
% 9.20/5.24        ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y5 ) @ ( times_times_nat @ A @ X4 ) )
% 9.20/5.24          & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
% 9.20/5.24            = ( gcd_gcd_nat @ A @ B ) ) )
% 9.20/5.24        | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y5 ) @ ( times_times_nat @ B @ X4 ) )
% 9.20/5.24          & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
% 9.20/5.24            = ( gcd_gcd_nat @ A @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9332_gcd__code__integer,axiom,
% 9.20/5.24      ( gcd_gcd_Code_integer
% 9.20/5.24      = ( ^ [K3: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9333_gcd__int__def,axiom,
% 9.20/5.24      ( gcd_gcd_int
% 9.20/5.24      = ( ^ [X: int,Y: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X ) ) @ ( nat2 @ ( abs_abs_int @ Y ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9334_bezw__aux,axiom,
% 9.20/5.24      ! [X3: nat,Y3: nat] :
% 9.20/5.24        ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
% 9.20/5.24        = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9335_OR__upper,axiom,
% 9.20/5.24      ! [X3: int,N2: nat,Y3: int] :
% 9.20/5.24        ( ( ord_less_eq_int @ zero_zero_int @ X3 )
% 9.20/5.24       => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.24         => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
% 9.20/5.24           => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9336_or__Suc__0__eq,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9337_Suc__0__or__eq,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ N2 )
% 9.20/5.24        = ( plus_plus_nat @ N2 @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9338_or__nat__rec,axiom,
% 9.20/5.24      ( bit_se1412395901928357646or_nat
% 9.20/5.24      = ( ^ [M2: nat,N: nat] :
% 9.20/5.24            ( plus_plus_nat
% 9.20/5.24            @ ( zero_n2687167440665602831ol_nat
% 9.20/5.24              @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
% 9.20/5.24                | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.20/5.24            @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9339_gcd__nat_Opelims,axiom,
% 9.20/5.24      ! [X3: nat,Xa2: nat,Y3: nat] :
% 9.20/5.24        ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ~ ( ( ( ( Xa2 = zero_zero_nat )
% 9.20/5.24                 => ( Y3 = X3 ) )
% 9.20/5.24                & ( ( Xa2 != zero_zero_nat )
% 9.20/5.24                 => ( Y3
% 9.20/5.24                    = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
% 9.20/5.24             => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9340_gcd__nat_Oidem,axiom,
% 9.20/5.24      ! [A: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ A @ A )
% 9.20/5.24        = A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9341_gcd__nat_Oleft__idem,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
% 9.20/5.24        = ( gcd_gcd_nat @ A @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9342_gcd__nat_Oright__idem,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
% 9.20/5.24        = ( gcd_gcd_nat @ A @ B ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9343_gcd__idem__nat,axiom,
% 9.20/5.24      ! [X3: nat] :
% 9.20/5.24        ( ( gcd_gcd_nat @ X3 @ X3 )
% 9.20/5.24        = X3 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9344_or__nat__unfold,axiom,
% 9.20/5.24      ( bit_se1412395901928357646or_nat
% 9.20/5.24      = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ N @ ( 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 @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9345_max__Suc__Suc,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 9.20/5.24        = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9346_max__0R,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_max_nat @ N2 @ zero_zero_nat )
% 9.20/5.24        = N2 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9347_max__0L,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_max_nat @ zero_zero_nat @ N2 )
% 9.20/5.24        = N2 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9348_max__nat_Oright__neutral,axiom,
% 9.20/5.24      ! [A: nat] :
% 9.20/5.24        ( ( ord_max_nat @ A @ zero_zero_nat )
% 9.20/5.24        = A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9349_max__nat_Oneutr__eq__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( zero_zero_nat
% 9.20/5.24          = ( ord_max_nat @ A @ B ) )
% 9.20/5.24        = ( ( A = zero_zero_nat )
% 9.20/5.24          & ( B = zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9350_max__nat_Oleft__neutral,axiom,
% 9.20/5.24      ! [A: nat] :
% 9.20/5.24        ( ( ord_max_nat @ zero_zero_nat @ A )
% 9.20/5.24        = A ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9351_max__nat_Oeq__neutr__iff,axiom,
% 9.20/5.24      ! [A: nat,B: nat] :
% 9.20/5.24        ( ( ( ord_max_nat @ A @ B )
% 9.20/5.24          = zero_zero_nat )
% 9.20/5.24        = ( ( A = zero_zero_nat )
% 9.20/5.24          & ( B = zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9352_max__Suc__numeral,axiom,
% 9.20/5.24      ! [N2: nat,K: num] :
% 9.20/5.24        ( ( ord_max_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24        = ( suc @ ( ord_max_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9353_max__numeral__Suc,axiom,
% 9.20/5.24      ! [K: num,N2: nat] :
% 9.20/5.24        ( ( ord_max_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 9.20/5.24        = ( suc @ ( ord_max_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9354_nat__add__max__right,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
% 9.20/5.24        = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9355_nat__add__max__left,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
% 9.20/5.24        = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N2 @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9356_nat__mult__max__left,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
% 9.20/5.24        = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9357_nat__mult__max__right,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
% 9.20/5.24        = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9358_nat__minus__add__max,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
% 9.20/5.24        = ( ord_max_nat @ N2 @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9359_max__Suc2,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_max_nat @ M @ ( suc @ N2 ) )
% 9.20/5.24        = ( case_nat_nat @ ( suc @ N2 )
% 9.20/5.24          @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N2 ) )
% 9.20/5.24          @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9360_max__Suc1,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_max_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24        = ( case_nat_nat @ ( suc @ N2 )
% 9.20/5.24          @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N2 @ M4 ) )
% 9.20/5.24          @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9361_divmod__integer__eq__cases,axiom,
% 9.20/5.24      ( code_divmod_integer
% 9.20/5.24      = ( ^ [K3: code_integer,L2: code_integer] :
% 9.20/5.24            ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
% 9.20/5.24            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
% 9.20/5.24              @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
% 9.20/5.24                @ ( if_Pro6119634080678213985nteger
% 9.20/5.24                  @ ( ( sgn_sgn_Code_integer @ K3 )
% 9.20/5.24                    = ( sgn_sgn_Code_integer @ L2 ) )
% 9.20/5.24                  @ ( code_divmod_abs @ K3 @ L2 )
% 9.20/5.24                  @ ( produc6916734918728496179nteger
% 9.20/5.24                    @ ^ [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 ) ) )
% 9.20/5.24                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9362_card_Ocomp__fun__commute__on,axiom,
% 9.20/5.24      ( ( comp_nat_nat_nat @ suc @ suc )
% 9.20/5.24      = ( comp_nat_nat_nat @ suc @ suc ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9363_root__powr__inverse,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ( root @ N2 @ X3 )
% 9.20/5.24            = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9364_drop__bit__negative__int__iff,axiom,
% 9.20/5.24      ! [N2: nat,K: int] :
% 9.20/5.24        ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
% 9.20/5.24        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9365_real__root__Suc__0,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
% 9.20/5.24        = X3 ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9366_root__0,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( root @ zero_zero_nat @ X3 )
% 9.20/5.24        = zero_zero_real ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9367_real__root__eq__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( root @ N2 @ X3 )
% 9.20/5.24            = ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( X3 = Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9368_drop__bit__of__Suc__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_se8570568707652914677it_nat @ N2 @ ( suc @ zero_zero_nat ) )
% 9.20/5.24        = ( zero_n2687167440665602831ol_nat @ ( N2 = zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9369_drop__bit__Suc__minus__bit0,axiom,
% 9.20/5.24      ! [N2: nat,K: num] :
% 9.20/5.24        ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
% 9.20/5.24        = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9370_real__root__eq__0__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( root @ N2 @ X3 )
% 9.20/5.24            = zero_zero_real )
% 9.20/5.24          = ( X3 = zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9371_real__root__less__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9372_real__root__le__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9373_real__root__eq__1__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( root @ N2 @ X3 )
% 9.20/5.24            = one_one_real )
% 9.20/5.24          = ( X3 = one_one_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9374_real__root__one,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( root @ N2 @ one_one_real )
% 9.20/5.24          = one_one_real ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9375_real__root__lt__0__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
% 9.20/5.24          = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9376_real__root__gt__0__iff,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9377_real__root__le__0__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
% 9.20/5.24          = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9378_real__root__ge__0__iff,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9379_real__root__lt__1__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ one_one_real )
% 9.20/5.24          = ( ord_less_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9380_real__root__gt__1__iff,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9381_real__root__le__1__iff,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ one_one_real )
% 9.20/5.24          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9382_real__root__ge__1__iff,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y3 ) )
% 9.20/5.24          = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9383_drop__bit__Suc__minus__bit1,axiom,
% 9.20/5.24      ! [N2: nat,K: num] :
% 9.20/5.24        ( ( bit_se8568078237143864401it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
% 9.20/5.24        = ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9384_real__root__pow__pos2,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
% 9.20/5.24            = X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9385_real__root__less__mono,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.24         => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9386_real__root__le__mono,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ X3 @ Y3 )
% 9.20/5.24         => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9387_real__root__power,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,K: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( root @ N2 @ ( power_power_real @ X3 @ K ) )
% 9.20/5.24          = ( power_power_real @ ( root @ N2 @ X3 ) @ K ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9388_real__root__abs,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( root @ N2 @ ( abs_abs_real @ X3 ) )
% 9.20/5.24          = ( abs_abs_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9389_sgn__root,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( sgn_sgn_real @ ( root @ N2 @ X3 ) )
% 9.20/5.24          = ( sgn_sgn_real @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9390_real__root__gt__zero,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9391_real__root__strict__decreasing,axiom,
% 9.20/5.24      ! [N2: nat,N7: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.24         => ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.24           => ( ord_less_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9392_root__abs__power,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y3 @ N2 ) ) )
% 9.20/5.24          = ( abs_abs_real @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9393_real__root__pos__pos,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9394_real__root__strict__increasing,axiom,
% 9.20/5.24      ! [N2: nat,N7: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_nat @ N2 @ N7 )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24           => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24             => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9395_real__root__decreasing,axiom,
% 9.20/5.24      ! [N2: nat,N7: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.24         => ( ( ord_less_eq_real @ one_one_real @ X3 )
% 9.20/5.24           => ( ord_less_eq_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9396_real__root__pow__pos,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
% 9.20/5.24            = X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9397_real__root__pos__unique,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
% 9.20/5.24         => ( ( ( power_power_real @ Y3 @ N2 )
% 9.20/5.24              = X3 )
% 9.20/5.24           => ( ( root @ N2 @ X3 )
% 9.20/5.24              = Y3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9398_real__root__power__cancel,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( ( root @ N2 @ ( power_power_real @ X3 @ N2 ) )
% 9.20/5.24            = X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9399_real__root__increasing,axiom,
% 9.20/5.24      ! [N2: nat,N7: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_eq_nat @ N2 @ N7 )
% 9.20/5.24         => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24           => ( ( ord_less_eq_real @ X3 @ one_one_real )
% 9.20/5.24             => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9400_root__sgn__power,axiom,
% 9.20/5.24      ! [N2: nat,Y3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N2 ) ) )
% 9.20/5.24          = Y3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9401_sgn__power__root,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X3 ) ) @ N2 ) )
% 9.20/5.24          = X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9402_ln__root,axiom,
% 9.20/5.24      ! [N2: nat,B: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.24         => ( ( ln_ln_real @ ( root @ N2 @ B ) )
% 9.20/5.24            = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9403_log__root,axiom,
% 9.20/5.24      ! [N2: nat,A: real,B: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ A )
% 9.20/5.24         => ( ( log @ B @ ( root @ N2 @ A ) )
% 9.20/5.24            = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9404_log__base__root,axiom,
% 9.20/5.24      ! [N2: nat,B: real,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ B )
% 9.20/5.24         => ( ( log @ ( root @ N2 @ B ) @ X3 )
% 9.20/5.24            = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9405_split__root,axiom,
% 9.20/5.24      ! [P: real > $o,N2: nat,X3: real] :
% 9.20/5.24        ( ( P @ ( root @ N2 @ X3 ) )
% 9.20/5.24        = ( ( ( N2 = zero_zero_nat )
% 9.20/5.24           => ( P @ zero_zero_real ) )
% 9.20/5.24          & ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24           => ! [Y: real] :
% 9.20/5.24                ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 9.20/5.24                  = X3 )
% 9.20/5.24               => ( P @ Y ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9406_Suc__funpow,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( compow_nat_nat @ N2 @ suc )
% 9.20/5.24        = ( plus_plus_nat @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9407_min__Suc__Suc,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
% 9.20/5.24        = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9408_min__0R,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ N2 @ zero_zero_nat )
% 9.20/5.24        = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9409_min__0L,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ zero_zero_nat @ N2 )
% 9.20/5.24        = zero_zero_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9410_min__Suc__numeral,axiom,
% 9.20/5.24      ! [N2: nat,K: num] :
% 9.20/5.24        ( ( ord_min_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
% 9.20/5.24        = ( suc @ ( ord_min_nat @ N2 @ ( pred_numeral @ K ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9411_min__numeral__Suc,axiom,
% 9.20/5.24      ! [K: num,N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ ( numeral_numeral_nat @ K ) @ ( suc @ N2 ) )
% 9.20/5.24        = ( suc @ ( ord_min_nat @ ( pred_numeral @ K ) @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9412_min__diff,axiom,
% 9.20/5.24      ! [M: nat,I: nat,N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
% 9.20/5.24        = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9413_nat__mult__min__left,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q4 )
% 9.20/5.24        = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9414_nat__mult__min__right,axiom,
% 9.20/5.24      ! [M: nat,N2: nat,Q4: nat] :
% 9.20/5.24        ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q4 ) )
% 9.20/5.24        = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9415_inf__nat__def,axiom,
% 9.20/5.24      inf_inf_nat = ord_min_nat ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9416_min__Suc2,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_min_nat @ M @ ( suc @ N2 ) )
% 9.20/5.24        = ( case_nat_nat @ zero_zero_nat
% 9.20/5.24          @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N2 ) )
% 9.20/5.24          @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9417_min__Suc1,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_min_nat @ ( suc @ N2 ) @ M )
% 9.20/5.24        = ( case_nat_nat @ zero_zero_nat
% 9.20/5.24          @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N2 @ M4 ) )
% 9.20/5.24          @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9418_inf__int__def,axiom,
% 9.20/5.24      inf_inf_int = ord_min_int ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9419_max__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.20/5.24      ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
% 9.20/5.24      @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
% 9.20/5.24      @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9420_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
% 9.20/5.24      ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
% 9.20/5.24      @ ^ [M2: nat,N: nat] :
% 9.20/5.24          ( ( dvd_dvd_nat @ M2 @ N )
% 9.20/5.24          & ( M2 != N ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9421_times__int_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( abs_Integ
% 9.20/5.24          @ ( produc27273713700761075at_nat
% 9.20/5.24            @ ^ [X: nat,Y: nat] :
% 9.20/5.24                ( produc2626176000494625587at_nat
% 9.20/5.24                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
% 9.20/5.24            @ Xa2
% 9.20/5.24            @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9422_int_Oabs__induct,axiom,
% 9.20/5.24      ! [P: int > $o,X3: int] :
% 9.20/5.24        ( ! [Y5: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y5 ) )
% 9.20/5.24       => ( P @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9423_eq__Abs__Integ,axiom,
% 9.20/5.24      ! [Z2: int] :
% 9.20/5.24        ~ ! [X4: nat,Y5: nat] :
% 9.20/5.24            ( Z2
% 9.20/5.24           != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9424_nat_Oabs__eq,axiom,
% 9.20/5.24      ! [X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( nat2 @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9425_zero__int__def,axiom,
% 9.20/5.24      ( zero_zero_int
% 9.20/5.24      = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9426_int__def,axiom,
% 9.20/5.24      ( semiri1314217659103216013at_int
% 9.20/5.24      = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9427_uminus__int_Oabs__eq,axiom,
% 9.20/5.24      ! [X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( abs_Integ
% 9.20/5.24          @ ( produc2626176000494625587at_nat
% 9.20/5.24            @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
% 9.20/5.24            @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9428_one__int__def,axiom,
% 9.20/5.24      ( one_one_int
% 9.20/5.24      = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9429_less__int_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( produc8739625826339149834_nat_o
% 9.20/5.24          @ ^ [X: nat,Y: nat] :
% 9.20/5.24              ( produc6081775807080527818_nat_o
% 9.20/5.24              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.20/5.24          @ Xa2
% 9.20/5.24          @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9430_less__eq__int_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( produc8739625826339149834_nat_o
% 9.20/5.24          @ ^ [X: nat,Y: nat] :
% 9.20/5.24              ( produc6081775807080527818_nat_o
% 9.20/5.24              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
% 9.20/5.24          @ Xa2
% 9.20/5.24          @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9431_plus__int_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( abs_Integ
% 9.20/5.24          @ ( produc27273713700761075at_nat
% 9.20/5.24            @ ^ [X: nat,Y: nat] :
% 9.20/5.24                ( produc2626176000494625587at_nat
% 9.20/5.24                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
% 9.20/5.24            @ Xa2
% 9.20/5.24            @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9432_minus__int_Oabs__eq,axiom,
% 9.20/5.24      ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
% 9.20/5.24        ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
% 9.20/5.24        = ( abs_Integ
% 9.20/5.24          @ ( produc27273713700761075at_nat
% 9.20/5.24            @ ^ [X: nat,Y: nat] :
% 9.20/5.24                ( produc2626176000494625587at_nat
% 9.20/5.24                @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
% 9.20/5.24            @ Xa2
% 9.20/5.24            @ X3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9433_max__rpair__set,axiom,
% 9.20/5.24      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9434_min__rpair__set,axiom,
% 9.20/5.24      fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9435_less__eq__int_Orep__eq,axiom,
% 9.20/5.24      ( ord_less_eq_int
% 9.20/5.24      = ( ^ [X: int,Xa4: int] :
% 9.20/5.24            ( produc8739625826339149834_nat_o
% 9.20/5.24            @ ^ [Y: nat,Z6: nat] :
% 9.20/5.24                ( produc6081775807080527818_nat_o
% 9.20/5.24                @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
% 9.20/5.24            @ ( rep_Integ @ X )
% 9.20/5.24            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9436_less__int_Orep__eq,axiom,
% 9.20/5.24      ( ord_less_int
% 9.20/5.24      = ( ^ [X: int,Xa4: int] :
% 9.20/5.24            ( produc8739625826339149834_nat_o
% 9.20/5.24            @ ^ [Y: nat,Z6: nat] :
% 9.20/5.24                ( produc6081775807080527818_nat_o
% 9.20/5.24                @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
% 9.20/5.24            @ ( rep_Integ @ X )
% 9.20/5.24            @ ( rep_Integ @ Xa4 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9437_prod__encode__def,axiom,
% 9.20/5.24      ( nat_prod_encode
% 9.20/5.24      = ( produc6842872674320459806at_nat
% 9.20/5.24        @ ^ [M2: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N ) ) @ M2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9438_le__prod__encode__2,axiom,
% 9.20/5.24      ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9439_le__prod__encode__1,axiom,
% 9.20/5.24      ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9440_nat_Orep__eq,axiom,
% 9.20/5.24      ( nat2
% 9.20/5.24      = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9441_prod__encode__prod__decode__aux,axiom,
% 9.20/5.24      ! [K: nat,M: nat] :
% 9.20/5.24        ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
% 9.20/5.24        = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9442_uminus__int__def,axiom,
% 9.20/5.24      ( uminus_uminus_int
% 9.20/5.24      = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
% 9.20/5.24        @ ( produc2626176000494625587at_nat
% 9.20/5.24          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9443_times__int__def,axiom,
% 9.20/5.24      ( times_times_int
% 9.20/5.24      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.20/5.24        @ ( produc27273713700761075at_nat
% 9.20/5.24          @ ^ [X: nat,Y: nat] :
% 9.20/5.24              ( produc2626176000494625587at_nat
% 9.20/5.24              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9444_minus__int__def,axiom,
% 9.20/5.24      ( minus_minus_int
% 9.20/5.24      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.20/5.24        @ ( produc27273713700761075at_nat
% 9.20/5.24          @ ^ [X: nat,Y: nat] :
% 9.20/5.24              ( produc2626176000494625587at_nat
% 9.20/5.24              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9445_plus__int__def,axiom,
% 9.20/5.24      ( plus_plus_int
% 9.20/5.24      = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
% 9.20/5.24        @ ( produc27273713700761075at_nat
% 9.20/5.24          @ ^ [X: nat,Y: nat] :
% 9.20/5.24              ( produc2626176000494625587at_nat
% 9.20/5.24              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9446_Gcd__remove0__nat,axiom,
% 9.20/5.24      ! [M7: set_nat] :
% 9.20/5.24        ( ( finite_finite_nat @ M7 )
% 9.20/5.24       => ( ( gcd_Gcd_nat @ M7 )
% 9.20/5.24          = ( gcd_Gcd_nat @ ( minus_minus_set_nat @ M7 @ ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9447_Gcd__dvd__nat,axiom,
% 9.20/5.24      ! [A: nat,A2: set_nat] :
% 9.20/5.24        ( ( member_nat @ A @ A2 )
% 9.20/5.24       => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9448_Gcd__greatest__nat,axiom,
% 9.20/5.24      ! [A2: set_nat,A: nat] :
% 9.20/5.24        ( ! [B4: nat] :
% 9.20/5.24            ( ( member_nat @ B4 @ A2 )
% 9.20/5.24           => ( dvd_dvd_nat @ A @ B4 ) )
% 9.20/5.24       => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9449_Gcd__nat__eq__one,axiom,
% 9.20/5.24      ! [N7: set_nat] :
% 9.20/5.24        ( ( member_nat @ one_one_nat @ N7 )
% 9.20/5.24       => ( ( gcd_Gcd_nat @ N7 )
% 9.20/5.24          = one_one_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9450_Gcd__in,axiom,
% 9.20/5.24      ! [A2: set_nat] :
% 9.20/5.24        ( ! [A4: nat,B4: nat] :
% 9.20/5.24            ( ( member_nat @ A4 @ A2 )
% 9.20/5.24           => ( ( member_nat @ B4 @ A2 )
% 9.20/5.24             => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A2 ) ) )
% 9.20/5.24       => ( ( A2 != bot_bot_set_nat )
% 9.20/5.24         => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9451_sub__BitM__One__eq,axiom,
% 9.20/5.24      ! [N2: num] :
% 9.20/5.24        ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
% 9.20/5.24        = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9452_num__of__nat_Osimps_I2_J,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( num_of_nat @ ( suc @ N2 ) )
% 9.20/5.24            = ( inc @ ( num_of_nat @ N2 ) ) ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( num_of_nat @ ( suc @ N2 ) )
% 9.20/5.24            = one ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9453_pred__nat__def,axiom,
% 9.20/5.24      ( pred_nat
% 9.20/5.24      = ( collec3392354462482085612at_nat
% 9.20/5.24        @ ( produc6081775807080527818_nat_o
% 9.20/5.24          @ ^ [M2: nat,N: nat] :
% 9.20/5.24              ( N
% 9.20/5.24              = ( suc @ M2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9454_abs__Gcd__eq,axiom,
% 9.20/5.24      ! [K5: set_int] :
% 9.20/5.24        ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
% 9.20/5.24        = ( gcd_Gcd_int @ K5 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9455_Gcd__dvd__int,axiom,
% 9.20/5.24      ! [A: int,A2: set_int] :
% 9.20/5.24        ( ( member_int @ A @ A2 )
% 9.20/5.24       => ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9456_Gcd__greatest__int,axiom,
% 9.20/5.24      ! [A2: set_int,A: int] :
% 9.20/5.24        ( ! [B4: int] :
% 9.20/5.24            ( ( member_int @ B4 @ A2 )
% 9.20/5.24           => ( dvd_dvd_int @ A @ B4 ) )
% 9.20/5.24       => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9457_Gcd__int__greater__eq__0,axiom,
% 9.20/5.24      ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9458_num__of__nat_Osimps_I1_J,axiom,
% 9.20/5.24      ( ( num_of_nat @ zero_zero_nat )
% 9.20/5.24      = one ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9459_numeral__num__of__nat,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( numeral_numeral_nat @ ( num_of_nat @ N2 ) )
% 9.20/5.24          = N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9460_num__of__nat__One,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ N2 @ one_one_nat )
% 9.20/5.24       => ( ( num_of_nat @ N2 )
% 9.20/5.24          = one ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9461_num__of__nat__double,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( num_of_nat @ ( plus_plus_nat @ N2 @ N2 ) )
% 9.20/5.24          = ( bit0 @ ( num_of_nat @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9462_num__of__nat__plus__distrib,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ M )
% 9.20/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
% 9.20/5.24            = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9463_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.24        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( X3
% 9.20/5.24                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24           => ( Xa2 = one_one_nat ) )
% 9.20/5.24         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24               => ( ( Deg2 = Xa2 )
% 9.20/5.24                  & ! [X4: vEBT_VEBT] :
% 9.20/5.24                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                     => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                  & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                  & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                  & ( case_o184042715313410164at_nat
% 9.20/5.24                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                      & ! [X: vEBT_VEBT] :
% 9.20/5.24                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                    @ ( produc6081775807080527818_nat_o
% 9.20/5.24                      @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                          & ! [I2: nat] :
% 9.20/5.24                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                          & ( ( Mi3 = Ma3 )
% 9.20/5.24                           => ! [X: vEBT_VEBT] :
% 9.20/5.24                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                          & ( ( Mi3 != Ma3 )
% 9.20/5.24                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                              & ! [X: nat] :
% 9.20/5.24                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                   => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                    @ Mima ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9464_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
% 9.20/5.24      ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
% 9.20/5.24        ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
% 9.20/5.24        = ( ( Deg = Deg4 )
% 9.20/5.24          & ! [X: vEBT_VEBT] :
% 9.20/5.24              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.20/5.24             => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24          & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24          & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
% 9.20/5.24            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24          & ( case_o184042715313410164at_nat
% 9.20/5.24            @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
% 9.20/5.24              & ! [X: vEBT_VEBT] :
% 9.20/5.24                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.20/5.24                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24            @ ( produc6081775807080527818_nat_o
% 9.20/5.24              @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                  ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                  & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.20/5.24                  & ! [I2: nat] :
% 9.20/5.24                      ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
% 9.20/5.24                        = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
% 9.20/5.24                  & ( ( Mi3 = Ma3 )
% 9.20/5.24                   => ! [X: vEBT_VEBT] :
% 9.20/5.24                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
% 9.20/5.24                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                  & ( ( Mi3 != Ma3 )
% 9.20/5.24                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
% 9.20/5.24                      & ! [X: nat] :
% 9.20/5.24                          ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
% 9.20/5.24                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
% 9.20/5.24                           => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                              & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24            @ Mima2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9465_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.24        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( X3
% 9.20/5.24                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24           => ( Y3
% 9.20/5.24              = ( Xa2 != one_one_nat ) ) )
% 9.20/5.24         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24               => ( Y3
% 9.20/5.24                  = ( ~ ( ( Deg2 = Xa2 )
% 9.20/5.24                        & ! [X: vEBT_VEBT] :
% 9.20/5.24                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                        & ( case_o184042715313410164at_nat
% 9.20/5.24                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                            & ! [X: vEBT_VEBT] :
% 9.20/5.24                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                          @ ( produc6081775807080527818_nat_o
% 9.20/5.24                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                & ! [I2: nat] :
% 9.20/5.24                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                                & ( ( Mi3 = Ma3 )
% 9.20/5.24                                 => ! [X: vEBT_VEBT] :
% 9.20/5.24                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                                & ( ( Mi3 != Ma3 )
% 9.20/5.24                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                                    & ! [X: nat] :
% 9.20/5.24                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                          @ Mima ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9466_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.24        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24       => ( ( ? [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( X3
% 9.20/5.24                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24           => ( Xa2 != one_one_nat ) )
% 9.20/5.24         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24               => ~ ( ( Deg2 = Xa2 )
% 9.20/5.24                    & ! [X2: vEBT_VEBT] :
% 9.20/5.24                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                       => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                    & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                    & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                    & ( case_o184042715313410164at_nat
% 9.20/5.24                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                        & ! [X: vEBT_VEBT] :
% 9.20/5.24                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                      @ ( produc6081775807080527818_nat_o
% 9.20/5.24                        @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                            & ! [I2: nat] :
% 9.20/5.24                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                            & ( ( Mi3 = Ma3 )
% 9.20/5.24                             => ! [X: vEBT_VEBT] :
% 9.20/5.24                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                            & ( ( Mi3 != Ma3 )
% 9.20/5.24                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                                & ! [X: nat] :
% 9.20/5.24                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                     => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                      @ Mima ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9467_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
% 9.20/5.24        ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24          = Y3 )
% 9.20/5.24       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24               => ( ( Y3
% 9.20/5.24                    = ( Xa2 = one_one_nat ) )
% 9.20/5.24                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
% 9.20/5.24           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24                 => ( ( Y3
% 9.20/5.24                      = ( ( Deg2 = Xa2 )
% 9.20/5.24                        & ! [X: vEBT_VEBT] :
% 9.20/5.24                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                        & ( case_o184042715313410164at_nat
% 9.20/5.24                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                            & ! [X: vEBT_VEBT] :
% 9.20/5.24                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                          @ ( produc6081775807080527818_nat_o
% 9.20/5.24                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                & ! [I2: nat] :
% 9.20/5.24                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                                & ( ( Mi3 = Ma3 )
% 9.20/5.24                                 => ! [X: vEBT_VEBT] :
% 9.20/5.24                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                                & ( ( Mi3 != Ma3 )
% 9.20/5.24                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                                    & ! [X: nat] :
% 9.20/5.24                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                          @ Mima ) ) )
% 9.20/5.24                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9468_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.24        ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.20/5.24                 => ( Xa2 != one_one_nat ) ) )
% 9.20/5.24           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.20/5.24                   => ~ ( ( Deg2 = Xa2 )
% 9.20/5.24                        & ! [X2: vEBT_VEBT] :
% 9.20/5.24                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                           => ( vEBT_VEBT_valid @ X2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                        & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                        & ( case_o184042715313410164at_nat
% 9.20/5.24                          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                            & ! [X: vEBT_VEBT] :
% 9.20/5.24                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                          @ ( produc6081775807080527818_nat_o
% 9.20/5.24                            @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                & ! [I2: nat] :
% 9.20/5.24                                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                                & ( ( Mi3 = Ma3 )
% 9.20/5.24                                 => ! [X: vEBT_VEBT] :
% 9.20/5.24                                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                                & ( ( Mi3 != Ma3 )
% 9.20/5.24                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                                    & ! [X: nat] :
% 9.20/5.24                                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                         => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                          @ Mima ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9469_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
% 9.20/5.24      ! [X3: vEBT_VEBT,Xa2: nat] :
% 9.20/5.24        ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
% 9.20/5.24       => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
% 9.20/5.24         => ( ! [Uu2: $o,Uv2: $o] :
% 9.20/5.24                ( ( X3
% 9.20/5.24                  = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
% 9.20/5.24               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) )
% 9.20/5.24                 => ( Xa2 = one_one_nat ) ) )
% 9.20/5.24           => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
% 9.20/5.24                  ( ( X3
% 9.20/5.24                    = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
% 9.20/5.24                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
% 9.20/5.24                   => ( ( Deg2 = Xa2 )
% 9.20/5.24                      & ! [X4: vEBT_VEBT] :
% 9.20/5.24                          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                         => ( vEBT_VEBT_valid @ X4 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
% 9.20/5.24                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                      & ( case_o184042715313410164at_nat
% 9.20/5.24                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
% 9.20/5.24                          & ! [X: vEBT_VEBT] :
% 9.20/5.24                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                        @ ( produc6081775807080527818_nat_o
% 9.20/5.24                          @ ^ [Mi3: nat,Ma3: nat] :
% 9.20/5.24                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
% 9.20/5.24                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                              & ! [I2: nat] :
% 9.20/5.24                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
% 9.20/5.24                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
% 9.20/5.24                              & ( ( Mi3 = Ma3 )
% 9.20/5.24                               => ! [X: vEBT_VEBT] :
% 9.20/5.24                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
% 9.20/5.24                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
% 9.20/5.24                              & ( ( Mi3 != Ma3 )
% 9.20/5.24                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
% 9.20/5.24                                  & ! [X: nat] :
% 9.20/5.24                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
% 9.20/5.24                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
% 9.20/5.24                                       => ( ( ord_less_nat @ Mi3 @ X )
% 9.20/5.24                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
% 9.20/5.24                        @ Mima ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9470_GreatestI__nat,axiom,
% 9.20/5.24      ! [P: nat > $o,K: nat,B: nat] :
% 9.20/5.24        ( ( P @ K )
% 9.20/5.24       => ( ! [Y5: nat] :
% 9.20/5.24              ( ( P @ Y5 )
% 9.20/5.24             => ( ord_less_eq_nat @ Y5 @ B ) )
% 9.20/5.24         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9471_Greatest__le__nat,axiom,
% 9.20/5.24      ! [P: nat > $o,K: nat,B: nat] :
% 9.20/5.24        ( ( P @ K )
% 9.20/5.24       => ( ! [Y5: nat] :
% 9.20/5.24              ( ( P @ Y5 )
% 9.20/5.24             => ( ord_less_eq_nat @ Y5 @ B ) )
% 9.20/5.24         => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9472_GreatestI__ex__nat,axiom,
% 9.20/5.24      ! [P: nat > $o,B: nat] :
% 9.20/5.24        ( ? [X_12: nat] : ( P @ X_12 )
% 9.20/5.24       => ( ! [Y5: nat] :
% 9.20/5.24              ( ( P @ Y5 )
% 9.20/5.24             => ( ord_less_eq_nat @ Y5 @ B ) )
% 9.20/5.24         => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9473_take__bit__num__simps_I1_J,axiom,
% 9.20/5.24      ! [M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ zero_zero_nat @ M )
% 9.20/5.24        = none_num ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9474_take__bit__num__simps_I2_J,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_take_bit_num @ ( suc @ N2 ) @ one )
% 9.20/5.24        = ( some_num @ one ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9475_take__bit__num__simps_I3_J,axiom,
% 9.20/5.24      ! [N2: nat,M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
% 9.20/5.24        = ( case_o6005452278849405969um_num @ none_num
% 9.20/5.24          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.20/5.24          @ ( bit_take_bit_num @ N2 @ M ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9476_take__bit__num__simps_I4_J,axiom,
% 9.20/5.24      ! [N2: nat,M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
% 9.20/5.24        = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9477_take__bit__num__simps_I6_J,axiom,
% 9.20/5.24      ! [R2: num,M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
% 9.20/5.24        = ( case_o6005452278849405969um_num @ none_num
% 9.20/5.24          @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.20/5.24          @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9478_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
% 9.20/5.24      ! [N2: nat,M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
% 9.20/5.24        = ( case_nat_option_num @ none_num
% 9.20/5.24          @ ^ [N: nat] :
% 9.20/5.24              ( case_o6005452278849405969um_num @ none_num
% 9.20/5.24              @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.20/5.24              @ ( bit_take_bit_num @ N @ M ) )
% 9.20/5.24          @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9479_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( bit_take_bit_num @ N2 @ one )
% 9.20/5.24        = ( case_nat_option_num @ none_num
% 9.20/5.24          @ ^ [N: nat] : ( some_num @ one )
% 9.20/5.24          @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
% 9.20/5.24      ! [N2: nat,M: num] :
% 9.20/5.24        ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
% 9.20/5.24        = ( case_nat_option_num @ none_num
% 9.20/5.24          @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
% 9.20/5.24          @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9481_take__bit__num__def,axiom,
% 9.20/5.24      ( bit_take_bit_num
% 9.20/5.24      = ( ^ [N: nat,M2: num] :
% 9.20/5.24            ( if_option_num
% 9.20/5.24            @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) )
% 9.20/5.24              = zero_zero_nat )
% 9.20/5.24            @ none_num
% 9.20/5.24            @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9482_Bit__Operations_Otake__bit__num__code,axiom,
% 9.20/5.24      ( bit_take_bit_num
% 9.20/5.24      = ( ^ [N: nat,M2: num] :
% 9.20/5.24            ( produc478579273971653890on_num
% 9.20/5.24            @ ^ [A3: nat,X: num] :
% 9.20/5.24                ( case_nat_option_num @ none_num
% 9.20/5.24                @ ^ [O: nat] :
% 9.20/5.24                    ( case_num_option_num @ ( some_num @ one )
% 9.20/5.24                    @ ^ [P6: num] :
% 9.20/5.24                        ( case_o6005452278849405969um_num @ none_num
% 9.20/5.24                        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
% 9.20/5.24                        @ ( bit_take_bit_num @ O @ P6 ) )
% 9.20/5.24                    @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
% 9.20/5.24                    @ X )
% 9.20/5.24                @ A3 )
% 9.20/5.24            @ ( product_Pair_nat_num @ N @ M2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9483_Rats__eq__int__div__nat,axiom,
% 9.20/5.24      ( field_5140801741446780682s_real
% 9.20/5.24      = ( collect_real
% 9.20/5.24        @ ^ [Uu3: real] :
% 9.20/5.24          ? [I2: int,N: nat] :
% 9.20/5.24            ( ( Uu3
% 9.20/5.24              = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N ) ) )
% 9.20/5.24            & ( N != zero_zero_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9484_Rats__dense__in__real,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.24       => ? [X4: real] :
% 9.20/5.24            ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 9.20/5.24            & ( ord_less_real @ X3 @ X4 )
% 9.20/5.24            & ( ord_less_real @ X4 @ Y3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9485_Rats__no__bot__less,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24      ? [X4: real] :
% 9.20/5.24        ( ( member_real @ X4 @ field_5140801741446780682s_real )
% 9.20/5.24        & ( ord_less_real @ X4 @ X3 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9486_rat__floor__lemma,axiom,
% 9.20/5.24      ! [A: int,B: int] :
% 9.20/5.24        ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
% 9.20/5.24        & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9487_image__minus__const__atLeastLessThan__nat,axiom,
% 9.20/5.24      ! [C: nat,Y3: nat,X3: nat] :
% 9.20/5.24        ( ( ( ord_less_nat @ C @ Y3 )
% 9.20/5.24         => ( ( image_nat_nat
% 9.20/5.24              @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 9.20/5.24              @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.20/5.24            = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
% 9.20/5.24        & ( ~ ( ord_less_nat @ C @ Y3 )
% 9.20/5.24         => ( ( ( ord_less_nat @ X3 @ Y3 )
% 9.20/5.24             => ( ( image_nat_nat
% 9.20/5.24                  @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 9.20/5.24                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.20/5.24                = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
% 9.20/5.24            & ( ~ ( ord_less_nat @ X3 @ Y3 )
% 9.20/5.24             => ( ( image_nat_nat
% 9.20/5.24                  @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
% 9.20/5.24                  @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
% 9.20/5.24                = bot_bot_set_nat ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9488_image__Suc__atLeastAtMost,axiom,
% 9.20/5.24      ! [I: nat,J: nat] :
% 9.20/5.24        ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I @ J ) )
% 9.20/5.24        = ( set_or1269000886237332187st_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9489_image__Suc__atLeastLessThan,axiom,
% 9.20/5.24      ! [I: nat,J: nat] :
% 9.20/5.24        ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
% 9.20/5.24        = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9490_less__rat,axiom,
% 9.20/5.24      ! [B: int,D: int,A: int,C: int] :
% 9.20/5.24        ( ( B != zero_zero_int )
% 9.20/5.24       => ( ( D != zero_zero_int )
% 9.20/5.24         => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
% 9.20/5.24            = ( 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 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9491_zero__notin__Suc__image,axiom,
% 9.20/5.24      ! [A2: set_nat] :
% 9.20/5.24        ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9492_Rat__induct__pos,axiom,
% 9.20/5.24      ! [P: rat > $o,Q4: rat] :
% 9.20/5.24        ( ! [A4: int,B4: int] :
% 9.20/5.24            ( ( ord_less_int @ zero_zero_int @ B4 )
% 9.20/5.24           => ( P @ ( fract @ A4 @ B4 ) ) )
% 9.20/5.24       => ( P @ Q4 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9493_quotient__of__eq,axiom,
% 9.20/5.24      ! [A: int,B: int,P4: int,Q4: int] :
% 9.20/5.24        ( ( ( quotient_of @ ( fract @ A @ B ) )
% 9.20/5.24          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24       => ( ( fract @ P4 @ Q4 )
% 9.20/5.24          = ( fract @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9494_normalize__eq,axiom,
% 9.20/5.24      ! [A: int,B: int,P4: int,Q4: int] :
% 9.20/5.24        ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
% 9.20/5.24          = ( product_Pair_int_int @ P4 @ Q4 ) )
% 9.20/5.24       => ( ( fract @ P4 @ Q4 )
% 9.20/5.24          = ( fract @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9495_quotient__of__Fract,axiom,
% 9.20/5.24      ! [A: int,B: int] :
% 9.20/5.24        ( ( quotient_of @ ( fract @ A @ B ) )
% 9.20/5.24        = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9496_image__Suc__lessThan,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24        = ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9497_image__Suc__atMost,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) )
% 9.20/5.24        = ( set_or1269000886237332187st_nat @ one_one_nat @ ( suc @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9498_atLeast0__atMost__Suc__eq__insert__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 9.20/5.24        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9499_atLeast0__lessThan__Suc__eq__insert__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( suc @ N2 ) )
% 9.20/5.24        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9500_lessThan__Suc__eq__insert__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_ord_lessThan_nat @ ( suc @ N2 ) )
% 9.20/5.24        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9501_atMost__Suc__eq__insert__0,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( set_ord_atMost_nat @ ( suc @ N2 ) )
% 9.20/5.24        = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9502_Fract__less__zero__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.20/5.24          = ( ord_less_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9503_zero__less__Fract__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.20/5.24          = ( ord_less_int @ zero_zero_int @ A ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9504_one__less__Fract__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.20/5.24          = ( ord_less_int @ B @ A ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9505_Fract__less__one__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.20/5.24          = ( ord_less_int @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9506_zero__le__Fract__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
% 9.20/5.24          = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9507_Fract__le__zero__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
% 9.20/5.24          = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9508_Fract__le__one__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
% 9.20/5.24          = ( ord_less_eq_int @ A @ B ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9509_one__le__Fract__iff,axiom,
% 9.20/5.24      ! [B: int,A: int] :
% 9.20/5.24        ( ( ord_less_int @ zero_zero_int @ B )
% 9.20/5.24       => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
% 9.20/5.24          = ( ord_less_eq_int @ B @ A ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9510_Gcd__abs__eq,axiom,
% 9.20/5.24      ! [K5: set_int] :
% 9.20/5.24        ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
% 9.20/5.24        = ( gcd_Gcd_int @ K5 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9511_Gcd__int__eq,axiom,
% 9.20/5.24      ! [N7: set_nat] :
% 9.20/5.24        ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N7 ) )
% 9.20/5.24        = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N7 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9512_Gcd__nat__abs__eq,axiom,
% 9.20/5.24      ! [K5: set_int] :
% 9.20/5.24        ( ( gcd_Gcd_nat
% 9.20/5.24          @ ( image_int_nat
% 9.20/5.24            @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
% 9.20/5.24            @ K5 ) )
% 9.20/5.24        = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9513_finite__int__iff__bounded,axiom,
% 9.20/5.24      ( finite_finite_int
% 9.20/5.24      = ( ^ [S7: set_int] :
% 9.20/5.24          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9514_finite__int__iff__bounded__le,axiom,
% 9.20/5.24      ( finite_finite_int
% 9.20/5.24      = ( ^ [S7: set_int] :
% 9.20/5.24          ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9515_infinite__int__iff__infinite__nat__abs,axiom,
% 9.20/5.24      ! [S3: set_int] :
% 9.20/5.24        ( ( ~ ( finite_finite_int @ S3 ) )
% 9.20/5.24        = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9516_Gcd__int__def,axiom,
% 9.20/5.24      ( gcd_Gcd_int
% 9.20/5.24      = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9517_range__mult,axiom,
% 9.20/5.24      ! [A: real] :
% 9.20/5.24        ( ( ( A = zero_zero_real )
% 9.20/5.24         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.20/5.24            = ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) )
% 9.20/5.24        & ( ( A != zero_zero_real )
% 9.20/5.24         => ( ( image_real_real @ ( times_times_real @ A ) @ top_top_set_real )
% 9.20/5.24            = top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9518_range__abs__Nats,axiom,
% 9.20/5.24      ( ( image_int_int @ abs_abs_int @ top_top_set_int )
% 9.20/5.24      = semiring_1_Nats_int ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9519_infinite__UNIV__int,axiom,
% 9.20/5.24      ~ ( finite_finite_int @ top_top_set_int ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9520_nat__not__finite,axiom,
% 9.20/5.24      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9521_infinite__UNIV__nat,axiom,
% 9.20/5.24      ~ ( finite_finite_nat @ top_top_set_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9522_int__in__range__abs,axiom,
% 9.20/5.24      ! [N2: nat] : ( member_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( image_int_int @ abs_abs_int @ top_top_set_int ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9523_UNIV__nat__eq,axiom,
% 9.20/5.24      ( top_top_set_nat
% 9.20/5.24      = ( insert_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ top_top_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9524_range__mod,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( image_nat_nat
% 9.20/5.24            @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N2 )
% 9.20/5.24            @ top_top_set_nat )
% 9.20/5.24          = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9525_root__def,axiom,
% 9.20/5.24      ( root
% 9.20/5.24      = ( ^ [N: nat,X: real] :
% 9.20/5.24            ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
% 9.20/5.24            @ ( the_in5290026491893676941l_real @ top_top_set_real
% 9.20/5.24              @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
% 9.20/5.24              @ X ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9526_DERIV__even__real__root,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24         => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.24           => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9527_DERIV__real__root__generic,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,D6: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( X3 != zero_zero_real )
% 9.20/5.24         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24             => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24               => ( D6
% 9.20/5.24                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
% 9.20/5.24           => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24               => ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.24                 => ( D6
% 9.20/5.24                    = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
% 9.20/5.24             => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24                 => ( D6
% 9.20/5.24                    = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9528_DERIV__neg__dec__right,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9529_DERIV__pos__inc__right,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9530_DERIV__pos__inc__left,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9531_DERIV__neg__dec__left,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9532_DERIV__neg__imp__decreasing,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.24             => ( ( ord_less_eq_real @ X4 @ B )
% 9.20/5.24               => ? [Y6: real] :
% 9.20/5.24                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                    & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
% 9.20/5.24         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9533_DERIV__pos__imp__increasing,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.24             => ( ( ord_less_eq_real @ X4 @ B )
% 9.20/5.24               => ? [Y6: real] :
% 9.20/5.24                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                    & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
% 9.20/5.24         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9534_has__real__derivative__pos__inc__right,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
% 9.20/5.24                   => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9535_has__real__derivative__neg__dec__right,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 9.20/5.24       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
% 9.20/5.24                   => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                     => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9536_has__real__derivative__neg__dec__left,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 9.20/5.24       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
% 9.20/5.24                   => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                     => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9537_has__real__derivative__pos__inc__left,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,S3: set_real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.20/5.24         => ? [D4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ D4 )
% 9.20/5.24              & ! [H4: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ H4 )
% 9.20/5.24                 => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
% 9.20/5.24                   => ( ( ord_less_real @ H4 @ D4 )
% 9.20/5.24                     => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9538_MVT2,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,F5: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.24             => ( ( ord_less_eq_real @ X4 @ B )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24         => ? [Z3: real] :
% 9.20/5.24              ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24              & ( ord_less_real @ Z3 @ B )
% 9.20/5.24              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.20/5.24                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9539_DERIV__local__const,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,D: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.20/5.24         => ( ! [Y5: real] :
% 9.20/5.24                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 9.20/5.24               => ( ( F @ X3 )
% 9.20/5.24                  = ( F @ Y5 ) ) )
% 9.20/5.24           => ( L = zero_zero_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9540_DERIV__ln,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9541_DERIV__local__max,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,D: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.20/5.24         => ( ! [Y5: real] :
% 9.20/5.24                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 9.20/5.24               => ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
% 9.20/5.24           => ( L = zero_zero_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9542_DERIV__local__min,axiom,
% 9.20/5.24      ! [F: real > real,L: real,X3: real,D: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ D )
% 9.20/5.24         => ( ! [Y5: real] :
% 9.20/5.24                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
% 9.20/5.24               => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
% 9.20/5.24           => ( L = zero_zero_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9543_DERIV__ln__divide,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9544_DERIV__pow,axiom,
% 9.20/5.24      ! [N2: nat,X3: real,S: set_real] :
% 9.20/5.24        ( has_fi5821293074295781190e_real
% 9.20/5.24        @ ^ [X: real] : ( power_power_real @ X @ N2 )
% 9.20/5.24        @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
% 9.20/5.24        @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9545_has__real__derivative__powr,axiom,
% 9.20/5.24      ! [Z2: real,R2: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ Z2 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real
% 9.20/5.24          @ ^ [Z6: real] : ( powr_real @ Z6 @ R2 )
% 9.20/5.24          @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
% 9.20/5.24          @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9546_DERIV__log,axiom,
% 9.20/5.24      ! [X3: real,B: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9547_DERIV__fun__powr,axiom,
% 9.20/5.24      ! [G: real > real,M: real,X3: real,R2: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 9.20/5.24         => ( has_fi5821293074295781190e_real
% 9.20/5.24            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
% 9.20/5.24            @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
% 9.20/5.24            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9548_DERIV__powr,axiom,
% 9.20/5.24      ! [G: real > real,M: real,X3: real,F: real > real,R2: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
% 9.20/5.24         => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
% 9.20/5.24           => ( has_fi5821293074295781190e_real
% 9.20/5.24              @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
% 9.20/5.24              @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
% 9.20/5.24              @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9549_arcosh__real__has__field__derivative,axiom,
% 9.20/5.24      ! [X3: real,A2: set_real] :
% 9.20/5.24        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9550_DERIV__real__sqrt,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9551_DERIV__real__sqrt__generic,axiom,
% 9.20/5.24      ! [X3: real,D6: real] :
% 9.20/5.24        ( ( X3 != zero_zero_real )
% 9.20/5.24       => ( ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24           => ( D6
% 9.20/5.24              = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24         => ( ( ( ord_less_real @ X3 @ zero_zero_real )
% 9.20/5.24             => ( D6
% 9.20/5.24                = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
% 9.20/5.24           => ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9552_artanh__real__has__field__derivative,axiom,
% 9.20/5.24      ! [X3: real,A2: set_real] :
% 9.20/5.24        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.24       => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9553_DERIV__real__root,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ X3 )
% 9.20/5.24         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9554_DERIV__arcsin,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9555_Maclaurin__all__le__objl,axiom,
% 9.20/5.24      ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
% 9.20/5.24        ( ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24            = F )
% 9.20/5.24          & ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.20/5.24       => ? [T6: real] :
% 9.20/5.24            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 9.20/5.24            & ( ( F @ X3 )
% 9.20/5.24              = ( plus_plus_real
% 9.20/5.24                @ ( groups6591440286371151544t_real
% 9.20/5.24                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.24                  @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9556_Maclaurin__all__le,axiom,
% 9.20/5.24      ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
% 9.20/5.24        ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24          = F )
% 9.20/5.24       => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24         => ? [T6: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 9.20/5.24              & ( ( F @ X3 )
% 9.20/5.24                = ( plus_plus_real
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9557_DERIV__odd__real__root,axiom,
% 9.20/5.24      ! [N2: nat,X3: real] :
% 9.20/5.24        ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24       => ( ( X3 != zero_zero_real )
% 9.20/5.24         => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9558_Maclaurin__minus,axiom,
% 9.20/5.24      ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ H2 @ zero_zero_real )
% 9.20/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24              = F )
% 9.20/5.24           => ( ! [M3: nat,T6: real] :
% 9.20/5.24                  ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                    & ( ord_less_eq_real @ H2 @ T6 )
% 9.20/5.24                    & ( ord_less_eq_real @ T6 @ zero_zero_real ) )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24             => ? [T6: real] :
% 9.20/5.24                  ( ( ord_less_real @ H2 @ T6 )
% 9.20/5.24                  & ( ord_less_real @ T6 @ zero_zero_real )
% 9.20/5.24                  & ( ( F @ H2 )
% 9.20/5.24                    = ( plus_plus_real
% 9.20/5.24                      @ ( groups6591440286371151544t_real
% 9.20/5.24                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.20/5.24                        @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9559_Maclaurin2,axiom,
% 9.20/5.24      ! [H2: real,Diff: nat > real > real,F: real > real,N2: nat] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.20/5.24       => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24            = F )
% 9.20/5.24         => ( ! [M3: nat,T6: real] :
% 9.20/5.24                ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                  & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.20/5.24                  & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24           => ? [T6: real] :
% 9.20/5.24                ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.20/5.24                & ( ord_less_eq_real @ T6 @ H2 )
% 9.20/5.24                & ( ( F @ H2 )
% 9.20/5.24                  = ( plus_plus_real
% 9.20/5.24                    @ ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9560_Maclaurin,axiom,
% 9.20/5.24      ! [H2: real,N2: nat,Diff: nat > real > real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ H2 )
% 9.20/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24              = F )
% 9.20/5.24           => ( ! [M3: nat,T6: real] :
% 9.20/5.24                  ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                    & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.20/5.24                    & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24             => ? [T6: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ T6 )
% 9.20/5.24                  & ( ord_less_real @ T6 @ H2 )
% 9.20/5.24                  & ( ( F @ H2 )
% 9.20/5.24                    = ( plus_plus_real
% 9.20/5.24                      @ ( groups6591440286371151544t_real
% 9.20/5.24                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
% 9.20/5.24                        @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H2 @ N2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9561_Maclaurin__all__lt,axiom,
% 9.20/5.24      ! [Diff: nat > real > real,F: real > real,N2: nat,X3: real] :
% 9.20/5.24        ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24          = F )
% 9.20/5.24       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24         => ( ( X3 != zero_zero_real )
% 9.20/5.24           => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24             => ? [T6: real] :
% 9.20/5.24                  ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
% 9.20/5.24                  & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 9.20/5.24                  & ( ( F @ X3 )
% 9.20/5.24                    = ( plus_plus_real
% 9.20/5.24                      @ ( groups6591440286371151544t_real
% 9.20/5.24                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.24                        @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9562_Maclaurin__bi__le,axiom,
% 9.20/5.24      ! [Diff: nat > real > real,F: real > real,N2: nat,X3: real] :
% 9.20/5.24        ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24          = F )
% 9.20/5.24       => ( ! [M3: nat,T6: real] :
% 9.20/5.24              ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                & ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) ) )
% 9.20/5.24             => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24         => ? [T6: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
% 9.20/5.24              & ( ( F @ X3 )
% 9.20/5.24                = ( plus_plus_real
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9563_Taylor,axiom,
% 9.20/5.24      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X3: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24            = F )
% 9.20/5.24         => ( ! [M3: nat,T6: real] :
% 9.20/5.24                ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                  & ( ord_less_eq_real @ A @ T6 )
% 9.20/5.24                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24           => ( ( ord_less_eq_real @ A @ C )
% 9.20/5.24             => ( ( ord_less_eq_real @ C @ B )
% 9.20/5.24               => ( ( ord_less_eq_real @ A @ X3 )
% 9.20/5.24                 => ( ( ord_less_eq_real @ X3 @ B )
% 9.20/5.24                   => ( ( X3 != C )
% 9.20/5.24                     => ? [T6: real] :
% 9.20/5.24                          ( ( ( ord_less_real @ X3 @ C )
% 9.20/5.24                           => ( ( ord_less_real @ X3 @ T6 )
% 9.20/5.24                              & ( ord_less_real @ T6 @ C ) ) )
% 9.20/5.24                          & ( ~ ( ord_less_real @ X3 @ C )
% 9.20/5.24                           => ( ( ord_less_real @ C @ T6 )
% 9.20/5.24                              & ( ord_less_real @ T6 @ X3 ) ) )
% 9.20/5.24                          & ( ( F @ X3 )
% 9.20/5.24                            = ( plus_plus_real
% 9.20/5.24                              @ ( groups6591440286371151544t_real
% 9.20/5.24                                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M2 ) )
% 9.20/5.24                                @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9564_Taylor__up,axiom,
% 9.20/5.24      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24            = F )
% 9.20/5.24         => ( ! [M3: nat,T6: real] :
% 9.20/5.24                ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                  & ( ord_less_eq_real @ A @ T6 )
% 9.20/5.24                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24           => ( ( ord_less_eq_real @ A @ C )
% 9.20/5.24             => ( ( ord_less_real @ C @ B )
% 9.20/5.24               => ? [T6: real] :
% 9.20/5.24                    ( ( ord_less_real @ C @ T6 )
% 9.20/5.24                    & ( ord_less_real @ T6 @ B )
% 9.20/5.24                    & ( ( F @ B )
% 9.20/5.24                      = ( plus_plus_real
% 9.20/5.24                        @ ( groups6591440286371151544t_real
% 9.20/5.24                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
% 9.20/5.24                          @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9565_Taylor__down,axiom,
% 9.20/5.24      ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( ( Diff @ zero_zero_nat )
% 9.20/5.24            = F )
% 9.20/5.24         => ( ! [M3: nat,T6: real] :
% 9.20/5.24                ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24                  & ( ord_less_eq_real @ A @ T6 )
% 9.20/5.24                  & ( ord_less_eq_real @ T6 @ B ) )
% 9.20/5.24               => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24           => ( ( ord_less_real @ A @ C )
% 9.20/5.24             => ( ( ord_less_eq_real @ C @ B )
% 9.20/5.24               => ? [T6: real] :
% 9.20/5.24                    ( ( ord_less_real @ A @ T6 )
% 9.20/5.24                    & ( ord_less_real @ T6 @ C )
% 9.20/5.24                    & ( ( F @ A )
% 9.20/5.24                      = ( plus_plus_real
% 9.20/5.24                        @ ( groups6591440286371151544t_real
% 9.20/5.24                          @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
% 9.20/5.24                          @ ( set_ord_lessThan_nat @ N2 ) )
% 9.20/5.24                        @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9566_Maclaurin__lemma2,axiom,
% 9.20/5.24      ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: real] :
% 9.20/5.24        ( ! [M3: nat,T6: real] :
% 9.20/5.24            ( ( ( ord_less_nat @ M3 @ N2 )
% 9.20/5.24              & ( ord_less_eq_real @ zero_zero_real @ T6 )
% 9.20/5.24              & ( ord_less_eq_real @ T6 @ H2 ) )
% 9.20/5.24           => ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T6 ) @ ( topolo2177554685111907308n_real @ T6 @ top_top_set_real ) ) )
% 9.20/5.24       => ( ( N2
% 9.20/5.24            = ( suc @ K ) )
% 9.20/5.24         => ! [M6: nat,T7: real] :
% 9.20/5.24              ( ( ( ord_less_nat @ M6 @ N2 )
% 9.20/5.24                & ( ord_less_eq_real @ zero_zero_real @ T7 )
% 9.20/5.24                & ( ord_less_eq_real @ T7 @ H2 ) )
% 9.20/5.24             => ( has_fi5821293074295781190e_real
% 9.20/5.24                @ ^ [U2: real] :
% 9.20/5.24                    ( minus_minus_real @ ( Diff @ M6 @ U2 )
% 9.20/5.24                    @ ( plus_plus_real
% 9.20/5.24                      @ ( groups6591440286371151544t_real
% 9.20/5.24                        @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M6 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
% 9.20/5.24                        @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M6 ) ) )
% 9.20/5.24                      @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M6 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M6 ) ) ) ) ) )
% 9.20/5.24                @ ( minus_minus_real @ ( Diff @ ( suc @ M6 ) @ T7 )
% 9.20/5.24                  @ ( plus_plus_real
% 9.20/5.24                    @ ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M6 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) )
% 9.20/5.24                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) )
% 9.20/5.24                @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9567_DERIV__arctan__series,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.24       => ( has_fi5821293074295781190e_real
% 9.20/5.24          @ ^ [X8: real] :
% 9.20/5.24              ( suminf_real
% 9.20/5.24              @ ^ [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 @ X8 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
% 9.20/5.24          @ ( suminf_real
% 9.20/5.24            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
% 9.20/5.24          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9568_DERIV__arccos,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9569_DERIV__power__series_H,axiom,
% 9.20/5.24      ! [R: real,F: nat > real,X0: real] :
% 9.20/5.24        ( ! [X4: real] :
% 9.20/5.24            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.20/5.24           => ( summable_real
% 9.20/5.24              @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X4 @ N ) ) ) )
% 9.20/5.24       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ R )
% 9.20/5.24           => ( has_fi5821293074295781190e_real
% 9.20/5.24              @ ^ [X: real] :
% 9.20/5.24                  ( suminf_real
% 9.20/5.24                  @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ ( suc @ N ) ) ) )
% 9.20/5.24              @ ( suminf_real
% 9.20/5.24                @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X0 @ N ) ) )
% 9.20/5.24              @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9570_arccos__less__arccos,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ Y3 )
% 9.20/5.24         => ( ( ord_less_eq_real @ Y3 @ one_one_real )
% 9.20/5.24           => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9571_arccos__less__mono,axiom,
% 9.20/5.24      ! [X3: real,Y3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.24       => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
% 9.20/5.24         => ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
% 9.20/5.24            = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9572_DERIV__isconst3,axiom,
% 9.20/5.24      ! [A: real,B: real,X3: real,Y3: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.20/5.24         => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.20/5.24           => ( ! [X4: real] :
% 9.20/5.24                  ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.20/5.24             => ( ( F @ X3 )
% 9.20/5.24                = ( F @ Y3 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9573_arccos__lt__bounded,axiom,
% 9.20/5.24      ! [Y3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
% 9.20/5.24       => ( ( ord_less_real @ Y3 @ one_one_real )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
% 9.20/5.24            & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9574_sin__arccos__nonzero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( ( sin_real @ ( arccos @ X3 ) )
% 9.20/5.24           != zero_zero_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9575_finite__greaterThanLessThan,axiom,
% 9.20/5.24      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9576_finite__greaterThanLessThan__int,axiom,
% 9.20/5.24      ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9577_card__greaterThanLessThan,axiom,
% 9.20/5.24      ! [L: nat,U: nat] :
% 9.20/5.24        ( ( finite_card_nat @ ( set_or5834768355832116004an_nat @ L @ U ) )
% 9.20/5.24        = ( minus_minus_nat @ U @ ( suc @ L ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9578_atLeastSucLessThan__greaterThanLessThan,axiom,
% 9.20/5.24      ! [L: nat,U: nat] :
% 9.20/5.24        ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
% 9.20/5.24        = ( set_or5834768355832116004an_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9579_LIM__fun__less__zero,axiom,
% 9.20/5.24      ! [F: real > real,L: real,C: real] :
% 9.20/5.24        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ L @ zero_zero_real )
% 9.20/5.24         => ? [R4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ R4 )
% 9.20/5.24              & ! [X2: real] :
% 9.20/5.24                  ( ( ( X2 != C )
% 9.20/5.24                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 9.20/5.24                 => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9580_LIM__fun__not__zero,axiom,
% 9.20/5.24      ! [F: real > real,L: real,C: real] :
% 9.20/5.24        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.20/5.24       => ( ( L != zero_zero_real )
% 9.20/5.24         => ? [R4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ R4 )
% 9.20/5.24              & ! [X2: real] :
% 9.20/5.24                  ( ( ( X2 != C )
% 9.20/5.24                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 9.20/5.24                 => ( ( F @ X2 )
% 9.20/5.24                   != zero_zero_real ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9581_LIM__fun__gt__zero,axiom,
% 9.20/5.24      ! [F: real > real,L: real,C: real] :
% 9.20/5.24        ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
% 9.20/5.24       => ( ( ord_less_real @ zero_zero_real @ L )
% 9.20/5.24         => ? [R4: real] :
% 9.20/5.24              ( ( ord_less_real @ zero_zero_real @ R4 )
% 9.20/5.24              & ! [X2: real] :
% 9.20/5.24                  ( ( ( X2 != C )
% 9.20/5.24                    & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
% 9.20/5.24                 => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9582_isCont__inverse__function2,axiom,
% 9.20/5.24      ! [A: real,X3: real,B: real,G: real > real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ B )
% 9.20/5.24         => ( ! [Z3: real] :
% 9.20/5.24                ( ( ord_less_eq_real @ A @ Z3 )
% 9.20/5.24               => ( ( ord_less_eq_real @ Z3 @ B )
% 9.20/5.24                 => ( ( G @ ( F @ Z3 ) )
% 9.20/5.24                    = Z3 ) ) )
% 9.20/5.24           => ( ! [Z3: real] :
% 9.20/5.24                  ( ( ord_less_eq_real @ A @ Z3 )
% 9.20/5.24                 => ( ( ord_less_eq_real @ Z3 @ B )
% 9.20/5.24                   => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 9.20/5.24             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9583_nth__sorted__list__of__set__greaterThanLessThan,axiom,
% 9.20/5.24      ! [N2: nat,J: nat,I: nat] :
% 9.20/5.24        ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I ) ) )
% 9.20/5.24       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) ) @ N2 )
% 9.20/5.24          = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9584_isCont__arcosh,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.24       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9585_DERIV__inverse__function,axiom,
% 9.20/5.24      ! [F: real > real,D6: real,G: real > real,X3: real,A: real,B: real] :
% 9.20/5.24        ( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
% 9.20/5.24       => ( ( D6 != zero_zero_real )
% 9.20/5.24         => ( ( ord_less_real @ A @ X3 )
% 9.20/5.24           => ( ( ord_less_real @ X3 @ B )
% 9.20/5.24             => ( ! [Y5: real] :
% 9.20/5.24                    ( ( ord_less_real @ A @ Y5 )
% 9.20/5.24                   => ( ( ord_less_real @ Y5 @ B )
% 9.20/5.24                     => ( ( F @ ( G @ Y5 ) )
% 9.20/5.24                        = Y5 ) ) )
% 9.20/5.24               => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9586_isCont__arccos,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9587_isCont__arcsin,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9588_LIM__less__bound,axiom,
% 9.20/5.24      ! [B: real,X3: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ B @ X3 )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X3 ) )
% 9.20/5.24             => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
% 9.20/5.24         => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
% 9.20/5.24           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9589_isCont__artanh,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9590_isCont__inverse__function,axiom,
% 9.20/5.24      ! [D: real,X3: real,G: real > real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ D )
% 9.20/5.24       => ( ! [Z3: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
% 9.20/5.24             => ( ( G @ ( F @ Z3 ) )
% 9.20/5.24                = Z3 ) )
% 9.20/5.24         => ( ! [Z3: real] :
% 9.20/5.24                ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
% 9.20/5.24               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
% 9.20/5.24           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9591_GMVT_H,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [Z3: real] :
% 9.20/5.24              ( ( ord_less_eq_real @ A @ Z3 )
% 9.20/5.24             => ( ( ord_less_eq_real @ Z3 @ B )
% 9.20/5.24               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
% 9.20/5.24         => ( ! [Z3: real] :
% 9.20/5.24                ( ( ord_less_eq_real @ A @ Z3 )
% 9.20/5.24               => ( ( ord_less_eq_real @ Z3 @ B )
% 9.20/5.24                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
% 9.20/5.24           => ( ! [Z3: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24                 => ( ( ord_less_real @ Z3 @ B )
% 9.20/5.24                   => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 9.20/5.24             => ( ! [Z3: real] :
% 9.20/5.24                    ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24                   => ( ( ord_less_real @ Z3 @ B )
% 9.20/5.24                     => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
% 9.20/5.24               => ? [C2: real] :
% 9.20/5.24                    ( ( ord_less_real @ A @ C2 )
% 9.20/5.24                    & ( ord_less_real @ C2 @ B )
% 9.20/5.24                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
% 9.20/5.24                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9592_summable__Leibniz_I2_J,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ( topolo6980174941875973593q_real @ A )
% 9.20/5.24         => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
% 9.20/5.24           => ! [N6: nat] :
% 9.20/5.24                ( member_real
% 9.20/5.24                @ ( suminf_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 9.20/5.24                @ ( set_or1222579329274155063t_real
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9593_summable__Leibniz_I3_J,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ( topolo6980174941875973593q_real @ A )
% 9.20/5.24         => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
% 9.20/5.24           => ! [N6: nat] :
% 9.20/5.24                ( member_real
% 9.20/5.24                @ ( suminf_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 9.20/5.24                @ ( set_or1222579329274155063t_real
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
% 9.20/5.24                  @ ( groups6591440286371151544t_real
% 9.20/5.24                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9594_summable__Leibniz_H_I5_J,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24           => ( filterlim_nat_real
% 9.20/5.24              @ ^ [N: nat] :
% 9.20/5.24                  ( groups6591440286371151544t_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 9.20/5.24              @ ( topolo2815343760600316023s_real
% 9.20/5.24                @ ( suminf_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 9.20/5.24              @ at_top_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9595_trivial__limit__sequentially,axiom,
% 9.20/5.24      at_top_nat != bot_bot_filter_nat ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9596_filterlim__Suc,axiom,
% 9.20/5.24      filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9597_mult__nat__left__at__top,axiom,
% 9.20/5.24      ! [C: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.24       => ( filterlim_nat_nat @ ( times_times_nat @ C ) @ at_top_nat @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9598_mult__nat__right__at__top,axiom,
% 9.20/5.24      ! [C: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ C )
% 9.20/5.24       => ( filterlim_nat_nat
% 9.20/5.24          @ ^ [X: nat] : ( times_times_nat @ X @ C )
% 9.20/5.24          @ at_top_nat
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9599_nested__sequence__unique,axiom,
% 9.20/5.24      ! [F: nat > real,G: nat > real] :
% 9.20/5.24        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
% 9.20/5.24           => ( ( filterlim_nat_real
% 9.20/5.24                @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( G @ N ) )
% 9.20/5.24                @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24                @ at_top_nat )
% 9.20/5.24             => ? [L4: real] :
% 9.20/5.24                  ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
% 9.20/5.24                  & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
% 9.20/5.24                  & ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G @ N6 ) )
% 9.20/5.24                  & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9600_LIMSEQ__inverse__zero,axiom,
% 9.20/5.24      ! [X7: nat > real] :
% 9.20/5.24        ( ! [R4: real] :
% 9.20/5.24          ? [N8: nat] :
% 9.20/5.24          ! [N3: nat] :
% 9.20/5.24            ( ( ord_less_eq_nat @ N8 @ N3 )
% 9.20/5.24           => ( ord_less_real @ R4 @ ( X7 @ N3 ) ) )
% 9.20/5.24       => ( filterlim_nat_real
% 9.20/5.24          @ ^ [N: nat] : ( inverse_inverse_real @ ( X7 @ N ) )
% 9.20/5.24          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9601_LIMSEQ__root__const,axiom,
% 9.20/5.24      ! [C: real] :
% 9.20/5.24        ( ( ord_less_real @ zero_zero_real @ C )
% 9.20/5.24       => ( filterlim_nat_real
% 9.20/5.24          @ ^ [N: nat] : ( root @ N @ C )
% 9.20/5.24          @ ( topolo2815343760600316023s_real @ one_one_real )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9602_LIMSEQ__inverse__real__of__nat,axiom,
% 9.20/5.24      ( filterlim_nat_real
% 9.20/5.24      @ ^ [N: nat] : ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) )
% 9.20/5.24      @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24      @ at_top_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9603_LIMSEQ__inverse__real__of__nat__add,axiom,
% 9.20/5.24      ! [R2: real] :
% 9.20/5.24        ( filterlim_nat_real
% 9.20/5.24        @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
% 9.20/5.24        @ ( topolo2815343760600316023s_real @ R2 )
% 9.20/5.24        @ at_top_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9604_increasing__LIMSEQ,axiom,
% 9.20/5.24      ! [F: nat > real,L: real] :
% 9.20/5.24        ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ L )
% 9.20/5.24         => ( ! [E: real] :
% 9.20/5.24                ( ( ord_less_real @ zero_zero_real @ E )
% 9.20/5.24               => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
% 9.20/5.24           => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9605_LIMSEQ__realpow__zero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.24       => ( ( ord_less_real @ X3 @ one_one_real )
% 9.20/5.24         => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9606_LIMSEQ__divide__realpow__zero,axiom,
% 9.20/5.24      ! [X3: real,A: real] :
% 9.20/5.24        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.24       => ( filterlim_nat_real
% 9.20/5.24          @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N ) )
% 9.20/5.24          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9607_LIMSEQ__abs__realpow__zero2,axiom,
% 9.20/5.24      ! [C: real] :
% 9.20/5.24        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.20/5.24       => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9608_LIMSEQ__abs__realpow__zero,axiom,
% 9.20/5.24      ! [C: real] :
% 9.20/5.24        ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
% 9.20/5.24       => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9609_LIMSEQ__inverse__realpow__zero,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_real @ one_one_real @ X3 )
% 9.20/5.24       => ( filterlim_nat_real
% 9.20/5.24          @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N ) )
% 9.20/5.24          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9610_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
% 9.20/5.24      ! [R2: real] :
% 9.20/5.24        ( filterlim_nat_real
% 9.20/5.24        @ ^ [N: nat] : ( plus_plus_real @ R2 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
% 9.20/5.24        @ ( topolo2815343760600316023s_real @ R2 )
% 9.20/5.24        @ at_top_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9611_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
% 9.20/5.24      ! [R2: real] :
% 9.20/5.24        ( filterlim_nat_real
% 9.20/5.24        @ ^ [N: nat] : ( times_times_real @ R2 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
% 9.20/5.24        @ ( topolo2815343760600316023s_real @ R2 )
% 9.20/5.24        @ at_top_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9612_summable,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24           => ( summable_real
% 9.20/5.24              @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9613_zeroseq__arctan__series,axiom,
% 9.20/5.24      ! [X3: real] :
% 9.20/5.24        ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
% 9.20/5.24       => ( filterlim_nat_real
% 9.20/5.24          @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) )
% 9.20/5.24          @ ( topolo2815343760600316023s_real @ zero_zero_real )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9614_summable__Leibniz_H_I3_J,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24           => ( filterlim_nat_real
% 9.20/5.24              @ ^ [N: nat] :
% 9.20/5.24                  ( groups6591440286371151544t_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.20/5.24              @ ( topolo2815343760600316023s_real
% 9.20/5.24                @ ( suminf_real
% 9.20/5.24                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 9.20/5.24              @ at_top_nat ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9615_summable__Leibniz_H_I2_J,axiom,
% 9.20/5.24      ! [A: nat > real,N2: nat] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24           => ( ord_less_eq_real
% 9.20/5.24              @ ( groups6591440286371151544t_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
% 9.20/5.24              @ ( suminf_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9616_sums__alternating__upper__lower,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24           => ? [L4: real] :
% 9.20/5.24                ( ! [N6: nat] :
% 9.20/5.24                    ( ord_less_eq_real
% 9.20/5.24                    @ ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
% 9.20/5.24                    @ L4 )
% 9.20/5.24                & ( filterlim_nat_real
% 9.20/5.24                  @ ^ [N: nat] :
% 9.20/5.24                      ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
% 9.20/5.24                  @ ( topolo2815343760600316023s_real @ L4 )
% 9.20/5.24                  @ at_top_nat )
% 9.20/5.24                & ! [N6: nat] :
% 9.20/5.24                    ( ord_less_eq_real @ L4
% 9.20/5.24                    @ ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
% 9.20/5.24                & ( filterlim_nat_real
% 9.20/5.24                  @ ^ [N: nat] :
% 9.20/5.24                      ( groups6591440286371151544t_real
% 9.20/5.24                      @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                      @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 9.20/5.24                  @ ( topolo2815343760600316023s_real @ L4 )
% 9.20/5.24                  @ at_top_nat ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9617_summable__Leibniz_I5_J,axiom,
% 9.20/5.24      ! [A: nat > real] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ( topolo6980174941875973593q_real @ A )
% 9.20/5.24         => ( filterlim_nat_real
% 9.20/5.24            @ ^ [N: nat] :
% 9.20/5.24                ( groups6591440286371151544t_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
% 9.20/5.24            @ ( topolo2815343760600316023s_real
% 9.20/5.24              @ ( suminf_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
% 9.20/5.24            @ at_top_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9618_summable__Leibniz_H_I4_J,axiom,
% 9.20/5.24      ! [A: nat > real,N2: nat] :
% 9.20/5.24        ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
% 9.20/5.24       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
% 9.20/5.24         => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
% 9.20/5.24           => ( ord_less_eq_real
% 9.20/5.24              @ ( suminf_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
% 9.20/5.24              @ ( groups6591440286371151544t_real
% 9.20/5.24                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
% 9.20/5.24                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9619_eventually__sequentially__Suc,axiom,
% 9.20/5.24      ! [P: nat > $o] :
% 9.20/5.24        ( ( eventually_nat
% 9.20/5.24          @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
% 9.20/5.24          @ at_top_nat )
% 9.20/5.24        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9620_eventually__sequentially__seg,axiom,
% 9.20/5.24      ! [P: nat > $o,K: nat] :
% 9.20/5.24        ( ( eventually_nat
% 9.20/5.24          @ ^ [N: nat] : ( P @ ( plus_plus_nat @ N @ K ) )
% 9.20/5.24          @ at_top_nat )
% 9.20/5.24        = ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9621_eventually__sequentiallyI,axiom,
% 9.20/5.24      ! [C: nat,P: nat > $o] :
% 9.20/5.24        ( ! [X4: nat] :
% 9.20/5.24            ( ( ord_less_eq_nat @ C @ X4 )
% 9.20/5.24           => ( P @ X4 ) )
% 9.20/5.24       => ( eventually_nat @ P @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9622_eventually__sequentially,axiom,
% 9.20/5.24      ! [P: nat > $o] :
% 9.20/5.24        ( ( eventually_nat @ P @ at_top_nat )
% 9.20/5.24        = ( ? [N5: nat] :
% 9.20/5.24            ! [N: nat] :
% 9.20/5.24              ( ( ord_less_eq_nat @ N5 @ N )
% 9.20/5.24             => ( P @ N ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9623_le__sequentially,axiom,
% 9.20/5.24      ! [F3: filter_nat] :
% 9.20/5.24        ( ( ord_le2510731241096832064er_nat @ F3 @ at_top_nat )
% 9.20/5.24        = ( ! [N5: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N5 ) @ F3 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9624_sequentially__offset,axiom,
% 9.20/5.24      ! [P: nat > $o,K: nat] :
% 9.20/5.24        ( ( eventually_nat @ P @ at_top_nat )
% 9.20/5.24       => ( eventually_nat
% 9.20/5.24          @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
% 9.20/5.24          @ at_top_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9625_DERIV__neg__imp__decreasing__at__top,axiom,
% 9.20/5.24      ! [B: real,F: real > real,Flim: real] :
% 9.20/5.24        ( ! [X4: real] :
% 9.20/5.24            ( ( ord_less_eq_real @ B @ X4 )
% 9.20/5.24           => ? [Y6: real] :
% 9.20/5.24                ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                & ( ord_less_real @ Y6 @ zero_zero_real ) ) )
% 9.20/5.24       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
% 9.20/5.24         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9626_finite__greaterThanAtMost,axiom,
% 9.20/5.24      ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9627_atLeastSucAtMost__greaterThanAtMost,axiom,
% 9.20/5.24      ! [L: nat,U: nat] :
% 9.20/5.24        ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
% 9.20/5.24        = ( set_or6659071591806873216st_nat @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9628_eventually__at__left__real,axiom,
% 9.20/5.24      ! [B: real,A: real] :
% 9.20/5.24        ( ( ord_less_real @ B @ A )
% 9.20/5.24       => ( eventually_real
% 9.20/5.24          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B @ A ) )
% 9.20/5.24          @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9629_nth__sorted__list__of__set__greaterThanAtMost,axiom,
% 9.20/5.24      ! [N2: nat,J: nat,I: nat] :
% 9.20/5.24        ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I ) )
% 9.20/5.24       => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) ) @ N2 )
% 9.20/5.24          = ( suc @ ( plus_plus_nat @ I @ N2 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9630_filterlim__pow__at__bot__even,axiom,
% 9.20/5.24      ! [N2: nat,F: real > real,F3: filter_real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 9.20/5.24         => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24           => ( filterlim_real_real
% 9.20/5.24              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
% 9.20/5.24              @ at_top_real
% 9.20/5.24              @ F3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9631_finite__greaterThanAtMost__int,axiom,
% 9.20/5.24      ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9632_DERIV__pos__imp__increasing__at__bot,axiom,
% 9.20/5.24      ! [B: real,F: real > real,Flim: real] :
% 9.20/5.24        ( ! [X4: real] :
% 9.20/5.24            ( ( ord_less_eq_real @ X4 @ B )
% 9.20/5.24           => ? [Y6: real] :
% 9.20/5.24                ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                & ( ord_less_real @ zero_zero_real @ Y6 ) ) )
% 9.20/5.24       => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
% 9.20/5.24         => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9633_filterlim__pow__at__bot__odd,axiom,
% 9.20/5.24      ! [N2: nat,F: real > real,F3: filter_real] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( filterlim_real_real @ F @ at_bot_real @ F3 )
% 9.20/5.24         => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
% 9.20/5.24           => ( filterlim_real_real
% 9.20/5.24              @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
% 9.20/5.24              @ at_bot_real
% 9.20/5.24              @ F3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9634_GMVT,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,G: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.24                & ( ord_less_eq_real @ X4 @ B ) )
% 9.20/5.24             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
% 9.20/5.24         => ( ! [X4: real] :
% 9.20/5.24                ( ( ( ord_less_real @ A @ X4 )
% 9.20/5.24                  & ( ord_less_real @ X4 @ B ) )
% 9.20/5.24               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.20/5.24           => ( ! [X4: real] :
% 9.20/5.24                  ( ( ( ord_less_eq_real @ A @ X4 )
% 9.20/5.24                    & ( ord_less_eq_real @ X4 @ B ) )
% 9.20/5.24                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
% 9.20/5.24             => ( ! [X4: real] :
% 9.20/5.24                    ( ( ( ord_less_real @ A @ X4 )
% 9.20/5.24                      & ( ord_less_real @ X4 @ B ) )
% 9.20/5.24                   => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
% 9.20/5.24               => ? [G_c: real,F_c: real,C2: real] :
% 9.20/5.24                    ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 9.20/5.24                    & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
% 9.20/5.24                    & ( ord_less_real @ A @ C2 )
% 9.20/5.24                    & ( ord_less_real @ C2 @ B )
% 9.20/5.24                    & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
% 9.20/5.24                      = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9635_MVT,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24         => ( ! [X4: real] :
% 9.20/5.24                ( ( ord_less_real @ A @ X4 )
% 9.20/5.24               => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24           => ? [L4: real,Z3: real] :
% 9.20/5.24                ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24                & ( ord_less_real @ Z3 @ B )
% 9.20/5.24                & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
% 9.20/5.24                & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.20/5.24                  = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9636_Rolle__deriv,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,F5: real > real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( ( F @ A )
% 9.20/5.24            = ( F @ B ) )
% 9.20/5.24         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24           => ( ! [X4: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ X4 )
% 9.20/5.24                 => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                   => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24             => ? [Z3: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24                  & ( ord_less_real @ Z3 @ B )
% 9.20/5.24                  & ( ( F5 @ Z3 )
% 9.20/5.24                    = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9637_mvt,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,F5: real > real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24         => ( ! [X4: real] :
% 9.20/5.24                ( ( ord_less_real @ A @ X4 )
% 9.20/5.24               => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24           => ~ ! [Xi: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ Xi )
% 9.20/5.24                 => ( ( ord_less_real @ Xi @ B )
% 9.20/5.24                   => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
% 9.20/5.24                     != ( F5 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9638_DERIV__pos__imp__increasing__open,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ord_less_real @ A @ X4 )
% 9.20/5.24             => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24               => ? [Y6: real] :
% 9.20/5.24                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                    & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
% 9.20/5.24         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24           => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9639_DERIV__neg__imp__decreasing__open,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ! [X4: real] :
% 9.20/5.24              ( ( ord_less_real @ A @ X4 )
% 9.20/5.24             => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24               => ? [Y6: real] :
% 9.20/5.24                    ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.24                    & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
% 9.20/5.24         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24           => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9640_DERIV__isconst__end,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24         => ( ! [X4: real] :
% 9.20/5.24                ( ( ord_less_real @ A @ X4 )
% 9.20/5.24               => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24           => ( ( F @ B )
% 9.20/5.24              = ( F @ A ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9641_DERIV__isconst2,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real,X3: real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24         => ( ! [X4: real] :
% 9.20/5.24                ( ( ord_less_real @ A @ X4 )
% 9.20/5.24               => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                 => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24           => ( ( ord_less_eq_real @ A @ X3 )
% 9.20/5.24             => ( ( ord_less_eq_real @ X3 @ B )
% 9.20/5.24               => ( ( F @ X3 )
% 9.20/5.24                  = ( F @ A ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9642_Rolle,axiom,
% 9.20/5.24      ! [A: real,B: real,F: real > real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( ( ( F @ A )
% 9.20/5.24            = ( F @ B ) )
% 9.20/5.24         => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
% 9.20/5.24           => ( ! [X4: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ X4 )
% 9.20/5.24                 => ( ( ord_less_real @ X4 @ B )
% 9.20/5.24                   => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
% 9.20/5.24             => ? [Z3: real] :
% 9.20/5.24                  ( ( ord_less_real @ A @ Z3 )
% 9.20/5.24                  & ( ord_less_real @ Z3 @ B )
% 9.20/5.24                  & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9643_INT__greaterThan__UNIV,axiom,
% 9.20/5.24      ( ( comple7806235888213564991et_nat @ ( image_nat_set_nat @ set_or1210151606488870762an_nat @ top_top_set_nat ) )
% 9.20/5.24      = bot_bot_set_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9644_greaterThan__0,axiom,
% 9.20/5.24      ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
% 9.20/5.24      = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9645_eventually__at__right__real,axiom,
% 9.20/5.24      ! [A: real,B: real] :
% 9.20/5.24        ( ( ord_less_real @ A @ B )
% 9.20/5.24       => ( eventually_real
% 9.20/5.24          @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
% 9.20/5.24          @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9646_greaterThan__Suc,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( set_or1210151606488870762an_nat @ ( suc @ K ) )
% 9.20/5.24        = ( minus_minus_set_nat @ ( set_or1210151606488870762an_nat @ K ) @ ( insert_nat @ ( suc @ K ) @ bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9647_atLeast__0,axiom,
% 9.20/5.24      ( ( set_ord_atLeast_nat @ zero_zero_nat )
% 9.20/5.24      = top_top_set_nat ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9648_atLeast__Suc__greaterThan,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.20/5.24        = ( set_or1210151606488870762an_nat @ K ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9649_atLeast__Suc,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( set_ord_atLeast_nat @ ( suc @ K ) )
% 9.20/5.24        = ( minus_minus_set_nat @ ( set_ord_atLeast_nat @ K ) @ ( insert_nat @ K @ bot_bot_set_nat ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9650_Gcd__eq__Max,axiom,
% 9.20/5.24      ! [M7: set_nat] :
% 9.20/5.24        ( ( finite_finite_nat @ M7 )
% 9.20/5.24       => ( ( M7 != bot_bot_set_nat )
% 9.20/5.24         => ( ~ ( member_nat @ zero_zero_nat @ M7 )
% 9.20/5.24           => ( ( gcd_Gcd_nat @ M7 )
% 9.20/5.24              = ( lattic8265883725875713057ax_nat
% 9.20/5.24                @ ( comple7806235888213564991et_nat
% 9.20/5.24                  @ ( image_nat_set_nat
% 9.20/5.24                    @ ^ [M2: nat] :
% 9.20/5.24                        ( collect_nat
% 9.20/5.24                        @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
% 9.20/5.24                    @ M7 ) ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9651_Max__divisors__self__nat,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( N2 != zero_zero_nat )
% 9.20/5.24       => ( ( lattic8265883725875713057ax_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N2 ) ) )
% 9.20/5.24          = N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9652_bdd__above__nat,axiom,
% 9.20/5.24      condit2214826472909112428ve_nat = finite_finite_nat ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9653_card__le__Suc__Max,axiom,
% 9.20/5.24      ! [S3: set_nat] :
% 9.20/5.24        ( ( finite_finite_nat @ S3 )
% 9.20/5.24       => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9654_Sup__nat__def,axiom,
% 9.20/5.24      ( complete_Sup_Sup_nat
% 9.20/5.24      = ( ^ [X5: set_nat] : ( if_nat @ ( X5 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X5 ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9655_divide__nat__def,axiom,
% 9.20/5.24      ( divide_divide_nat
% 9.20/5.24      = ( ^ [M2: nat,N: nat] :
% 9.20/5.24            ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
% 9.20/5.24            @ ( lattic8265883725875713057ax_nat
% 9.20/5.24              @ ( collect_nat
% 9.20/5.24                @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9656_gcd__is__Max__divisors__nat,axiom,
% 9.20/5.24      ! [N2: nat,M: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( ( gcd_gcd_nat @ M @ N2 )
% 9.20/5.24          = ( lattic8265883725875713057ax_nat
% 9.20/5.24            @ ( collect_nat
% 9.20/5.24              @ ^ [D5: nat] :
% 9.20/5.24                  ( ( dvd_dvd_nat @ D5 @ M )
% 9.20/5.24                  & ( dvd_dvd_nat @ D5 @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9657_Max__divisors__self__int,axiom,
% 9.20/5.24      ! [N2: int] :
% 9.20/5.24        ( ( N2 != zero_zero_int )
% 9.20/5.24       => ( ( lattic8263393255366662781ax_int
% 9.20/5.24            @ ( collect_int
% 9.20/5.24              @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ N2 ) ) )
% 9.20/5.24          = ( abs_abs_int @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9658_gcd__is__Max__divisors__int,axiom,
% 9.20/5.24      ! [N2: int,M: int] :
% 9.20/5.24        ( ( N2 != zero_zero_int )
% 9.20/5.24       => ( ( gcd_gcd_int @ M @ N2 )
% 9.20/5.24          = ( lattic8263393255366662781ax_int
% 9.20/5.24            @ ( collect_int
% 9.20/5.24              @ ^ [D5: int] :
% 9.20/5.24                  ( ( dvd_dvd_int @ D5 @ M )
% 9.20/5.24                  & ( dvd_dvd_int @ D5 @ N2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9659_uniformity__complex__def,axiom,
% 9.20/5.24      ( topolo896644834953643431omplex
% 9.20/5.24      = ( comple8358262395181532106omplex
% 9.20/5.24        @ ( image_5971271580939081552omplex
% 9.20/5.24          @ ^ [E3: real] :
% 9.20/5.24              ( princi3496590319149328850omplex
% 9.20/5.24              @ ( collec8663557070575231912omplex
% 9.20/5.24                @ ( produc6771430404735790350plex_o
% 9.20/5.24                  @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
% 9.20/5.24          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9660_uniformity__real__def,axiom,
% 9.20/5.24      ( topolo1511823702728130853y_real
% 9.20/5.24      = ( comple2936214249959783750l_real
% 9.20/5.24        @ ( image_2178119161166701260l_real
% 9.20/5.24          @ ^ [E3: real] :
% 9.20/5.24              ( princi6114159922880469582l_real
% 9.20/5.24              @ ( collec3799799289383736868l_real
% 9.20/5.24                @ ( produc5414030515140494994real_o
% 9.20/5.24                  @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
% 9.20/5.24          @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9661_eventually__prod__sequentially,axiom,
% 9.20/5.24      ! [P: product_prod_nat_nat > $o] :
% 9.20/5.24        ( ( eventu1038000079068216329at_nat @ P @ ( prod_filter_nat_nat @ at_top_nat @ at_top_nat ) )
% 9.20/5.24        = ( ? [N5: nat] :
% 9.20/5.24            ! [M2: nat] :
% 9.20/5.24              ( ( ord_less_eq_nat @ N5 @ M2 )
% 9.20/5.24             => ! [N: nat] :
% 9.20/5.24                  ( ( ord_less_eq_nat @ N5 @ N )
% 9.20/5.24                 => ( P @ ( product_Pair_nat_nat @ N @ M2 ) ) ) ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9662_mono__Suc,axiom,
% 9.20/5.24      order_mono_nat_nat @ suc ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9663_mono__times__nat,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( order_mono_nat_nat @ ( times_times_nat @ N2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9664_mono__ge2__power__minus__self,axiom,
% 9.20/5.24      ! [K: nat] :
% 9.20/5.24        ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
% 9.20/5.24       => ( order_mono_nat_nat
% 9.20/5.24          @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9665_less__eq,axiom,
% 9.20/5.24      ! [M: nat,N2: nat] :
% 9.20/5.24        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
% 9.20/5.24        = ( ord_less_nat @ M @ N2 ) ) ).
% 9.20/5.24  
% 9.20/5.24  thf(fact_9666_inj__sgn__power,axiom,
% 9.20/5.24      ! [N2: nat] :
% 9.20/5.24        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.24       => ( inj_on_real_real
% 9.20/5.24          @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
% 9.20/5.24          @ top_top_set_real ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9667_log__inj,axiom,
% 9.20/5.25      ! [B: real] :
% 9.20/5.25        ( ( ord_less_real @ one_one_real @ B )
% 9.20/5.25       => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9668_inj__Suc,axiom,
% 9.20/5.25      ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9669_inj__on__diff__nat,axiom,
% 9.20/5.25      ! [N7: set_nat,K: nat] :
% 9.20/5.25        ( ! [N3: nat] :
% 9.20/5.25            ( ( member_nat @ N3 @ N7 )
% 9.20/5.25           => ( ord_less_eq_nat @ K @ N3 ) )
% 9.20/5.25       => ( inj_on_nat_nat
% 9.20/5.25          @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
% 9.20/5.25          @ N7 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9670_inj__on__set__encode,axiom,
% 9.20/5.25      inj_on_set_nat_nat @ nat_set_encode @ ( collect_set_nat @ finite_finite_nat ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9671_pred__nat__trancl__eq__le,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
% 9.20/5.25        = ( ord_less_eq_nat @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9672_measure__function__int,axiom,
% 9.20/5.25      fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9673_powr__real__of__int_H,axiom,
% 9.20/5.25      ! [X3: real,N2: int] :
% 9.20/5.25        ( ( ord_less_eq_real @ zero_zero_real @ X3 )
% 9.20/5.25       => ( ( ( X3 != zero_zero_real )
% 9.20/5.25            | ( ord_less_int @ zero_zero_int @ N2 ) )
% 9.20/5.25         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
% 9.20/5.25            = ( power_int_real @ X3 @ N2 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9674_atLeastLessThan__add__Un,axiom,
% 9.20/5.25      ! [I: nat,J: nat,K: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.25       => ( ( set_or4665077453230672383an_nat @ I @ ( plus_plus_nat @ J @ K ) )
% 9.20/5.25          = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9675_min__weak__def,axiom,
% 9.20/5.25      ( fun_min_weak
% 9.20/5.25      = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9676_sup__int__def,axiom,
% 9.20/5.25      sup_sup_int = ord_max_int ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9677_sup__nat__def,axiom,
% 9.20/5.25      sup_sup_nat = ord_max_nat ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9678_max__weak__def,axiom,
% 9.20/5.25      ( fun_max_weak
% 9.20/5.25      = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9679_pos__deriv__imp__strict__mono,axiom,
% 9.20/5.25      ! [F: real > real,F5: real > real] :
% 9.20/5.25        ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
% 9.20/5.25       => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
% 9.20/5.25         => ( order_7092887310737990675l_real @ F ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9680_of__nat__eq__id,axiom,
% 9.20/5.25      semiri1316708129612266289at_nat = id_nat ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9681_infinite__enumerate,axiom,
% 9.20/5.25      ! [S3: set_nat] :
% 9.20/5.25        ( ~ ( finite_finite_nat @ S3 )
% 9.20/5.25       => ? [R4: nat > nat] :
% 9.20/5.25            ( ( order_5726023648592871131at_nat @ R4 )
% 9.20/5.25            & ! [N6: nat] : ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9682_strict__mono__imp__increasing,axiom,
% 9.20/5.25      ! [F: nat > nat,N2: nat] :
% 9.20/5.25        ( ( order_5726023648592871131at_nat @ F )
% 9.20/5.25       => ( ord_less_eq_nat @ N2 @ ( F @ N2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9683_less__int__def,axiom,
% 9.20/5.25      ( ord_less_int
% 9.20/5.25      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.20/5.25        @ ( produc8739625826339149834_nat_o
% 9.20/5.25          @ ^ [X: nat,Y: nat] :
% 9.20/5.25              ( produc6081775807080527818_nat_o
% 9.20/5.25              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9684_nat__def,axiom,
% 9.20/5.25      ( nat2
% 9.20/5.25      = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9685_less__eq__int__def,axiom,
% 9.20/5.25      ( ord_less_eq_int
% 9.20/5.25      = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
% 9.20/5.25        @ ( produc8739625826339149834_nat_o
% 9.20/5.25          @ ^ [X: nat,Y: nat] :
% 9.20/5.25              ( produc6081775807080527818_nat_o
% 9.20/5.25              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9686_positive__rat,axiom,
% 9.20/5.25      ! [A: int,B: int] :
% 9.20/5.25        ( ( positive @ ( fract @ A @ B ) )
% 9.20/5.25        = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9687_less__rat__def,axiom,
% 9.20/5.25      ( ord_less_rat
% 9.20/5.25      = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9688_Rat_Opositive_Orep__eq,axiom,
% 9.20/5.25      ( positive
% 9.20/5.25      = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9689_Rat_Opositive__def,axiom,
% 9.20/5.25      ( positive
% 9.20/5.25      = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
% 9.20/5.25        @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9690_sorted__list__of__set__greaterThanAtMost,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ ( suc @ I ) @ J )
% 9.20/5.25       => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I @ J ) )
% 9.20/5.25          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9691_sorted__list__of__set__greaterThanLessThan,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_nat @ ( suc @ I ) @ J )
% 9.20/5.25       => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I @ J ) )
% 9.20/5.25          = ( cons_nat @ ( suc @ I ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I ) @ J ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9692_sorted__list__of__set__lessThan__Suc,axiom,
% 9.20/5.25      ! [K: nat] :
% 9.20/5.25        ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
% 9.20/5.25        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9693_sorted__list__of__set__atMost__Suc,axiom,
% 9.20/5.25      ! [K: nat] :
% 9.20/5.25        ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
% 9.20/5.25        = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9694_list__encode_Oelims,axiom,
% 9.20/5.25      ! [X3: list_nat,Y3: nat] :
% 9.20/5.25        ( ( ( nat_list_encode @ X3 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( ( X3 = nil_nat )
% 9.20/5.25           => ( Y3 != zero_zero_nat ) )
% 9.20/5.25         => ~ ! [X4: nat,Xs3: list_nat] :
% 9.20/5.25                ( ( X3
% 9.20/5.25                  = ( cons_nat @ X4 @ Xs3 ) )
% 9.20/5.25               => ( Y3
% 9.20/5.25                 != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9695_list__encode_Osimps_I1_J,axiom,
% 9.20/5.25      ( ( nat_list_encode @ nil_nat )
% 9.20/5.25      = zero_zero_nat ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9696_list__encode_Osimps_I2_J,axiom,
% 9.20/5.25      ! [X3: nat,Xs: list_nat] :
% 9.20/5.25        ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs ) )
% 9.20/5.25        = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9697_upto__aux__rec,axiom,
% 9.20/5.25      ( upto_aux
% 9.20/5.25      = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9698_upto_Opsimps,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I @ J ) )
% 9.20/5.25       => ( ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25           => ( ( upto @ I @ J )
% 9.20/5.25              = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) )
% 9.20/5.25          & ( ~ ( ord_less_eq_int @ I @ J )
% 9.20/5.25           => ( ( upto @ I @ J )
% 9.20/5.25              = nil_int ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9699_upto_Opelims,axiom,
% 9.20/5.25      ! [X3: int,Xa2: int,Y3: list_int] :
% 9.20/5.25        ( ( ( upto @ X3 @ Xa2 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
% 9.20/5.25         => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 9.20/5.25                 => ( Y3
% 9.20/5.25                    = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 9.20/5.25                & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 9.20/5.25                 => ( Y3 = nil_int ) ) )
% 9.20/5.25             => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9700_upto__empty,axiom,
% 9.20/5.25      ! [J: int,I: int] :
% 9.20/5.25        ( ( ord_less_int @ J @ I )
% 9.20/5.25       => ( ( upto @ I @ J )
% 9.20/5.25          = nil_int ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9701_upto__Nil2,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( nil_int
% 9.20/5.25          = ( upto @ I @ J ) )
% 9.20/5.25        = ( ord_less_int @ J @ I ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9702_upto__Nil,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( ( upto @ I @ J )
% 9.20/5.25          = nil_int )
% 9.20/5.25        = ( ord_less_int @ J @ I ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9703_upto__single,axiom,
% 9.20/5.25      ! [I: int] :
% 9.20/5.25        ( ( upto @ I @ I )
% 9.20/5.25        = ( cons_int @ I @ nil_int ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9704_nth__upto,axiom,
% 9.20/5.25      ! [I: int,K: nat,J: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) @ J )
% 9.20/5.25       => ( ( nth_int @ ( upto @ I @ J ) @ K )
% 9.20/5.25          = ( plus_plus_int @ I @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9705_length__upto,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( size_size_list_int @ ( upto @ I @ J ) )
% 9.20/5.25        = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I ) @ one_one_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9706_upto__rec__numeral_I1_J,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25            = nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9707_upto__rec__numeral_I2_J,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25            = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25         => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25            = nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9708_upto__rec__numeral_I3_J,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25            = ( 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 @ N2 ) ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
% 9.20/5.25            = nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9709_upto__rec__numeral_I4_J,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25            = ( 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 @ N2 ) ) ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25         => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25            = nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9710_atLeastAtMost__upto,axiom,
% 9.20/5.25      ( set_or1266510415728281911st_int
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ J3 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9711_distinct__upto,axiom,
% 9.20/5.25      ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9712_upto__aux__def,axiom,
% 9.20/5.25      ( upto_aux
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( append_int @ ( upto @ I2 @ J3 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9713_upto__code,axiom,
% 9.20/5.25      ( upto
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( upto_aux @ I2 @ J3 @ nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9714_upto__split2,axiom,
% 9.20/5.25      ! [I: int,J: int,K: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25       => ( ( ord_less_eq_int @ J @ K )
% 9.20/5.25         => ( ( upto @ I @ K )
% 9.20/5.25            = ( append_int @ ( upto @ I @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9715_upto__split1,axiom,
% 9.20/5.25      ! [I: int,J: int,K: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25       => ( ( ord_less_eq_int @ J @ K )
% 9.20/5.25         => ( ( upto @ I @ K )
% 9.20/5.25            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9716_atLeastLessThan__upto,axiom,
% 9.20/5.25      ( set_or4662586982721622107an_int
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9717_greaterThanAtMost__upto,axiom,
% 9.20/5.25      ( set_or6656581121297822940st_int
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9718_upto__rec1,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25       => ( ( upto @ I @ J )
% 9.20/5.25          = ( cons_int @ I @ ( upto @ ( plus_plus_int @ I @ one_one_int ) @ J ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9719_upto_Osimps,axiom,
% 9.20/5.25      ( upto
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9720_upto_Oelims,axiom,
% 9.20/5.25      ! [X3: int,Xa2: int,Y3: list_int] :
% 9.20/5.25        ( ( ( upto @ X3 @ Xa2 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
% 9.20/5.25           => ( Y3
% 9.20/5.25              = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
% 9.20/5.25          & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
% 9.20/5.25           => ( Y3 = nil_int ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9721_upto__rec2,axiom,
% 9.20/5.25      ! [I: int,J: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25       => ( ( upto @ I @ J )
% 9.20/5.25          = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9722_greaterThanLessThan__upto,axiom,
% 9.20/5.25      ( set_or5832277885323065728an_int
% 9.20/5.25      = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9723_upto__split3,axiom,
% 9.20/5.25      ! [I: int,J: int,K: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ I @ J )
% 9.20/5.25       => ( ( ord_less_eq_int @ J @ K )
% 9.20/5.25         => ( ( upto @ I @ K )
% 9.20/5.25            = ( append_int @ ( upto @ I @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9724_list__encode_Opelims,axiom,
% 9.20/5.25      ! [X3: list_nat,Y3: nat] :
% 9.20/5.25        ( ( ( nat_list_encode @ X3 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
% 9.20/5.25         => ( ( ( X3 = nil_nat )
% 9.20/5.25             => ( ( Y3 = zero_zero_nat )
% 9.20/5.25               => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
% 9.20/5.25           => ~ ! [X4: nat,Xs3: list_nat] :
% 9.20/5.25                  ( ( X3
% 9.20/5.25                    = ( cons_nat @ X4 @ Xs3 ) )
% 9.20/5.25                 => ( ( Y3
% 9.20/5.25                      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
% 9.20/5.25                   => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9725_remdups__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( remdups_nat @ ( upt @ M @ N2 ) )
% 9.20/5.25        = ( upt @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9726_hd__upt,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_nat @ I @ J )
% 9.20/5.25       => ( ( hd_nat @ ( upt @ I @ J ) )
% 9.20/5.25          = I ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9727_drop__upt,axiom,
% 9.20/5.25      ! [M: nat,I: nat,J: nat] :
% 9.20/5.25        ( ( drop_nat @ M @ ( upt @ I @ J ) )
% 9.20/5.25        = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9728_length__upt,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( size_size_list_nat @ ( upt @ I @ J ) )
% 9.20/5.25        = ( minus_minus_nat @ J @ I ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9729_take__upt,axiom,
% 9.20/5.25      ! [I: nat,M: nat,N2: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
% 9.20/5.25       => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
% 9.20/5.25          = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9730_upt__conv__Nil,axiom,
% 9.20/5.25      ! [J: nat,I: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ J @ I )
% 9.20/5.25       => ( ( upt @ I @ J )
% 9.20/5.25          = nil_nat ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9731_sorted__list__of__set__range,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
% 9.20/5.25        = ( upt @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9732_upt__eq__Nil__conv,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ( upt @ I @ J )
% 9.20/5.25          = nil_nat )
% 9.20/5.25        = ( ( J = zero_zero_nat )
% 9.20/5.25          | ( ord_less_eq_nat @ J @ I ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9733_nth__upt,axiom,
% 9.20/5.25      ! [I: nat,K: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
% 9.20/5.25       => ( ( nth_nat @ ( upt @ I @ J ) @ K )
% 9.20/5.25          = ( plus_plus_nat @ I @ K ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9734_upt__rec__numeral,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.25         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.25            = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.25         => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
% 9.20/5.25            = nil_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9735_upt__0,axiom,
% 9.20/5.25      ! [I: nat] :
% 9.20/5.25        ( ( upt @ I @ zero_zero_nat )
% 9.20/5.25        = nil_nat ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9736_upt__conv__Cons__Cons,axiom,
% 9.20/5.25      ! [M: nat,N2: nat,Ns: list_nat,Q4: nat] :
% 9.20/5.25        ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
% 9.20/5.25          = ( upt @ M @ Q4 ) )
% 9.20/5.25        = ( ( cons_nat @ N2 @ Ns )
% 9.20/5.25          = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9737_map__add__upt,axiom,
% 9.20/5.25      ! [N2: nat,M: nat] :
% 9.20/5.25        ( ( map_nat_nat
% 9.20/5.25          @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N2 )
% 9.20/5.25          @ ( upt @ zero_zero_nat @ M ) )
% 9.20/5.25        = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9738_map__Suc__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
% 9.20/5.25        = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9739_atMost__upto,axiom,
% 9.20/5.25      ( set_ord_atMost_nat
% 9.20/5.25      = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9740_greaterThanAtMost__upt,axiom,
% 9.20/5.25      ( set_or6659071591806873216st_nat
% 9.20/5.25      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9741_atLeast__upt,axiom,
% 9.20/5.25      ( set_ord_lessThan_nat
% 9.20/5.25      = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9742_atLeastAtMost__upt,axiom,
% 9.20/5.25      ( set_or1269000886237332187st_nat
% 9.20/5.25      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9743_atLeastLessThan__upt,axiom,
% 9.20/5.25      ( set_or4665077453230672383an_nat
% 9.20/5.25      = ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9744_greaterThanLessThan__upt,axiom,
% 9.20/5.25      ( set_or5834768355832116004an_nat
% 9.20/5.25      = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9745_distinct__upt,axiom,
% 9.20/5.25      ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9746_upt__conv__Cons,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_nat @ I @ J )
% 9.20/5.25       => ( ( upt @ I @ J )
% 9.20/5.25          = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9747_map__decr__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( map_nat_nat
% 9.20/5.25          @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
% 9.20/5.25          @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
% 9.20/5.25        = ( upt @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9748_upt__add__eq__append,axiom,
% 9.20/5.25      ! [I: nat,J: nat,K: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.25       => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
% 9.20/5.25          = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9749_upt__eq__Cons__conv,axiom,
% 9.20/5.25      ! [I: nat,J: nat,X3: nat,Xs: list_nat] :
% 9.20/5.25        ( ( ( upt @ I @ J )
% 9.20/5.25          = ( cons_nat @ X3 @ Xs ) )
% 9.20/5.25        = ( ( ord_less_nat @ I @ J )
% 9.20/5.25          & ( I = X3 )
% 9.20/5.25          & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
% 9.20/5.25            = Xs ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9750_upt__rec,axiom,
% 9.20/5.25      ( upt
% 9.20/5.25      = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9751_upt__Suc,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.25         => ( ( upt @ I @ ( suc @ J ) )
% 9.20/5.25            = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
% 9.20/5.25        & ( ~ ( ord_less_eq_nat @ I @ J )
% 9.20/5.25         => ( ( upt @ I @ ( suc @ J ) )
% 9.20/5.25            = nil_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9752_upt__Suc__append,axiom,
% 9.20/5.25      ! [I: nat,J: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ I @ J )
% 9.20/5.25       => ( ( upt @ I @ ( suc @ J ) )
% 9.20/5.25          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9753_sum__list__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ M @ N2 )
% 9.20/5.25       => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
% 9.20/5.25          = ( groups3542108847815614940at_nat
% 9.20/5.25            @ ^ [X: nat] : X
% 9.20/5.25            @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9754_card__length__sum__list__rec,axiom,
% 9.20/5.25      ! [M: nat,N7: nat] :
% 9.20/5.25        ( ( ord_less_eq_nat @ one_one_nat @ M )
% 9.20/5.25       => ( ( finite_card_list_nat
% 9.20/5.25            @ ( collect_list_nat
% 9.20/5.25              @ ^ [L2: list_nat] :
% 9.20/5.25                  ( ( ( size_size_list_nat @ L2 )
% 9.20/5.25                    = M )
% 9.20/5.25                  & ( ( groups4561878855575611511st_nat @ L2 )
% 9.20/5.25                    = N7 ) ) ) )
% 9.20/5.25          = ( plus_plus_nat
% 9.20/5.25            @ ( finite_card_list_nat
% 9.20/5.25              @ ( collect_list_nat
% 9.20/5.25                @ ^ [L2: list_nat] :
% 9.20/5.25                    ( ( ( size_size_list_nat @ L2 )
% 9.20/5.25                      = ( minus_minus_nat @ M @ one_one_nat ) )
% 9.20/5.25                    & ( ( groups4561878855575611511st_nat @ L2 )
% 9.20/5.25                      = N7 ) ) ) )
% 9.20/5.25            @ ( finite_card_list_nat
% 9.20/5.25              @ ( collect_list_nat
% 9.20/5.25                @ ^ [L2: list_nat] :
% 9.20/5.25                    ( ( ( size_size_list_nat @ L2 )
% 9.20/5.25                      = M )
% 9.20/5.25                    & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
% 9.20/5.25                      = N7 ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9755_card__length__sum__list,axiom,
% 9.20/5.25      ! [M: nat,N7: nat] :
% 9.20/5.25        ( ( finite_card_list_nat
% 9.20/5.25          @ ( collect_list_nat
% 9.20/5.25            @ ^ [L2: list_nat] :
% 9.20/5.25                ( ( ( size_size_list_nat @ L2 )
% 9.20/5.25                  = M )
% 9.20/5.25                & ( ( groups4561878855575611511st_nat @ L2 )
% 9.20/5.25                  = N7 ) ) ) )
% 9.20/5.25        = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9756_tl__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] :
% 9.20/5.25        ( ( tl_nat @ ( upt @ M @ N2 ) )
% 9.20/5.25        = ( upt @ ( suc @ M ) @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9757_sorted__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9758_sorted__wrt__upt,axiom,
% 9.20/5.25      ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9759_sorted__wrt__less__idx,axiom,
% 9.20/5.25      ! [Ns: list_nat,I: nat] :
% 9.20/5.25        ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
% 9.20/5.25       => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ns ) )
% 9.20/5.25         => ( ord_less_eq_nat @ I @ ( nth_nat @ Ns @ I ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9760_sorted__upto,axiom,
% 9.20/5.25      ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9761_sorted__wrt__upto,axiom,
% 9.20/5.25      ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9762_pairs__le__eq__Sigma,axiom,
% 9.20/5.25      ! [M: nat] :
% 9.20/5.25        ( ( collec3392354462482085612at_nat
% 9.20/5.25          @ ( produc6081775807080527818_nat_o
% 9.20/5.25            @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
% 9.20/5.25        = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
% 9.20/5.25          @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9763_natLess__def,axiom,
% 9.20/5.25      ( bNF_Ca8459412986667044542atLess
% 9.20/5.25      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9764_vimage__Suc__insert__Suc,axiom,
% 9.20/5.25      ! [N2: nat,A2: set_nat] :
% 9.20/5.25        ( ( vimage_nat_nat @ suc @ ( insert_nat @ ( suc @ N2 ) @ A2 ) )
% 9.20/5.25        = ( insert_nat @ N2 @ ( vimage_nat_nat @ suc @ A2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9765_vimage__Suc__insert__0,axiom,
% 9.20/5.25      ! [A2: set_nat] :
% 9.20/5.25        ( ( vimage_nat_nat @ suc @ ( insert_nat @ zero_zero_nat @ A2 ) )
% 9.20/5.25        = ( vimage_nat_nat @ suc @ A2 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9766_finite__vimage__Suc__iff,axiom,
% 9.20/5.25      ! [F3: set_nat] :
% 9.20/5.25        ( ( finite_finite_nat @ ( vimage_nat_nat @ suc @ F3 ) )
% 9.20/5.25        = ( finite_finite_nat @ F3 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9767_set__decode__div__2,axiom,
% 9.20/5.25      ! [X3: nat] :
% 9.20/5.25        ( ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
% 9.20/5.25        = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9768_set__encode__vimage__Suc,axiom,
% 9.20/5.25      ! [A2: set_nat] :
% 9.20/5.25        ( ( nat_set_encode @ ( vimage_nat_nat @ suc @ A2 ) )
% 9.20/5.25        = ( divide_divide_nat @ ( nat_set_encode @ A2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9769_Restr__natLeq,axiom,
% 9.20/5.25      ! [N2: nat] :
% 9.20/5.25        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.20/5.25          @ ( produc457027306803732586at_nat
% 9.20/5.25            @ ( collect_nat
% 9.20/5.25              @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) )
% 9.20/5.25            @ ^ [Uu3: nat] :
% 9.20/5.25                ( collect_nat
% 9.20/5.25                @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) )
% 9.20/5.25        = ( collec3392354462482085612at_nat
% 9.20/5.25          @ ( produc6081775807080527818_nat_o
% 9.20/5.25            @ ^ [X: nat,Y: nat] :
% 9.20/5.25                ( ( ord_less_nat @ X @ N2 )
% 9.20/5.25                & ( ord_less_nat @ Y @ N2 )
% 9.20/5.25                & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9770_natLeq__def,axiom,
% 9.20/5.25      ( bNF_Ca8665028551170535155natLeq
% 9.20/5.25      = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9771_Restr__natLeq2,axiom,
% 9.20/5.25      ! [N2: nat] :
% 9.20/5.25        ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
% 9.20/5.25          @ ( produc457027306803732586at_nat @ ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
% 9.20/5.25            @ ^ [Uu3: nat] : ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 ) ) )
% 9.20/5.25        = ( collec3392354462482085612at_nat
% 9.20/5.25          @ ( produc6081775807080527818_nat_o
% 9.20/5.25            @ ^ [X: nat,Y: nat] :
% 9.20/5.25                ( ( ord_less_nat @ X @ N2 )
% 9.20/5.25                & ( ord_less_nat @ Y @ N2 )
% 9.20/5.25                & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9772_natLeq__underS__less,axiom,
% 9.20/5.25      ! [N2: nat] :
% 9.20/5.25        ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
% 9.20/5.25        = ( collect_nat
% 9.20/5.25          @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9773_Arg__bounded,axiom,
% 9.20/5.25      ! [Z2: complex] :
% 9.20/5.25        ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 9.20/5.25        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9774_Arg__correct,axiom,
% 9.20/5.25      ! [Z2: complex] :
% 9.20/5.25        ( ( Z2 != zero_zero_complex )
% 9.20/5.25       => ( ( ( sgn_sgn_complex @ Z2 )
% 9.20/5.25            = ( cis @ ( arg @ Z2 ) ) )
% 9.20/5.25          & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
% 9.20/5.25          & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9775_Field__natLeq__on,axiom,
% 9.20/5.25      ! [N2: nat] :
% 9.20/5.25        ( ( field_nat
% 9.20/5.25          @ ( collec3392354462482085612at_nat
% 9.20/5.25            @ ( produc6081775807080527818_nat_o
% 9.20/5.25              @ ^ [X: nat,Y: nat] :
% 9.20/5.25                  ( ( ord_less_nat @ X @ N2 )
% 9.20/5.25                  & ( ord_less_nat @ Y @ N2 )
% 9.20/5.25                  & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
% 9.20/5.25        = ( collect_nat
% 9.20/5.25          @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9776_cis__Arg__unique,axiom,
% 9.20/5.25      ! [Z2: complex,X3: real] :
% 9.20/5.25        ( ( ( sgn_sgn_complex @ Z2 )
% 9.20/5.25          = ( cis @ X3 ) )
% 9.20/5.25       => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
% 9.20/5.25         => ( ( ord_less_eq_real @ X3 @ pi )
% 9.20/5.25           => ( ( arg @ Z2 )
% 9.20/5.25              = X3 ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9777_bij__betw__roots__unity,axiom,
% 9.20/5.25      ! [N2: nat] :
% 9.20/5.25        ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.25       => ( bij_betw_nat_complex
% 9.20/5.25          @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
% 9.20/5.25          @ ( set_ord_lessThan_nat @ N2 )
% 9.20/5.25          @ ( collect_complex
% 9.20/5.25            @ ^ [Z6: complex] :
% 9.20/5.25                ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.25                = one_one_complex ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9778_bij__betw__nth__root__unity,axiom,
% 9.20/5.25      ! [C: complex,N2: nat] :
% 9.20/5.25        ( ( C != zero_zero_complex )
% 9.20/5.25       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
% 9.20/5.25         => ( bij_be1856998921033663316omplex @ ( times_times_complex @ ( times_times_complex @ ( real_V4546457046886955230omplex @ ( root @ N2 @ ( real_V1022390504157884413omplex @ C ) ) ) @ ( cis @ ( divide_divide_real @ ( arg @ C ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) )
% 9.20/5.25            @ ( collect_complex
% 9.20/5.25              @ ^ [Z6: complex] :
% 9.20/5.25                  ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.25                  = one_one_complex ) )
% 9.20/5.25            @ ( collect_complex
% 9.20/5.25              @ ^ [Z6: complex] :
% 9.20/5.25                  ( ( power_power_complex @ Z6 @ N2 )
% 9.20/5.25                  = C ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9779_bij__betw__Suc,axiom,
% 9.20/5.25      ! [M7: set_nat,N7: set_nat] :
% 9.20/5.25        ( ( bij_betw_nat_nat @ suc @ M7 @ N7 )
% 9.20/5.25        = ( ( image_nat_nat @ suc @ M7 )
% 9.20/5.25          = N7 ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9780_Arg__def,axiom,
% 9.20/5.25      ( arg
% 9.20/5.25      = ( ^ [Z6: complex] :
% 9.20/5.25            ( if_real @ ( Z6 = zero_zero_complex ) @ zero_zero_real
% 9.20/5.25            @ ( fChoice_real
% 9.20/5.25              @ ^ [A3: real] :
% 9.20/5.25                  ( ( ( sgn_sgn_complex @ Z6 )
% 9.20/5.25                    = ( cis @ A3 ) )
% 9.20/5.25                  & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
% 9.20/5.25                  & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9781_not__negative__int__iff,axiom,
% 9.20/5.25      ! [K: int] :
% 9.20/5.25        ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
% 9.20/5.25        = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9782_not__nonnegative__int__iff,axiom,
% 9.20/5.25      ! [K: int] :
% 9.20/5.25        ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
% 9.20/5.25        = ( ord_less_int @ K @ zero_zero_int ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9783_wf__int__ge__less__than2,axiom,
% 9.20/5.25      ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9784_wf__int__ge__less__than,axiom,
% 9.20/5.25      ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9785_wf__less,axiom,
% 9.20/5.25      wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9786_and__not__num__eq__None__iff,axiom,
% 9.20/5.25      ! [M: num,N2: num] :
% 9.20/5.25        ( ( ( bit_and_not_num @ M @ N2 )
% 9.20/5.25          = none_num )
% 9.20/5.25        = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
% 9.20/5.25          = zero_zero_int ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9787_and__not__num_Osimps_I1_J,axiom,
% 9.20/5.25      ( ( bit_and_not_num @ one @ one )
% 9.20/5.25      = none_num ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9788_and__not__num_Osimps_I3_J,axiom,
% 9.20/5.25      ! [N2: num] :
% 9.20/5.25        ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
% 9.20/5.25        = none_num ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9789_and__not__num_Oelims,axiom,
% 9.20/5.25      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.20/5.25        ( ( ( bit_and_not_num @ X3 @ Xa2 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( ( X3 = one )
% 9.20/5.25           => ( ( Xa2 = one )
% 9.20/5.25             => ( Y3 != none_num ) ) )
% 9.20/5.25         => ( ( ( X3 = one )
% 9.20/5.25             => ( ? [N3: num] :
% 9.20/5.25                    ( Xa2
% 9.20/5.25                    = ( bit0 @ N3 ) )
% 9.20/5.25               => ( Y3
% 9.20/5.25                 != ( some_num @ one ) ) ) )
% 9.20/5.25           => ( ( ( X3 = one )
% 9.20/5.25               => ( ? [N3: num] :
% 9.20/5.25                      ( Xa2
% 9.20/5.25                      = ( bit1 @ N3 ) )
% 9.20/5.25                 => ( Y3 != none_num ) ) )
% 9.20/5.25             => ( ! [M3: num] :
% 9.20/5.25                    ( ( X3
% 9.20/5.25                      = ( bit0 @ M3 ) )
% 9.20/5.25                   => ( ( Xa2 = one )
% 9.20/5.25                     => ( Y3
% 9.20/5.25                       != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 9.20/5.25               => ( ! [M3: num] :
% 9.20/5.25                      ( ( X3
% 9.20/5.25                        = ( bit0 @ M3 ) )
% 9.20/5.25                     => ! [N3: num] :
% 9.20/5.25                          ( ( Xa2
% 9.20/5.25                            = ( bit0 @ N3 ) )
% 9.20/5.25                         => ( Y3
% 9.20/5.25                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                 => ( ! [M3: num] :
% 9.20/5.25                        ( ( X3
% 9.20/5.25                          = ( bit0 @ M3 ) )
% 9.20/5.25                       => ! [N3: num] :
% 9.20/5.25                            ( ( Xa2
% 9.20/5.25                              = ( bit1 @ N3 ) )
% 9.20/5.25                           => ( Y3
% 9.20/5.25                             != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                   => ( ! [M3: num] :
% 9.20/5.25                          ( ( X3
% 9.20/5.25                            = ( bit1 @ M3 ) )
% 9.20/5.25                         => ( ( Xa2 = one )
% 9.20/5.25                           => ( Y3
% 9.20/5.25                             != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 9.20/5.25                     => ( ! [M3: num] :
% 9.20/5.25                            ( ( X3
% 9.20/5.25                              = ( bit1 @ M3 ) )
% 9.20/5.25                           => ! [N3: num] :
% 9.20/5.25                                ( ( Xa2
% 9.20/5.25                                  = ( bit0 @ N3 ) )
% 9.20/5.25                               => ( Y3
% 9.20/5.25                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.20/5.25                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.20/5.25                                    @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                       => ~ ! [M3: num] :
% 9.20/5.25                              ( ( X3
% 9.20/5.25                                = ( bit1 @ M3 ) )
% 9.20/5.25                             => ! [N3: num] :
% 9.20/5.25                                  ( ( Xa2
% 9.20/5.25                                    = ( bit1 @ N3 ) )
% 9.20/5.25                                 => ( Y3
% 9.20/5.25                                   != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9790_and__num_Oelims,axiom,
% 9.20/5.25      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.20/5.25        ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( ( X3 = one )
% 9.20/5.25           => ( ( Xa2 = one )
% 9.20/5.25             => ( Y3
% 9.20/5.25               != ( some_num @ one ) ) ) )
% 9.20/5.25         => ( ( ( X3 = one )
% 9.20/5.25             => ( ? [N3: num] :
% 9.20/5.25                    ( Xa2
% 9.20/5.25                    = ( bit0 @ N3 ) )
% 9.20/5.25               => ( Y3 != none_num ) ) )
% 9.20/5.25           => ( ( ( X3 = one )
% 9.20/5.25               => ( ? [N3: num] :
% 9.20/5.25                      ( Xa2
% 9.20/5.25                      = ( bit1 @ N3 ) )
% 9.20/5.25                 => ( Y3
% 9.20/5.25                   != ( some_num @ one ) ) ) )
% 9.20/5.25             => ( ( ? [M3: num] :
% 9.20/5.25                      ( X3
% 9.20/5.25                      = ( bit0 @ M3 ) )
% 9.20/5.25                 => ( ( Xa2 = one )
% 9.20/5.25                   => ( Y3 != none_num ) ) )
% 9.20/5.25               => ( ! [M3: num] :
% 9.20/5.25                      ( ( X3
% 9.20/5.25                        = ( bit0 @ M3 ) )
% 9.20/5.25                     => ! [N3: num] :
% 9.20/5.25                          ( ( Xa2
% 9.20/5.25                            = ( bit0 @ N3 ) )
% 9.20/5.25                         => ( Y3
% 9.20/5.25                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                 => ( ! [M3: num] :
% 9.20/5.25                        ( ( X3
% 9.20/5.25                          = ( bit0 @ M3 ) )
% 9.20/5.25                       => ! [N3: num] :
% 9.20/5.25                            ( ( Xa2
% 9.20/5.25                              = ( bit1 @ N3 ) )
% 9.20/5.25                           => ( Y3
% 9.20/5.25                             != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                   => ( ( ? [M3: num] :
% 9.20/5.25                            ( X3
% 9.20/5.25                            = ( bit1 @ M3 ) )
% 9.20/5.25                       => ( ( Xa2 = one )
% 9.20/5.25                         => ( Y3
% 9.20/5.25                           != ( some_num @ one ) ) ) )
% 9.20/5.25                     => ( ! [M3: num] :
% 9.20/5.25                            ( ( X3
% 9.20/5.25                              = ( bit1 @ M3 ) )
% 9.20/5.25                           => ! [N3: num] :
% 9.20/5.25                                ( ( Xa2
% 9.20/5.25                                  = ( bit0 @ N3 ) )
% 9.20/5.25                               => ( Y3
% 9.20/5.25                                 != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                       => ~ ! [M3: num] :
% 9.20/5.25                              ( ( X3
% 9.20/5.25                                = ( bit1 @ M3 ) )
% 9.20/5.25                             => ! [N3: num] :
% 9.20/5.25                                  ( ( Xa2
% 9.20/5.25                                    = ( bit1 @ N3 ) )
% 9.20/5.25                                 => ( Y3
% 9.20/5.25                                   != ( case_o6005452278849405969um_num @ ( some_num @ one )
% 9.20/5.25                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
% 9.20/5.25                                      @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9791_and__num_Osimps_I4_J,axiom,
% 9.20/5.25      ! [M: num] :
% 9.20/5.25        ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
% 9.20/5.25        = none_num ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9792_and__num_Osimps_I2_J,axiom,
% 9.20/5.25      ! [N2: num] :
% 9.20/5.25        ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
% 9.20/5.25        = none_num ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9793_xor__num_Oelims,axiom,
% 9.20/5.25      ! [X3: num,Xa2: num,Y3: option_num] :
% 9.20/5.25        ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
% 9.20/5.25          = Y3 )
% 9.20/5.25       => ( ( ( X3 = one )
% 9.20/5.25           => ( ( Xa2 = one )
% 9.20/5.25             => ( Y3 != none_num ) ) )
% 9.20/5.25         => ( ( ( X3 = one )
% 9.20/5.25             => ! [N3: num] :
% 9.20/5.25                  ( ( Xa2
% 9.20/5.25                    = ( bit0 @ N3 ) )
% 9.20/5.25                 => ( Y3
% 9.20/5.25                   != ( some_num @ ( bit1 @ N3 ) ) ) ) )
% 9.20/5.25           => ( ( ( X3 = one )
% 9.20/5.25               => ! [N3: num] :
% 9.20/5.25                    ( ( Xa2
% 9.20/5.25                      = ( bit1 @ N3 ) )
% 9.20/5.25                   => ( Y3
% 9.20/5.25                     != ( some_num @ ( bit0 @ N3 ) ) ) ) )
% 9.20/5.25             => ( ! [M3: num] :
% 9.20/5.25                    ( ( X3
% 9.20/5.25                      = ( bit0 @ M3 ) )
% 9.20/5.25                   => ( ( Xa2 = one )
% 9.20/5.25                     => ( Y3
% 9.20/5.25                       != ( some_num @ ( bit1 @ M3 ) ) ) ) )
% 9.20/5.25               => ( ! [M3: num] :
% 9.20/5.25                      ( ( X3
% 9.20/5.25                        = ( bit0 @ M3 ) )
% 9.20/5.25                     => ! [N3: num] :
% 9.20/5.25                          ( ( Xa2
% 9.20/5.25                            = ( bit0 @ N3 ) )
% 9.20/5.25                         => ( Y3
% 9.20/5.25                           != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) )
% 9.20/5.25                 => ( ! [M3: num] :
% 9.20/5.25                        ( ( X3
% 9.20/5.25                          = ( bit0 @ M3 ) )
% 9.20/5.25                       => ! [N3: num] :
% 9.20/5.25                            ( ( Xa2
% 9.20/5.25                              = ( bit1 @ N3 ) )
% 9.20/5.25                           => ( Y3
% 9.20/5.25                             != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
% 9.20/5.25                   => ( ! [M3: num] :
% 9.20/5.25                          ( ( X3
% 9.20/5.25                            = ( bit1 @ M3 ) )
% 9.20/5.25                         => ( ( Xa2 = one )
% 9.20/5.25                           => ( Y3
% 9.20/5.25                             != ( some_num @ ( bit0 @ M3 ) ) ) ) )
% 9.20/5.25                     => ( ! [M3: num] :
% 9.20/5.25                            ( ( X3
% 9.20/5.25                              = ( bit1 @ M3 ) )
% 9.20/5.25                           => ! [N3: num] :
% 9.20/5.25                                ( ( Xa2
% 9.20/5.25                                  = ( bit0 @ N3 ) )
% 9.20/5.25                               => ( Y3
% 9.20/5.25                                 != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
% 9.20/5.25                       => ~ ! [M3: num] :
% 9.20/5.25                              ( ( X3
% 9.20/5.25                                = ( bit1 @ M3 ) )
% 9.20/5.25                             => ! [N3: num] :
% 9.20/5.25                                  ( ( Xa2
% 9.20/5.25                                    = ( bit1 @ N3 ) )
% 9.20/5.25                                 => ( Y3
% 9.20/5.25                                   != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(fact_9794_xor__num_Osimps_I1_J,axiom,
% 9.20/5.25      ( ( bit_un2480387367778600638or_num @ one @ one )
% 9.20/5.25      = none_num ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
% 9.20/5.25      ! [X3: int,Y3: int] :
% 9.20/5.25        ( ( if_int @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
% 9.20/5.25      ! [X3: int,Y3: int] :
% 9.20/5.25        ( ( if_int @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
% 9.20/5.25      ! [X3: nat,Y3: nat] :
% 9.20/5.25        ( ( if_nat @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
% 9.20/5.25      ! [X3: nat,Y3: nat] :
% 9.20/5.25        ( ( if_nat @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
% 9.20/5.25      ! [X3: rat,Y3: rat] :
% 9.20/5.25        ( ( if_rat @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
% 9.20/5.25      ! [X3: rat,Y3: rat] :
% 9.20/5.25        ( ( if_rat @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
% 9.20/5.25      ! [X3: real,Y3: real] :
% 9.20/5.25        ( ( if_real @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
% 9.20/5.25      ! [X3: real,Y3: real] :
% 9.20/5.25        ( ( if_real @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
% 9.20/5.25      ! [P: real > $o] :
% 9.20/5.25        ( ( P @ ( fChoice_real @ P ) )
% 9.20/5.25        = ( ? [X5: real] : ( P @ X5 ) ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.20/5.25      ! [X3: complex,Y3: complex] :
% 9.20/5.25        ( ( if_complex @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
% 9.20/5.25      ! [X3: complex,Y3: complex] :
% 9.20/5.25        ( ( if_complex @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.20/5.25      ! [X3: extended_enat,Y3: extended_enat] :
% 9.20/5.25        ( ( if_Extended_enat @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
% 9.20/5.25      ! [X3: extended_enat,Y3: extended_enat] :
% 9.20/5.25        ( ( if_Extended_enat @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.20/5.25      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.25        ( ( if_Code_integer @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
% 9.20/5.25      ! [X3: code_integer,Y3: code_integer] :
% 9.20/5.25        ( ( if_Code_integer @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: set_int,Y3: set_int] :
% 9.20/5.25        ( ( if_set_int @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: set_int,Y3: set_int] :
% 9.20/5.25        ( ( if_set_int @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: list_int,Y3: list_int] :
% 9.20/5.25        ( ( if_list_int @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: list_int,Y3: list_int] :
% 9.20/5.25        ( ( if_list_int @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.20/5.25      ! [X3: list_nat,Y3: list_nat] :
% 9.20/5.25        ( ( if_list_nat @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
% 9.20/5.25      ! [X3: list_nat,Y3: list_nat] :
% 9.20/5.25        ( ( if_list_nat @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: int > int,Y3: int > int] :
% 9.20/5.25        ( ( if_int_int @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: int > int,Y3: int > int] :
% 9.20/5.25        ( ( if_int_int @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.20/5.25      ! [X3: option_num,Y3: option_num] :
% 9.20/5.25        ( ( if_option_num @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
% 9.20/5.25      ! [X3: option_num,Y3: option_num] :
% 9.20/5.25        ( ( if_option_num @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 9.20/5.25        ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
% 9.20/5.25      ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
% 9.20/5.25        ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.20/5.25      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.20/5.25        ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
% 9.20/5.25      ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
% 9.20/5.25        ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.20/5.25      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.20/5.25        ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
% 9.20/5.25      ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
% 9.20/5.25        ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.20/5.25      ! [P: $o] :
% 9.20/5.25        ( ( P = $true )
% 9.20/5.25        | ( P = $false ) ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.20/5.25      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.20/5.25        ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
% 9.20/5.25        = Y3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
% 9.20/5.25      ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
% 9.20/5.25        ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
% 9.20/5.25        = X3 ) ).
% 9.20/5.25  
% 9.20/5.25  thf(conj_0,hypothesis,
% 9.20/5.25      ! [X2: nat] :
% 9.20/5.25        ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X2 )
% 9.20/5.25       => thesis ) ).
% 9.20/5.25  
% 9.20/5.25  thf(conj_1,conjecture,
% 9.20/5.25      thesis ).
% 9.20/5.25  ------------------------
% 9.21/5.74  % (22505)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22391 for (2995ds/18Mi)
% 9.21/5.74  % (22501)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_22391 for (2995ds/27Mi)
% 9.21/5.74  % (22499)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22391 for (2995ds/183Mi)
% 9.21/5.74  % (22502)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22391 for (2995ds/2Mi)
% 9.21/5.74  % (22500)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_22391 for (2995ds/4Mi)
% 9.21/5.74  % (22503)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_22391 for (2995ds/2Mi)
% 9.21/5.75  % (22504)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_22391 for (2995ds/275Mi)
% 9.21/5.75  % (22503)Instruction limit reached!
% 9.21/5.75  % (22503)------------------------------
% 9.21/5.75  % (22503)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.75  % (22503)Termination reason: Unknown
% 9.21/5.75  % (22503)Termination phase: shuffling
% 9.21/5.75  
% 9.21/5.75  % (22503)Memory used [KB]: 13048
% 9.21/5.75  % (22503)Time elapsed: 0.003 s
% 9.21/5.75  % (22503)Instructions burned: 2 (million)
% 9.21/5.75  % (22503)------------------------------
% 9.21/5.75  % (22503)------------------------------
% 9.21/5.75  % (22502)Instruction limit reached!
% 9.21/5.75  % (22502)------------------------------
% 9.21/5.75  % (22502)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.75  % (22502)Termination reason: Unknown
% 9.21/5.75  % (22502)Termination phase: shuffling
% 9.21/5.75  
% 9.21/5.75  % (22502)Memory used [KB]: 13048
% 9.21/5.75  % (22502)Time elapsed: 0.004 s
% 9.21/5.75  % (22502)Instructions burned: 2 (million)
% 9.21/5.75  % (22502)------------------------------
% 9.21/5.75  % (22502)------------------------------
% 9.21/5.75  % (22500)Instruction limit reached!
% 9.21/5.75  % (22500)------------------------------
% 9.21/5.75  % (22500)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.75  % (22500)Termination reason: Unknown
% 9.21/5.75  % (22500)Termination phase: shuffling
% 9.21/5.75  
% 9.21/5.75  % (22500)Memory used [KB]: 13176
% 9.21/5.75  % (22500)Time elapsed: 0.007 s
% 9.21/5.75  % (22500)Instructions burned: 4 (million)
% 9.21/5.75  % (22500)------------------------------
% 9.21/5.75  % (22500)------------------------------
% 9.21/5.75  % (22505)Instruction limit reached!
% 9.21/5.75  % (22505)------------------------------
% 9.21/5.75  % (22505)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.75  % (22505)Termination reason: Unknown
% 9.21/5.75  % (22505)Termination phase: shuffling
% 9.21/5.75  
% 9.21/5.75  % (22505)Memory used [KB]: 13944
% 9.21/5.75  % (22505)Time elapsed: 0.015 s
% 9.21/5.75  % (22505)Instructions burned: 19 (million)
% 9.21/5.75  % (22505)------------------------------
% 9.21/5.75  % (22505)------------------------------
% 9.21/5.76  % (22509)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_22391 for (2995ds/3Mi)
% 9.21/5.76  % (22509)Instruction limit reached!
% 9.21/5.76  % (22509)------------------------------
% 9.21/5.76  % (22509)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.76  % (22509)Termination reason: Unknown
% 9.21/5.76  % (22509)Termination phase: shuffling
% 9.21/5.76  
% 9.21/5.76  % (22509)Memory used [KB]: 13176
% 9.21/5.76  % (22509)Time elapsed: 0.004 s
% 9.21/5.76  % (22509)Instructions burned: 3 (million)
% 9.21/5.76  % (22509)------------------------------
% 9.21/5.76  % (22509)------------------------------
% 9.21/5.77  % (22506)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22391 for (2995ds/3Mi)
% 9.21/5.77  % (22507)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_22391 for (2995ds/37Mi)
% 9.21/5.77  % (22506)Instruction limit reached!
% 9.21/5.77  % (22506)------------------------------
% 9.21/5.77  % (22506)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.77  % (22506)Termination reason: Unknown
% 9.21/5.77  % (22506)Termination phase: shuffling
% 9.21/5.77  
% 9.21/5.77  % (22508)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_22391 for (2995ds/15Mi)
% 9.21/5.77  % (22506)Memory used [KB]: 13048
% 9.21/5.77  % (22506)Time elapsed: 0.005 s
% 9.21/5.77  % (22506)Instructions burned: 3 (million)
% 9.21/5.77  % (22506)------------------------------
% 9.21/5.77  % (22506)------------------------------
% 9.21/5.77  % (22501)Instruction limit reached!
% 9.21/5.77  % (22501)------------------------------
% 9.21/5.77  % (22501)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.77  % (22501)Termination reason: Unknown
% 9.21/5.77  % (22501)Termination phase: shuffling
% 9.21/5.77  
% 9.21/5.77  % (22501)Memory used [KB]: 14072
% 9.21/5.77  % (22501)Time elapsed: 0.029 s
% 9.21/5.77  % (22501)Instructions burned: 27 (million)
% 9.21/5.77  % (22501)------------------------------
% 9.21/5.77  % (22501)------------------------------
% 9.21/5.77  % (22510)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_22391 for (2994ds/1041Mi)
% 9.21/5.79  % (22508)Instruction limit reached!
% 9.21/5.79  % (22508)------------------------------
% 9.21/5.79  % (22508)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.79  % (22508)Termination reason: Unknown
% 9.21/5.79  % (22508)Termination phase: shuffling
% 9.21/5.79  
% 9.21/5.79  % (22511)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/7Mi)
% 9.21/5.79  % (22508)Memory used [KB]: 13944
% 9.21/5.79  % (22508)Time elapsed: 0.021 s
% 9.21/5.79  % (22508)Instructions burned: 15 (million)
% 9.21/5.79  % (22508)------------------------------
% 9.21/5.79  % (22508)------------------------------
% 9.21/5.79  % (22512)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/16Mi)
% 9.21/5.80  % (22511)Instruction limit reached!
% 9.21/5.80  % (22511)------------------------------
% 9.21/5.80  % (22511)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.80  % (22511)Termination reason: Unknown
% 9.21/5.80  % (22511)Termination phase: shuffling
% 9.21/5.80  
% 9.21/5.80  % (22511)Memory used [KB]: 13432
% 9.21/5.80  % (22511)Time elapsed: 0.011 s
% 9.21/5.80  % (22511)Instructions burned: 7 (million)
% 9.21/5.80  % (22511)------------------------------
% 9.21/5.80  % (22511)------------------------------
% 9.21/5.80  % (22507)Instruction limit reached!
% 9.21/5.80  % (22507)------------------------------
% 9.21/5.80  % (22507)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.80  % (22507)Termination reason: Unknown
% 9.21/5.80  % (22507)Termination phase: shuffling
% 9.21/5.80  
% 9.21/5.80  % (22507)Memory used [KB]: 14200
% 9.21/5.80  % (22507)Time elapsed: 0.033 s
% 9.21/5.80  % (22507)Instructions burned: 37 (million)
% 9.21/5.80  % (22507)------------------------------
% 9.21/5.80  % (22507)------------------------------
% 9.21/5.81  % (22513)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_22391 for (2994ds/3Mi)
% 9.21/5.81  % (22512)Instruction limit reached!
% 9.21/5.81  % (22512)------------------------------
% 9.21/5.81  % (22512)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.81  % (22512)Termination reason: Unknown
% 9.21/5.81  % (22512)Termination phase: shuffling
% 9.21/5.81  
% 9.21/5.81  % (22512)Memory used [KB]: 13944
% 9.21/5.81  % (22512)Time elapsed: 0.022 s
% 9.21/5.81  % (22512)Instructions burned: 16 (million)
% 9.21/5.81  % (22512)------------------------------
% 9.21/5.81  % (22512)------------------------------
% 9.21/5.81  % (22513)Instruction limit reached!
% 9.21/5.81  % (22513)------------------------------
% 9.21/5.81  % (22513)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.81  % (22513)Termination reason: Unknown
% 9.21/5.81  % (22513)Termination phase: shuffling
% 9.21/5.81  
% 9.21/5.81  % (22513)Memory used [KB]: 13176
% 9.21/5.81  % (22513)Time elapsed: 0.006 s
% 9.21/5.81  % (22513)Instructions burned: 3 (million)
% 9.21/5.81  % (22513)------------------------------
% 9.21/5.81  % (22513)------------------------------
% 9.21/5.82  % (22514)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_22391 for (2994ds/3Mi)
% 9.21/5.82  % (22515)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/7Mi)
% 9.21/5.82  % (22514)Instruction limit reached!
% 9.21/5.82  % (22514)------------------------------
% 9.21/5.82  % (22514)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.82  % (22514)Termination reason: Unknown
% 9.21/5.82  % (22514)Termination phase: shuffling
% 9.21/5.82  
% 9.21/5.82  % (22514)Memory used [KB]: 13048
% 9.21/5.82  % (22514)Time elapsed: 0.005 s
% 9.21/5.82  % (22514)Instructions burned: 3 (million)
% 9.21/5.82  % (22514)------------------------------
% 9.21/5.82  % (22514)------------------------------
% 9.21/5.83  % (22515)Instruction limit reached!
% 9.21/5.83  % (22515)------------------------------
% 9.21/5.83  % (22515)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.83  % (22515)Termination reason: Unknown
% 9.21/5.83  % (22515)Termination phase: shuffling
% 9.21/5.83  
% 9.21/5.83  % (22515)Memory used [KB]: 13432
% 9.21/5.83  % (22515)Time elapsed: 0.010 s
% 9.21/5.83  % (22515)Instructions burned: 7 (million)
% 9.21/5.83  % (22515)------------------------------
% 9.21/5.83  % (22515)------------------------------
% 9.21/5.83  % (22516)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/3Mi)
% 9.21/5.83  % (22517)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/4Mi)
% 9.21/5.83  % (22516)Instruction limit reached!
% 9.21/5.83  % (22516)------------------------------
% 9.21/5.83  % (22516)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.83  % (22516)Termination reason: Unknown
% 9.21/5.83  % (22516)Termination phase: shuffling
% 9.21/5.83  
% 9.21/5.83  % (22516)Memory used [KB]: 13048
% 9.21/5.83  % (22516)Time elapsed: 0.005 s
% 9.21/5.83  % (22516)Instructions burned: 3 (million)
% 9.21/5.83  % (22516)------------------------------
% 9.21/5.83  % (22516)------------------------------
% 9.21/5.84  % (22517)Instruction limit reached!
% 9.21/5.84  % (22517)------------------------------
% 9.21/5.84  % (22517)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.84  % (22517)Termination reason: Unknown
% 9.21/5.84  % (22517)Termination phase: shuffling
% 9.21/5.84  
% 9.21/5.84  % (22517)Memory used [KB]: 13176
% 9.21/5.84  % (22517)Time elapsed: 0.006 s
% 9.21/5.84  % (22517)Instructions burned: 4 (million)
% 9.21/5.84  % (22517)------------------------------
% 9.21/5.84  % (22517)------------------------------
% 9.21/5.84  % (22518)lrs+1002_1:1_anc=all_dependent:au=on:cbe=off:fde=unused:ntd=on:i=18:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/18Mi)
% 9.21/5.84  % (22519)lrs+10_1:1_e2e=on:sd=1:sgt=8:ss=axioms:i=710:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/710Mi)
% 9.21/5.85  % (22520)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/6Mi)
% 9.21/5.86  % (22521)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_22391 for (2994ds/902Mi)
% 9.21/5.86  % (22518)Instruction limit reached!
% 9.21/5.86  % (22518)------------------------------
% 9.21/5.86  % (22518)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.86  % (22518)Termination reason: Unknown
% 9.21/5.86  % (22518)Termination phase: shuffling
% 9.21/5.86  
% 9.21/5.86  % (22518)Memory used [KB]: 13944
% 9.21/5.86  % (22518)Time elapsed: 0.023 s
% 9.21/5.86  % (22518)Instructions burned: 18 (million)
% 9.21/5.86  % (22518)------------------------------
% 9.21/5.86  % (22518)------------------------------
% 9.21/5.86  % (22520)Instruction limit reached!
% 9.21/5.86  % (22520)------------------------------
% 9.21/5.86  % (22520)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.86  % (22520)Termination reason: Unknown
% 9.21/5.86  % (22520)Termination phase: shuffling
% 9.21/5.86  
% 9.21/5.86  % (22520)Memory used [KB]: 13304
% 9.21/5.86  % (22520)Time elapsed: 0.010 s
% 9.21/5.86  % (22520)Instructions burned: 6 (million)
% 9.21/5.86  % (22520)------------------------------
% 9.21/5.86  % (22520)------------------------------
% 9.21/5.87  % (22499)Instruction limit reached!
% 9.21/5.87  % (22499)------------------------------
% 9.21/5.87  % (22499)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.87  % (22499)Termination reason: Unknown
% 9.21/5.87  % (22499)Termination phase: shuffling
% 9.21/5.87  
% 9.21/5.87  % (22499)Memory used [KB]: 16375
% 9.21/5.87  % (22499)Time elapsed: 0.124 s
% 9.21/5.87  % (22499)Instructions burned: 183 (million)
% 9.21/5.87  % (22499)------------------------------
% 9.21/5.87  % (22499)------------------------------
% 9.21/5.88  % (22522)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_22391 for (2994ds/21Mi)
% 9.21/5.88  % (22523)dis+10_1:1_cnfonf=lazy_gen:fe=off:i=5:si=on:rtra=on_0 on DTF2THF_22391 for (2994ds/5Mi)
% 9.21/5.88  % (22524)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_22391 for (2993ds/6Mi)
% 9.21/5.88  % (22523)Instruction limit reached!
% 9.21/5.88  % (22523)------------------------------
% 9.21/5.88  % (22523)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.88  % (22523)Termination reason: Unknown
% 9.21/5.88  % (22523)Termination phase: shuffling
% 9.21/5.88  
% 9.21/5.88  % (22523)Memory used [KB]: 13304
% 9.21/5.88  % (22523)Time elapsed: 0.007 s
% 9.21/5.88  % (22523)Instructions burned: 5 (million)
% 9.21/5.88  % (22523)------------------------------
% 9.21/5.88  % (22523)------------------------------
% 9.21/5.89  % (22524)Instruction limit reached!
% 9.21/5.89  % (22524)------------------------------
% 9.21/5.89  % (22524)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.89  % (22524)Termination reason: Unknown
% 9.21/5.89  % (22524)Termination phase: shuffling
% 9.21/5.89  
% 9.21/5.89  % (22524)Memory used [KB]: 13432
% 9.21/5.89  % (22524)Time elapsed: 0.006 s
% 9.21/5.89  % (22524)Instructions burned: 6 (million)
% 9.21/5.89  % (22524)------------------------------
% 9.21/5.89  % (22524)------------------------------
% 9.21/5.90  % (22522)Instruction limit reached!
% 9.21/5.90  % (22522)------------------------------
% 9.21/5.90  % (22522)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.90  % (22522)Termination reason: Unknown
% 9.21/5.90  % (22522)Termination phase: shuffling
% 9.21/5.90  
% 9.21/5.90  % (22522)Memory used [KB]: 13944
% 9.21/5.90  % (22522)Time elapsed: 0.018 s
% 9.21/5.90  % (22522)Instructions burned: 22 (million)
% 9.21/5.90  % (22522)------------------------------
% 9.21/5.90  % (22522)------------------------------
% 9.21/5.90  % (22525)lrs+1002_1:128_au=on:c=on:fsr=off:piset=equals:i=377:si=on:rtra=on_0 on DTF2THF_22391 for (2993ds/377Mi)
% 9.21/5.90  % (22526)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_22391 for (2993ds/779Mi)
% 9.21/5.91  % (22504)Instruction limit reached!
% 9.21/5.91  % (22504)------------------------------
% 9.21/5.91  % (22504)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.91  % (22504)Termination reason: Unknown
% 9.21/5.91  % (22504)Termination phase: shuffling
% 9.21/5.91  
% 9.21/5.91  % (22504)Memory used [KB]: 17782
% 9.21/5.91  % (22504)Time elapsed: 0.164 s
% 9.21/5.91  % (22504)Instructions burned: 275 (million)
% 9.21/5.91  % (22504)------------------------------
% 9.21/5.91  % (22504)------------------------------
% 9.21/5.91  % (22528)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_22391 for (2993ds/19Mi)
% 9.21/5.92  % (22529)lrs+1010_1:1_au=on:s2a=on:sd=1:sgt=50:ss=axioms:i=879:si=on:rtra=on_0 on DTF2THF_22391 for (2993ds/879Mi)
% 9.21/5.92  % (22528)Instruction limit reached!
% 9.21/5.92  % (22528)------------------------------
% 9.21/5.92  % (22528)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.92  % (22528)Termination reason: Unknown
% 9.21/5.92  % (22528)Termination phase: shuffling
% 9.21/5.92  
% 9.21/5.92  % (22528)Memory used [KB]: 13944
% 9.21/5.92  % (22528)Time elapsed: 0.016 s
% 9.21/5.92  % (22528)Instructions burned: 21 (million)
% 9.21/5.92  % (22528)------------------------------
% 9.21/5.92  % (22528)------------------------------
% 9.21/5.93  % (22530)dis+1002_1:128_acc=on:er=filter:i=17:si=on:rtra=on_0 on DTF2THF_22391 for (2993ds/17Mi)
% 9.21/5.95  % (22530)Instruction limit reached!
% 9.21/5.95  % (22530)------------------------------
% 9.21/5.95  % (22530)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.95  % (22530)Termination reason: Unknown
% 9.21/5.95  % (22530)Termination phase: shuffling
% 9.21/5.95  
% 9.21/5.95  % (22530)Memory used [KB]: 13944
% 9.21/5.95  % (22530)Time elapsed: 0.015 s
% 9.21/5.95  % (22530)Instructions burned: 17 (million)
% 9.21/5.95  % (22530)------------------------------
% 9.21/5.95  % (22530)------------------------------
% 9.21/5.96  % (22531)ott+21_1:1_apa=on:au=on:cnfonf=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22391 for (2993ds/3Mi)
% 9.21/5.96  % (22531)Instruction limit reached!
% 9.21/5.96  % (22531)------------------------------
% 9.21/5.96  % (22531)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.96  % (22531)Termination reason: Unknown
% 9.21/5.96  % (22531)Termination phase: shuffling
% 9.21/5.96  
% 9.21/5.96  % (22531)Memory used [KB]: 13176
% 9.21/5.96  % (22531)Time elapsed: 0.004 s
% 9.21/5.96  % (22531)Instructions burned: 3 (million)
% 9.21/5.96  % (22531)------------------------------
% 9.21/5.96  % (22531)------------------------------
% 9.21/5.97  % (22532)lrs+1010_1:8_cnfonf=off:hud=1:inj=on:tnu=5:i=30:si=on:rtra=on_0 on DTF2THF_22391 for (2992ds/30Mi)
% 9.21/5.99  % (22532)Instruction limit reached!
% 9.21/5.99  % (22532)------------------------------
% 9.21/5.99  % (22532)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 9.21/5.99  % (22532)Termination reason: Unknown
% 9.21/5.99  % (22532)Termination phase: shuffling
% 9.21/5.99  
% 9.21/5.99  % (22532)Memory used [KB]: 14072
% 9.21/5.99  % (22532)Time elapsed: 0.020 s
% 9.21/5.99  % (22532)Instructions burned: 32 (million)
% 9.21/5.99  % (22532)------------------------------
% 9.21/5.99  % (22532)------------------------------
% 9.21/6.00  % (22533)dis+10_1:1_ixr=off:plsq=on:plsqc=1:plsqr=32,1:s2a=on:i=127:si=on:rtra=on_0 on DTF2THF_22391 for (2992ds/127Mi)
% 11.59/6.06  % (22525)Instruction limit reached!
% 11.59/6.06  % (22525)------------------------------
% 11.59/6.06  % (22525)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.06  % (22525)Termination reason: Unknown
% 11.59/6.06  % (22525)Termination phase: shuffling
% 11.59/6.06  
% 11.59/6.06  % (22525)Memory used [KB]: 19061
% 11.59/6.06  % (22525)Time elapsed: 0.160 s
% 11.59/6.06  % (22525)Instructions burned: 377 (million)
% 11.59/6.06  % (22525)------------------------------
% 11.59/6.06  % (22525)------------------------------
% 11.59/6.06  % (22533)Instruction limit reached!
% 11.59/6.06  % (22533)------------------------------
% 11.59/6.06  % (22533)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.06  % (22533)Termination reason: Unknown
% 11.59/6.06  % (22533)Termination phase: shuffling
% 11.59/6.06  
% 11.59/6.06  % (22533)Memory used [KB]: 15479
% 11.59/6.06  % (22533)Time elapsed: 0.055 s
% 11.59/6.06  % (22533)Instructions burned: 127 (million)
% 11.59/6.06  % (22533)------------------------------
% 11.59/6.06  % (22533)------------------------------
% 11.59/6.07  % (22534)lrs+1002_1:1_au=on:cbe=off:cnfonf=conj_eager:cond=on:hi=on:i=100:si=on:rtra=on_0 on DTF2THF_22391 for (2991ds/100Mi)
% 11.59/6.07  % (22535)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_22391 for (2991ds/3Mi)
% 11.59/6.07  % (22535)Instruction limit reached!
% 11.59/6.07  % (22535)------------------------------
% 11.59/6.07  % (22535)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.07  % (22535)Termination reason: Unknown
% 11.59/6.07  % (22535)Termination phase: shuffling
% 11.59/6.07  
% 11.59/6.07  % (22535)Memory used [KB]: 13048
% 11.59/6.07  % (22535)Time elapsed: 0.003 s
% 11.59/6.07  % (22535)Instructions burned: 3 (million)
% 11.59/6.07  % (22535)------------------------------
% 11.59/6.07  % (22535)------------------------------
% 11.59/6.08  % (22510)Instruction limit reached!
% 11.59/6.08  % (22510)------------------------------
% 11.59/6.08  % (22510)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.08  % (22510)Termination reason: Unknown
% 11.59/6.08  % (22510)Termination phase: shuffling
% 11.59/6.08  
% 11.59/6.08  % (22510)Memory used [KB]: 34413
% 11.59/6.08  % (22510)Time elapsed: 0.306 s
% 11.59/6.08  % (22510)Instructions burned: 1042 (million)
% 11.59/6.08  % (22510)------------------------------
% 11.59/6.08  % (22510)------------------------------
% 11.59/6.08  % (22536)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_22391 for (2991ds/20Mi)
% 11.59/6.09  % (22537)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_22391 for (2991ds/86Mi)
% 11.59/6.10  % (22536)Instruction limit reached!
% 11.59/6.10  % (22536)------------------------------
% 11.59/6.10  % (22536)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.10  % (22536)Termination reason: Unknown
% 11.59/6.10  % (22536)Termination phase: shuffling
% 11.59/6.10  
% 11.59/6.10  % (22536)Memory used [KB]: 13944
% 11.59/6.10  % (22536)Time elapsed: 0.016 s
% 11.59/6.10  % (22536)Instructions burned: 20 (million)
% 11.59/6.10  % (22536)------------------------------
% 11.59/6.10  % (22536)------------------------------
% 11.59/6.11  % (22534)Instruction limit reached!
% 11.59/6.11  % (22534)------------------------------
% 11.59/6.11  % (22534)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 11.59/6.11  % (22534)Termination reason: Unknown
% 11.59/6.11  % (22534)Termination phase: shuffling
% 11.59/6.11  
% 11.59/6.11  % (22534)Memory used [KB]: 15095
% 11.59/6.11  % (22534)Time elapsed: 0.041 s
% 11.59/6.11  % (22534)Instructions burned: 103 (million)
% 11.59/6.11  % (22534)------------------------------
% 11.59/6.11  % (22534)------------------------------
% 11.59/6.11  % (22538)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_22391 for (2991ds/107Mi)
% 12.30/6.12  % (22539)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_22391 for (2991ds/39Mi)
% 12.30/6.13  % (22519)Instruction limit reached!
% 12.30/6.13  % (22519)------------------------------
% 12.30/6.13  % (22519)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.13  % (22519)Termination reason: Unknown
% 12.30/6.13  % (22519)Termination phase: shuffling
% 12.30/6.13  
% 12.30/6.13  % (22519)Memory used [KB]: 26097
% 12.30/6.13  % (22519)Time elapsed: 0.283 s
% 12.30/6.13  % (22519)Instructions burned: 710 (million)
% 12.30/6.13  % (22519)------------------------------
% 12.30/6.13  % (22519)------------------------------
% 12.30/6.13  % (22526)Instruction limit reached!
% 12.30/6.13  % (22526)------------------------------
% 12.30/6.13  % (22526)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.13  % (22526)Termination reason: Unknown
% 12.30/6.13  % (22526)Termination phase: shuffling
% 12.30/6.13  
% 12.30/6.13  % (22526)Memory used [KB]: 26737
% 12.30/6.13  % (22526)Time elapsed: 0.231 s
% 12.30/6.13  % (22526)Instructions burned: 780 (million)
% 12.30/6.13  % (22526)------------------------------
% 12.30/6.13  % (22526)------------------------------
% 12.30/6.13  % (22537)Instruction limit reached!
% 12.30/6.13  % (22537)------------------------------
% 12.30/6.13  % (22537)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.13  % (22537)Termination reason: Unknown
% 12.30/6.13  % (22537)Termination phase: shuffling
% 12.30/6.13  
% 12.30/6.13  % (22537)Memory used [KB]: 14967
% 12.30/6.13  % (22537)Time elapsed: 0.037 s
% 12.30/6.13  % (22537)Instructions burned: 88 (million)
% 12.30/6.13  % (22537)------------------------------
% 12.30/6.13  % (22537)------------------------------
% 12.30/6.14  % (22540)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_22391 for (2991ds/448Mi)
% 12.30/6.14  % (22542)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_22391 for (2991ds/98Mi)
% 12.30/6.14  % (22541)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_22391 for (2991ds/46Mi)
% 12.30/6.14  % (22539)Instruction limit reached!
% 12.30/6.14  % (22539)------------------------------
% 12.30/6.14  % (22539)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.14  % (22539)Termination reason: Unknown
% 12.30/6.14  % (22539)Termination phase: shuffling
% 12.30/6.14  
% 12.30/6.14  % (22539)Memory used [KB]: 14328
% 12.30/6.14  % (22539)Time elapsed: 0.022 s
% 12.30/6.14  % (22539)Instructions burned: 41 (million)
% 12.30/6.14  % (22539)------------------------------
% 12.30/6.14  % (22539)------------------------------
% 12.30/6.15  % (22543)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_22391 for (2991ds/507Mi)
% 12.30/6.16  % (22538)Instruction limit reached!
% 12.30/6.16  % (22538)------------------------------
% 12.30/6.16  % (22538)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.16  % (22538)Termination reason: Unknown
% 12.30/6.16  % (22538)Termination phase: shuffling
% 12.30/6.16  
% 12.30/6.16  % (22538)Memory used [KB]: 15095
% 12.30/6.16  % (22538)Time elapsed: 0.047 s
% 12.30/6.16  % (22538)Instructions burned: 107 (million)
% 12.30/6.16  % (22538)------------------------------
% 12.30/6.16  % (22538)------------------------------
% 12.30/6.16  % (22541)Instruction limit reached!
% 12.30/6.16  % (22541)------------------------------
% 12.30/6.16  % (22541)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.16  % (22541)Termination reason: Unknown
% 12.30/6.16  % (22541)Termination phase: shuffling
% 12.30/6.16  
% 12.30/6.16  % (22541)Memory used [KB]: 14328
% 12.30/6.16  % (22541)Time elapsed: 0.025 s
% 12.30/6.16  % (22541)Instructions burned: 48 (million)
% 12.30/6.16  % (22541)------------------------------
% 12.30/6.16  % (22541)------------------------------
% 12.30/6.17  % (22544)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_22391 for (2991ds/149Mi)
% 12.30/6.17  % (22542)Instruction limit reached!
% 12.30/6.17  % (22542)------------------------------
% 12.30/6.17  % (22542)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.17  % (22542)Termination reason: Unknown
% 12.30/6.17  % (22542)Termination phase: shuffling
% 12.30/6.17  
% 12.30/6.17  % (22542)Memory used [KB]: 14967
% 12.30/6.17  % (22542)Time elapsed: 0.036 s
% 12.30/6.17  % (22542)Instructions burned: 98 (million)
% 12.30/6.17  % (22542)------------------------------
% 12.30/6.17  % (22542)------------------------------
% 12.30/6.18  % (22545)lrs+21_1:16_au=on:bd=off:piset=and:i=39:si=on:rtra=on_0 on DTF2THF_22391 for (2991ds/39Mi)
% 12.30/6.19  % (22546)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_22391 for (2990ds/6Mi)
% 12.30/6.19  % (22546)Instruction limit reached!
% 12.30/6.19  % (22546)------------------------------
% 12.30/6.19  % (22546)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.19  % (22546)Termination reason: Unknown
% 12.30/6.19  % (22546)Termination phase: shuffling
% 12.30/6.19  
% 12.30/6.19  % (22546)Memory used [KB]: 13432
% 12.30/6.19  % (22546)Time elapsed: 0.007 s
% 12.30/6.19  % (22546)Instructions burned: 6 (million)
% 12.30/6.19  % (22546)------------------------------
% 12.30/6.19  % (22546)------------------------------
% 12.30/6.19  % (22521)Instruction limit reached!
% 12.30/6.19  % (22521)------------------------------
% 12.30/6.19  % (22521)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.30/6.19  % (22521)Termination reason: Unknown
% 12.30/6.19  % (22521)Termination phase: Property scanning
% 12.30/6.19  
% 12.30/6.19  % (22521)Memory used [KB]: 27121
% 12.30/6.19  % (22521)Time elapsed: 0.342 s
% 12.30/6.19  % (22521)Instructions burned: 903 (million)
% 12.30/6.19  % (22521)------------------------------
% 12.30/6.19  % (22521)------------------------------
% 12.39/6.20  % (22545)Instruction limit reached!
% 12.39/6.20  % (22545)------------------------------
% 12.39/6.20  % (22545)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.20  % (22545)Termination reason: Unknown
% 12.39/6.20  % (22545)Termination phase: shuffling
% 12.39/6.20  
% 12.39/6.20  % (22545)Memory used [KB]: 14200
% 12.39/6.20  % (22545)Time elapsed: 0.023 s
% 12.39/6.20  % (22545)Instructions burned: 40 (million)
% 12.39/6.20  % (22545)------------------------------
% 12.39/6.20  % (22545)------------------------------
% 12.39/6.20  % (22547)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_22391 for (2990ds/1087Mi)
% 12.39/6.21  % (22548)dis+10_1:1_aac=none:cs=on:i=47:si=on:rtra=on_0 on DTF2THF_22391 for (2990ds/47Mi)
% 12.39/6.21  % (22549)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_22391 for (2990ds/44Mi)
% 12.39/6.23  % (22544)Instruction limit reached!
% 12.39/6.23  % (22544)------------------------------
% 12.39/6.23  % (22544)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.23  % (22544)Termination reason: Unknown
% 12.39/6.23  % (22544)Termination phase: shuffling
% 12.39/6.23  
% 12.39/6.23  % (22544)Memory used [KB]: 15863
% 12.39/6.23  % (22544)Time elapsed: 0.062 s
% 12.39/6.23  % (22544)Instructions burned: 150 (million)
% 12.39/6.23  % (22544)------------------------------
% 12.39/6.23  % (22544)------------------------------
% 12.39/6.23  % (22549)Instruction limit reached!
% 12.39/6.23  % (22549)------------------------------
% 12.39/6.23  % (22549)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.23  % (22549)Termination reason: Unknown
% 12.39/6.23  % (22549)Termination phase: shuffling
% 12.39/6.23  
% 12.39/6.23  % (22549)Memory used [KB]: 14328
% 12.39/6.23  % (22549)Time elapsed: 0.021 s
% 12.39/6.23  % (22549)Instructions burned: 45 (million)
% 12.39/6.23  % (22549)------------------------------
% 12.39/6.23  % (22549)------------------------------
% 12.39/6.23  % (22548)Instruction limit reached!
% 12.39/6.23  % (22548)------------------------------
% 12.39/6.23  % (22548)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.23  % (22548)Termination reason: Unknown
% 12.39/6.23  % (22548)Termination phase: shuffling
% 12.39/6.23  
% 12.39/6.23  % (22548)Memory used [KB]: 14328
% 12.39/6.23  % (22548)Time elapsed: 0.025 s
% 12.39/6.23  % (22548)Instructions burned: 47 (million)
% 12.39/6.23  % (22548)------------------------------
% 12.39/6.23  % (22548)------------------------------
% 12.39/6.24  % (22551)lrs+10_7:1_cnfonf=lazy_gen:fe=off:nwc=5.0:plsq=on:i=62:si=on:rtra=on_0 on DTF2THF_22391 for (2990ds/62Mi)
% 12.39/6.24  % (22550)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_22391 for (2990ds/432Mi)
% 12.39/6.24  % (22529)Instruction limit reached!
% 12.39/6.24  % (22529)------------------------------
% 12.39/6.24  % (22529)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.24  % (22529)Termination reason: Unknown
% 12.39/6.24  % (22529)Termination phase: Property scanning
% 12.39/6.24  
% 12.39/6.24  % (22529)Memory used [KB]: 26993
% 12.39/6.24  % (22529)Time elapsed: 0.322 s
% 12.39/6.24  % (22529)Instructions burned: 880 (million)
% 12.39/6.24  % (22529)------------------------------
% 12.39/6.24  % (22529)------------------------------
% 12.39/6.24  % (22552)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_22391 for (2990ds/31Mi)
% 12.39/6.25  % (22553)dis+10_1:128_hud=1:i=96:si=on:rtra=on_0 on DTF2THF_22391 for (2990ds/96Mi)
% 12.39/6.26  % (22552)Instruction limit reached!
% 12.39/6.26  % (22552)------------------------------
% 12.39/6.26  % (22552)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.26  % (22552)Termination reason: Unknown
% 12.39/6.26  % (22552)Termination phase: shuffling
% 12.39/6.26  
% 12.39/6.26  % (22552)Memory used [KB]: 14072
% 12.39/6.26  % (22552)Time elapsed: 0.018 s
% 12.39/6.26  % (22552)Instructions burned: 31 (million)
% 12.39/6.26  % (22552)------------------------------
% 12.39/6.26  % (22552)------------------------------
% 12.39/6.27  % (22551)Instruction limit reached!
% 12.39/6.27  % (22551)------------------------------
% 12.39/6.27  % (22551)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.27  % (22551)Termination reason: Unknown
% 12.39/6.27  % (22551)Termination phase: shuffling
% 12.39/6.27  
% 12.39/6.27  % (22551)Memory used [KB]: 14583
% 12.39/6.27  % (22551)Time elapsed: 0.027 s
% 12.39/6.27  % (22551)Instructions burned: 64 (million)
% 12.39/6.27  % (22551)------------------------------
% 12.39/6.27  % (22551)------------------------------
% 12.39/6.27  % (22554)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_22391 for (2990ds/453Mi)
% 12.39/6.27  % (22540)Instruction limit reached!
% 12.39/6.27  % (22540)------------------------------
% 12.39/6.27  % (22540)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.27  % (22540)Termination reason: Unknown
% 12.39/6.27  % (22540)Termination phase: shuffling
% 12.39/6.27  
% 12.39/6.27  % (22540)Memory used [KB]: 22131
% 12.39/6.27  % (22540)Time elapsed: 0.137 s
% 12.39/6.27  % (22540)Instructions burned: 450 (million)
% 12.39/6.27  % (22540)------------------------------
% 12.39/6.27  % (22540)------------------------------
% 12.39/6.28  % (22555)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_22391 for (2989ds/71Mi)
% 12.39/6.29  % (22556)dis+1002_1:1_bd=off:fsr=off:ntd=on:s2a=on:ss=axioms:i=83:si=on:rtra=on_0 on DTF2THF_22391 for (2989ds/83Mi)
% 12.39/6.29  % (22553)Instruction limit reached!
% 12.39/6.29  % (22553)------------------------------
% 12.39/6.29  % (22553)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 12.39/6.29  % (22553)Termination reason: Unknown
% 12.39/6.29  % (22553)Termination phase: shuffling
% 12.39/6.29  
% 12.39/6.29  % (22553)Memory used [KB]: 15095
% 12.39/6.29  % (22553)Time elapsed: 0.041 s
% 12.39/6.29  % (22553)Instructions burned: 98 (million)
% 12.39/6.29  % (22553)------------------------------
% 12.39/6.29  % (22553)------------------------------
% 13.24/6.30  % (22557)lrs+10_1:128_cnfonf=lazy_not_gen_be_off:hud=1:ins=1:i=11:si=on:rtra=on_0 on DTF2THF_22391 for (2989ds/11Mi)
% 13.24/6.31  % (22555)Instruction limit reached!
% 13.24/6.31  % (22555)------------------------------
% 13.24/6.31  % (22555)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.31  % (22555)Termination reason: Unknown
% 13.24/6.31  % (22555)Termination phase: shuffling
% 13.24/6.31  
% 13.24/6.31  % (22555)Memory used [KB]: 14583
% 13.24/6.31  % (22555)Time elapsed: 0.032 s
% 13.24/6.31  % (22555)Instructions burned: 72 (million)
% 13.24/6.31  % (22555)------------------------------
% 13.24/6.31  % (22555)------------------------------
% 13.24/6.31  % (22557)Instruction limit reached!
% 13.24/6.31  % (22557)------------------------------
% 13.24/6.31  % (22557)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.31  % (22557)Termination reason: Unknown
% 13.24/6.31  % (22557)Termination phase: shuffling
% 13.24/6.31  
% 13.24/6.31  % (22557)Memory used [KB]: 13816
% 13.24/6.31  % (22557)Time elapsed: 0.012 s
% 13.24/6.31  % (22557)Instructions burned: 11 (million)
% 13.24/6.31  % (22557)------------------------------
% 13.24/6.31  % (22557)------------------------------
% 13.24/6.32  % (22558)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_22391 for (2989ds/41Mi)
% 13.24/6.32  % (22556)Instruction limit reached!
% 13.24/6.32  % (22556)------------------------------
% 13.24/6.32  % (22556)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.32  % (22556)Termination reason: Unknown
% 13.24/6.32  % (22556)Termination phase: shuffling
% 13.24/6.32  
% 13.24/6.32  % (22556)Memory used [KB]: 14839
% 13.24/6.32  % (22556)Time elapsed: 0.036 s
% 13.24/6.32  % (22556)Instructions burned: 84 (million)
% 13.24/6.32  % (22556)------------------------------
% 13.24/6.32  % (22556)------------------------------
% 13.24/6.32  % (22559)lrs+1004_1:1_chr=on:prag=on:i=6:si=on:rtra=on_0 on DTF2THF_22391 for (2989ds/6Mi)
% 13.24/6.33  % (22559)Instruction limit reached!
% 13.24/6.33  % (22559)------------------------------
% 13.24/6.33  % (22559)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.33  % (22559)Termination reason: Unknown
% 13.24/6.33  % (22559)Termination phase: shuffling
% 13.24/6.33  
% 13.24/6.33  % (22559)Memory used [KB]: 13304
% 13.24/6.33  % (22559)Time elapsed: 0.006 s
% 13.24/6.33  % (22559)Instructions burned: 6 (million)
% 13.24/6.33  % (22559)------------------------------
% 13.24/6.33  % (22559)------------------------------
% 13.24/6.33  % (22560)dis+10_1:1_avsq=on:cnfonf=off:e2e=on:nm=32:i=150:si=on:rtra=on_0 on DTF2THF_22391 for (2989ds/150Mi)
% 13.24/6.34  % (22558)Instruction limit reached!
% 13.24/6.34  % (22558)------------------------------
% 13.24/6.34  % (22558)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.34  % (22558)Termination reason: Unknown
% 13.24/6.34  % (22558)Termination phase: shuffling
% 13.24/6.34  
% 13.24/6.34  % (22558)Memory used [KB]: 14328
% 13.24/6.34  % (22558)Time elapsed: 0.021 s
% 13.24/6.34  % (22558)Instructions burned: 42 (million)
% 13.24/6.34  % (22558)------------------------------
% 13.24/6.34  % (22558)------------------------------
% 13.24/6.34  % (22561)ott+1010_1:1_au=on:s2a=on:sd=7:ss=axioms:st=1.5:i=343:si=on:rtra=on_0 on DTF2THF_22391 for (2989ds/343Mi)
% 13.24/6.34  % (22543)Instruction limit reached!
% 13.24/6.34  % (22543)------------------------------
% 13.24/6.34  % (22543)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.34  % (22543)Termination reason: Unknown
% 13.24/6.34  % (22543)Termination phase: shuffling
% 13.24/6.34  
% 13.24/6.34  % (22543)Memory used [KB]: 20724
% 13.24/6.34  % (22543)Time elapsed: 0.192 s
% 13.24/6.34  % (22543)Instructions burned: 507 (million)
% 13.24/6.34  % (22543)------------------------------
% 13.24/6.34  % (22543)------------------------------
% 13.24/6.35  % (22562)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_22391 for (2989ds/109Mi)
% 13.24/6.36  % (22563)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_22391 for (2989ds/253Mi)
% 13.24/6.37  % (22550)Instruction limit reached!
% 13.24/6.37  % (22550)------------------------------
% 13.24/6.37  % (22550)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.37  % (22550)Termination reason: Unknown
% 13.24/6.37  % (22550)Termination phase: shuffling
% 13.24/6.37  
% 13.24/6.37  % (22550)Memory used [KB]: 21875
% 13.24/6.37  % (22550)Time elapsed: 0.132 s
% 13.24/6.37  % (22550)Instructions burned: 432 (million)
% 13.24/6.37  % (22550)------------------------------
% 13.24/6.37  % (22550)------------------------------
% 13.24/6.38  % (22564)lrs+10_1:1_au=on:cnfonf=off:sd=1:sgt=16:ss=axioms:i=140:si=on:rtra=on_0 on DTF2THF_22391 for (2988ds/140Mi)
% 13.24/6.39  % (22560)Instruction limit reached!
% 13.24/6.39  % (22560)------------------------------
% 13.24/6.39  % (22560)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.39  % (22560)Termination reason: Unknown
% 13.24/6.39  % (22560)Termination phase: shuffling
% 13.24/6.39  
% 13.24/6.39  % (22560)Memory used [KB]: 15863
% 13.24/6.39  % (22560)Time elapsed: 0.058 s
% 13.24/6.39  % (22560)Instructions burned: 150 (million)
% 13.24/6.39  % (22560)------------------------------
% 13.24/6.39  % (22560)------------------------------
% 13.24/6.39  % (22562)Instruction limit reached!
% 13.24/6.39  % (22562)------------------------------
% 13.24/6.39  % (22562)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.24/6.39  % (22562)Termination reason: Unknown
% 13.24/6.39  % (22562)Termination phase: shuffling
% 13.24/6.39  
% 13.24/6.39  % (22562)Memory used [KB]: 15223
% 13.24/6.39  % (22562)Time elapsed: 0.044 s
% 13.24/6.39  % (22562)Instructions burned: 109 (million)
% 13.24/6.39  % (22562)------------------------------
% 13.24/6.39  % (22562)------------------------------
% 13.85/6.40  % (22565)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_22391 for (2988ds/614Mi)
% 13.85/6.40  % (22566)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_22391 for (2988ds/202Mi)
% 13.85/6.41  % (22554)Instruction limit reached!
% 13.85/6.41  % (22554)------------------------------
% 13.85/6.41  % (22554)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/6.41  % (22554)Termination reason: Unknown
% 13.85/6.41  % (22554)Termination phase: shuffling
% 13.85/6.41  
% 13.85/6.41  % (22554)Memory used [KB]: 22259
% 13.85/6.41  % (22554)Time elapsed: 0.139 s
% 13.85/6.41  % (22554)Instructions burned: 456 (million)
% 13.85/6.41  % (22554)------------------------------
% 13.85/6.41  % (22554)------------------------------
% 13.85/6.42  % (22567)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_22391 for (2988ds/339Mi)
% 13.85/6.44  % (22564)Instruction limit reached!
% 13.85/6.44  % (22564)------------------------------
% 13.85/6.44  % (22564)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/6.44  % (22564)Termination reason: Unknown
% 13.85/6.44  % (22564)Termination phase: shuffling
% 13.85/6.44  
% 13.85/6.44  % (22564)Memory used [KB]: 15607
% 13.85/6.44  % (22564)Time elapsed: 0.055 s
% 13.85/6.44  % (22564)Instructions burned: 142 (million)
% 13.85/6.44  % (22564)------------------------------
% 13.85/6.44  % (22564)------------------------------
% 13.85/6.45  % (22568)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_22391 for (2988ds/283Mi)
% 13.85/6.45  % (22563)Instruction limit reached!
% 13.85/6.45  % (22563)------------------------------
% 13.85/6.45  % (22563)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/6.45  % (22563)Termination reason: Unknown
% 13.85/6.45  % (22563)Termination phase: shuffling
% 13.85/6.45  
% 13.85/6.45  % (22563)Memory used [KB]: 17398
% 13.85/6.45  % (22563)Time elapsed: 0.094 s
% 13.85/6.45  % (22563)Instructions burned: 253 (million)
% 13.85/6.45  % (22563)------------------------------
% 13.85/6.45  % (22563)------------------------------
% 13.85/6.46  % (22561)Instruction limit reached!
% 13.85/6.46  % (22561)------------------------------
% 13.85/6.46  % (22561)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/6.46  % (22561)Termination reason: Unknown
% 13.85/6.46  % (22561)Termination phase: shuffling
% 13.85/6.46  
% 13.85/6.46  % (22561)Memory used [KB]: 18677
% 13.85/6.46  % (22561)Time elapsed: 0.122 s
% 13.85/6.46  % (22561)Instructions burned: 344 (million)
% 13.85/6.46  % (22561)------------------------------
% 13.85/6.46  % (22561)------------------------------
% 13.85/6.46  % (22569)dis+2_1:1_cha=on:hud=7:nwc=3.0:prag=on:tnu=2:i=134:si=on:rtra=on_0 on DTF2THF_22391 for (2988ds/134Mi)
% 13.85/6.47  % (22570)dis+1010_1:64_cnfonf=lazy_simp:fd=off:i=298:si=on:rtra=on_0 on DTF2THF_22391 for (2987ds/298Mi)
% 13.85/6.48  % (22566)Instruction limit reached!
% 13.85/6.48  % (22566)------------------------------
% 13.85/6.48  % (22566)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 13.85/6.48  % (22566)Termination reason: Unknown
% 13.85/6.48  % (22566)Termination phase: shuffling
% 13.85/6.48  
% 13.85/6.48  % (22566)Memory used [KB]: 16630
% 13.85/6.48  % (22566)Time elapsed: 0.076 s
% 13.85/6.48  % (22566)Instructions burned: 202 (million)
% 13.85/6.48  % (22566)------------------------------
% 13.85/6.48  % (22566)------------------------------
% 13.85/6.49  % (22571)lrs+2_1:8_erd=off:plsq=on:plsqr=32,1:sos=on:i=192:si=on:rtra=on_0 on DTF2THF_22391 for (2987ds/192Mi)
% 14.40/6.51  % (22569)Instruction limit reached!
% 14.40/6.51  % (22569)------------------------------
% 14.40/6.51  % (22569)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.51  % (22569)Termination reason: Unknown
% 14.40/6.51  % (22569)Termination phase: shuffling
% 14.40/6.51  
% 14.40/6.51  % (22569)Memory used [KB]: 15607
% 14.40/6.51  % (22569)Time elapsed: 0.054 s
% 14.40/6.51  % (22569)Instructions burned: 135 (million)
% 14.40/6.51  % (22569)------------------------------
% 14.40/6.51  % (22569)------------------------------
% 14.40/6.52  % (22572)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_22391 for (2987ds/258Mi)
% 14.40/6.54  % (22568)Instruction limit reached!
% 14.40/6.54  % (22568)------------------------------
% 14.40/6.54  % (22568)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.54  % (22568)Termination reason: Unknown
% 14.40/6.54  % (22568)Termination phase: shuffling
% 14.40/6.54  
% 14.40/6.54  % (22568)Memory used [KB]: 19829
% 14.40/6.54  % (22568)Time elapsed: 0.091 s
% 14.40/6.54  % (22568)Instructions burned: 285 (million)
% 14.40/6.54  % (22568)------------------------------
% 14.40/6.54  % (22568)------------------------------
% 14.40/6.54  % (22567)Instruction limit reached!
% 14.40/6.54  % (22567)------------------------------
% 14.40/6.54  % (22567)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.54  % (22567)Termination reason: Unknown
% 14.40/6.54  % (22567)Termination phase: shuffling
% 14.40/6.54  
% 14.40/6.54  % (22567)Memory used [KB]: 18549
% 14.40/6.54  % (22567)Time elapsed: 0.122 s
% 14.40/6.54  % (22567)Instructions burned: 340 (million)
% 14.40/6.54  % (22567)------------------------------
% 14.40/6.54  % (22567)------------------------------
% 14.40/6.55  % (22573)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_22391 for (2987ds/90Mi)
% 14.40/6.55  % (22574)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_22391 for (2987ds/231Mi)
% 14.40/6.56  % (22547)Instruction limit reached!
% 14.40/6.56  % (22547)------------------------------
% 14.40/6.56  % (22547)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.56  % (22547)Termination reason: Unknown
% 14.40/6.56  % (22547)Termination phase: Property scanning
% 14.40/6.56  
% 14.40/6.56  % (22547)Memory used [KB]: 27121
% 14.40/6.56  % (22547)Time elapsed: 0.357 s
% 14.40/6.56  % (22547)Instructions burned: 1088 (million)
% 14.40/6.56  % (22547)------------------------------
% 14.40/6.56  % (22547)------------------------------
% 14.40/6.56  % (22571)Instruction limit reached!
% 14.40/6.56  % (22571)------------------------------
% 14.40/6.56  % (22571)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.56  % (22571)Termination reason: Unknown
% 14.40/6.56  % (22571)Termination phase: shuffling
% 14.40/6.56  
% 14.40/6.56  % (22571)Memory used [KB]: 16502
% 14.40/6.56  % (22571)Time elapsed: 0.074 s
% 14.40/6.56  % (22571)Instructions burned: 194 (million)
% 14.40/6.56  % (22571)------------------------------
% 14.40/6.56  % (22571)------------------------------
% 14.40/6.57  % (22570)Instruction limit reached!
% 14.40/6.57  % (22570)------------------------------
% 14.40/6.57  % (22570)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.57  % (22570)Termination reason: Unknown
% 14.40/6.57  % (22570)Termination phase: shuffling
% 14.40/6.57  
% 14.40/6.57  % (22570)Memory used [KB]: 19957
% 14.40/6.57  % (22570)Time elapsed: 0.096 s
% 14.40/6.57  % (22570)Instructions burned: 300 (million)
% 14.40/6.57  % (22570)------------------------------
% 14.40/6.57  % (22570)------------------------------
% 14.40/6.57  % (22575)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_22391 for (2987ds/458Mi)
% 14.40/6.57  % (22576)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_22391 for (2987ds/31Mi)
% 14.40/6.58  % (22577)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_22391 for (2986ds/297Mi)
% 14.40/6.59  % (22573)Instruction limit reached!
% 14.40/6.59  % (22573)------------------------------
% 14.40/6.59  % (22573)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.59  % (22573)Termination reason: Unknown
% 14.40/6.59  % (22573)Termination phase: shuffling
% 14.40/6.59  
% 14.40/6.59  % (22573)Memory used [KB]: 14967
% 14.40/6.59  % (22573)Time elapsed: 0.042 s
% 14.40/6.59  % (22573)Instructions burned: 92 (million)
% 14.40/6.59  % (22573)------------------------------
% 14.40/6.59  % (22573)------------------------------
% 14.40/6.59  % (22576)Instruction limit reached!
% 14.40/6.59  % (22576)------------------------------
% 14.40/6.59  % (22576)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 14.40/6.59  % (22576)Termination reason: Unknown
% 14.40/6.59  % (22576)Termination phase: shuffling
% 14.40/6.59  
% 14.40/6.59  % (22576)Memory used [KB]: 14072
% 14.40/6.59  % (22576)Time elapsed: 0.019 s
% 14.40/6.59  % (22576)Instructions burned: 31 (million)
% 14.40/6.59  % (22576)------------------------------
% 14.40/6.59  % (22576)------------------------------
% 15.34/6.60  % (22579)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_22391 for (2986ds/182Mi)
% 15.34/6.60  % (22578)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_22391 for (2986ds/987Mi)
% 15.34/6.62  % (22565)Instruction limit reached!
% 15.34/6.62  % (22565)------------------------------
% 15.34/6.62  % (22565)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.62  % (22565)Termination reason: Unknown
% 15.34/6.62  % (22565)Termination phase: shuffling
% 15.34/6.62  
% 15.34/6.62  % (22565)Memory used [KB]: 24434
% 15.34/6.62  % (22565)Time elapsed: 0.218 s
% 15.34/6.62  % (22565)Instructions burned: 616 (million)
% 15.34/6.62  % (22565)------------------------------
% 15.34/6.62  % (22565)------------------------------
% 15.34/6.63  % (22574)Instruction limit reached!
% 15.34/6.63  % (22574)------------------------------
% 15.34/6.63  % (22574)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.63  % (22574)Termination reason: Unknown
% 15.34/6.63  % (22574)Termination phase: shuffling
% 15.34/6.63  
% 15.34/6.63  % (22574)Memory used [KB]: 17014
% 15.34/6.63  % (22574)Time elapsed: 0.076 s
% 15.34/6.63  % (22574)Instructions burned: 233 (million)
% 15.34/6.63  % (22574)------------------------------
% 15.34/6.63  % (22574)------------------------------
% 15.34/6.63  % (22580)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_22391 for (2986ds/182Mi)
% 15.34/6.63  % (22572)Instruction limit reached!
% 15.34/6.63  % (22572)------------------------------
% 15.34/6.63  % (22572)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.63  % (22572)Termination reason: Unknown
% 15.34/6.63  % (22572)Termination phase: shuffling
% 15.34/6.63  
% 15.34/6.63  % (22572)Memory used [KB]: 17398
% 15.34/6.63  % (22572)Time elapsed: 0.107 s
% 15.34/6.63  % (22572)Instructions burned: 259 (million)
% 15.34/6.63  % (22572)------------------------------
% 15.34/6.63  % (22572)------------------------------
% 15.34/6.64  % (22581)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_22391 for (2986ds/154Mi)
% 15.34/6.64  % (22582)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_22391 for (2986ds/79Mi)
% 15.34/6.67  % (22579)Instruction limit reached!
% 15.34/6.67  % (22579)------------------------------
% 15.34/6.67  % (22579)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.67  % (22579)Termination reason: Unknown
% 15.34/6.67  % (22579)Termination phase: shuffling
% 15.34/6.67  
% 15.34/6.67  % (22579)Memory used [KB]: 16375
% 15.34/6.67  % (22579)Time elapsed: 0.070 s
% 15.34/6.67  % (22579)Instructions burned: 182 (million)
% 15.34/6.67  % (22579)------------------------------
% 15.34/6.67  % (22579)------------------------------
% 15.34/6.67  % (22577)Instruction limit reached!
% 15.34/6.67  % (22577)------------------------------
% 15.34/6.67  % (22577)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.67  % (22577)Termination reason: Unknown
% 15.34/6.67  % (22577)Termination phase: shuffling
% 15.34/6.67  
% 15.34/6.67  % (22577)Memory used [KB]: 19957
% 15.34/6.67  % (22577)Time elapsed: 0.095 s
% 15.34/6.67  % (22577)Instructions burned: 298 (million)
% 15.34/6.67  % (22577)------------------------------
% 15.34/6.67  % (22577)------------------------------
% 15.34/6.68  % (22582)Instruction limit reached!
% 15.34/6.68  % (22582)------------------------------
% 15.34/6.68  % (22582)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.68  % (22582)Termination reason: Unknown
% 15.34/6.68  % (22582)Termination phase: shuffling
% 15.34/6.68  
% 15.34/6.68  % (22582)Memory used [KB]: 14711
% 15.34/6.68  % (22582)Time elapsed: 0.036 s
% 15.34/6.68  % (22582)Instructions burned: 80 (million)
% 15.34/6.68  % (22582)------------------------------
% 15.34/6.68  % (22582)------------------------------
% 15.34/6.68  % (22592)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_22391 for (2985ds/216Mi)
% 15.34/6.68  % (22594)lrs+1002_1:1_fe=off:hud=10:ins=2:sac=on:sp=const_frequency:i=114:si=on:rtra=on_0 on DTF2THF_22391 for (2985ds/114Mi)
% 15.34/6.69  % (22596)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_22391 for (2985ds/922Mi)
% 15.34/6.69  % (22581)Instruction limit reached!
% 15.34/6.69  % (22581)------------------------------
% 15.34/6.69  % (22581)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 15.34/6.69  % (22581)Termination reason: Unknown
% 15.34/6.69  % (22581)Termination phase: shuffling
% 15.34/6.69  
% 15.34/6.69  % (22581)Memory used [KB]: 15863
% 15.34/6.69  % (22581)Time elapsed: 0.053 s
% 15.34/6.69  % (22581)Instructions burned: 154 (million)
% 15.34/6.69  % (22581)------------------------------
% 15.34/6.69  % (22581)------------------------------
% 16.07/6.70  % (22580)Instruction limit reached!
% 16.07/6.70  % (22580)------------------------------
% 16.07/6.70  % (22580)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.70  % (22580)Termination reason: Unknown
% 16.07/6.70  % (22580)Termination phase: shuffling
% 16.07/6.70  
% 16.07/6.70  % (22580)Memory used [KB]: 16375
% 16.07/6.70  % (22580)Time elapsed: 0.071 s
% 16.07/6.70  % (22580)Instructions burned: 184 (million)
% 16.07/6.70  % (22580)------------------------------
% 16.07/6.70  % (22580)------------------------------
% 16.07/6.70  % (22602)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_22391 for (2985ds/182Mi)
% 16.07/6.71  % (22607)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_22391 for (2985ds/164Mi)
% 16.07/6.73  % (22594)Instruction limit reached!
% 16.07/6.73  % (22594)------------------------------
% 16.07/6.73  % (22594)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.73  % (22594)Termination reason: Unknown
% 16.07/6.73  % (22594)Termination phase: shuffling
% 16.07/6.73  
% 16.07/6.73  % (22594)Memory used [KB]: 15223
% 16.07/6.73  % (22594)Time elapsed: 0.048 s
% 16.07/6.73  % (22594)Instructions burned: 114 (million)
% 16.07/6.73  % (22594)------------------------------
% 16.07/6.73  % (22594)------------------------------
% 16.07/6.73  % (22575)Instruction limit reached!
% 16.07/6.73  % (22575)------------------------------
% 16.07/6.73  % (22575)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.73  % (22575)Termination reason: Unknown
% 16.07/6.73  % (22575)Termination phase: shuffling
% 16.07/6.73  
% 16.07/6.73  % (22575)Memory used [KB]: 20212
% 16.07/6.73  % (22575)Time elapsed: 0.166 s
% 16.07/6.73  % (22575)Instructions burned: 460 (million)
% 16.07/6.73  % (22575)------------------------------
% 16.07/6.73  % (22575)------------------------------
% 16.07/6.74  % (22621)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_22391 for (2985ds/50Mi)
% 16.07/6.75  % (22624)dis-30_1:1_acc=on:bd=off:fsr=off:plsq=on:plsqr=32,1:i=368:si=on:rtra=on_0 on DTF2THF_22391 for (2985ds/368Mi)
% 16.07/6.75  % (22592)Instruction limit reached!
% 16.07/6.75  % (22592)------------------------------
% 16.07/6.75  % (22592)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.75  % (22592)Termination reason: Unknown
% 16.07/6.75  % (22592)Termination phase: shuffling
% 16.07/6.75  
% 16.07/6.75  % (22592)Memory used [KB]: 16758
% 16.07/6.75  % (22592)Time elapsed: 0.072 s
% 16.07/6.75  % (22592)Instructions burned: 217 (million)
% 16.07/6.75  % (22592)------------------------------
% 16.07/6.75  % (22592)------------------------------
% 16.07/6.76  % (22602)Instruction limit reached!
% 16.07/6.76  % (22602)------------------------------
% 16.07/6.76  % (22602)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.76  % (22602)Termination reason: Unknown
% 16.07/6.76  % (22602)Termination phase: shuffling
% 16.07/6.76  
% 16.07/6.76  % (22602)Memory used [KB]: 16247
% 16.07/6.76  % (22602)Time elapsed: 0.063 s
% 16.07/6.76  % (22602)Instructions burned: 185 (million)
% 16.07/6.76  % (22602)------------------------------
% 16.07/6.76  % (22602)------------------------------
% 16.07/6.76  % (22630)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_22391 for (2984ds/1021Mi)
% 16.07/6.77  % (22621)Instruction limit reached!
% 16.07/6.77  % (22621)------------------------------
% 16.07/6.77  % (22621)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.77  % (22621)Termination reason: Unknown
% 16.07/6.77  % (22621)Termination phase: shuffling
% 16.07/6.77  
% 16.07/6.77  % (22621)Memory used [KB]: 14456
% 16.07/6.77  % (22621)Time elapsed: 0.024 s
% 16.07/6.77  % (22621)Instructions burned: 52 (million)
% 16.07/6.77  % (22621)------------------------------
% 16.07/6.77  % (22621)------------------------------
% 16.07/6.77  % (22607)Instruction limit reached!
% 16.07/6.77  % (22607)------------------------------
% 16.07/6.77  % (22607)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.77  % (22607)Termination reason: Unknown
% 16.07/6.77  % (22607)Termination phase: shuffling
% 16.07/6.77  
% 16.07/6.77  % (22607)Memory used [KB]: 15991
% 16.07/6.77  % (22607)Time elapsed: 0.056 s
% 16.07/6.77  % (22607)Instructions burned: 164 (million)
% 16.07/6.77  % (22607)------------------------------
% 16.07/6.77  % (22607)------------------------------
% 16.07/6.77  % (22635)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_22391 for (2984ds/267Mi)
% 16.07/6.78  % (22637)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_22391 for (2984ds/204Mi)
% 16.07/6.78  % (22638)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_22391 for (2984ds/26Mi)
% 16.07/6.79  % (22638)Instruction limit reached!
% 16.07/6.79  % (22638)------------------------------
% 16.07/6.79  % (22638)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.07/6.79  % (22638)Termination reason: Unknown
% 16.07/6.79  % (22638)Termination phase: shuffling
% 16.07/6.79  
% 16.07/6.79  % (22638)Memory used [KB]: 14072
% 16.07/6.79  % (22638)Time elapsed: 0.017 s
% 16.07/6.79  % (22638)Instructions burned: 26 (million)
% 16.07/6.79  % (22638)------------------------------
% 16.07/6.79  % (22638)------------------------------
% 16.67/6.80  % (22649)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_22391 for (2984ds/53Mi)
% 16.67/6.83  % (22649)Instruction limit reached!
% 16.67/6.83  % (22649)------------------------------
% 16.67/6.83  % (22649)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.67/6.83  % (22649)Termination reason: Unknown
% 16.67/6.83  % (22649)Termination phase: shuffling
% 16.67/6.83  
% 16.67/6.83  % (22649)Memory used [KB]: 14456
% 16.67/6.83  % (22649)Time elapsed: 0.025 s
% 16.67/6.83  % (22649)Instructions burned: 56 (million)
% 16.67/6.83  % (22649)------------------------------
% 16.67/6.83  % (22649)------------------------------
% 16.67/6.84  % (22661)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_22391 for (2984ds/4436Mi)
% 16.67/6.85  % (22637)Instruction limit reached!
% 16.67/6.85  % (22637)------------------------------
% 16.67/6.85  % (22637)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.67/6.85  % (22637)Termination reason: Unknown
% 16.67/6.85  % (22637)Termination phase: shuffling
% 16.67/6.85  
% 16.67/6.85  % (22637)Memory used [KB]: 16630
% 16.67/6.85  % (22637)Time elapsed: 0.078 s
% 16.67/6.85  % (22637)Instructions burned: 206 (million)
% 16.67/6.85  % (22637)------------------------------
% 16.67/6.85  % (22637)------------------------------
% 16.67/6.86  % (22635)Instruction limit reached!
% 16.67/6.86  % (22635)------------------------------
% 16.67/6.86  % (22635)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.67/6.86  % (22635)Termination reason: Unknown
% 16.67/6.86  % (22635)Termination phase: shuffling
% 16.67/6.86  
% 16.67/6.86  % (22635)Memory used [KB]: 19573
% 16.67/6.86  % (22635)Time elapsed: 0.087 s
% 16.67/6.86  % (22635)Instructions burned: 267 (million)
% 16.67/6.86  % (22635)------------------------------
% 16.67/6.86  % (22635)------------------------------
% 16.67/6.87  % (22675)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_22391 for (2983ds/1539Mi)
% 16.67/6.87  % (22681)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_22391 for (2983ds/408Mi)
% 16.67/6.88  % (22624)Instruction limit reached!
% 16.67/6.88  % (22624)------------------------------
% 16.67/6.88  % (22624)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 16.67/6.88  % (22624)Termination reason: Unknown
% 16.67/6.88  % (22624)Termination phase: shuffling
% 16.67/6.88  
% 16.67/6.88  % (22624)Memory used [KB]: 19061
% 16.67/6.88  % (22624)Time elapsed: 0.134 s
% 16.67/6.88  % (22624)Instructions burned: 369 (million)
% 16.67/6.88  % (22624)------------------------------
% 16.67/6.88  % (22624)------------------------------
% 16.67/6.89  % (22684)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_22391 for (2983ds/5205Mi)
% 17.03/6.90  % (22578)Instruction limit reached!
% 17.03/6.90  % (22578)------------------------------
% 17.03/6.90  % (22578)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.03/6.90  % (22578)Termination reason: Unknown
% 17.03/6.90  % (22578)Termination phase: shuffling
% 17.03/6.90  
% 17.03/6.90  % (22578)Memory used [KB]: 33517
% 17.03/6.90  % (22578)Time elapsed: 0.301 s
% 17.03/6.90  % (22578)Instructions burned: 988 (million)
% 17.03/6.90  % (22578)------------------------------
% 17.03/6.90  % (22578)------------------------------
% 17.03/6.92  % (22704)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_22391 for (2983ds/1913Mi)
% 17.03/7.01  % (22596)Instruction limit reached!
% 17.03/7.01  % (22596)------------------------------
% 17.03/7.01  % (22596)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.03/7.01  % (22596)Termination reason: Unknown
% 17.03/7.01  % (22596)Termination phase: Property scanning
% 17.03/7.01  
% 17.03/7.01  % (22596)Memory used [KB]: 27121
% 17.03/7.01  % (22596)Time elapsed: 0.322 s
% 17.03/7.01  % (22596)Instructions burned: 923 (million)
% 17.03/7.01  % (22596)------------------------------
% 17.03/7.01  % (22596)------------------------------
% 17.03/7.02  % (22681)Instruction limit reached!
% 17.03/7.02  % (22681)------------------------------
% 17.03/7.02  % (22681)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 17.03/7.02  % (22681)Termination reason: Unknown
% 17.03/7.02  % (22681)Termination phase: shuffling
% 17.03/7.02  
% 17.03/7.02  % (22681)Memory used [KB]: 19701
% 17.03/7.02  % (22681)Time elapsed: 0.150 s
% 17.03/7.02  % (22681)Instructions burned: 409 (million)
% 17.03/7.02  % (22681)------------------------------
% 17.03/7.02  % (22681)------------------------------
% 17.03/7.02  % (22747)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_22391 for (2982ds/4350Mi)
% 18.68/7.03  % (22751)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_22391 for (2982ds/68Mi)
% 18.68/7.06  % (22751)Instruction limit reached!
% 18.68/7.06  % (22751)------------------------------
% 18.68/7.06  % (22751)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/7.06  % (22751)Termination reason: Unknown
% 18.68/7.06  % (22751)Termination phase: shuffling
% 18.68/7.06  
% 18.68/7.06  % (22751)Memory used [KB]: 14583
% 18.68/7.06  % (22751)Time elapsed: 0.032 s
% 18.68/7.06  % (22751)Instructions burned: 70 (million)
% 18.68/7.06  % (22751)------------------------------
% 18.68/7.06  % (22751)------------------------------
% 18.68/7.07  % (22768)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_22391 for (2982ds/358Mi)
% 18.68/7.11  % (22630)Instruction limit reached!
% 18.68/7.11  % (22630)------------------------------
% 18.68/7.11  % (22630)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 18.68/7.11  % (22630)Termination reason: Unknown
% 18.68/7.11  % (22630)Termination phase: Property scanning
% 18.68/7.11  
% 18.68/7.11  % (22630)Memory used [KB]: 27121
% 18.68/7.11  % (22630)Time elapsed: 0.347 s
% 18.68/7.11  % (22630)Instructions burned: 1024 (million)
% 18.68/7.11  % (22630)------------------------------
% 18.68/7.11  % (22630)------------------------------
% 18.68/7.12  % (22779)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_22391 for (2981ds/31Mi)
% 19.24/7.14  % (22779)Instruction limit reached!
% 19.24/7.14  % (22779)------------------------------
% 19.24/7.14  % (22779)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.24/7.14  % (22779)Termination reason: Unknown
% 19.24/7.14  % (22779)Termination phase: shuffling
% 19.24/7.14  
% 19.24/7.14  % (22779)Memory used [KB]: 14072
% 19.24/7.14  % (22779)Time elapsed: 0.019 s
% 19.24/7.14  % (22779)Instructions burned: 31 (million)
% 19.24/7.14  % (22779)------------------------------
% 19.24/7.14  % (22779)------------------------------
% 19.24/7.15  % (22785)lrs+10_1:1_cnfonf=lazy_simp:sgt=5:sos=on:ss=axioms:i=279:si=on:rtra=on_0 on DTF2THF_22391 for (2981ds/279Mi)
% 19.55/7.20  % (22768)Instruction limit reached!
% 19.55/7.20  % (22768)------------------------------
% 19.55/7.20  % (22768)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.55/7.20  % (22768)Termination reason: Unknown
% 19.55/7.20  % (22768)Termination phase: shuffling
% 19.55/7.20  
% 19.55/7.20  % (22768)Memory used [KB]: 18933
% 19.55/7.20  % (22768)Time elapsed: 0.128 s
% 19.55/7.20  % (22768)Instructions burned: 360 (million)
% 19.55/7.20  % (22768)------------------------------
% 19.55/7.20  % (22768)------------------------------
% 19.55/7.21  % (22801)lrs+10_8:1_au=on:fde=none:slsq=on:i=258:si=on:rtra=on_0 on DTF2THF_22391 for (2980ds/258Mi)
% 19.55/7.24  % (22785)Instruction limit reached!
% 19.55/7.24  % (22785)------------------------------
% 19.55/7.24  % (22785)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.55/7.24  % (22785)Termination reason: Unknown
% 19.55/7.24  % (22785)Termination phase: shuffling
% 19.55/7.24  
% 19.55/7.24  % (22785)Memory used [KB]: 19701
% 19.55/7.24  % (22785)Time elapsed: 0.089 s
% 19.55/7.24  % (22785)Instructions burned: 281 (million)
% 19.55/7.24  % (22785)------------------------------
% 19.55/7.24  % (22785)------------------------------
% 19.55/7.25  % (22824)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_22391 for (2980ds/26Mi)
% 19.55/7.27  % (22824)Instruction limit reached!
% 19.55/7.27  % (22824)------------------------------
% 19.55/7.27  % (22824)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 19.55/7.27  % (22824)Termination reason: Unknown
% 19.55/7.27  % (22824)Termination phase: shuffling
% 19.55/7.27  
% 19.55/7.27  % (22824)Memory used [KB]: 14072
% 19.55/7.27  % (22824)Time elapsed: 0.017 s
% 19.55/7.27  % (22824)Instructions burned: 26 (million)
% 19.55/7.27  % (22824)------------------------------
% 19.55/7.27  % (22824)------------------------------
% 19.55/7.28  % (22835)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_22391 for (2980ds/995Mi)
% 20.23/7.31  % (22801)Instruction limit reached!
% 20.23/7.31  % (22801)------------------------------
% 20.23/7.31  % (22801)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.23/7.31  % (22801)Termination reason: Unknown
% 20.23/7.31  % (22801)Termination phase: shuffling
% 20.23/7.31  
% 20.23/7.31  % (22801)Memory used [KB]: 17398
% 20.23/7.31  % (22801)Time elapsed: 0.096 s
% 20.23/7.31  % (22801)Instructions burned: 260 (million)
% 20.23/7.31  % (22801)------------------------------
% 20.23/7.31  % (22801)------------------------------
% 20.23/7.32  % (22837)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_22391 for (2979ds/373Mi)
% 20.23/7.36  % (22675)Instruction limit reached!
% 20.23/7.36  % (22675)------------------------------
% 20.23/7.36  % (22675)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.23/7.36  % (22675)Termination reason: Unknown
% 20.23/7.36  % (22675)Termination phase: SInE selection
% 20.23/7.36  
% 20.23/7.36  % (22675)Memory used [KB]: 27249
% 20.23/7.36  % (22675)Time elapsed: 0.491 s
% 20.23/7.36  % (22675)Instructions burned: 1540 (million)
% 20.23/7.36  % (22675)------------------------------
% 20.23/7.36  % (22675)------------------------------
% 20.23/7.36  % (22661)First to succeed.
% 20.23/7.37  % (22838)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_22391 for (2978ds/253Mi)
% 20.23/7.37  % (22661)Refutation found. Thanks to Tanya!
% 20.23/7.37  % SZS status Theorem for DTF2THF_22391
% 20.23/7.37  % SZS output start Proof for DTF2THF_22391
% 20.23/7.37  thf(type_def_5, type, rat: $tType).
% 20.23/7.37  thf(type_def_6, type, int: $tType).
% 20.23/7.37  thf(type_def_8, type, real: $tType).
% 20.23/7.37  thf(type_def_9, type, set_Pr1261947904930325089at_nat: $tType).
% 20.23/7.37  thf(type_def_10, type, nat: $tType).
% 20.23/7.37  thf(type_def_11, type, complex: $tType).
% 20.23/7.37  thf(type_def_12, type, product_prod_int_int: $tType).
% 20.23/7.37  thf(type_def_13, type, num: $tType).
% 20.23/7.37  thf(type_def_14, type, option_num: $tType).
% 20.23/7.37  thf(type_def_15, type, code_integer: $tType).
% 20.23/7.37  thf(type_def_16, type, produc6271795597528267376eger_o: $tType).
% 20.23/7.37  thf(type_def_17, type, produc8923325533196201883nteger: $tType).
% 20.23/7.37  thf(type_def_18, type, set_fi4554929511873752355omplex: $tType).
% 20.23/7.37  thf(type_def_19, type, filter6041513312241820739omplex: $tType).
% 20.23/7.37  thf(type_def_20, type, set_fi7789364187291644575l_real: $tType).
% 20.23/7.37  thf(type_def_21, type, filter2146258269922977983l_real: $tType).
% 20.23/7.37  thf(type_def_22, type, set_nat: $tType).
% 20.23/7.37  thf(type_def_23, type, set_set_nat: $tType).
% 20.23/7.37  thf(type_def_24, type, filter_real: $tType).
% 20.23/7.37  thf(type_def_25, type, product_prod_nat_nat: $tType).
% 20.23/7.37  thf(type_def_26, type, filter_nat: $tType).
% 20.23/7.37  thf(type_def_27, type, filter1242075044329608583at_nat: $tType).
% 20.23/7.37  thf(type_def_28, type, set_Pr5085853215250843933omplex: $tType).
% 20.23/7.37  thf(type_def_29, type, set_Pr6218003697084177305l_real: $tType).
% 20.23/7.37  thf(type_def_30, type, set_complex: $tType).
% 20.23/7.37  thf(type_def_31, type, set_list_nat: $tType).
% 20.23/7.37  thf(type_def_32, type, set_o: $tType).
% 20.23/7.37  thf(type_def_33, type, set_int: $tType).
% 20.23/7.37  thf(type_def_34, type, set_list_o: $tType).
% 20.23/7.37  thf(type_def_35, type, set_list_complex: $tType).
% 20.23/7.37  thf(type_def_36, type, set_list_int: $tType).
% 20.23/7.37  thf(type_def_37, type, set_li5450038453877631591at_nat: $tType).
% 20.23/7.37  thf(type_def_38, type, set_list_VEBT_VEBT: $tType).
% 20.23/7.37  thf(type_def_39, type, set_num: $tType).
% 20.23/7.37  thf(type_def_40, type, set_Pr4329608150637261639at_nat: $tType).
% 20.23/7.37  thf(type_def_41, type, set_rat: $tType).
% 20.23/7.37  thf(type_def_42, type, set_real: $tType).
% 20.23/7.37  thf(type_def_43, type, set_set_complex: $tType).
% 20.23/7.37  thf(type_def_44, type, set_set_int: $tType).
% 20.23/7.37  thf(type_def_45, type, set_se7855581050983116737at_nat: $tType).
% 20.23/7.37  thf(type_def_46, type, set_VEBT_VEBT: $tType).
% 20.23/7.37  thf(type_def_47, type, set_Pr8693737435421807431at_nat: $tType).
% 20.23/7.37  thf(type_def_48, type, produc1319942482725812455at_nat: $tType).
% 20.23/7.37  thf(type_def_49, type, list_nat: $tType).
% 20.23/7.37  thf(type_def_50, type, extended_enat: $tType).
% 20.23/7.37  thf(type_def_51, type, produc3843707927480180839at_nat: $tType).
% 20.23/7.37  thf(type_def_52, type, vEBT_VEBT: $tType).
% 20.23/7.37  thf(type_def_53, type, list_o: $tType).
% 20.23/7.37  thf(type_def_54, type, list_int: $tType).
% 20.23/7.37  thf(type_def_55, type, set_Pr958786334691620121nt_int: $tType).
% 20.23/7.37  thf(type_def_56, type, list_complex: $tType).
% 20.23/7.37  thf(type_def_57, type, list_P6011104703257516679at_nat: $tType).
% 20.23/7.37  thf(type_def_58, type, list_real: $tType).
% 20.23/7.37  thf(type_def_59, type, list_set_nat: $tType).
% 20.23/7.37  thf(type_def_60, type, list_VEBT_VEBT: $tType).
% 20.23/7.37  thf(type_def_61, type, list_P7333126701944960589_nat_o: $tType).
% 20.23/7.37  thf(type_def_62, type, list_P3521021558325789923at_int: $tType).
% 20.23/7.37  thf(type_def_63, type, list_P5647936690300460905T_VEBT: $tType).
% 20.23/7.37  thf(type_def_64, type, option_o: $tType).
% 20.23/7.37  thf(type_def_65, type, option_int: $tType).
% 20.23/7.37  thf(type_def_66, type, option_nat: $tType).
% 20.23/7.37  thf(type_def_67, type, list_num: $tType).
% 20.23/7.37  thf(type_def_68, type, option4927543243414619207at_nat: $tType).
% 20.23/7.37  thf(type_def_69, type, option_real: $tType).
% 20.23/7.37  thf(type_def_70, type, option_set_nat: $tType).
% 20.23/7.37  thf(type_def_71, type, option_VEBT_VEBT: $tType).
% 20.23/7.37  thf(type_def_72, type, list_C878401137130745250e_term: $tType).
% 20.23/7.37  thf(type_def_73, type, set_Co9149898834107579976e_term: $tType).
% 20.23/7.37  thf(type_def_74, type, list_i8448526496819171953e_term: $tType).
% 20.23/7.37  thf(type_def_75, type, set_in3461395444621081367e_term: $tType).
% 20.23/7.37  thf(type_def_76, type, list_P1316552470764441098e_term: $tType).
% 20.23/7.37  thf(type_def_77, type, set_Pr7604974323444597168e_term: $tType).
% 20.23/7.37  thf(type_def_78, type, list_P1743416141875011707e_term: $tType).
% 20.23/7.37  thf(type_def_79, type, set_Pr3642885161833720865e_term: $tType).
% 20.23/7.37  thf(type_def_80, type, list_list_nat: $tType).
% 20.23/7.37  thf(type_def_81, type, list_P5311841565141990158nteger: $tType).
% 20.23/7.37  thf(type_def_82, type, set_Pr8056137968301705908nteger: $tType).
% 20.23/7.37  thf(type_def_83, type, list_P8915022641806594461nt_int: $tType).
% 20.23/7.37  thf(type_def_84, type, set_Pr1872883991513573699nt_int: $tType).
% 20.23/7.37  thf(type_def_85, type, list_P7828571989066258726nteger: $tType).
% 20.23/7.37  thf(type_def_86, type, set_Pr1281608226676607948nteger: $tType).
% 20.23/7.37  thf(type_def_87, type, list_P651320350408439699nt_int: $tType).
% 20.23/7.37  thf(type_def_88, type, set_Pr9222295170931077689nt_int: $tType).
% 20.23/7.37  thf(type_def_89, type, list_P4002435161011370285od_o_o: $tType).
% 20.23/7.37  thf(type_def_90, type, set_Product_prod_o_o: $tType).
% 20.23/7.37  thf(type_def_91, type, list_P3795440434834930179_o_int: $tType).
% 20.23/7.37  thf(type_def_92, type, set_Pr8834758594704517033_o_int: $tType).
% 20.23/7.37  thf(type_def_93, type, list_P6285523579766656935_o_nat: $tType).
% 20.23/7.37  thf(type_def_94, type, set_Pr2101469702781467981_o_nat: $tType).
% 20.23/7.37  thf(type_def_95, type, list_P5232166724548748803o_real: $tType).
% 20.23/7.37  thf(type_def_96, type, set_Pr6573716822653411497o_real: $tType).
% 20.23/7.37  thf(type_def_97, type, list_P7495141550334521929T_VEBT: $tType).
% 20.23/7.37  thf(type_def_98, type, set_Pr7543698050874017315T_VEBT: $tType).
% 20.23/7.37  thf(type_def_99, type, list_P5578671422887162913nteger: $tType).
% 20.23/7.37  thf(type_def_100, type, set_Pr4811707699266497531nteger: $tType).
% 20.23/7.37  thf(type_def_101, type, list_P5707943133018811711nt_int: $tType).
% 20.23/7.37  thf(type_def_102, type, list_P6863124054624500543t_real: $tType).
% 20.23/7.37  thf(type_def_103, type, set_Pr3538720872664544793t_real: $tType).
% 20.23/7.37  thf(type_def_104, type, list_P7524865323317820941T_VEBT: $tType).
% 20.23/7.37  thf(type_def_105, type, set_Pr8044002425091019955T_VEBT: $tType).
% 20.23/7.37  thf(type_def_106, type, list_P8469869581646625389at_nat: $tType).
% 20.23/7.37  thf(type_def_107, type, list_P3595434254542482545real_o: $tType).
% 20.23/7.37  thf(type_def_108, type, set_Pr4936984352647145239real_o: $tType).
% 20.23/7.37  thf(type_def_109, type, list_P4344331454722006975al_int: $tType).
% 20.23/7.37  thf(type_def_110, type, set_Pr1019928272762051225al_int: $tType).
% 20.23/7.37  thf(type_def_111, type, list_P6834414599653733731al_nat: $tType).
% 20.23/7.37  thf(type_def_112, type, set_Pr3510011417693777981al_nat: $tType).
% 20.23/7.37  thf(type_def_113, type, list_P8689742595348180415l_real: $tType).
% 20.23/7.37  thf(type_def_114, type, list_P877281246627933069T_VEBT: $tType).
% 20.23/7.37  thf(type_def_115, type, set_Pr6019664923565264691T_VEBT: $tType).
% 20.23/7.37  thf(type_def_116, type, list_P6254988961118846195et_nat: $tType).
% 20.23/7.37  thf(type_def_117, type, set_Pr5488025237498180813et_nat: $tType).
% 20.23/7.37  thf(type_def_118, type, list_P5464809261938338413at_nat: $tType).
% 20.23/7.37  thf(type_def_119, type, list_P3126845725202233233VEBT_o: $tType).
% 20.23/7.37  thf(type_def_120, type, set_Pr3175402225741728619VEBT_o: $tType).
% 20.23/7.37  thf(type_def_121, type, list_P4547456442757143711BT_int: $tType).
% 20.23/7.37  thf(type_def_122, type, set_Pr5066593544530342725BT_int: $tType).
% 20.23/7.37  thf(type_def_123, type, list_P7037539587688870467BT_nat: $tType).
% 20.23/7.37  thf(type_def_124, type, set_Pr7556676689462069481BT_nat: $tType).
% 20.23/7.37  thf(type_def_125, type, list_P2623026923184700063T_real: $tType).
% 20.23/7.37  thf(type_def_126, type, set_Pr7765410600122031685T_real: $tType).
% 20.23/7.37  thf(type_def_127, type, list_P7413028617227757229T_VEBT: $tType).
% 20.23/7.37  thf(type_def_128, type, set_Pr6192946355708809607T_VEBT: $tType).
% 20.23/7.37  thf(type_def_129, type, option6357759511663192854e_term: $tType).
% 20.23/7.37  thf(type_def_130, type, produc6241069584506657477e_term: $tType).
% 20.23/7.37  thf(type_def_131, type, produc8551481072490612790e_term: $tType).
% 20.23/7.37  thf(type_def_132, type, produc8763457246119570046nteger: $tType).
% 20.23/7.37  thf(type_def_133, type, produc7773217078559923341nt_int: $tType).
% 20.23/7.37  thf(type_def_134, type, produc1908205239877642774nteger: $tType).
% 20.23/7.37  thf(type_def_135, type, produc2285326912895808259nt_int: $tType).
% 20.23/7.37  thf(type_def_136, type, set_Pr6227168374412355847list_o: $tType).
% 20.23/7.37  thf(type_def_137, type, set_Pr765067013931698361st_int: $tType).
% 20.23/7.37  thf(type_def_138, type, set_Pr3451248702717554689st_nat: $tType).
% 20.23/7.37  thf(type_def_139, type, set_Pr1916528119006554503T_VEBT: $tType).
% 20.23/7.37  thf(type_def_140, type, product_prod_o_o: $tType).
% 20.23/7.37  thf(type_def_141, type, product_prod_o_int: $tType).
% 20.23/7.37  thf(type_def_142, type, product_prod_o_nat: $tType).
% 20.23/7.37  thf(type_def_143, type, produc2504756804600209347T_VEBT: $tType).
% 20.23/7.37  thf(type_def_144, type, product_prod_nat_o: $tType).
% 20.23/7.37  thf(type_def_145, type, product_prod_nat_int: $tType).
% 20.23/7.37  thf(type_def_146, type, produc8025551001238799321T_VEBT: $tType).
% 20.23/7.37  thf(type_def_147, type, produc334124729049499915VEBT_o: $tType).
% 20.23/7.37  thf(type_def_148, type, produc4894624898956917775BT_int: $tType).
% 20.23/7.37  thf(type_def_149, type, produc9072475918466114483BT_nat: $tType).
% 20.23/7.37  thf(type_def_150, type, produc8243902056947475879T_VEBT: $tType).
% 20.23/7.37  thf(type_def_151, type, set_Code_integer: $tType).
% 20.23/7.37  thf(type_def_152, type, produc4953844613479565601on_nat: $tType).
% 20.23/7.37  thf(type_def_153, type, produc2233624965454879586on_nat: $tType).
% 20.23/7.37  thf(type_def_154, type, produc8306885398267862888on_nat: $tType).
% 20.23/7.37  thf(type_def_155, type, produc3447558737645232053on_num: $tType).
% 20.23/7.37  thf(type_def_156, type, produc7036089656553540234on_num: $tType).
% 20.23/7.37  thf(type_def_157, type, produc1193250871479095198on_num: $tType).
% 20.23/7.37  thf(type_def_158, type, produc6121120109295599847at_nat: $tType).
% 20.23/7.37  thf(type_def_159, type, produc5491161045314408544at_nat: $tType).
% 20.23/7.37  thf(type_def_160, type, produc5542196010084753463at_nat: $tType).
% 20.23/7.37  thf(type_def_161, type, product_prod_o_real: $tType).
% 20.23/7.37  thf(type_def_162, type, produc679980390762269497t_real: $tType).
% 20.23/7.37  thf(type_def_163, type, produc1531783533982839933T_VEBT: $tType).
% 20.23/7.37  thf(type_def_164, type, produc7102631898165422375list_o: $tType).
% 20.23/7.37  thf(type_def_165, type, produc1186641810826059865st_int: $tType).
% 20.23/7.37  thf(type_def_166, type, produc1828647624359046049st_nat: $tType).
% 20.23/7.37  thf(type_def_167, type, produc9211091688327510695T_VEBT: $tType).
% 20.23/7.37  thf(type_def_168, type, product_prod_nat_num: $tType).
% 20.23/7.37  thf(type_def_169, type, product_prod_num_num: $tType).
% 20.23/7.37  thf(type_def_170, type, produc859450856879609959at_nat: $tType).
% 20.23/7.37  thf(type_def_171, type, product_prod_real_o: $tType).
% 20.23/7.37  thf(type_def_172, type, produc8786904178792722361al_int: $tType).
% 20.23/7.37  thf(type_def_173, type, produc3741383161447143261al_nat: $tType).
% 20.23/7.37  thf(type_def_174, type, produc2422161461964618553l_real: $tType).
% 20.23/7.37  thf(type_def_175, type, produc3757001726724277373T_VEBT: $tType).
% 20.23/7.37  thf(type_def_176, type, produc7819656566062154093et_nat: $tType).
% 20.23/7.37  thf(type_def_177, type, produc5170161368751668367T_real: $tType).
% 20.23/7.37  thf(type_def_178, type, produc4411394909380815293omplex: $tType).
% 20.23/7.37  thf(func_def_0, type, produc1319942482725812455at_nat: $tType).
% 20.23/7.37  thf(func_def_1, type, produc5542196010084753463at_nat: $tType).
% 20.23/7.37  thf(func_def_2, type, list_P7828571989066258726nteger: $tType).
% 20.23/7.37  thf(func_def_3, type, set_Pr1281608226676607948nteger: $tType).
% 20.23/7.37  thf(func_def_4, type, produc1908205239877642774nteger: $tType).
% 20.23/7.37  thf(func_def_5, type, produc5491161045314408544at_nat: $tType).
% 20.23/7.37  thf(func_def_6, type, list_P651320350408439699nt_int: $tType).
% 20.23/7.37  thf(func_def_7, type, set_Pr9222295170931077689nt_int: $tType).
% 20.23/7.37  thf(func_def_8, type, produc2285326912895808259nt_int: $tType).
% 20.23/7.37  thf(func_def_9, type, list_P5311841565141990158nteger: $tType).
% 20.23/7.37  thf(func_def_10, type, set_Pr8056137968301705908nteger: $tType).
% 20.23/7.37  thf(func_def_11, type, produc8763457246119570046nteger: $tType).
% 20.23/7.37  thf(func_def_12, type, list_P1316552470764441098e_term: $tType).
% 20.23/7.37  thf(func_def_13, type, set_Pr7604974323444597168e_term: $tType).
% 20.23/7.37  thf(func_def_14, type, list_P1743416141875011707e_term: $tType).
% 20.23/7.37  thf(func_def_15, type, set_Pr3642885161833720865e_term: $tType).
% 20.23/7.37  thf(func_def_16, type, list_P8915022641806594461nt_int: $tType).
% 20.23/7.37  thf(func_def_17, type, set_Pr1872883991513573699nt_int: $tType).
% 20.23/7.37  thf(func_def_18, type, produc7773217078559923341nt_int: $tType).
% 20.23/7.37  thf(func_def_19, type, list_P5464809261938338413at_nat: $tType).
% 20.23/7.37  thf(func_def_20, type, set_Pr4329608150637261639at_nat: $tType).
% 20.23/7.37  thf(func_def_21, type, produc1193250871479095198on_num: $tType).
% 20.23/7.37  thf(func_def_22, type, produc8306885398267862888on_nat: $tType).
% 20.23/7.37  thf(func_def_23, type, produc6121120109295599847at_nat: $tType).
% 20.23/7.37  thf(func_def_24, type, produc7036089656553540234on_num: $tType).
% 20.23/7.37  thf(func_def_25, type, produc2233624965454879586on_nat: $tType).
% 20.23/7.37  thf(func_def_26, type, produc3843707927480180839at_nat: $tType).
% 20.23/7.37  thf(func_def_27, type, list_P8469869581646625389at_nat: $tType).
% 20.23/7.37  thf(func_def_28, type, list_C878401137130745250e_term: $tType).
% 20.23/7.37  thf(func_def_29, type, set_Pr8693737435421807431at_nat: $tType).
% 20.23/7.37  thf(func_def_30, type, set_Co9149898834107579976e_term: $tType).
% 20.23/7.37  thf(func_def_31, type, list_i8448526496819171953e_term: $tType).
% 20.23/7.37  thf(func_def_32, type, set_Pr1916528119006554503T_VEBT: $tType).
% 20.23/7.37  thf(func_def_33, type, produc859450856879609959at_nat: $tType).
% 20.23/7.37  thf(func_def_34, type, set_in3461395444621081367e_term: $tType).
% 20.23/7.37  thf(func_def_35, type, produc9211091688327510695T_VEBT: $tType).
% 20.23/7.37  thf(func_def_36, type, produc6241069584506657477e_term: $tType).
% 20.23/7.37  thf(func_def_37, type, set_fi4554929511873752355omplex: $tType).
% 20.23/7.37  thf(func_def_38, type, set_Pr3451248702717554689st_nat: $tType).
% 20.23/7.37  thf(func_def_39, type, set_Pr765067013931698361st_int: $tType).
% 20.23/7.37  thf(func_def_40, type, list_P7413028617227757229T_VEBT: $tType).
% 20.23/7.37  thf(func_def_41, type, produc8551481072490612790e_term: $tType).
% 20.23/7.37  thf(func_def_42, type, set_Pr6192946355708809607T_VEBT: $tType).
% 20.23/7.37  thf(func_def_43, type, list_P6254988961118846195et_nat: $tType).
% 20.23/7.37  thf(func_def_44, type, list_P5578671422887162913nteger: $tType).
% 20.23/7.37  thf(func_def_45, type, set_Pr5488025237498180813et_nat: $tType).
% 20.23/7.37  thf(func_def_46, type, set_Pr4811707699266497531nteger: $tType).
% 20.23/7.37  thf(func_def_47, type, option6357759511663192854e_term: $tType).
% 20.23/7.37  thf(func_def_48, type, produc3447558737645232053on_num: $tType).
% 20.23/7.37  thf(func_def_49, type, produc4953844613479565601on_nat: $tType).
% 20.23/7.37  thf(func_def_50, type, set_fi7789364187291644575l_real: $tType).
% 20.23/7.37  thf(func_def_51, type, filter6041513312241820739omplex: $tType).
% 20.23/7.37  thf(func_def_52, type, list_P2623026923184700063T_real: $tType).
% 20.23/7.37  thf(func_def_53, type, list_P877281246627933069T_VEBT: $tType).
% 20.23/7.37  thf(func_def_54, type, produc1828647624359046049st_nat: $tType).
% 20.23/7.37  thf(func_def_55, type, produc1186641810826059865st_int: $tType).
% 20.23/7.37  thf(func_def_56, type, set_Pr7765410600122031685T_real: $tType).
% 20.23/7.37  thf(func_def_57, type, set_Pr6019664923565264691T_VEBT: $tType).
% 20.23/7.37  thf(func_def_58, type, list_P7037539587688870467BT_nat: $tType).
% 20.23/7.37  thf(func_def_59, type, list_P4547456442757143711BT_int: $tType).
% 20.23/7.37  thf(func_def_60, type, list_P5647936690300460905T_VEBT: $tType).
% 20.23/7.37  thf(func_def_61, type, list_P7524865323317820941T_VEBT: $tType).
% 20.23/7.37  thf(func_def_62, type, set_Pr6227168374412355847list_o: $tType).
% 20.23/7.37  thf(func_def_63, type, set_li5450038453877631591at_nat: $tType).
% 20.23/7.37  thf(func_def_64, type, produc8243902056947475879T_VEBT: $tType).
% 20.23/7.37  thf(func_def_65, type, set_Pr7556676689462069481BT_nat: $tType).
% 20.23/7.37  thf(func_def_66, type, set_Pr5066593544530342725BT_int: $tType).
% 20.23/7.37  thf(func_def_67, type, set_Pr8044002425091019955T_VEBT: $tType).
% 20.23/7.37  thf(func_def_68, type, set_se7855581050983116737at_nat: $tType).
% 20.23/7.37  thf(func_def_69, type, set_Pr5085853215250843933omplex: $tType).
% 20.23/7.37  thf(func_def_70, type, produc7819656566062154093et_nat: $tType).
% 20.23/7.37  thf(func_def_71, type, produc8923325533196201883nteger: $tType).
% 20.23/7.37  thf(func_def_72, type, list_P3126845725202233233VEBT_o: $tType).
% 20.23/7.37  thf(func_def_73, type, list_P7495141550334521929T_VEBT: $tType).
% 20.23/7.37  thf(func_def_74, type, filter2146258269922977983l_real: $tType).
% 20.23/7.37  thf(func_def_75, type, set_Pr3175402225741728619VEBT_o: $tType).
% 20.23/7.37  thf(func_def_76, type, set_Pr7543698050874017315T_VEBT: $tType).
% 20.23/7.37  thf(func_def_77, type, option4927543243414619207at_nat: $tType).
% 20.23/7.37  thf(func_def_78, type, list_P8689742595348180415l_real: $tType).
% 20.23/7.37  thf(func_def_79, type, filter1242075044329608583at_nat: $tType).
% 20.23/7.37  thf(func_def_80, type, set_Pr6218003697084177305l_real: $tType).
% 20.23/7.37  thf(func_def_81, type, list_P6834414599653733731al_nat: $tType).
% 20.23/7.37  thf(func_def_82, type, list_P4344331454722006975al_int: $tType).
% 20.23/7.37  thf(func_def_83, type, list_P6863124054624500543t_real: $tType).
% 20.23/7.37  thf(func_def_84, type, produc5170161368751668367T_real: $tType).
% 20.23/7.37  thf(func_def_85, type, produc3757001726724277373T_VEBT: $tType).
% 20.23/7.37  thf(func_def_86, type, set_Pr3510011417693777981al_nat: $tType).
% 20.23/7.37  thf(func_def_87, type, set_Pr1019928272762051225al_int: $tType).
% 20.23/7.37  thf(func_def_88, type, set_Pr3538720872664544793t_real: $tType).
% 20.23/7.37  thf(func_def_89, type, produc7102631898165422375list_o: $tType).
% 20.23/7.37  thf(func_def_90, type, list_P6011104703257516679at_nat: $tType).
% 20.23/7.37  thf(func_def_91, type, list_P3521021558325789923at_int: $tType).
% 20.23/7.37  thf(func_def_92, type, list_P5707943133018811711nt_int: $tType).
% 20.23/7.37  thf(func_def_93, type, produc9072475918466114483BT_nat: $tType).
% 20.23/7.37  thf(func_def_94, type, produc4894624898956917775BT_int: $tType).
% 20.23/7.37  thf(func_def_95, type, produc8025551001238799321T_VEBT: $tType).
% 20.23/7.37  thf(func_def_96, type, produc1531783533982839933T_VEBT: $tType).
% 20.23/7.37  thf(func_def_97, type, set_Pr1261947904930325089at_nat: $tType).
% 20.23/7.37  thf(func_def_98, type, set_Pr958786334691620121nt_int: $tType).
% 20.23/7.37  thf(func_def_99, type, produc4411394909380815293omplex: $tType).
% 20.23/7.37  thf(func_def_100, type, list_P3595434254542482545real_o: $tType).
% 20.23/7.37  thf(func_def_101, type, list_P5232166724548748803o_real: $tType).
% 20.23/7.37  thf(func_def_102, type, set_Pr4936984352647145239real_o: $tType).
% 20.23/7.37  thf(func_def_103, type, set_Pr6573716822653411497o_real: $tType).
% 20.23/7.37  thf(func_def_104, type, list_P7333126701944960589_nat_o: $tType).
% 20.23/7.37  thf(func_def_105, type, list_P6285523579766656935_o_nat: $tType).
% 20.23/7.37  thf(func_def_106, type, list_P3795440434834930179_o_int: $tType).
% 20.23/7.37  thf(func_def_107, type, set_list_VEBT_VEBT: $tType).
% 20.23/7.37  thf(func_def_108, type, produc334124729049499915VEBT_o: $tType).
% 20.23/7.37  thf(func_def_109, type, produc2504756804600209347T_VEBT: $tType).
% 20.23/7.37  thf(func_def_110, type, set_Pr2101469702781467981_o_nat: $tType).
% 20.23/7.37  thf(func_def_111, type, set_Pr8834758594704517033_o_int: $tType).
% 20.23/7.37  thf(func_def_112, type, produc6271795597528267376eger_o: $tType).
% 20.23/7.37  thf(func_def_113, type, produc2422161461964618553l_real: $tType).
% 20.23/7.37  thf(func_def_114, type, produc3741383161447143261al_nat: $tType).
% 20.23/7.37  thf(func_def_115, type, produc8786904178792722361al_int: $tType).
% 20.23/7.37  thf(func_def_116, type, produc679980390762269497t_real: $tType).
% 20.23/7.37  thf(func_def_117, type, product_prod_num_num: $tType).
% 20.23/7.37  thf(func_def_118, type, product_prod_nat_num: $tType).
% 20.23/7.37  thf(func_def_119, type, product_prod_nat_nat: $tType).
% 20.23/7.37  thf(func_def_120, type, product_prod_nat_int: $tType).
% 20.23/7.37  thf(func_def_121, type, product_prod_int_int: $tType).
% 20.23/7.37  thf(func_def_122, type, list_P4002435161011370285od_o_o: $tType).
% 20.23/7.37  thf(func_def_123, type, set_list_complex: $tType).
% 20.23/7.37  thf(func_def_124, type, set_Product_prod_o_o: $tType).
% 20.23/7.37  thf(func_def_125, type, set_set_complex: $tType).
% 20.23/7.37  thf(func_def_126, type, option_VEBT_VEBT: $tType).
% 20.23/7.37  thf(func_def_127, type, option_set_nat: $tType).
% 20.23/7.37  thf(func_def_128, type, product_prod_real_o: $tType).
% 20.23/7.37  thf(func_def_129, type, product_prod_o_real: $tType).
% 20.23/7.37  thf(func_def_130, type, list_list_nat: $tType).
% 20.23/7.37  thf(func_def_131, type, list_VEBT_VEBT: $tType).
% 20.23/7.37  thf(func_def_132, type, set_list_nat: $tType).
% 20.23/7.37  thf(func_def_133, type, set_list_int: $tType).
% 20.23/7.37  thf(func_def_134, type, product_prod_nat_o: $tType).
% 20.23/7.37  thf(func_def_135, type, product_prod_o_nat: $tType).
% 20.23/7.37  thf(func_def_136, type, product_prod_o_int: $tType).
% 20.23/7.37  thf(func_def_137, type, list_set_nat: $tType).
% 20.23/7.37  thf(func_def_138, type, set_VEBT_VEBT: $tType).
% 20.23/7.37  thf(func_def_139, type, set_set_nat: $tType).
% 20.23/7.37  thf(func_def_140, type, set_set_int: $tType).
% 20.23/7.37  thf(func_def_141, type, set_Code_integer: $tType).
% 20.23/7.37  thf(func_def_142, type, list_complex: $tType).
% 20.23/7.37  thf(func_def_143, type, set_list_o: $tType).
% 20.23/7.37  thf(func_def_144, type, product_prod_o_o: $tType).
% 20.23/7.37  thf(func_def_145, type, set_complex: $tType).
% 20.23/7.37  thf(func_def_146, type, option_real: $tType).
% 20.23/7.37  thf(func_def_147, type, filter_real: $tType).
% 20.23/7.37  thf(func_def_148, type, option_num: $tType).
% 20.23/7.37  thf(func_def_149, type, option_nat: $tType).
% 20.23/7.37  thf(func_def_150, type, option_int: $tType).
% 20.23/7.37  thf(func_def_151, type, filter_nat: $tType).
% 20.23/7.37  thf(func_def_152, type, list_real: $tType).
% 20.23/7.37  thf(func_def_153, type, set_real: $tType).
% 20.23/7.37  thf(func_def_154, type, list_num: $tType).
% 20.23/7.37  thf(func_def_155, type, list_nat: $tType).
% 20.23/7.37  thf(func_def_156, type, list_int: $tType).
% 20.23/7.37  thf(func_def_157, type, vEBT_VEBT: $tType).
% 20.23/7.37  thf(func_def_158, type, set_rat: $tType).
% 20.23/7.37  thf(func_def_159, type, set_num: $tType).
% 20.23/7.37  thf(func_def_160, type, set_nat: $tType).
% 20.23/7.37  thf(func_def_161, type, set_int: $tType).
% 20.23/7.37  thf(func_def_162, type, code_integer: $tType).
% 20.23/7.37  thf(func_def_163, type, option_o: $tType).
% 20.23/7.37  thf(func_def_164, type, extended_enat: $tType).
% 20.23/7.37  thf(func_def_165, type, list_o: $tType).
% 20.23/7.37  thf(func_def_166, type, complex: $tType).
% 20.23/7.37  thf(func_def_167, type, set_o: $tType).
% 20.23/7.37  thf(func_def_168, type, real: $tType).
% 20.23/7.37  thf(func_def_169, type, rat: $tType).
% 20.23/7.37  thf(func_def_170, type, num: $tType).
% 20.23/7.37  thf(func_def_171, type, nat: $tType).
% 20.23/7.37  thf(func_def_172, type, int: $tType).
% 20.23/7.37  thf(func_def_173, type, archim2889992004027027881ng_rat: rat > int).
% 20.23/7.37  thf(func_def_174, type, archim7802044766580827645g_real: real > int).
% 20.23/7.37  thf(func_def_175, type, archim3151403230148437115or_rat: rat > int).
% 20.23/7.37  thf(func_def_176, type, archim6058952711729229775r_real: real > int).
% 20.23/7.37  thf(func_def_177, type, archimedean_frac_rat: rat > rat).
% 20.23/7.37  thf(func_def_178, type, archim2898591450579166408c_real: real > real).
% 20.23/7.37  thf(func_def_179, type, archim7778729529865785530nd_rat: rat > int).
% 20.23/7.37  thf(func_def_180, type, archim8280529875227126926d_real: real > int).
% 20.23/7.37  thf(func_def_181, type, bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_182, type, bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_183, type, binomial: nat > nat > nat).
% 20.23/7.37  thf(func_def_184, type, gbinomial_complex: complex > nat > complex).
% 20.23/7.37  thf(func_def_185, type, gbinomial_int: int > nat > int).
% 20.23/7.37  thf(func_def_186, type, gbinomial_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_187, type, gbinomial_rat: rat > nat > rat).
% 20.23/7.37  thf(func_def_188, type, gbinomial_real: real > nat > real).
% 20.23/7.37  thf(func_def_189, type, bit_and_int_rel: product_prod_int_int > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_190, type, bit_and_not_num: num > num > option_num).
% 20.23/7.37  thf(func_def_191, type, bit_concat_bit: nat > int > int > int).
% 20.23/7.37  thf(func_def_192, type, bit_ri7919022796975470100ot_int: int > int).
% 20.23/7.37  thf(func_def_193, type, bit_ri6519982836138164636nteger: nat > code_integer > code_integer).
% 20.23/7.37  thf(func_def_194, type, bit_ri631733984087533419it_int: nat > int > int).
% 20.23/7.37  thf(func_def_195, type, bit_se725231765392027082nd_int: int > int > int).
% 20.23/7.37  thf(func_def_196, type, bit_se727722235901077358nd_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_197, type, bit_se8568078237143864401it_int: nat > int > int).
% 20.23/7.37  thf(func_def_198, type, bit_se8570568707652914677it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_199, type, bit_se1345352211410354436nteger: nat > code_integer > code_integer).
% 20.23/7.37  thf(func_def_200, type, bit_se2159334234014336723it_int: nat > int > int).
% 20.23/7.37  thf(func_def_201, type, bit_se2161824704523386999it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_202, type, bit_se2119862282449309892nteger: nat > code_integer).
% 20.23/7.37  thf(func_def_203, type, bit_se2000444600071755411sk_int: nat > int).
% 20.23/7.37  thf(func_def_204, type, bit_se2002935070580805687sk_nat: nat > nat).
% 20.23/7.37  thf(func_def_205, type, bit_se1409905431419307370or_int: int > int > int).
% 20.23/7.37  thf(func_def_206, type, bit_se1412395901928357646or_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_207, type, bit_se545348938243370406it_int: nat > int > int).
% 20.23/7.37  thf(func_def_208, type, bit_se547839408752420682it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_209, type, bit_se2793503036327961859nteger: nat > code_integer > code_integer).
% 20.23/7.37  thf(func_def_210, type, bit_se7879613467334960850it_int: nat > int > int).
% 20.23/7.37  thf(func_def_211, type, bit_se7882103937844011126it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_212, type, bit_se1745604003318907178nteger: nat > code_integer > code_integer).
% 20.23/7.37  thf(func_def_213, type, bit_se2923211474154528505it_int: nat > int > int).
% 20.23/7.37  thf(func_def_214, type, bit_se2925701944663578781it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_215, type, bit_se8260200283734997820nteger: nat > code_integer > code_integer).
% 20.23/7.37  thf(func_def_216, type, bit_se4203085406695923979it_int: nat > int > int).
% 20.23/7.37  thf(func_def_217, type, bit_se4205575877204974255it_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_218, type, bit_se3222712562003087583nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_219, type, bit_se6526347334894502574or_int: int > int > int).
% 20.23/7.37  thf(func_def_220, type, bit_se6528837805403552850or_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_221, type, bit_se9216721137139052372nteger: code_integer > nat > $o).
% 20.23/7.37  thf(func_def_222, type, bit_se1146084159140164899it_int: int > nat > $o).
% 20.23/7.37  thf(func_def_223, type, bit_se1148574629649215175it_nat: nat > nat > $o).
% 20.23/7.37  thf(func_def_224, type, bit_take_bit_num: nat > num > option_num).
% 20.23/7.37  thf(func_def_225, type, bit_un7362597486090784418nd_num: num > num > option_num).
% 20.23/7.37  thf(func_def_226, type, bit_un2480387367778600638or_num: num > num > option_num).
% 20.23/7.37  thf(func_def_227, type, code_bit_cut_integer: code_integer > produc6271795597528267376eger_o).
% 20.23/7.37  thf(func_def_228, type, code_divmod_abs: code_integer > code_integer > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_229, type, code_divmod_integer: code_integer > code_integer > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_230, type, code_int_of_integer: code_integer > int).
% 20.23/7.37  thf(func_def_231, type, code_integer_of_int: int > code_integer).
% 20.23/7.37  thf(func_def_232, type, code_nat_of_integer: code_integer > nat).
% 20.23/7.37  thf(func_def_233, type, comple8358262395181532106omplex: set_fi4554929511873752355omplex > filter6041513312241820739omplex).
% 20.23/7.37  thf(func_def_234, type, comple2936214249959783750l_real: set_fi7789364187291644575l_real > filter2146258269922977983l_real).
% 20.23/7.37  thf(func_def_235, type, complete_Inf_Inf_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_236, type, comple7806235888213564991et_nat: set_set_nat > set_nat).
% 20.23/7.37  thf(func_def_237, type, complete_Sup_Sup_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_238, type, arg: complex > real).
% 20.23/7.37  thf(func_def_239, type, cis: real > complex).
% 20.23/7.37  thf(func_def_240, type, complex2: real > real > complex).
% 20.23/7.37  thf(func_def_241, type, condit2214826472909112428ve_nat: set_nat > $o).
% 20.23/7.37  thf(func_def_242, type, differ6690327859849518006l_real: (real > real) > filter_real > $o).
% 20.23/7.37  thf(func_def_243, type, has_de1759254742604945161l_real: (real > real) > (real > real) > filter_real > $o).
% 20.23/7.37  thf(func_def_244, type, has_fi5821293074295781190e_real: (real > real) > real > filter_real > $o).
% 20.23/7.37  thf(func_def_245, type, adjust_div: product_prod_int_int > int).
% 20.23/7.37  thf(func_def_246, type, divmod_nat: nat > nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_247, type, eucl_rel_int: int > int > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_248, type, unique6319869463603278526ux_int: product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_249, type, unique6322359934112328802ux_nat: product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_250, type, unique3479559517661332726nteger: num > num > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_251, type, unique5052692396658037445od_int: num > num > product_prod_int_int).
% 20.23/7.37  thf(func_def_252, type, unique5055182867167087721od_nat: num > num > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_253, type, unique4921790084139445826nteger: num > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_254, type, unique5024387138958732305ep_int: num > product_prod_int_int > product_prod_int_int).
% 20.23/7.37  thf(func_def_255, type, unique5026877609467782581ep_nat: num > product_prod_nat_nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_256, type, comm_s8582702949713902594nteger: code_integer > nat > code_integer).
% 20.23/7.37  thf(func_def_257, type, comm_s2602460028002588243omplex: complex > nat > complex).
% 20.23/7.37  thf(func_def_258, type, comm_s4660882817536571857er_int: int > nat > int).
% 20.23/7.37  thf(func_def_259, type, comm_s4663373288045622133er_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_260, type, comm_s4028243227959126397er_rat: rat > nat > rat).
% 20.23/7.37  thf(func_def_261, type, comm_s7457072308508201937r_real: real > nat > real).
% 20.23/7.37  thf(func_def_262, type, semiri3624122377584611663nteger: nat > code_integer).
% 20.23/7.37  thf(func_def_263, type, semiri5044797733671781792omplex: nat > complex).
% 20.23/7.37  thf(func_def_264, type, semiri1406184849735516958ct_int: nat > int).
% 20.23/7.37  thf(func_def_265, type, semiri1408675320244567234ct_nat: nat > nat).
% 20.23/7.37  thf(func_def_266, type, semiri773545260158071498ct_rat: nat > rat).
% 20.23/7.37  thf(func_def_267, type, semiri2265585572941072030t_real: nat > real).
% 20.23/7.37  thf(func_def_268, type, inverse_inverse_rat: rat > rat).
% 20.23/7.37  thf(func_def_269, type, inverse_inverse_real: real > real).
% 20.23/7.37  thf(func_def_270, type, at_bot_real: filter_real).
% 20.23/7.37  thf(func_def_271, type, at_top_nat: filter_nat).
% 20.23/7.37  thf(func_def_272, type, at_top_real: filter_real).
% 20.23/7.37  thf(func_def_273, type, eventually_nat: (nat > $o) > filter_nat > $o).
% 20.23/7.37  thf(func_def_274, type, eventu1038000079068216329at_nat: (product_prod_nat_nat > $o) > filter1242075044329608583at_nat > $o).
% 20.23/7.37  thf(func_def_275, type, eventually_real: (real > $o) > filter_real > $o).
% 20.23/7.37  thf(func_def_276, type, filterlim_nat_nat: (nat > nat) > filter_nat > filter_nat > $o).
% 20.23/7.37  thf(func_def_277, type, filterlim_nat_real: (nat > real) > filter_real > filter_nat > $o).
% 20.23/7.37  thf(func_def_278, type, filterlim_real_real: (real > real) > filter_real > filter_real > $o).
% 20.23/7.37  thf(func_def_279, type, princi3496590319149328850omplex: set_Pr5085853215250843933omplex > filter6041513312241820739omplex).
% 20.23/7.37  thf(func_def_280, type, princi6114159922880469582l_real: set_Pr6218003697084177305l_real > filter2146258269922977983l_real).
% 20.23/7.37  thf(func_def_281, type, prod_filter_nat_nat: filter_nat > filter_nat > filter1242075044329608583at_nat).
% 20.23/7.37  thf(func_def_282, type, finite_card_complex: set_complex > nat).
% 20.23/7.37  thf(func_def_283, type, finite_card_list_nat: set_list_nat > nat).
% 20.23/7.37  thf(func_def_284, type, finite_card_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_285, type, finite_card_set_nat: set_set_nat > nat).
% 20.23/7.37  thf(func_def_286, type, finite_finite_o: set_o > $o).
% 20.23/7.37  thf(func_def_287, type, finite3207457112153483333omplex: set_complex > $o).
% 20.23/7.37  thf(func_def_288, type, finite_finite_int: set_int > $o).
% 20.23/7.37  thf(func_def_289, type, finite_finite_list_o: set_list_o > $o).
% 20.23/7.37  thf(func_def_290, type, finite8712137658972009173omplex: set_list_complex > $o).
% 20.23/7.37  thf(func_def_291, type, finite3922522038869484883st_int: set_list_int > $o).
% 20.23/7.37  thf(func_def_292, type, finite8100373058378681591st_nat: set_list_nat > $o).
% 20.23/7.37  thf(func_def_293, type, finite500796754983035824at_nat: set_li5450038453877631591at_nat > $o).
% 20.23/7.37  thf(func_def_294, type, finite3004134309566078307T_VEBT: set_list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_295, type, finite_finite_nat: set_nat > $o).
% 20.23/7.37  thf(func_def_296, type, finite_finite_num: set_num > $o).
% 20.23/7.37  thf(func_def_297, type, finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o).
% 20.23/7.37  thf(func_def_298, type, finite4343798906461161616at_nat: set_Pr4329608150637261639at_nat > $o).
% 20.23/7.37  thf(func_def_299, type, finite_finite_rat: set_rat > $o).
% 20.23/7.37  thf(func_def_300, type, finite_finite_real: set_real > $o).
% 20.23/7.37  thf(func_def_301, type, finite6551019134538273531omplex: set_set_complex > $o).
% 20.23/7.37  thf(func_def_302, type, finite6197958912794628473et_int: set_set_int > $o).
% 20.23/7.37  thf(func_def_303, type, finite1152437895449049373et_nat: set_set_nat > $o).
% 20.23/7.37  thf(func_def_304, type, finite9047747110432174090at_nat: set_se7855581050983116737at_nat > $o).
% 20.23/7.37  thf(func_def_305, type, finite5795047828879050333T_VEBT: set_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_306, type, bij_be1856998921033663316omplex: (complex > complex) > set_complex > set_complex > $o).
% 20.23/7.37  thf(func_def_307, type, bij_betw_nat_complex: (nat > complex) > set_nat > set_complex > $o).
% 20.23/7.37  thf(func_def_308, type, bij_betw_nat_nat: (nat > nat) > set_nat > set_nat > $o).
% 20.23/7.37  thf(func_def_309, type, comp_C8797469213163452608nteger: ((code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_310, type, comp_C1593894019821074884nteger: (code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger) > (code_integer > code_integer) > code_integer > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_311, type, comp_int_nat_int: (int > nat) > (int > int) > int > nat).
% 20.23/7.37  thf(func_def_312, type, comp_nat_nat_nat: (nat > nat) > (nat > nat) > nat > nat).
% 20.23/7.37  thf(func_def_313, type, id_o: $o > $o).
% 20.23/7.37  thf(func_def_314, type, id_nat: nat > nat).
% 20.23/7.37  thf(func_def_315, type, inj_on_nat_nat: (nat > nat) > set_nat > $o).
% 20.23/7.37  thf(func_def_316, type, inj_on_real_real: (real > real) > set_real > $o).
% 20.23/7.37  thf(func_def_317, type, inj_on_set_nat_nat: (set_nat > nat) > set_set_nat > $o).
% 20.23/7.37  thf(func_def_318, type, map_fu434086159418415080_int_o: (int > product_prod_nat_nat) > ((product_prod_nat_nat > $o) > int > $o) > (product_prod_nat_nat > product_prod_nat_nat > $o) > int > int > $o).
% 20.23/7.37  thf(func_def_319, type, map_fu4960017516451851995nt_int: (int > product_prod_nat_nat) > ((product_prod_nat_nat > product_prod_nat_nat) > int > int) > (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > int > int > int).
% 20.23/7.37  thf(func_def_320, type, map_fu4826362097070443709at_o_o: (int > product_prod_nat_nat) > ($o > $o) > (product_prod_nat_nat > $o) > int > $o).
% 20.23/7.37  thf(func_def_321, type, map_fu2345160673673942751at_nat: (int > product_prod_nat_nat) > (nat > nat) > (product_prod_nat_nat > nat) > int > nat).
% 20.23/7.37  thf(func_def_322, type, map_fu3667384564859982768at_int: (int > product_prod_nat_nat) > (product_prod_nat_nat > int) > (product_prod_nat_nat > product_prod_nat_nat) > int > int).
% 20.23/7.37  thf(func_def_323, type, map_fu898904425404107465nt_o_o: (rat > product_prod_int_int) > ($o > $o) > (product_prod_int_int > $o) > rat > $o).
% 20.23/7.37  thf(func_def_324, type, strict1292158309912662752at_nat: (nat > nat) > set_nat > $o).
% 20.23/7.37  thf(func_def_325, type, the_in5290026491893676941l_real: set_real > (real > real) > real > real).
% 20.23/7.37  thf(func_def_326, type, fun_is_measure_int: (int > nat) > $o).
% 20.23/7.37  thf(func_def_327, type, fun_max_strict: set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_328, type, fun_max_weak: set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_329, type, fun_min_strict: set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_330, type, fun_min_weak: set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_331, type, fun_pair_leq: set_Pr8693737435421807431at_nat).
% 20.23/7.37  thf(func_def_332, type, fun_pair_less: set_Pr8693737435421807431at_nat).
% 20.23/7.37  thf(func_def_333, type, fun_re2478310338295953701at_nat: produc1319942482725812455at_nat > $o).
% 20.23/7.37  thf(func_def_334, type, gcd_Gcd_int: set_int > int).
% 20.23/7.37  thf(func_def_335, type, gcd_Gcd_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_336, type, bezw: nat > nat > product_prod_int_int).
% 20.23/7.37  thf(func_def_337, type, bezw_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_338, type, gcd_gcd_Code_integer: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_339, type, gcd_gcd_int: int > int > int).
% 20.23/7.37  thf(func_def_340, type, gcd_gcd_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_341, type, gcd_nat_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_342, type, abs_abs_Code_integer: code_integer > code_integer).
% 20.23/7.37  thf(func_def_343, type, abs_abs_complex: complex > complex).
% 20.23/7.37  thf(func_def_344, type, abs_abs_int: int > int).
% 20.23/7.37  thf(func_def_345, type, abs_abs_rat: rat > rat).
% 20.23/7.37  thf(func_def_346, type, abs_abs_real: real > real).
% 20.23/7.37  thf(func_def_347, type, minus_minus_int_o: (int > $o) > (int > $o) > int > $o).
% 20.23/7.37  thf(func_def_348, type, minus_1139252259498527702_nat_o: (list_nat > $o) > (list_nat > $o) > list_nat > $o).
% 20.23/7.37  thf(func_def_349, type, minus_minus_nat_o: (nat > $o) > (nat > $o) > nat > $o).
% 20.23/7.37  thf(func_def_350, type, minus_2270307095948843157_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_351, type, minus_minus_real_o: (real > $o) > (real > $o) > real > $o).
% 20.23/7.37  thf(func_def_352, type, minus_6910147592129066416_nat_o: (set_nat > $o) > (set_nat > $o) > set_nat > $o).
% 20.23/7.37  thf(func_def_353, type, minus_8373710615458151222nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_354, type, minus_minus_complex: complex > complex > complex).
% 20.23/7.37  thf(func_def_355, type, minus_minus_int: int > int > int).
% 20.23/7.37  thf(func_def_356, type, minus_minus_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_357, type, minus_minus_rat: rat > rat > rat).
% 20.23/7.37  thf(func_def_358, type, minus_minus_real: real > real > real).
% 20.23/7.37  thf(func_def_359, type, minus_811609699411566653omplex: set_complex > set_complex > set_complex).
% 20.23/7.37  thf(func_def_360, type, minus_minus_set_int: set_int > set_int > set_int).
% 20.23/7.37  thf(func_def_361, type, minus_7954133019191499631st_nat: set_list_nat > set_list_nat > set_list_nat).
% 20.23/7.37  thf(func_def_362, type, minus_minus_set_nat: set_nat > set_nat > set_nat).
% 20.23/7.37  thf(func_def_363, type, minus_1356011639430497352at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_364, type, minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_365, type, minus_minus_set_real: set_real > set_real > set_real).
% 20.23/7.37  thf(func_def_366, type, minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_367, type, one_one_Code_integer: code_integer).
% 20.23/7.37  thf(func_def_368, type, one_one_complex: complex).
% 20.23/7.37  thf(func_def_369, type, one_on7984719198319812577d_enat: extended_enat).
% 20.23/7.37  thf(func_def_370, type, one_one_int: int).
% 20.23/7.37  thf(func_def_371, type, one_one_nat: nat).
% 20.23/7.37  thf(func_def_372, type, one_one_rat: rat).
% 20.23/7.37  thf(func_def_373, type, one_one_real: real).
% 20.23/7.37  thf(func_def_374, type, plus_p5714425477246183910nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_375, type, plus_plus_complex: complex > complex > complex).
% 20.23/7.37  thf(func_def_376, type, plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat).
% 20.23/7.37  thf(func_def_377, type, plus_plus_int: int > int > int).
% 20.23/7.37  thf(func_def_378, type, plus_plus_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_379, type, plus_plus_num: num > num > num).
% 20.23/7.37  thf(func_def_380, type, plus_plus_rat: rat > rat > rat).
% 20.23/7.37  thf(func_def_381, type, plus_plus_real: real > real > real).
% 20.23/7.37  thf(func_def_382, type, sgn_sgn_Code_integer: code_integer > code_integer).
% 20.23/7.37  thf(func_def_383, type, sgn_sgn_complex: complex > complex).
% 20.23/7.37  thf(func_def_384, type, sgn_sgn_int: int > int).
% 20.23/7.37  thf(func_def_385, type, sgn_sgn_rat: rat > rat).
% 20.23/7.37  thf(func_def_386, type, sgn_sgn_real: real > real).
% 20.23/7.37  thf(func_def_387, type, times_3573771949741848930nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_388, type, times_times_complex: complex > complex > complex).
% 20.23/7.37  thf(func_def_389, type, times_7803423173614009249d_enat: extended_enat > extended_enat > extended_enat).
% 20.23/7.37  thf(func_def_390, type, times_times_int: int > int > int).
% 20.23/7.37  thf(func_def_391, type, times_times_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_392, type, times_times_rat: rat > rat > rat).
% 20.23/7.37  thf(func_def_393, type, times_times_real: real > real > real).
% 20.23/7.37  thf(func_def_394, type, uminus_uminus_int_o: (int > $o) > int > $o).
% 20.23/7.37  thf(func_def_395, type, uminus5770388063884162150_nat_o: (list_nat > $o) > list_nat > $o).
% 20.23/7.37  thf(func_def_396, type, uminus_uminus_nat_o: (nat > $o) > nat > $o).
% 20.23/7.37  thf(func_def_397, type, uminus8676089048583255045_nat_o: (product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_398, type, uminus_uminus_real_o: (real > $o) > real > $o).
% 20.23/7.37  thf(func_def_399, type, uminus6401447641752708672_nat_o: (set_nat > $o) > set_nat > $o).
% 20.23/7.37  thf(func_def_400, type, uminus1351360451143612070nteger: code_integer > code_integer).
% 20.23/7.37  thf(func_def_401, type, uminus1482373934393186551omplex: complex > complex).
% 20.23/7.37  thf(func_def_402, type, uminus_uminus_int: int > int).
% 20.23/7.37  thf(func_def_403, type, uminus_uminus_rat: rat > rat).
% 20.23/7.37  thf(func_def_404, type, uminus_uminus_real: real > real).
% 20.23/7.37  thf(func_def_405, type, uminus1532241313380277803et_int: set_int > set_int).
% 20.23/7.37  thf(func_def_406, type, uminus3195874150345416415st_nat: set_list_nat > set_list_nat).
% 20.23/7.37  thf(func_def_407, type, uminus5710092332889474511et_nat: set_nat > set_nat).
% 20.23/7.37  thf(func_def_408, type, uminus6524753893492686040at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_409, type, uminus935396558254630718at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_410, type, uminus612125837232591019t_real: set_real > set_real).
% 20.23/7.37  thf(func_def_411, type, uminus613421341184616069et_nat: set_set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_412, type, zero_z3403309356797280102nteger: code_integer).
% 20.23/7.37  thf(func_def_413, type, zero_zero_complex: complex).
% 20.23/7.37  thf(func_def_414, type, zero_z5237406670263579293d_enat: extended_enat).
% 20.23/7.37  thf(func_def_415, type, zero_zero_int: int).
% 20.23/7.37  thf(func_def_416, type, zero_zero_nat: nat).
% 20.23/7.37  thf(func_def_417, type, zero_zero_rat: rat).
% 20.23/7.37  thf(func_def_418, type, zero_zero_real: real).
% 20.23/7.37  thf(func_def_419, type, groups8507830703676809646_o_nat: ($o > nat) > set_o > nat).
% 20.23/7.37  thf(func_def_420, type, groups7754918857620584856omplex: (complex > complex) > set_complex > complex).
% 20.23/7.37  thf(func_def_421, type, groups5690904116761175830ex_int: (complex > int) > set_complex > int).
% 20.23/7.37  thf(func_def_422, type, groups5693394587270226106ex_nat: (complex > nat) > set_complex > nat).
% 20.23/7.37  thf(func_def_423, type, groups5058264527183730370ex_rat: (complex > rat) > set_complex > rat).
% 20.23/7.37  thf(func_def_424, type, groups5808333547571424918x_real: (complex > real) > set_complex > real).
% 20.23/7.37  thf(func_def_425, type, groups7873554091576472773nteger: (int > code_integer) > set_int > code_integer).
% 20.23/7.37  thf(func_def_426, type, groups3049146728041665814omplex: (int > complex) > set_int > complex).
% 20.23/7.37  thf(func_def_427, type, groups4538972089207619220nt_int: (int > int) > set_int > int).
% 20.23/7.37  thf(func_def_428, type, groups4541462559716669496nt_nat: (int > nat) > set_int > nat).
% 20.23/7.37  thf(func_def_429, type, groups3906332499630173760nt_rat: (int > rat) > set_int > rat).
% 20.23/7.37  thf(func_def_430, type, groups8778361861064173332t_real: (int > real) > set_int > real).
% 20.23/7.37  thf(func_def_431, type, groups7501900531339628137nteger: (nat > code_integer) > set_nat > code_integer).
% 20.23/7.37  thf(func_def_432, type, groups2073611262835488442omplex: (nat > complex) > set_nat > complex).
% 20.23/7.37  thf(func_def_433, type, groups7108830773950497114d_enat: (nat > extended_enat) > set_nat > extended_enat).
% 20.23/7.37  thf(func_def_434, type, groups3539618377306564664at_int: (nat > int) > set_nat > int).
% 20.23/7.37  thf(func_def_435, type, groups3542108847815614940at_nat: (nat > nat) > set_nat > nat).
% 20.23/7.37  thf(func_def_436, type, groups2906978787729119204at_rat: (nat > rat) > set_nat > rat).
% 20.23/7.37  thf(func_def_437, type, groups6591440286371151544t_real: (nat > real) > set_nat > real).
% 20.23/7.37  thf(func_def_438, type, groups977919841031483927at_nat: (product_prod_nat_nat > nat) > set_Pr1261947904930325089at_nat > nat).
% 20.23/7.37  thf(func_def_439, type, groups3860910324918113789at_nat: (produc3843707927480180839at_nat > nat) > set_Pr4329608150637261639at_nat > nat).
% 20.23/7.37  thf(func_def_440, type, groups7713935264441627589nteger: (real > code_integer) > set_real > code_integer).
% 20.23/7.37  thf(func_def_441, type, groups5754745047067104278omplex: (real > complex) > set_real > complex).
% 20.23/7.37  thf(func_def_442, type, groups1932886352136224148al_int: (real > int) > set_real > int).
% 20.23/7.37  thf(func_def_443, type, groups1935376822645274424al_nat: (real > nat) > set_real > nat).
% 20.23/7.37  thf(func_def_444, type, groups1300246762558778688al_rat: (real > rat) > set_real > rat).
% 20.23/7.37  thf(func_def_445, type, groups8097168146408367636l_real: (real > real) > set_real > real).
% 20.23/7.37  thf(func_def_446, type, groups8294997508430121362at_nat: (set_nat > nat) > set_set_nat > nat).
% 20.23/7.37  thf(func_def_447, type, groups771621172384141258BT_nat: (vEBT_VEBT > nat) > set_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_448, type, groups8682486955453173170nteger: (complex > code_integer) > set_complex > code_integer).
% 20.23/7.37  thf(func_def_449, type, groups3708469109370488835omplex: (complex > complex) > set_complex > complex).
% 20.23/7.37  thf(func_def_450, type, groups858564598930262913ex_int: (complex > int) > set_complex > int).
% 20.23/7.37  thf(func_def_451, type, groups861055069439313189ex_nat: (complex > nat) > set_complex > nat).
% 20.23/7.37  thf(func_def_452, type, groups225925009352817453ex_rat: (complex > rat) > set_complex > rat).
% 20.23/7.37  thf(func_def_453, type, groups766887009212190081x_real: (complex > real) > set_complex > real).
% 20.23/7.37  thf(func_def_454, type, groups3827104343326376752nteger: (int > code_integer) > set_int > code_integer).
% 20.23/7.37  thf(func_def_455, type, groups7440179247065528705omplex: (int > complex) > set_int > complex).
% 20.23/7.37  thf(func_def_456, type, groups1705073143266064639nt_int: (int > int) > set_int > int).
% 20.23/7.37  thf(func_def_457, type, groups1707563613775114915nt_nat: (int > nat) > set_int > nat).
% 20.23/7.37  thf(func_def_458, type, groups1072433553688619179nt_rat: (int > rat) > set_int > rat).
% 20.23/7.37  thf(func_def_459, type, groups2316167850115554303t_real: (int > real) > set_int > real).
% 20.23/7.37  thf(func_def_460, type, groups3455450783089532116nteger: (nat > code_integer) > set_nat > code_integer).
% 20.23/7.37  thf(func_def_461, type, groups6464643781859351333omplex: (nat > complex) > set_nat > complex).
% 20.23/7.37  thf(func_def_462, type, groups705719431365010083at_int: (nat > int) > set_nat > int).
% 20.23/7.37  thf(func_def_463, type, groups708209901874060359at_nat: (nat > nat) > set_nat > nat).
% 20.23/7.37  thf(func_def_464, type, groups73079841787564623at_rat: (nat > rat) > set_nat > rat).
% 20.23/7.37  thf(func_def_465, type, groups129246275422532515t_real: (nat > real) > set_nat > real).
% 20.23/7.37  thf(func_def_466, type, groups6225526099057966256nteger: (real > code_integer) > set_real > code_integer).
% 20.23/7.37  thf(func_def_467, type, groups713298508707869441omplex: (real > complex) > set_real > complex).
% 20.23/7.37  thf(func_def_468, type, groups4694064378042380927al_int: (real > int) > set_real > int).
% 20.23/7.37  thf(func_def_469, type, groups4696554848551431203al_nat: (real > nat) > set_real > nat).
% 20.23/7.37  thf(func_def_470, type, groups4061424788464935467al_rat: (real > rat) > set_real > rat).
% 20.23/7.37  thf(func_def_471, type, groups1681761925125756287l_real: (real > real) > set_real > real).
% 20.23/7.37  thf(func_def_472, type, groups9116527308978886569_o_int: ($o > int) > int > list_o > int).
% 20.23/7.37  thf(func_def_473, type, groups9119017779487936845_o_nat: ($o > nat) > nat > list_o > nat).
% 20.23/7.37  thf(func_def_474, type, groups4561878855575611511st_nat: list_nat > nat).
% 20.23/7.37  thf(func_def_475, type, the_int: (int > $o) > int).
% 20.23/7.37  thf(func_def_476, type, the_real: (real > $o) > real).
% 20.23/7.37  thf(func_def_477, type, if_int_int: $o > (int > int) > (int > int) > int > int).
% 20.23/7.37  thf(func_def_478, type, if_Code_integer: $o > code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_479, type, if_complex: $o > complex > complex > complex).
% 20.23/7.37  thf(func_def_480, type, if_Extended_enat: $o > extended_enat > extended_enat > extended_enat).
% 20.23/7.37  thf(func_def_481, type, if_int: $o > int > int > int).
% 20.23/7.37  thf(func_def_482, type, if_list_int: $o > list_int > list_int > list_int).
% 20.23/7.37  thf(func_def_483, type, if_list_nat: $o > list_nat > list_nat > list_nat).
% 20.23/7.37  thf(func_def_484, type, if_nat: $o > nat > nat > nat).
% 20.23/7.37  thf(func_def_485, type, if_option_num: $o > option_num > option_num > option_num).
% 20.23/7.37  thf(func_def_486, type, if_Pro5737122678794959658eger_o: $o > produc6271795597528267376eger_o > produc6271795597528267376eger_o > produc6271795597528267376eger_o).
% 20.23/7.37  thf(func_def_487, type, if_Pro6119634080678213985nteger: $o > produc8923325533196201883nteger > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_488, type, if_Pro3027730157355071871nt_int: $o > product_prod_int_int > product_prod_int_int > product_prod_int_int).
% 20.23/7.37  thf(func_def_489, type, if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_490, type, if_rat: $o > rat > rat > rat).
% 20.23/7.37  thf(func_def_491, type, if_real: $o > real > real > real).
% 20.23/7.37  thf(func_def_492, type, if_set_int: $o > set_int > set_int > set_int).
% 20.23/7.37  thf(func_def_493, type, abs_Integ: product_prod_nat_nat > int).
% 20.23/7.37  thf(func_def_494, type, rep_Integ: int > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_495, type, int_ge_less_than: int > set_Pr958786334691620121nt_int).
% 20.23/7.37  thf(func_def_496, type, int_ge_less_than2: int > set_Pr958786334691620121nt_int).
% 20.23/7.37  thf(func_def_497, type, nat2: int > nat).
% 20.23/7.37  thf(func_def_498, type, power_int_real: real > int > real).
% 20.23/7.37  thf(func_def_499, type, ring_18347121197199848620nteger: int > code_integer).
% 20.23/7.37  thf(func_def_500, type, ring_17405671764205052669omplex: int > complex).
% 20.23/7.37  thf(func_def_501, type, ring_1_of_int_int: int > int).
% 20.23/7.37  thf(func_def_502, type, ring_1_of_int_rat: int > rat).
% 20.23/7.37  thf(func_def_503, type, ring_1_of_int_real: int > real).
% 20.23/7.37  thf(func_def_504, type, inf_inf_int: int > int > int).
% 20.23/7.37  thf(func_def_505, type, inf_inf_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_506, type, inf_in2572325071724192079at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_507, type, semila1623282765462674594er_nat: (nat > nat > nat) > nat > (nat > nat > $o) > (nat > nat > $o) > $o).
% 20.23/7.37  thf(func_def_508, type, sup_sup_int: int > int > int).
% 20.23/7.37  thf(func_def_509, type, sup_sup_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_510, type, sup_sup_set_nat: set_nat > set_nat > set_nat).
% 20.23/7.37  thf(func_def_511, type, sup_su5525570899277871387at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_512, type, lattic8263393255366662781ax_int: set_int > int).
% 20.23/7.37  thf(func_def_513, type, lattic8265883725875713057ax_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_514, type, lattic5364784637807008409ex_nat: (complex > nat) > set_complex > complex).
% 20.23/7.37  thf(func_def_515, type, lattic1922116423962787043ex_num: (complex > num) > set_complex > complex).
% 20.23/7.37  thf(func_def_516, type, lattic4729654577720512673ex_rat: (complex > rat) > set_complex > complex).
% 20.23/7.37  thf(func_def_517, type, lattic8794016678065449205x_real: (complex > real) > set_complex > complex).
% 20.23/7.37  thf(func_def_518, type, lattic5003618458639192673nt_num: (int > num) > set_int > int).
% 20.23/7.37  thf(func_def_519, type, lattic7811156612396918303nt_rat: (int > rat) > set_int > int).
% 20.23/7.37  thf(func_def_520, type, lattic2675449441010098035t_real: (int > real) > set_int > int).
% 20.23/7.37  thf(func_def_521, type, lattic4004264746738138117at_num: (nat > num) > set_nat > nat).
% 20.23/7.37  thf(func_def_522, type, lattic6811802900495863747at_rat: (nat > rat) > set_nat > nat).
% 20.23/7.37  thf(func_def_523, type, lattic488527866317076247t_real: (nat > real) > set_nat > nat).
% 20.23/7.37  thf(func_def_524, type, lattic5055836439445974935al_nat: (real > nat) > set_real > real).
% 20.23/7.37  thf(func_def_525, type, lattic1613168225601753569al_num: (real > num) > set_real > real).
% 20.23/7.37  thf(func_def_526, type, lattic4420706379359479199al_rat: (real > rat) > set_real > real).
% 20.23/7.37  thf(func_def_527, type, lattic8440615504127631091l_real: (real > real) > set_real > real).
% 20.23/7.37  thf(func_def_528, type, append_int: list_int > list_int > list_int).
% 20.23/7.37  thf(func_def_529, type, append_nat: list_nat > list_nat > list_nat).
% 20.23/7.37  thf(func_def_530, type, count_list_o: list_o > $o > nat).
% 20.23/7.37  thf(func_def_531, type, count_list_complex: list_complex > complex > nat).
% 20.23/7.37  thf(func_def_532, type, count_list_int: list_int > int > nat).
% 20.23/7.37  thf(func_def_533, type, count_list_nat: list_nat > nat > nat).
% 20.23/7.37  thf(func_def_534, type, count_4203492906077236349at_nat: list_P6011104703257516679at_nat > product_prod_nat_nat > nat).
% 20.23/7.37  thf(func_def_535, type, count_list_real: list_real > real > nat).
% 20.23/7.37  thf(func_def_536, type, count_list_set_nat: list_set_nat > set_nat > nat).
% 20.23/7.37  thf(func_def_537, type, count_list_VEBT_VEBT: list_VEBT_VEBT > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_538, type, distinct_int: list_int > $o).
% 20.23/7.37  thf(func_def_539, type, distinct_nat: list_nat > $o).
% 20.23/7.37  thf(func_def_540, type, drop_nat: nat > list_nat > list_nat).
% 20.23/7.37  thf(func_def_541, type, enumerate_o: nat > list_o > list_P7333126701944960589_nat_o).
% 20.23/7.37  thf(func_def_542, type, enumerate_int: nat > list_int > list_P3521021558325789923at_int).
% 20.23/7.37  thf(func_def_543, type, enumerate_nat: nat > list_nat > list_P6011104703257516679at_nat).
% 20.23/7.37  thf(func_def_544, type, enumerate_VEBT_VEBT: nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 20.23/7.37  thf(func_def_545, type, find_o: ($o > $o) > list_o > option_o).
% 20.23/7.37  thf(func_def_546, type, find_int: (int > $o) > list_int > option_int).
% 20.23/7.37  thf(func_def_547, type, find_nat: (nat > $o) > list_nat > option_nat).
% 20.23/7.37  thf(func_def_548, type, find_num: (num > $o) > list_num > option_num).
% 20.23/7.37  thf(func_def_549, type, find_P8199882355184865565at_nat: (product_prod_nat_nat > $o) > list_P6011104703257516679at_nat > option4927543243414619207at_nat).
% 20.23/7.37  thf(func_def_550, type, find_real: (real > $o) > list_real > option_real).
% 20.23/7.37  thf(func_def_551, type, find_set_nat: (set_nat > $o) > list_set_nat > option_set_nat).
% 20.23/7.37  thf(func_def_552, type, find_VEBT_VEBT: (vEBT_VEBT > $o) > list_VEBT_VEBT > option_VEBT_VEBT).
% 20.23/7.37  thf(func_def_553, type, gen_length_o: nat > list_o > nat).
% 20.23/7.37  thf(func_def_554, type, gen_length_int: nat > list_int > nat).
% 20.23/7.37  thf(func_def_555, type, gen_length_nat: nat > list_nat > nat).
% 20.23/7.37  thf(func_def_556, type, gen_length_VEBT_VEBT: nat > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_557, type, linord2614967742042102400et_nat: set_nat > list_nat).
% 20.23/7.37  thf(func_def_558, type, cons_int: int > list_int > list_int).
% 20.23/7.37  thf(func_def_559, type, cons_nat: nat > list_nat > list_nat).
% 20.23/7.37  thf(func_def_560, type, nil_int: list_int).
% 20.23/7.37  thf(func_def_561, type, nil_nat: list_nat).
% 20.23/7.37  thf(func_def_562, type, hd_nat: list_nat > nat).
% 20.23/7.37  thf(func_def_563, type, map_nat_nat: (nat > nat) > list_nat > list_nat).
% 20.23/7.37  thf(func_def_564, type, set_Co8062243466402858685e_term: list_C878401137130745250e_term > set_Co9149898834107579976e_term).
% 20.23/7.37  thf(func_def_565, type, set_in5217446777445088012e_term: list_i8448526496819171953e_term > set_in3461395444621081367e_term).
% 20.23/7.37  thf(func_def_566, type, set_Pr8342322266483756581e_term: list_P1316552470764441098e_term > set_Pr7604974323444597168e_term).
% 20.23/7.37  thf(func_def_567, type, set_Pr16608062948090134e_term: list_P1743416141875011707e_term > set_Pr3642885161833720865e_term).
% 20.23/7.37  thf(func_def_568, type, set_o2: list_o > set_o).
% 20.23/7.37  thf(func_def_569, type, set_complex2: list_complex > set_complex).
% 20.23/7.37  thf(func_def_570, type, set_int2: list_int > set_int).
% 20.23/7.37  thf(func_def_571, type, set_list_nat2: list_list_nat > set_list_nat).
% 20.23/7.37  thf(func_def_572, type, set_nat2: list_nat > set_nat).
% 20.23/7.37  thf(func_def_573, type, set_num2: list_num > set_num).
% 20.23/7.37  thf(func_def_574, type, set_Pr2999063419360598313nteger: list_P5311841565141990158nteger > set_Pr8056137968301705908nteger).
% 20.23/7.37  thf(func_def_575, type, set_Pr1633835384712236856nt_int: list_P8915022641806594461nt_int > set_Pr1872883991513573699nt_int).
% 20.23/7.37  thf(func_def_576, type, set_Pr2135590979564877377nteger: list_P7828571989066258726nteger > set_Pr1281608226676607948nteger).
% 20.23/7.37  thf(func_def_577, type, set_Pr4943052134776177454nt_int: list_P651320350408439699nt_int > set_Pr9222295170931077689nt_int).
% 20.23/7.37  thf(func_def_578, type, set_Product_prod_o_o2: list_P4002435161011370285od_o_o > set_Product_prod_o_o).
% 20.23/7.37  thf(func_def_579, type, set_Pr2828948584524939422_o_int: list_P3795440434834930179_o_int > set_Pr8834758594704517033_o_int).
% 20.23/7.37  thf(func_def_580, type, set_Pr7006799604034136130_o_nat: list_P6285523579766656935_o_nat > set_Pr2101469702781467981_o_nat).
% 20.23/7.37  thf(func_def_581, type, set_Pr2600826154070092190o_real: list_P5232166724548748803o_real > set_Pr6573716822653411497o_real).
% 20.23/7.37  thf(func_def_582, type, set_Pr655345902815428824T_VEBT: list_P7495141550334521929T_VEBT > set_Pr7543698050874017315T_VEBT).
% 20.23/7.37  thf(func_def_583, type, set_Pr920681315882439344nteger: list_P5578671422887162913nteger > set_Pr4811707699266497531nteger).
% 20.23/7.37  thf(func_def_584, type, set_Pr2470121279949933262nt_int: list_P5707943133018811711nt_int > set_Pr958786334691620121nt_int).
% 20.23/7.37  thf(func_def_585, type, set_Pr112895574167722958t_real: list_P6863124054624500543t_real > set_Pr3538720872664544793t_real).
% 20.23/7.37  thf(func_def_586, type, set_Pr8714266321650254504T_VEBT: list_P7524865323317820941T_VEBT > set_Pr8044002425091019955T_VEBT).
% 20.23/7.37  thf(func_def_587, type, set_Pr5648618587558075414at_nat: list_P6011104703257516679at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_588, type, set_Pr5518436109238095868at_nat: list_P8469869581646625389at_nat > set_Pr8693737435421807431at_nat).
% 20.23/7.37  thf(func_def_589, type, set_Pr5196769464307566348real_o: list_P3595434254542482545real_o > set_Pr4936984352647145239real_o).
% 20.23/7.37  thf(func_def_590, type, set_Pr8219819362198175822al_int: list_P4344331454722006975al_int > set_Pr1019928272762051225al_int).
% 20.23/7.37  thf(func_def_591, type, set_Pr3174298344852596722al_nat: list_P6834414599653733731al_nat > set_Pr3510011417693777981al_nat).
% 20.23/7.37  thf(func_def_592, type, set_Pr5999470521830281550l_real: list_P8689742595348180415l_real > set_Pr6218003697084177305l_real).
% 20.23/7.37  thf(func_def_593, type, set_Pr8897343066327330088T_VEBT: list_P877281246627933069T_VEBT > set_Pr6019664923565264691T_VEBT).
% 20.23/7.37  thf(func_def_594, type, set_Pr9040384385603167362et_nat: list_P6254988961118846195et_nat > set_Pr5488025237498180813et_nat).
% 20.23/7.37  thf(func_def_595, type, set_Pr3765526544606949372at_nat: list_P5464809261938338413at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_596, type, set_Pr7708085864119495200VEBT_o: list_P3126845725202233233VEBT_o > set_Pr3175402225741728619VEBT_o).
% 20.23/7.37  thf(func_def_597, type, set_Pr2853735649769556538BT_int: list_P4547456442757143711BT_int > set_Pr5066593544530342725BT_int).
% 20.23/7.37  thf(func_def_598, type, set_Pr7031586669278753246BT_nat: list_P7037539587688870467BT_nat > set_Pr7556676689462069481BT_nat).
% 20.23/7.37  thf(func_def_599, type, set_Pr1087130671499945274T_real: list_P2623026923184700063T_real > set_Pr7765410600122031685T_real).
% 20.23/7.37  thf(func_def_600, type, set_Pr9182192707038809660T_VEBT: list_P7413028617227757229T_VEBT > set_Pr6192946355708809607T_VEBT).
% 20.23/7.37  thf(func_def_601, type, set_real2: list_real > set_real).
% 20.23/7.37  thf(func_def_602, type, set_set_nat2: list_set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_603, type, set_VEBT_VEBT2: list_VEBT_VEBT > set_VEBT_VEBT).
% 20.23/7.37  thf(func_def_604, type, size_list_VEBT_VEBT: (vEBT_VEBT > nat) > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_605, type, tl_nat: list_nat > list_nat).
% 20.23/7.37  thf(func_def_606, type, list_u4743598893156345252e_term: list_C878401137130745250e_term > nat > (code_integer > option6357759511663192854e_term) > list_C878401137130745250e_term).
% 20.23/7.37  thf(func_def_607, type, list_u8946639151299769843e_term: list_i8448526496819171953e_term > nat > (int > option6357759511663192854e_term) > list_i8448526496819171953e_term).
% 20.23/7.37  thf(func_def_608, type, list_u877304756163299468e_term: list_P1316552470764441098e_term > nat > (produc6241069584506657477e_term > option6357759511663192854e_term) > list_P1316552470764441098e_term).
% 20.23/7.37  thf(func_def_609, type, list_u3533491785856317309e_term: list_P1743416141875011707e_term > nat > (produc8551481072490612790e_term > option6357759511663192854e_term) > list_P1743416141875011707e_term).
% 20.23/7.37  thf(func_def_610, type, list_update_o: list_o > nat > $o > list_o).
% 20.23/7.37  thf(func_def_611, type, list_update_int: list_int > nat > int > list_int).
% 20.23/7.37  thf(func_def_612, type, list_update_nat: list_nat > nat > nat > list_nat).
% 20.23/7.37  thf(func_def_613, type, list_u1133519416628930960nteger: list_P5311841565141990158nteger > nat > produc8763457246119570046nteger > list_P5311841565141990158nteger).
% 20.23/7.37  thf(func_def_614, type, list_u4780935413889332127nt_int: list_P8915022641806594461nt_int > nat > produc7773217078559923341nt_int > list_P8915022641806594461nt_int).
% 20.23/7.37  thf(func_def_615, type, list_u234853988314817064nteger: list_P7828571989066258726nteger > nat > produc1908205239877642774nteger > list_P7828571989066258726nteger).
% 20.23/7.37  thf(func_def_616, type, list_u7736365598306452245nt_int: list_P651320350408439699nt_int > nat > produc2285326912895808259nt_int > list_P651320350408439699nt_int).
% 20.23/7.37  thf(func_def_617, type, list_u2254550707601501961nteger: list_P5578671422887162913nteger > nat > produc8923325533196201883nteger > list_P5578671422887162913nteger).
% 20.23/7.37  thf(func_def_618, type, list_u3002344382305578791nt_int: list_P5707943133018811711nt_int > nat > product_prod_int_int > list_P5707943133018811711nt_int).
% 20.23/7.37  thf(func_def_619, type, list_u6180841689913720943at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 20.23/7.37  thf(func_def_620, type, list_u4696772448584712917at_nat: list_P5464809261938338413at_nat > nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat).
% 20.23/7.37  thf(func_def_621, type, list_update_real: list_real > nat > real > list_real).
% 20.23/7.37  thf(func_def_622, type, list_update_set_nat: list_set_nat > nat > set_nat > list_set_nat).
% 20.23/7.37  thf(func_def_623, type, list_u1324408373059187874T_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_624, type, listrel1_o: set_Product_prod_o_o > set_Pr6227168374412355847list_o).
% 20.23/7.37  thf(func_def_625, type, listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int).
% 20.23/7.37  thf(func_def_626, type, listrel1_nat: set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat).
% 20.23/7.37  thf(func_def_627, type, listrel1_VEBT_VEBT: set_Pr6192946355708809607T_VEBT > set_Pr1916528119006554503T_VEBT).
% 20.23/7.37  thf(func_def_628, type, listrel1p_int: (int > int > $o) > list_int > list_int > $o).
% 20.23/7.37  thf(func_def_629, type, nth_o: list_o > nat > $o).
% 20.23/7.37  thf(func_def_630, type, nth_int: list_int > nat > int).
% 20.23/7.37  thf(func_def_631, type, nth_list_nat: list_list_nat > nat > list_nat).
% 20.23/7.37  thf(func_def_632, type, nth_nat: list_nat > nat > nat).
% 20.23/7.37  thf(func_def_633, type, nth_num: list_num > nat > num).
% 20.23/7.37  thf(func_def_634, type, nth_Product_prod_o_o: list_P4002435161011370285od_o_o > nat > product_prod_o_o).
% 20.23/7.37  thf(func_def_635, type, nth_Pr1649062631805364268_o_int: list_P3795440434834930179_o_int > nat > product_prod_o_int).
% 20.23/7.37  thf(func_def_636, type, nth_Pr5826913651314560976_o_nat: list_P6285523579766656935_o_nat > nat > product_prod_o_nat).
% 20.23/7.37  thf(func_def_637, type, nth_Pr6777367263587873994T_VEBT: list_P7495141550334521929T_VEBT > nat > produc2504756804600209347T_VEBT).
% 20.23/7.37  thf(func_def_638, type, nth_Pr112076138515278198_nat_o: list_P7333126701944960589_nat_o > nat > product_prod_nat_o).
% 20.23/7.37  thf(func_def_639, type, nth_Pr3440142176431000676at_int: list_P3521021558325789923at_int > nat > product_prod_nat_int).
% 20.23/7.37  thf(func_def_640, type, nth_Pr7617993195940197384at_nat: list_P6011104703257516679at_nat > nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_641, type, nth_Pr744662078594809490T_VEBT: list_P5647936690300460905T_VEBT > nat > produc8025551001238799321T_VEBT).
% 20.23/7.37  thf(func_def_642, type, nth_Pr4606735188037164562VEBT_o: list_P3126845725202233233VEBT_o > nat > produc334124729049499915VEBT_o).
% 20.23/7.37  thf(func_def_643, type, nth_Pr6837108013167703752BT_int: list_P4547456442757143711BT_int > nat > produc4894624898956917775BT_int).
% 20.23/7.37  thf(func_def_644, type, nth_Pr1791586995822124652BT_nat: list_P7037539587688870467BT_nat > nat > produc9072475918466114483BT_nat).
% 20.23/7.37  thf(func_def_645, type, nth_Pr4953567300277697838T_VEBT: list_P7413028617227757229T_VEBT > nat > produc8243902056947475879T_VEBT).
% 20.23/7.37  thf(func_def_646, type, nth_real: list_real > nat > real).
% 20.23/7.37  thf(func_def_647, type, nth_set_nat: list_set_nat > nat > set_nat).
% 20.23/7.37  thf(func_def_648, type, nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT).
% 20.23/7.37  thf(func_def_649, type, product_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 20.23/7.37  thf(func_def_650, type, product_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 20.23/7.37  thf(func_def_651, type, product_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 20.23/7.37  thf(func_def_652, type, product_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 20.23/7.37  thf(func_def_653, type, product_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 20.23/7.37  thf(func_def_654, type, produc7156399406898700509T_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 20.23/7.37  thf(func_def_655, type, product_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 20.23/7.37  thf(func_def_656, type, produc7292646706713671643BT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 20.23/7.37  thf(func_def_657, type, produc7295137177222721919BT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 20.23/7.37  thf(func_def_658, type, produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 20.23/7.37  thf(func_def_659, type, remdups_nat: list_nat > list_nat).
% 20.23/7.37  thf(func_def_660, type, replicate_o: nat > $o > list_o).
% 20.23/7.37  thf(func_def_661, type, replicate_int: nat > int > list_int).
% 20.23/7.37  thf(func_def_662, type, replicate_nat: nat > nat > list_nat).
% 20.23/7.37  thf(func_def_663, type, replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat).
% 20.23/7.37  thf(func_def_664, type, replic2264142908078655527at_nat: nat > produc3843707927480180839at_nat > list_P5464809261938338413at_nat).
% 20.23/7.37  thf(func_def_665, type, replicate_real: nat > real > list_real).
% 20.23/7.37  thf(func_def_666, type, replicate_set_nat: nat > set_nat > list_set_nat).
% 20.23/7.37  thf(func_def_667, type, replicate_VEBT_VEBT: nat > vEBT_VEBT > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_668, type, sorted_wrt_int: (int > int > $o) > list_int > $o).
% 20.23/7.37  thf(func_def_669, type, sorted_wrt_nat: (nat > nat > $o) > list_nat > $o).
% 20.23/7.37  thf(func_def_670, type, take_nat: nat > list_nat > list_nat).
% 20.23/7.37  thf(func_def_671, type, upt: nat > nat > list_nat).
% 20.23/7.37  thf(func_def_672, type, upto: int > int > list_int).
% 20.23/7.37  thf(func_def_673, type, upto_aux: int > int > list_int > list_int).
% 20.23/7.37  thf(func_def_674, type, upto_rel: product_prod_int_int > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_675, type, zip_Co8729459035503499408nteger: list_C878401137130745250e_term > list_P5578671422887162913nteger > list_P5311841565141990158nteger).
% 20.23/7.37  thf(func_def_676, type, zip_in8766932505889695135nt_int: list_i8448526496819171953e_term > list_P5707943133018811711nt_int > list_P8915022641806594461nt_int).
% 20.23/7.37  thf(func_def_677, type, zip_Pr8292346330294042792nteger: list_P1316552470764441098e_term > list_P5578671422887162913nteger > list_P7828571989066258726nteger).
% 20.23/7.37  thf(func_def_678, type, zip_Pr4168994715204986005nt_int: list_P1743416141875011707e_term > list_P5707943133018811711nt_int > list_P651320350408439699nt_int).
% 20.23/7.37  thf(func_def_679, type, zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o).
% 20.23/7.37  thf(func_def_680, type, zip_o_int: list_o > list_int > list_P3795440434834930179_o_int).
% 20.23/7.37  thf(func_def_681, type, zip_o_nat: list_o > list_nat > list_P6285523579766656935_o_nat).
% 20.23/7.37  thf(func_def_682, type, zip_o_real: list_o > list_real > list_P5232166724548748803o_real).
% 20.23/7.37  thf(func_def_683, type, zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT).
% 20.23/7.37  thf(func_def_684, type, zip_int_int: list_int > list_int > list_P5707943133018811711nt_int).
% 20.23/7.37  thf(func_def_685, type, zip_int_real: list_int > list_real > list_P6863124054624500543t_real).
% 20.23/7.37  thf(func_def_686, type, zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT).
% 20.23/7.37  thf(func_def_687, type, zip_nat_o: list_nat > list_o > list_P7333126701944960589_nat_o).
% 20.23/7.37  thf(func_def_688, type, zip_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat).
% 20.23/7.37  thf(func_def_689, type, zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT).
% 20.23/7.37  thf(func_def_690, type, zip_Pr4664179122662387191at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P8469869581646625389at_nat).
% 20.23/7.37  thf(func_def_691, type, zip_real_o: list_real > list_o > list_P3595434254542482545real_o).
% 20.23/7.37  thf(func_def_692, type, zip_real_int: list_real > list_int > list_P4344331454722006975al_int).
% 20.23/7.37  thf(func_def_693, type, zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat).
% 20.23/7.37  thf(func_def_694, type, zip_real_real: list_real > list_real > list_P8689742595348180415l_real).
% 20.23/7.37  thf(func_def_695, type, zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT).
% 20.23/7.37  thf(func_def_696, type, zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat).
% 20.23/7.37  thf(func_def_697, type, zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o).
% 20.23/7.37  thf(func_def_698, type, zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int).
% 20.23/7.37  thf(func_def_699, type, zip_VEBT_VEBT_nat: list_VEBT_VEBT > list_nat > list_P7037539587688870467BT_nat).
% 20.23/7.37  thf(func_def_700, type, zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real).
% 20.23/7.37  thf(func_def_701, type, zip_VE537291747668921783T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT).
% 20.23/7.37  thf(func_def_702, type, suc: nat > nat).
% 20.23/7.37  thf(func_def_703, type, compow_nat_nat: nat > (nat > nat) > nat > nat).
% 20.23/7.37  thf(func_def_704, type, case_nat_o: $o > (nat > $o) > nat > $o).
% 20.23/7.37  thf(func_def_705, type, case_nat_nat: nat > (nat > nat) > nat > nat).
% 20.23/7.37  thf(func_def_706, type, case_nat_option_num: option_num > (nat > option_num) > nat > option_num).
% 20.23/7.37  thf(func_def_707, type, pred: nat > nat).
% 20.23/7.37  thf(func_def_708, type, semiring_1_Nats_int: set_int).
% 20.23/7.37  thf(func_def_709, type, semiri4939895301339042750nteger: nat > code_integer).
% 20.23/7.37  thf(func_def_710, type, semiri8010041392384452111omplex: nat > complex).
% 20.23/7.37  thf(func_def_711, type, semiri4216267220026989637d_enat: nat > extended_enat).
% 20.23/7.37  thf(func_def_712, type, semiri1314217659103216013at_int: nat > int).
% 20.23/7.37  thf(func_def_713, type, semiri1316708129612266289at_nat: nat > nat).
% 20.23/7.37  thf(func_def_714, type, semiri681578069525770553at_rat: nat > rat).
% 20.23/7.37  thf(func_def_715, type, semiri5074537144036343181t_real: nat > real).
% 20.23/7.37  thf(func_def_716, type, semiri2816024913162550771omplex: (complex > complex) > nat > complex > complex).
% 20.23/7.37  thf(func_def_717, type, semiri8420488043553186161ux_int: (int > int) > nat > int > int).
% 20.23/7.37  thf(func_def_718, type, semiri8422978514062236437ux_nat: (nat > nat) > nat > nat > nat).
% 20.23/7.37  thf(func_def_719, type, semiri7787848453975740701ux_rat: (rat > rat) > nat > rat > rat).
% 20.23/7.37  thf(func_def_720, type, semiri7260567687927622513x_real: (real > real) > nat > real > real).
% 20.23/7.37  thf(func_def_721, type, size_size_list_o: list_o > nat).
% 20.23/7.37  thf(func_def_722, type, size_s3451745648224563538omplex: list_complex > nat).
% 20.23/7.37  thf(func_def_723, type, size_size_list_int: list_int > nat).
% 20.23/7.37  thf(func_def_724, type, size_s3023201423986296836st_nat: list_list_nat > nat).
% 20.23/7.37  thf(func_def_725, type, size_size_list_nat: list_nat > nat).
% 20.23/7.37  thf(func_def_726, type, size_size_list_num: list_num > nat).
% 20.23/7.37  thf(func_def_727, type, size_s1515746228057227161od_o_o: list_P4002435161011370285od_o_o > nat).
% 20.23/7.37  thf(func_def_728, type, size_s2953683556165314199_o_int: list_P3795440434834930179_o_int > nat).
% 20.23/7.37  thf(func_def_729, type, size_s5443766701097040955_o_nat: list_P6285523579766656935_o_nat > nat).
% 20.23/7.37  thf(func_def_730, type, size_s4313452262239582901T_VEBT: list_P7495141550334521929T_VEBT > nat).
% 20.23/7.37  thf(func_def_731, type, size_s6491369823275344609_nat_o: list_P7333126701944960589_nat_o > nat).
% 20.23/7.37  thf(func_def_732, type, size_s2970893825323803983at_int: list_P3521021558325789923at_int > nat).
% 20.23/7.37  thf(func_def_733, type, size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat).
% 20.23/7.37  thf(func_def_734, type, size_s4762443039079500285T_VEBT: list_P5647936690300460905T_VEBT > nat).
% 20.23/7.37  thf(func_def_735, type, size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat).
% 20.23/7.37  thf(func_def_736, type, size_s3661962791536183091BT_int: list_P4547456442757143711BT_int > nat).
% 20.23/7.37  thf(func_def_737, type, size_s6152045936467909847BT_nat: list_P7037539587688870467BT_nat > nat).
% 20.23/7.37  thf(func_def_738, type, size_s7466405169056248089T_VEBT: list_P7413028617227757229T_VEBT > nat).
% 20.23/7.37  thf(func_def_739, type, size_size_list_real: list_real > nat).
% 20.23/7.37  thf(func_def_740, type, size_s3254054031482475050et_nat: list_set_nat > nat).
% 20.23/7.37  thf(func_def_741, type, size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_742, type, size_size_num: num > nat).
% 20.23/7.37  thf(func_def_743, type, size_size_option_nat: option_nat > nat).
% 20.23/7.37  thf(func_def_744, type, size_size_option_num: option_num > nat).
% 20.23/7.37  thf(func_def_745, type, size_s170228958280169651at_nat: option4927543243414619207at_nat > nat).
% 20.23/7.37  thf(func_def_746, type, size_size_VEBT_VEBT: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_747, type, nat_list_encode: list_nat > nat).
% 20.23/7.37  thf(func_def_748, type, nat_list_encode_rel: list_nat > list_nat > $o).
% 20.23/7.37  thf(func_def_749, type, nat_prod_decode_aux: nat > nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_750, type, nat_pr5047031295181774490ux_rel: product_prod_nat_nat > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_751, type, nat_prod_encode: product_prod_nat_nat > nat).
% 20.23/7.37  thf(func_def_752, type, nat_set_decode: nat > set_nat).
% 20.23/7.37  thf(func_def_753, type, nat_set_encode: set_nat > nat).
% 20.23/7.37  thf(func_def_754, type, nat_triangle: nat > nat).
% 20.23/7.37  thf(func_def_755, type, root: nat > real > real).
% 20.23/7.37  thf(func_def_756, type, sqrt: real > real).
% 20.23/7.37  thf(func_def_757, type, bitM: num > num).
% 20.23/7.37  thf(func_def_758, type, inc: num > num).
% 20.23/7.37  thf(func_def_759, type, neg_nu7757733837767384882nteger: code_integer > code_integer).
% 20.23/7.37  thf(func_def_760, type, neg_nu6511756317524482435omplex: complex > complex).
% 20.23/7.37  thf(func_def_761, type, neg_nu3811975205180677377ec_int: int > int).
% 20.23/7.37  thf(func_def_762, type, neg_nu3179335615603231917ec_rat: rat > rat).
% 20.23/7.37  thf(func_def_763, type, neg_nu6075765906172075777c_real: real > real).
% 20.23/7.37  thf(func_def_764, type, neg_numeral_sub_int: num > num > int).
% 20.23/7.37  thf(func_def_765, type, bit0: num > num).
% 20.23/7.37  thf(func_def_766, type, bit1: num > num).
% 20.23/7.37  thf(func_def_767, type, one: num).
% 20.23/7.37  thf(func_def_768, type, case_num_option_num: option_num > (num > option_num) > (num > option_num) > num > option_num).
% 20.23/7.37  thf(func_def_769, type, size_num: num > nat).
% 20.23/7.37  thf(func_def_770, type, num_of_nat: nat > num).
% 20.23/7.37  thf(func_def_771, type, numera6620942414471956472nteger: num > code_integer).
% 20.23/7.37  thf(func_def_772, type, numera6690914467698888265omplex: num > complex).
% 20.23/7.37  thf(func_def_773, type, numera1916890842035813515d_enat: num > extended_enat).
% 20.23/7.37  thf(func_def_774, type, numeral_numeral_int: num > int).
% 20.23/7.37  thf(func_def_775, type, numeral_numeral_nat: num > nat).
% 20.23/7.37  thf(func_def_776, type, numeral_numeral_rat: num > rat).
% 20.23/7.37  thf(func_def_777, type, numeral_numeral_real: num > real).
% 20.23/7.37  thf(func_def_778, type, pred_numeral: num > nat).
% 20.23/7.37  thf(func_def_779, type, none_int: option_int).
% 20.23/7.37  thf(func_def_780, type, none_nat: option_nat).
% 20.23/7.37  thf(func_def_781, type, none_num: option_num).
% 20.23/7.37  thf(func_def_782, type, none_P5556105721700978146at_nat: option4927543243414619207at_nat).
% 20.23/7.37  thf(func_def_783, type, none_real: option_real).
% 20.23/7.37  thf(func_def_784, type, none_set_nat: option_set_nat).
% 20.23/7.37  thf(func_def_785, type, none_VEBT_VEBT: option_VEBT_VEBT).
% 20.23/7.37  thf(func_def_786, type, some_o: $o > option_o).
% 20.23/7.37  thf(func_def_787, type, some_int: int > option_int).
% 20.23/7.37  thf(func_def_788, type, some_nat: nat > option_nat).
% 20.23/7.37  thf(func_def_789, type, some_num: num > option_num).
% 20.23/7.37  thf(func_def_790, type, some_P7363390416028606310at_nat: product_prod_nat_nat > option4927543243414619207at_nat).
% 20.23/7.37  thf(func_def_791, type, some_VEBT_VEBT: vEBT_VEBT > option_VEBT_VEBT).
% 20.23/7.37  thf(func_def_792, type, case_o184042715313410164at_nat: $o > (product_prod_nat_nat > $o) > option4927543243414619207at_nat > $o).
% 20.23/7.37  thf(func_def_793, type, case_option_num_num: num > (num > num) > option_num > num).
% 20.23/7.37  thf(func_def_794, type, case_o6005452278849405969um_num: option_num > (num > option_num) > option_num > option_num).
% 20.23/7.37  thf(func_def_795, type, map_option_num_num: (num > num) > option_num > option_num).
% 20.23/7.37  thf(func_def_796, type, size_option_nat: (nat > nat) > option_nat > nat).
% 20.23/7.37  thf(func_def_797, type, size_option_num: (num > nat) > option_num > nat).
% 20.23/7.37  thf(func_def_798, type, size_o8335143837870341156at_nat: (product_prod_nat_nat > nat) > option4927543243414619207at_nat > nat).
% 20.23/7.37  thf(func_def_799, type, the_nat: option_nat > nat).
% 20.23/7.37  thf(func_def_800, type, the_num: option_num > num).
% 20.23/7.37  thf(func_def_801, type, the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_802, type, order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat).
% 20.23/7.37  thf(func_def_803, type, bot_bo5358457235160185703eger_o: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 20.23/7.37  thf(func_def_804, type, bot_bo1403522918969695512_int_o: (int > option6357759511663192854e_term) > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_805, type, bot_bo3000040243691356879eger_o: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o).
% 20.23/7.37  thf(func_def_806, type, bot_bo8662317086119403298_int_o: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_807, type, bot_bot_int_int_o: int > int > $o).
% 20.23/7.37  thf(func_def_808, type, bot_bot_int_o: int > $o).
% 20.23/7.37  thf(func_def_809, type, bot_bot_list_nat_o: list_nat > $o).
% 20.23/7.37  thf(func_def_810, type, bot_bot_nat_nat_o: nat > nat > $o).
% 20.23/7.37  thf(func_def_811, type, bot_bot_nat_o: nat > $o).
% 20.23/7.37  thf(func_def_812, type, bot_bo482883023278783056_nat_o: product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_813, type, bot_bot_real_o: real > $o).
% 20.23/7.37  thf(func_def_814, type, bot_bot_set_nat_o: set_nat > $o).
% 20.23/7.37  thf(func_def_815, type, bot_bo4199563552545308370d_enat: extended_enat).
% 20.23/7.37  thf(func_def_816, type, bot_bot_filter_nat: filter_nat).
% 20.23/7.37  thf(func_def_817, type, bot_bot_nat: nat).
% 20.23/7.37  thf(func_def_818, type, bot_bot_set_complex: set_complex).
% 20.23/7.37  thf(func_def_819, type, bot_bot_set_int: set_int).
% 20.23/7.37  thf(func_def_820, type, bot_bot_set_list_nat: set_list_nat).
% 20.23/7.37  thf(func_def_821, type, bot_bot_set_nat: set_nat).
% 20.23/7.37  thf(func_def_822, type, bot_bot_set_num: set_num).
% 20.23/7.37  thf(func_def_823, type, bot_bo3145834390647256904nteger: set_Pr8056137968301705908nteger).
% 20.23/7.37  thf(func_def_824, type, bot_bo4508923176915781079nt_int: set_Pr1872883991513573699nt_int).
% 20.23/7.37  thf(func_def_825, type, bot_bo5443222936135328352nteger: set_Pr1281608226676607948nteger).
% 20.23/7.37  thf(func_def_826, type, bot_bo572930865798478029nt_int: set_Pr9222295170931077689nt_int).
% 20.23/7.37  thf(func_def_827, type, bot_bo1796632182523588997nt_int: set_Pr958786334691620121nt_int).
% 20.23/7.37  thf(func_def_828, type, bot_bo2099793752762293965at_nat: set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_829, type, bot_bo228742789529271731at_nat: set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_830, type, bot_bot_set_rat: set_rat).
% 20.23/7.37  thf(func_def_831, type, bot_bot_set_real: set_real).
% 20.23/7.37  thf(func_def_832, type, bot_bot_set_set_int: set_set_int).
% 20.23/7.37  thf(func_def_833, type, bot_bot_set_set_nat: set_set_nat).
% 20.23/7.37  thf(func_def_834, type, bot_bo8194388402131092736T_VEBT: set_VEBT_VEBT).
% 20.23/7.37  thf(func_def_835, type, ord_less_int_o: (int > $o) > (int > $o) > $o).
% 20.23/7.37  thf(func_def_836, type, ord_less_nat_o: (nat > $o) > (nat > $o) > $o).
% 20.23/7.37  thf(func_def_837, type, ord_le549003669493604880_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 20.23/7.37  thf(func_def_838, type, ord_less_real_o: (real > $o) > (real > $o) > $o).
% 20.23/7.37  thf(func_def_839, type, ord_less_set_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 20.23/7.37  thf(func_def_840, type, ord_le6747313008572928689nteger: code_integer > code_integer > $o).
% 20.23/7.37  thf(func_def_841, type, ord_le72135733267957522d_enat: extended_enat > extended_enat > $o).
% 20.23/7.37  thf(func_def_842, type, ord_less_int: int > int > $o).
% 20.23/7.37  thf(func_def_843, type, ord_less_nat: nat > nat > $o).
% 20.23/7.37  thf(func_def_844, type, ord_less_num: num > num > $o).
% 20.23/7.37  thf(func_def_845, type, ord_less_rat: rat > rat > $o).
% 20.23/7.37  thf(func_def_846, type, ord_less_real: real > real > $o).
% 20.23/7.37  thf(func_def_847, type, ord_le1307284697595431911nteger: set_Code_integer > set_Code_integer > $o).
% 20.23/7.37  thf(func_def_848, type, ord_less_set_complex: set_complex > set_complex > $o).
% 20.23/7.37  thf(func_def_849, type, ord_less_set_int: set_int > set_int > $o).
% 20.23/7.37  thf(func_def_850, type, ord_less_set_nat: set_nat > set_nat > $o).
% 20.23/7.37  thf(func_def_851, type, ord_less_set_num: set_num > set_num > $o).
% 20.23/7.37  thf(func_def_852, type, ord_le7866589430770878221at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 20.23/7.37  thf(func_def_853, type, ord_le2604355607129572851at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 20.23/7.37  thf(func_def_854, type, ord_less_set_rat: set_rat > set_rat > $o).
% 20.23/7.37  thf(func_def_855, type, ord_less_set_real: set_real > set_real > $o).
% 20.23/7.37  thf(func_def_856, type, ord_less_set_set_int: set_set_int > set_set_int > $o).
% 20.23/7.37  thf(func_def_857, type, ord_less_set_set_nat: set_set_nat > set_set_nat > $o).
% 20.23/7.37  thf(func_def_858, type, ord_le3636971675376928563eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 20.23/7.37  thf(func_def_859, type, ord_le2124322318746777828_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 20.23/7.37  thf(func_def_860, type, ord_le4340812435750786203eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > $o).
% 20.23/7.37  thf(func_def_861, type, ord_le5643404153117327598_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > $o).
% 20.23/7.37  thf(func_def_862, type, ord_le6741204236512500942_int_o: (int > int > $o) > (int > int > $o) > $o).
% 20.23/7.37  thf(func_def_863, type, ord_less_eq_int_o: (int > $o) > (int > $o) > $o).
% 20.23/7.37  thf(func_def_864, type, ord_le2646555220125990790_nat_o: (nat > nat > $o) > (nat > nat > $o) > $o).
% 20.23/7.37  thf(func_def_865, type, ord_less_eq_nat_o: (nat > $o) > (nat > $o) > $o).
% 20.23/7.37  thf(func_def_866, type, ord_le704812498762024988_nat_o: (product_prod_nat_nat > $o) > (product_prod_nat_nat > $o) > $o).
% 20.23/7.37  thf(func_def_867, type, ord_less_eq_real_o: (real > $o) > (real > $o) > $o).
% 20.23/7.37  thf(func_def_868, type, ord_le3964352015994296041_nat_o: (set_nat > $o) > (set_nat > $o) > $o).
% 20.23/7.37  thf(func_def_869, type, ord_le3102999989581377725nteger: code_integer > code_integer > $o).
% 20.23/7.37  thf(func_def_870, type, ord_le2932123472753598470d_enat: extended_enat > extended_enat > $o).
% 20.23/7.37  thf(func_def_871, type, ord_le2510731241096832064er_nat: filter_nat > filter_nat > $o).
% 20.23/7.37  thf(func_def_872, type, ord_less_eq_int: int > int > $o).
% 20.23/7.37  thf(func_def_873, type, ord_less_eq_nat: nat > nat > $o).
% 20.23/7.37  thf(func_def_874, type, ord_less_eq_num: num > num > $o).
% 20.23/7.37  thf(func_def_875, type, ord_less_eq_rat: rat > rat > $o).
% 20.23/7.37  thf(func_def_876, type, ord_less_eq_real: real > real > $o).
% 20.23/7.37  thf(func_def_877, type, ord_less_eq_set_o: set_o > set_o > $o).
% 20.23/7.37  thf(func_def_878, type, ord_le7084787975880047091nteger: set_Code_integer > set_Code_integer > $o).
% 20.23/7.37  thf(func_def_879, type, ord_le211207098394363844omplex: set_complex > set_complex > $o).
% 20.23/7.37  thf(func_def_880, type, ord_less_eq_set_int: set_int > set_int > $o).
% 20.23/7.37  thf(func_def_881, type, ord_le6045566169113846134st_nat: set_list_nat > set_list_nat > $o).
% 20.23/7.37  thf(func_def_882, type, ord_less_eq_set_nat: set_nat > set_nat > $o).
% 20.23/7.37  thf(func_def_883, type, ord_less_eq_set_num: set_num > set_num > $o).
% 20.23/7.37  thf(func_def_884, type, ord_le3216752416896350996nteger: set_Pr8056137968301705908nteger > set_Pr8056137968301705908nteger > $o).
% 20.23/7.37  thf(func_def_885, type, ord_le135402666524580259nt_int: set_Pr1872883991513573699nt_int > set_Pr1872883991513573699nt_int > $o).
% 20.23/7.37  thf(func_def_886, type, ord_le653643898420964396nteger: set_Pr1281608226676607948nteger > set_Pr1281608226676607948nteger > $o).
% 20.23/7.37  thf(func_def_887, type, ord_le8725513860283290265nt_int: set_Pr9222295170931077689nt_int > set_Pr9222295170931077689nt_int > $o).
% 20.23/7.37  thf(func_def_888, type, ord_le2843351958646193337nt_int: set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int > $o).
% 20.23/7.37  thf(func_def_889, type, ord_le3146513528884898305at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o).
% 20.23/7.37  thf(func_def_890, type, ord_le1268244103169919719at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > $o).
% 20.23/7.37  thf(func_def_891, type, ord_less_eq_set_rat: set_rat > set_rat > $o).
% 20.23/7.37  thf(func_def_892, type, ord_less_eq_set_real: set_real > set_real > $o).
% 20.23/7.37  thf(func_def_893, type, ord_le4403425263959731960et_int: set_set_int > set_set_int > $o).
% 20.23/7.37  thf(func_def_894, type, ord_le6893508408891458716et_nat: set_set_nat > set_set_nat > $o).
% 20.23/7.37  thf(func_def_895, type, ord_le4337996190870823476T_VEBT: set_VEBT_VEBT > set_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_896, type, ord_max_int: int > int > int).
% 20.23/7.37  thf(func_def_897, type, ord_max_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_898, type, ord_min_int: int > int > int).
% 20.23/7.37  thf(func_def_899, type, ord_min_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_900, type, order_Greatest_nat: (nat > $o) > nat).
% 20.23/7.37  thf(func_def_901, type, order_mono_nat_nat: (nat > nat) > $o).
% 20.23/7.37  thf(func_def_902, type, order_5726023648592871131at_nat: (nat > nat) > $o).
% 20.23/7.37  thf(func_def_903, type, order_7092887310737990675l_real: (real > real) > $o).
% 20.23/7.37  thf(func_def_904, type, top_top_set_int: set_int).
% 20.23/7.37  thf(func_def_905, type, top_top_set_nat: set_nat).
% 20.23/7.37  thf(func_def_906, type, top_top_set_real: set_real).
% 20.23/7.37  thf(func_def_907, type, power_8256067586552552935nteger: code_integer > nat > code_integer).
% 20.23/7.37  thf(func_def_908, type, power_power_complex: complex > nat > complex).
% 20.23/7.37  thf(func_def_909, type, power_8040749407984259932d_enat: extended_enat > nat > extended_enat).
% 20.23/7.37  thf(func_def_910, type, power_power_int: int > nat > int).
% 20.23/7.37  thf(func_def_911, type, power_power_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_912, type, power_power_rat: rat > nat > rat).
% 20.23/7.37  thf(func_def_913, type, power_power_real: real > nat > real).
% 20.23/7.37  thf(func_def_914, type, produc6137756002093451184nteger: (code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > produc8763457246119570046nteger).
% 20.23/7.37  thf(func_def_915, type, produc4305682042979456191nt_int: (int > option6357759511663192854e_term) > product_prod_int_int > produc7773217078559923341nt_int).
% 20.23/7.37  thf(func_def_916, type, produc4035269172776083154on_nat: (nat > nat > $o) > produc4953844613479565601on_nat > produc2233624965454879586on_nat).
% 20.23/7.37  thf(func_def_917, type, produc8929957630744042906on_nat: (nat > nat > nat) > produc4953844613479565601on_nat > produc8306885398267862888on_nat).
% 20.23/7.37  thf(func_def_918, type, produc3576312749637752826on_num: (num > num > $o) > produc3447558737645232053on_num > produc7036089656553540234on_num).
% 20.23/7.37  thf(func_def_919, type, produc5778274026573060048on_num: (num > num > num) > produc3447558737645232053on_num > produc1193250871479095198on_num).
% 20.23/7.37  thf(func_def_920, type, produc8603105652947943368nteger: (produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > produc1908205239877642774nteger).
% 20.23/7.37  thf(func_def_921, type, produc5700946648718959541nt_int: (produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > produc2285326912895808259nt_int).
% 20.23/7.37  thf(func_def_922, type, produc3994169339658061776at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > produc6121120109295599847at_nat > produc5491161045314408544at_nat).
% 20.23/7.37  thf(func_def_923, type, produc2899441246263362727at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > produc6121120109295599847at_nat > produc5542196010084753463at_nat).
% 20.23/7.37  thf(func_def_924, type, product_Pair_o_o: $o > $o > product_prod_o_o).
% 20.23/7.37  thf(func_def_925, type, product_Pair_o_int: $o > int > product_prod_o_int).
% 20.23/7.37  thf(func_def_926, type, product_Pair_o_nat: $o > nat > product_prod_o_nat).
% 20.23/7.37  thf(func_def_927, type, product_Pair_o_real: $o > real > product_prod_o_real).
% 20.23/7.37  thf(func_def_928, type, produc2982872950893828659T_VEBT: $o > vEBT_VEBT > produc2504756804600209347T_VEBT).
% 20.23/7.37  thf(func_def_929, type, produc6677183202524767010eger_o: code_integer > $o > produc6271795597528267376eger_o).
% 20.23/7.37  thf(func_def_930, type, produc1086072967326762835nteger: code_integer > code_integer > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_931, type, product_Pair_int_int: int > int > product_prod_int_int).
% 20.23/7.37  thf(func_def_932, type, produc801115645435158769t_real: int > real > produc679980390762269497t_real).
% 20.23/7.37  thf(func_def_933, type, produc3329399203697025711T_VEBT: int > vEBT_VEBT > produc1531783533982839933T_VEBT).
% 20.23/7.37  thf(func_def_934, type, produc8435520187683070743list_o: list_o > list_o > produc7102631898165422375list_o).
% 20.23/7.37  thf(func_def_935, type, produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int).
% 20.23/7.37  thf(func_def_936, type, produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat).
% 20.23/7.37  thf(func_def_937, type, produc3897820843166775703T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > produc9211091688327510695T_VEBT).
% 20.23/7.37  thf(func_def_938, type, product_Pair_nat_o: nat > $o > product_prod_nat_o).
% 20.23/7.37  thf(func_def_939, type, product_Pair_nat_int: nat > int > product_prod_nat_int).
% 20.23/7.37  thf(func_def_940, type, product_Pair_nat_nat: nat > nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_941, type, product_Pair_nat_num: nat > num > product_prod_nat_num).
% 20.23/7.37  thf(func_def_942, type, produc599794634098209291T_VEBT: nat > vEBT_VEBT > produc8025551001238799321T_VEBT).
% 20.23/7.37  thf(func_def_943, type, product_Pair_num_num: num > num > product_prod_num_num).
% 20.23/7.37  thf(func_def_944, type, produc5098337634421038937on_nat: option_nat > option_nat > produc4953844613479565601on_nat).
% 20.23/7.37  thf(func_def_945, type, produc8585076106096196333on_num: option_num > option_num > produc3447558737645232053on_num).
% 20.23/7.37  thf(func_def_946, type, produc488173922507101015at_nat: option4927543243414619207at_nat > option4927543243414619207at_nat > produc6121120109295599847at_nat).
% 20.23/7.37  thf(func_def_947, type, produc6161850002892822231at_nat: product_prod_nat_nat > product_prod_nat_nat > produc859450856879609959at_nat).
% 20.23/7.37  thf(func_def_948, type, product_Pair_real_o: real > $o > product_prod_real_o).
% 20.23/7.37  thf(func_def_949, type, produc3179012173361985393al_int: real > int > produc8786904178792722361al_int).
% 20.23/7.37  thf(func_def_950, type, produc3181502643871035669al_nat: real > nat > produc3741383161447143261al_nat).
% 20.23/7.37  thf(func_def_951, type, produc4511245868158468465l_real: real > real > produc2422161461964618553l_real).
% 20.23/7.37  thf(func_def_952, type, produc6931449550656315951T_VEBT: real > vEBT_VEBT > produc3757001726724277373T_VEBT).
% 20.23/7.37  thf(func_def_953, type, produc4532415448927165861et_nat: set_nat > set_nat > produc7819656566062154093et_nat).
% 20.23/7.37  thf(func_def_954, type, produc2922128104949294807at_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > produc3843707927480180839at_nat).
% 20.23/7.37  thf(func_def_955, type, produc9060074326276436823at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > produc1319942482725812455at_nat).
% 20.23/7.37  thf(func_def_956, type, produc8721562602347293563VEBT_o: vEBT_VEBT > $o > produc334124729049499915VEBT_o).
% 20.23/7.37  thf(func_def_957, type, produc736041933913180425BT_int: vEBT_VEBT > int > produc4894624898956917775BT_int).
% 20.23/7.37  thf(func_def_958, type, produc738532404422230701BT_nat: vEBT_VEBT > nat > produc9072475918466114483BT_nat).
% 20.23/7.37  thf(func_def_959, type, produc8117437818029410057T_real: vEBT_VEBT > real > produc5170161368751668367T_real).
% 20.23/7.37  thf(func_def_960, type, produc537772716801021591T_VEBT: vEBT_VEBT > vEBT_VEBT > produc8243902056947475879T_VEBT).
% 20.23/7.37  thf(func_def_961, type, produc457027306803732586at_nat: set_nat > (nat > set_nat) > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_962, type, produc6499014454317279255nteger: (code_integer > code_integer) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_963, type, produc127349428274296955eger_o: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc8763457246119570046nteger > $o).
% 20.23/7.37  thf(func_def_964, type, produc3906647086178084059nteger: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > produc8763457246119570046nteger) > produc8763457246119570046nteger > produc8763457246119570046nteger).
% 20.23/7.37  thf(func_def_965, type, produc8604463032469472703et_int: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_int) > produc8763457246119570046nteger > set_int).
% 20.23/7.37  thf(func_def_966, type, produc3558942015123893603et_nat: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_nat) > produc8763457246119570046nteger > set_nat).
% 20.23/7.37  thf(func_def_967, type, produc815715089573277247t_real: ((code_integer > option6357759511663192854e_term) > produc8923325533196201883nteger > set_real) > produc8763457246119570046nteger > set_real).
% 20.23/7.37  thf(func_def_968, type, produc2558449545302689196_int_o: ((int > option6357759511663192854e_term) > product_prod_int_int > $o) > produc7773217078559923341nt_int > $o).
% 20.23/7.37  thf(func_def_969, type, produc5122537100556696953nt_int: ((int > option6357759511663192854e_term) > product_prod_int_int > produc7773217078559923341nt_int) > produc7773217078559923341nt_int > produc7773217078559923341nt_int).
% 20.23/7.37  thf(func_def_970, type, produc8289552606927098482et_nat: ((int > option6357759511663192854e_term) > product_prod_int_int > set_nat) > produc7773217078559923341nt_int > set_nat).
% 20.23/7.37  thf(func_def_971, type, produc8709739885379107790t_real: ((int > option6357759511663192854e_term) > product_prod_int_int > set_real) > produc7773217078559923341nt_int > set_real).
% 20.23/7.37  thf(func_def_972, type, produc6253627499356882019eger_o: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > $o) > produc1908205239877642774nteger > $o).
% 20.23/7.37  thf(func_def_973, type, produc6512950862096126219nteger: ((produc6241069584506657477e_term > option6357759511663192854e_term) > produc8923325533196201883nteger > produc1908205239877642774nteger) > produc1908205239877642774nteger > produc1908205239877642774nteger).
% 20.23/7.37  thf(func_def_974, type, produc1573362020775583542_int_o: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > $o) > produc2285326912895808259nt_int > $o).
% 20.23/7.37  thf(func_def_975, type, produc8492565224438309093nt_int: ((produc8551481072490612790e_term > option6357759511663192854e_term) > product_prod_int_int > produc2285326912895808259nt_int) > produc2285326912895808259nt_int > produc2285326912895808259nt_int).
% 20.23/7.37  thf(func_def_976, type, produc1553301316500091796er_int: (code_integer > code_integer > int) > produc8923325533196201883nteger > int).
% 20.23/7.37  thf(func_def_977, type, produc1555791787009142072er_nat: (code_integer > code_integer > nat) > produc8923325533196201883nteger > nat).
% 20.23/7.37  thf(func_def_978, type, produc9125791028180074456eger_o: (code_integer > code_integer > produc6271795597528267376eger_o) > produc8923325533196201883nteger > produc6271795597528267376eger_o).
% 20.23/7.37  thf(func_def_979, type, produc6916734918728496179nteger: (code_integer > code_integer > produc8923325533196201883nteger) > produc8923325533196201883nteger > produc8923325533196201883nteger).
% 20.23/7.37  thf(func_def_980, type, produc6771430404735790350plex_o: (complex > complex > $o) > produc4411394909380815293omplex > $o).
% 20.23/7.37  thf(func_def_981, type, produc4947309494688390418_int_o: (int > int > $o) > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_982, type, produc4245557441103728435nt_int: (int > int > product_prod_int_int) > product_prod_int_int > product_prod_int_int).
% 20.23/7.37  thf(func_def_983, type, produc73460835934605544et_int: (int > int > set_int) > product_prod_int_int > set_int).
% 20.23/7.37  thf(func_def_984, type, produc4251311855443802252et_nat: (int > int > set_nat) > product_prod_int_int > set_nat).
% 20.23/7.37  thf(func_def_985, type, produc1656060378719767003at_nat: (int > int > set_Pr1261947904930325089at_nat) > product_prod_int_int > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_986, type, produc6452406959799940328t_real: (int > int > set_real) > product_prod_int_int > set_real).
% 20.23/7.37  thf(func_def_987, type, produc5233655623923918146et_nat: (int > int > set_set_nat) > product_prod_int_int > set_set_nat).
% 20.23/7.37  thf(func_def_988, type, produc8739625826339149834_nat_o: (nat > nat > product_prod_nat_nat > $o) > product_prod_nat_nat > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_989, 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).
% 20.23/7.37  thf(func_def_990, type, produc6081775807080527818_nat_o: (nat > nat > $o) > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_991, type, produc1917071388513777916omplex: (nat > nat > complex) > product_prod_nat_nat > complex).
% 20.23/7.37  thf(func_def_992, type, produc2676513652042109336d_enat: (nat > nat > extended_enat) > product_prod_nat_nat > extended_enat).
% 20.23/7.37  thf(func_def_993, type, produc6840382203811409530at_int: (nat > nat > int) > product_prod_nat_nat > int).
% 20.23/7.37  thf(func_def_994, type, produc6842872674320459806at_nat: (nat > nat > nat) > product_prod_nat_nat > nat).
% 20.23/7.37  thf(func_def_995, type, produc2626176000494625587at_nat: (nat > nat > product_prod_nat_nat) > product_prod_nat_nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_996, type, produc6207742614233964070at_rat: (nat > nat > rat) > product_prod_nat_nat > rat).
% 20.23/7.37  thf(func_def_997, type, produc1703576794950452218t_real: (nat > nat > real) > product_prod_nat_nat > real).
% 20.23/7.37  thf(func_def_998, type, produc478579273971653890on_num: (nat > num > option_num) > product_prod_nat_num > option_num).
% 20.23/7.37  thf(func_def_999, type, produc5414030515140494994real_o: (real > real > $o) > produc2422161461964618553l_real > $o).
% 20.23/7.37  thf(func_def_1000, type, product_fst_int_int: product_prod_int_int > int).
% 20.23/7.37  thf(func_def_1001, type, product_fst_nat_nat: product_prod_nat_nat > nat).
% 20.23/7.37  thf(func_def_1002, type, product_snd_int_int: product_prod_int_int > int).
% 20.23/7.37  thf(func_def_1003, type, product_snd_nat_nat: product_prod_nat_nat > nat).
% 20.23/7.37  thf(func_def_1004, type, fract: int > int > rat).
% 20.23/7.37  thf(func_def_1005, type, frct: product_prod_int_int > rat).
% 20.23/7.37  thf(func_def_1006, type, rep_Rat: rat > product_prod_int_int).
% 20.23/7.37  thf(func_def_1007, type, field_5140801741446780682s_real: set_real).
% 20.23/7.37  thf(func_def_1008, type, normalize: product_prod_int_int > product_prod_int_int).
% 20.23/7.37  thf(func_def_1009, type, of_int: int > rat).
% 20.23/7.37  thf(func_def_1010, type, positive: rat > $o).
% 20.23/7.37  thf(func_def_1011, type, quotient_of: rat > product_prod_int_int).
% 20.23/7.37  thf(func_def_1012, type, real_V3694042436643373181omplex: complex > complex > real).
% 20.23/7.37  thf(func_def_1013, type, real_V975177566351809787t_real: real > real > real).
% 20.23/7.37  thf(func_def_1014, type, real_V1022390504157884413omplex: complex > real).
% 20.23/7.37  thf(func_def_1015, type, real_V7735802525324610683m_real: real > real).
% 20.23/7.37  thf(func_def_1016, type, real_V4546457046886955230omplex: real > complex).
% 20.23/7.37  thf(func_def_1017, type, real_V1803761363581548252l_real: real > real).
% 20.23/7.37  thf(func_def_1018, type, field_nat: set_Pr1261947904930325089at_nat > set_nat).
% 20.23/7.37  thf(func_def_1019, type, divide6298287555418463151nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_1020, type, divide1717551699836669952omplex: complex > complex > complex).
% 20.23/7.37  thf(func_def_1021, type, divide_divide_int: int > int > int).
% 20.23/7.37  thf(func_def_1022, type, divide_divide_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_1023, type, divide_divide_rat: rat > rat > rat).
% 20.23/7.37  thf(func_def_1024, type, divide_divide_real: real > real > real).
% 20.23/7.37  thf(func_def_1025, type, dvd_dvd_Code_integer: code_integer > code_integer > $o).
% 20.23/7.37  thf(func_def_1026, type, dvd_dvd_complex: complex > complex > $o).
% 20.23/7.37  thf(func_def_1027, type, dvd_dvd_int: int > int > $o).
% 20.23/7.37  thf(func_def_1028, type, dvd_dvd_nat: nat > nat > $o).
% 20.23/7.37  thf(func_def_1029, type, dvd_dvd_rat: rat > rat > $o).
% 20.23/7.37  thf(func_def_1030, type, dvd_dvd_real: real > real > $o).
% 20.23/7.37  thf(func_def_1031, type, modulo364778990260209775nteger: code_integer > code_integer > code_integer).
% 20.23/7.37  thf(func_def_1032, type, modulo_modulo_int: int > int > int).
% 20.23/7.37  thf(func_def_1033, type, modulo_modulo_nat: nat > nat > nat).
% 20.23/7.37  thf(func_def_1034, type, zero_n356916108424825756nteger: $o > code_integer).
% 20.23/7.37  thf(func_def_1035, type, zero_n1201886186963655149omplex: $o > complex).
% 20.23/7.37  thf(func_def_1036, type, zero_n2684676970156552555ol_int: $o > int).
% 20.23/7.37  thf(func_def_1037, type, zero_n2687167440665602831ol_nat: $o > nat).
% 20.23/7.37  thf(func_def_1038, type, zero_n2052037380579107095ol_rat: $o > rat).
% 20.23/7.37  thf(func_def_1039, type, zero_n3304061248610475627l_real: $o > real).
% 20.23/7.37  thf(func_def_1040, type, suminf_complex: (nat > complex) > complex).
% 20.23/7.37  thf(func_def_1041, type, suminf_int: (nat > int) > int).
% 20.23/7.37  thf(func_def_1042, type, suminf_nat: (nat > nat) > nat).
% 20.23/7.37  thf(func_def_1043, type, suminf_real: (nat > real) > real).
% 20.23/7.37  thf(func_def_1044, type, summable_complex: (nat > complex) > $o).
% 20.23/7.37  thf(func_def_1045, type, summable_int: (nat > int) > $o).
% 20.23/7.37  thf(func_def_1046, type, summable_nat: (nat > nat) > $o).
% 20.23/7.37  thf(func_def_1047, type, summable_real: (nat > real) > $o).
% 20.23/7.37  thf(func_def_1048, type, sums_complex: (nat > complex) > complex > $o).
% 20.23/7.37  thf(func_def_1049, type, sums_int: (nat > int) > int > $o).
% 20.23/7.37  thf(func_def_1050, type, sums_nat: (nat > nat) > nat > $o).
% 20.23/7.37  thf(func_def_1051, type, sums_real: (nat > real) > real > $o).
% 20.23/7.37  thf(func_def_1052, type, collect_o: ($o > $o) > set_o).
% 20.23/7.37  thf(func_def_1053, type, collect_Code_integer: (code_integer > $o) > set_Code_integer).
% 20.23/7.37  thf(func_def_1054, type, collect_complex: (complex > $o) > set_complex).
% 20.23/7.37  thf(func_def_1055, type, collect_int: (int > $o) > set_int).
% 20.23/7.37  thf(func_def_1056, type, collect_list_o: (list_o > $o) > set_list_o).
% 20.23/7.37  thf(func_def_1057, type, collect_list_complex: (list_complex > $o) > set_list_complex).
% 20.23/7.37  thf(func_def_1058, type, collect_list_int: (list_int > $o) > set_list_int).
% 20.23/7.37  thf(func_def_1059, type, collect_list_nat: (list_nat > $o) > set_list_nat).
% 20.23/7.37  thf(func_def_1060, type, collec3343600615725829874at_nat: (list_P6011104703257516679at_nat > $o) > set_li5450038453877631591at_nat).
% 20.23/7.37  thf(func_def_1061, type, collec5608196760682091941T_VEBT: (list_VEBT_VEBT > $o) > set_list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1062, type, collect_nat: (nat > $o) > set_nat).
% 20.23/7.37  thf(func_def_1063, type, collect_num: (num > $o) > set_num).
% 20.23/7.37  thf(func_def_1064, type, collec8663557070575231912omplex: (produc4411394909380815293omplex > $o) > set_Pr5085853215250843933omplex).
% 20.23/7.37  thf(func_def_1065, type, collec213857154873943460nt_int: (product_prod_int_int > $o) > set_Pr958786334691620121nt_int).
% 20.23/7.37  thf(func_def_1066, type, collec3392354462482085612at_nat: (product_prod_nat_nat > $o) > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_1067, type, collec3799799289383736868l_real: (produc2422161461964618553l_real > $o) > set_Pr6218003697084177305l_real).
% 20.23/7.37  thf(func_def_1068, type, collec6321179662152712658at_nat: (produc3843707927480180839at_nat > $o) > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_1069, type, collect_rat: (rat > $o) > set_rat).
% 20.23/7.37  thf(func_def_1070, type, collect_real: (real > $o) > set_real).
% 20.23/7.37  thf(func_def_1071, type, collect_set_complex: (set_complex > $o) > set_set_complex).
% 20.23/7.37  thf(func_def_1072, type, collect_set_int: (set_int > $o) > set_set_int).
% 20.23/7.37  thf(func_def_1073, type, collect_set_nat: (set_nat > $o) > set_set_nat).
% 20.23/7.37  thf(func_def_1074, type, collec5514110066124741708at_nat: (set_Pr1261947904930325089at_nat > $o) > set_se7855581050983116737at_nat).
% 20.23/7.37  thf(func_def_1075, type, collect_VEBT_VEBT: (vEBT_VEBT > $o) > set_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1076, type, pow_nat: set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1077, type, image_int_int: (int > int) > set_int > set_int).
% 20.23/7.37  thf(func_def_1078, type, image_int_nat: (int > nat) > set_int > set_nat).
% 20.23/7.37  thf(func_def_1079, type, image_nat_int: (nat > int) > set_nat > set_int).
% 20.23/7.37  thf(func_def_1080, type, image_nat_nat: (nat > nat) > set_nat > set_nat).
% 20.23/7.37  thf(func_def_1081, type, image_nat_set_nat: (nat > set_nat) > set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1082, type, image_5971271580939081552omplex: (real > filter6041513312241820739omplex) > set_real > set_fi4554929511873752355omplex).
% 20.23/7.37  thf(func_def_1083, type, image_2178119161166701260l_real: (real > filter2146258269922977983l_real) > set_real > set_fi7789364187291644575l_real).
% 20.23/7.37  thf(func_def_1084, type, image_real_real: (real > real) > set_real > set_real).
% 20.23/7.37  thf(func_def_1085, type, insert_complex: complex > set_complex > set_complex).
% 20.23/7.37  thf(func_def_1086, type, insert_int: int > set_int > set_int).
% 20.23/7.37  thf(func_def_1087, type, insert_list_nat: list_nat > set_list_nat > set_list_nat).
% 20.23/7.37  thf(func_def_1088, type, insert_nat: nat > set_nat > set_nat).
% 20.23/7.37  thf(func_def_1089, type, insert_num: num > set_num > set_num).
% 20.23/7.37  thf(func_def_1090, type, insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_1091, type, insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_1092, type, insert_rat: rat > set_rat > set_rat).
% 20.23/7.37  thf(func_def_1093, type, insert_real: real > set_real > set_real).
% 20.23/7.37  thf(func_def_1094, type, insert_set_nat: set_nat > set_set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1095, type, insert_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > set_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1096, type, the_elem_int: set_int > int).
% 20.23/7.37  thf(func_def_1097, type, the_elem_nat: set_nat > nat).
% 20.23/7.37  thf(func_def_1098, type, the_el2281957884133575798at_nat: set_Pr1261947904930325089at_nat > product_prod_nat_nat).
% 20.23/7.37  thf(func_def_1099, type, the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat).
% 20.23/7.37  thf(func_def_1100, type, the_elem_real: set_real > real).
% 20.23/7.37  thf(func_def_1101, type, vimage_nat_nat: (nat > nat) > set_nat > set_nat).
% 20.23/7.37  thf(func_def_1102, type, set_fo1517530859248394432omplex: (nat > complex > complex) > nat > nat > complex > complex).
% 20.23/7.37  thf(func_def_1103, type, set_fo2581907887559384638at_int: (nat > int > int) > nat > nat > int > int).
% 20.23/7.37  thf(func_def_1104, type, set_fo2584398358068434914at_nat: (nat > nat > nat) > nat > nat > nat > nat).
% 20.23/7.37  thf(func_def_1105, type, set_fo1949268297981939178at_rat: (nat > rat > rat) > nat > nat > rat > rat).
% 20.23/7.37  thf(func_def_1106, type, set_fo3111899725591712190t_real: (nat > real > real) > nat > nat > real > real).
% 20.23/7.37  thf(func_def_1107, type, set_or1266510415728281911st_int: int > int > set_int).
% 20.23/7.37  thf(func_def_1108, type, set_or1269000886237332187st_nat: nat > nat > set_nat).
% 20.23/7.37  thf(func_def_1109, type, set_or7049704709247886629st_num: num > num > set_num).
% 20.23/7.37  thf(func_def_1110, type, set_or633870826150836451st_rat: rat > rat > set_rat).
% 20.23/7.37  thf(func_def_1111, type, set_or1222579329274155063t_real: real > real > set_real).
% 20.23/7.37  thf(func_def_1112, type, set_or370866239135849197et_int: set_int > set_int > set_set_int).
% 20.23/7.37  thf(func_def_1113, type, set_or4548717258645045905et_nat: set_nat > set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1114, type, set_or4662586982721622107an_int: int > int > set_int).
% 20.23/7.37  thf(func_def_1115, type, set_or4665077453230672383an_nat: nat > nat > set_nat).
% 20.23/7.37  thf(func_def_1116, type, set_ord_atLeast_nat: nat > set_nat).
% 20.23/7.37  thf(func_def_1117, type, set_ord_atMost_int: int > set_int).
% 20.23/7.37  thf(func_def_1118, type, set_ord_atMost_nat: nat > set_nat).
% 20.23/7.37  thf(func_def_1119, type, set_ord_atMost_num: num > set_num).
% 20.23/7.37  thf(func_def_1120, type, set_ord_atMost_rat: rat > set_rat).
% 20.23/7.37  thf(func_def_1121, type, set_ord_atMost_real: real > set_real).
% 20.23/7.37  thf(func_def_1122, type, set_or58775011639299419et_int: set_int > set_set_int).
% 20.23/7.37  thf(func_def_1123, type, set_or4236626031148496127et_nat: set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1124, type, set_or6656581121297822940st_int: int > int > set_int).
% 20.23/7.37  thf(func_def_1125, type, set_or6659071591806873216st_nat: nat > nat > set_nat).
% 20.23/7.37  thf(func_def_1126, type, set_or5832277885323065728an_int: int > int > set_int).
% 20.23/7.37  thf(func_def_1127, type, set_or5834768355832116004an_nat: nat > nat > set_nat).
% 20.23/7.37  thf(func_def_1128, type, set_or1633881224788618240n_real: real > real > set_real).
% 20.23/7.37  thf(func_def_1129, type, set_or1210151606488870762an_nat: nat > set_nat).
% 20.23/7.37  thf(func_def_1130, type, set_or5849166863359141190n_real: real > set_real).
% 20.23/7.37  thf(func_def_1131, type, set_ord_lessThan_int: int > set_int).
% 20.23/7.37  thf(func_def_1132, type, set_ord_lessThan_nat: nat > set_nat).
% 20.23/7.37  thf(func_def_1133, type, set_ord_lessThan_num: num > set_num).
% 20.23/7.37  thf(func_def_1134, type, set_ord_lessThan_rat: rat > set_rat).
% 20.23/7.37  thf(func_def_1135, type, set_or5984915006950818249n_real: real > set_real).
% 20.23/7.37  thf(func_def_1136, type, set_or890127255671739683et_nat: set_nat > set_set_nat).
% 20.23/7.37  thf(func_def_1137, type, topolo4422821103128117721l_real: filter_real > (real > real) > $o).
% 20.23/7.37  thf(func_def_1138, type, topolo5044208981011980120l_real: set_real > (real > real) > $o).
% 20.23/7.37  thf(func_def_1139, type, topolo4899668324122417113eq_int: (nat > int) > $o).
% 20.23/7.37  thf(func_def_1140, type, topolo4902158794631467389eq_nat: (nat > nat) > $o).
% 20.23/7.37  thf(func_def_1141, type, topolo1459490580787246023eq_num: (nat > num) > $o).
% 20.23/7.37  thf(func_def_1142, type, topolo4267028734544971653eq_rat: (nat > rat) > $o).
% 20.23/7.37  thf(func_def_1143, type, topolo6980174941875973593q_real: (nat > real) > $o).
% 20.23/7.37  thf(func_def_1144, type, topolo3100542954746470799et_int: (nat > set_int) > $o).
% 20.23/7.37  thf(func_def_1145, type, topolo2177554685111907308n_real: real > set_real > filter_real).
% 20.23/7.37  thf(func_def_1146, type, topolo2815343760600316023s_real: real > filter_real).
% 20.23/7.37  thf(func_def_1147, type, topolo4055970368930404560y_real: (nat > real) > $o).
% 20.23/7.37  thf(func_def_1148, type, topolo896644834953643431omplex: filter6041513312241820739omplex).
% 20.23/7.37  thf(func_def_1149, type, topolo1511823702728130853y_real: filter2146258269922977983l_real).
% 20.23/7.37  thf(func_def_1150, type, arccos: real > real).
% 20.23/7.37  thf(func_def_1151, type, arcosh_real: real > real).
% 20.23/7.37  thf(func_def_1152, type, arcsin: real > real).
% 20.23/7.37  thf(func_def_1153, type, arctan: real > real).
% 20.23/7.37  thf(func_def_1154, type, arsinh_real: real > real).
% 20.23/7.37  thf(func_def_1155, type, artanh_real: real > real).
% 20.23/7.37  thf(func_def_1156, type, cos_complex: complex > complex).
% 20.23/7.37  thf(func_def_1157, type, cos_real: real > real).
% 20.23/7.37  thf(func_def_1158, type, cos_coeff: nat > real).
% 20.23/7.37  thf(func_def_1159, type, cosh_real: real > real).
% 20.23/7.37  thf(func_def_1160, type, cot_real: real > real).
% 20.23/7.37  thf(func_def_1161, type, diffs_complex: (nat > complex) > nat > complex).
% 20.23/7.37  thf(func_def_1162, type, diffs_int: (nat > int) > nat > int).
% 20.23/7.37  thf(func_def_1163, type, diffs_rat: (nat > rat) > nat > rat).
% 20.23/7.37  thf(func_def_1164, type, diffs_real: (nat > real) > nat > real).
% 20.23/7.37  thf(func_def_1165, type, exp_complex: complex > complex).
% 20.23/7.37  thf(func_def_1166, type, exp_real: real > real).
% 20.23/7.37  thf(func_def_1167, type, ln_ln_real: real > real).
% 20.23/7.37  thf(func_def_1168, type, log: real > real > real).
% 20.23/7.37  thf(func_def_1169, type, pi: real).
% 20.23/7.37  thf(func_def_1170, type, powr_real: real > real > real).
% 20.23/7.37  thf(func_def_1171, type, sin_complex: complex > complex).
% 20.23/7.37  thf(func_def_1172, type, sin_real: real > real).
% 20.23/7.37  thf(func_def_1173, type, sin_coeff: nat > real).
% 20.23/7.37  thf(func_def_1174, type, sinh_real: real > real).
% 20.23/7.37  thf(func_def_1175, type, tan_complex: complex > complex).
% 20.23/7.37  thf(func_def_1176, type, tan_real: real > real).
% 20.23/7.37  thf(func_def_1177, type, tanh_complex: complex > complex).
% 20.23/7.37  thf(func_def_1178, type, tanh_real: real > real).
% 20.23/7.37  thf(func_def_1179, type, transi2905341329935302413cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_1180, type, transi6264000038957366511cl_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_1181, type, vEBT_Leaf: $o > $o > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1182, type, vEBT_Node: option4927543243414619207at_nat > nat > list_VEBT_VEBT > vEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1183, type, vEBT_size_VEBT: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1184, type, vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1185, type, vEBT_VEBT_high: nat > nat > nat).
% 20.23/7.37  thf(func_def_1186, type, vEBT_V5917875025757280293ildren: nat > list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1187, type, vEBT_VEBT_low: nat > nat > nat).
% 20.23/7.37  thf(func_def_1188, type, vEBT_VEBT_membermima: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1189, type, vEBT_V4351362008482014158ma_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 20.23/7.37  thf(func_def_1190, type, vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1191, type, vEBT_V5765760719290551771er_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 20.23/7.37  thf(func_def_1192, type, vEBT_VEBT_valid: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1193, type, vEBT_VEBT_valid_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 20.23/7.37  thf(func_def_1194, type, vEBT_invar_vebt: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1195, type, vEBT_set_vebt: vEBT_VEBT > set_nat).
% 20.23/7.37  thf(func_def_1196, type, vEBT_vebt_buildup: nat > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1197, type, vEBT_v4011308405150292612up_rel: nat > nat > $o).
% 20.23/7.37  thf(func_def_1198, type, vEBT_VEBT_bit_concat: nat > nat > nat > nat).
% 20.23/7.37  thf(func_def_1199, type, vEBT_VEBT_minNull: vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1200, type, vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1201, type, vEBT_VEBT_set_vebt: vEBT_VEBT > set_nat).
% 20.23/7.37  thf(func_def_1202, type, vEBT_vebt_member: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1203, type, vEBT_vebt_member_rel: produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o).
% 20.23/7.37  thf(func_def_1204, type, vEBT_VEBT_add: option_nat > option_nat > option_nat).
% 20.23/7.37  thf(func_def_1205, type, vEBT_VEBT_greater: option_nat > option_nat > $o).
% 20.23/7.37  thf(func_def_1206, type, vEBT_VEBT_less: option_nat > option_nat > $o).
% 20.23/7.37  thf(func_def_1207, type, vEBT_VEBT_lesseq: option_nat > option_nat > $o).
% 20.23/7.37  thf(func_def_1208, type, vEBT_VEBT_max_in_set: set_nat > nat > $o).
% 20.23/7.37  thf(func_def_1209, type, vEBT_VEBT_min_in_set: set_nat > nat > $o).
% 20.23/7.37  thf(func_def_1210, type, vEBT_VEBT_mul: option_nat > option_nat > option_nat).
% 20.23/7.37  thf(func_def_1211, type, vEBT_V4262088993061758097ft_nat: (nat > nat > nat) > option_nat > option_nat > option_nat).
% 20.23/7.37  thf(func_def_1212, type, vEBT_V819420779217536731ft_num: (num > num > num) > option_num > option_num > option_num).
% 20.23/7.37  thf(func_def_1213, type, vEBT_V1502963449132264192at_nat: (product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat).
% 20.23/7.37  thf(func_def_1214, type, vEBT_VEBT_power: option_nat > option_nat > option_nat).
% 20.23/7.37  thf(func_def_1215, type, vEBT_vebt_maxt: vEBT_VEBT > option_nat).
% 20.23/7.37  thf(func_def_1216, type, vEBT_vebt_maxt_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1217, type, vEBT_vebt_mint: vEBT_VEBT > option_nat).
% 20.23/7.37  thf(func_def_1218, type, vEBT_vebt_mint_rel: vEBT_VEBT > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1219, type, accp_list_nat: (list_nat > list_nat > $o) > list_nat > $o).
% 20.23/7.37  thf(func_def_1220, type, accp_nat: (nat > nat > $o) > nat > $o).
% 20.23/7.37  thf(func_def_1221, type, accp_P1096762738010456898nt_int: (product_prod_int_int > product_prod_int_int > $o) > product_prod_int_int > $o).
% 20.23/7.37  thf(func_def_1222, type, accp_P4275260045618599050at_nat: (product_prod_nat_nat > product_prod_nat_nat > $o) > product_prod_nat_nat > $o).
% 20.23/7.37  thf(func_def_1223, type, accp_P2887432264394892906BT_nat: (produc9072475918466114483BT_nat > produc9072475918466114483BT_nat > $o) > produc9072475918466114483BT_nat > $o).
% 20.23/7.37  thf(func_def_1224, type, accp_VEBT_VEBT: (vEBT_VEBT > vEBT_VEBT > $o) > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1225, type, max_ex8135407076693332796at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_1226, type, min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat).
% 20.23/7.37  thf(func_def_1227, type, pred_nat: set_Pr1261947904930325089at_nat).
% 20.23/7.37  thf(func_def_1228, type, wf_int: set_Pr958786334691620121nt_int > $o).
% 20.23/7.37  thf(func_def_1229, type, wf_nat: set_Pr1261947904930325089at_nat > $o).
% 20.23/7.37  thf(func_def_1230, type, fChoice_real: (real > $o) > real).
% 20.23/7.37  thf(func_def_1231, type, member1535805642427569193e_term: (code_integer > option6357759511663192854e_term) > set_Co9149898834107579976e_term > $o).
% 20.23/7.37  thf(func_def_1232, type, member8845023287901829240e_term: (int > option6357759511663192854e_term) > set_in3461395444621081367e_term > $o).
% 20.23/7.37  thf(func_def_1233, type, member4242434998011752849e_term: (produc6241069584506657477e_term > option6357759511663192854e_term) > set_Pr7604974323444597168e_term > $o).
% 20.23/7.37  thf(func_def_1234, type, member3222579708246209666e_term: (produc8551481072490612790e_term > option6357759511663192854e_term) > set_Pr3642885161833720865e_term > $o).
% 20.23/7.37  thf(func_def_1235, type, member_o: $o > set_o > $o).
% 20.23/7.37  thf(func_def_1236, type, member_complex: complex > set_complex > $o).
% 20.23/7.37  thf(func_def_1237, type, member_int: int > set_int > $o).
% 20.23/7.37  thf(func_def_1238, type, member_list_o: list_o > set_list_o > $o).
% 20.23/7.37  thf(func_def_1239, type, member_list_int: list_int > set_list_int > $o).
% 20.23/7.37  thf(func_def_1240, type, member_list_nat: list_nat > set_list_nat > $o).
% 20.23/7.37  thf(func_def_1241, type, member2936631157270082147T_VEBT: list_VEBT_VEBT > set_list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1242, type, member_nat: nat > set_nat > $o).
% 20.23/7.37  thf(func_def_1243, type, member_num: num > set_num > $o).
% 20.23/7.37  thf(func_def_1244, type, member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o).
% 20.23/7.37  thf(func_def_1245, type, member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o).
% 20.23/7.37  thf(func_def_1246, type, member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o).
% 20.23/7.37  thf(func_def_1247, type, member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o).
% 20.23/7.37  thf(func_def_1248, type, member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o).
% 20.23/7.37  thf(func_def_1249, type, member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o).
% 20.23/7.37  thf(func_def_1250, type, member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o).
% 20.23/7.37  thf(func_def_1251, type, member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o).
% 20.23/7.37  thf(func_def_1252, type, member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o).
% 20.23/7.37  thf(func_def_1253, type, member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $o).
% 20.23/7.37  thf(func_def_1254, type, member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o).
% 20.23/7.37  thf(func_def_1255, type, member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o).
% 20.23/7.37  thf(func_def_1256, type, member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o).
% 20.23/7.37  thf(func_def_1257, type, member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o).
% 20.23/7.37  thf(func_def_1258, type, member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o).
% 20.23/7.37  thf(func_def_1259, type, member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o).
% 20.23/7.37  thf(func_def_1260, type, member4439316823752958928T_VEBT: produc9211091688327510695T_VEBT > set_Pr1916528119006554503T_VEBT > $o).
% 20.23/7.37  thf(func_def_1261, type, member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o).
% 20.23/7.37  thf(func_def_1262, type, member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o).
% 20.23/7.37  thf(func_def_1263, type, member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o).
% 20.23/7.37  thf(func_def_1264, type, member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o).
% 20.23/7.37  thf(func_def_1265, type, member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o).
% 20.23/7.37  thf(func_def_1266, type, member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o).
% 20.23/7.37  thf(func_def_1267, type, member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o).
% 20.23/7.37  thf(func_def_1268, type, member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o).
% 20.23/7.37  thf(func_def_1269, type, member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o).
% 20.23/7.37  thf(func_def_1270, type, member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o).
% 20.23/7.37  thf(func_def_1271, type, member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $o).
% 20.23/7.37  thf(func_def_1272, type, member373505688050248522BT_nat: produc9072475918466114483BT_nat > set_Pr7556676689462069481BT_nat > $o).
% 20.23/7.37  thf(func_def_1273, type, member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $o).
% 20.23/7.37  thf(func_def_1274, type, member568628332442017744T_VEBT: produc8243902056947475879T_VEBT > set_Pr6192946355708809607T_VEBT > $o).
% 20.23/7.37  thf(func_def_1275, type, member_rat: rat > set_rat > $o).
% 20.23/7.37  thf(func_def_1276, type, member_real: real > set_real > $o).
% 20.23/7.37  thf(func_def_1277, type, member_set_int: set_int > set_set_int > $o).
% 20.23/7.37  thf(func_def_1278, type, member_set_nat: set_nat > set_set_nat > $o).
% 20.23/7.37  thf(func_def_1279, type, member_VEBT_VEBT: vEBT_VEBT > set_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1280, type, deg: nat).
% 20.23/7.37  thf(func_def_1281, type, m: nat).
% 20.23/7.37  thf(func_def_1282, type, ma: nat).
% 20.23/7.37  thf(func_def_1283, type, maxs: nat).
% 20.23/7.37  thf(func_def_1284, type, mi: nat).
% 20.23/7.37  thf(func_def_1285, type, n: nat).
% 20.23/7.37  thf(func_def_1286, type, summary: vEBT_VEBT).
% 20.23/7.37  thf(func_def_1288, type, treeList: list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1307, type, sP0: (nat > vEBT_VEBT > $o) > nat > $o).
% 20.23/7.37  thf(func_def_1308, type, sP1: nat > (nat > vEBT_VEBT > $o) > $o).
% 20.23/7.37  thf(func_def_1309, type, sP2: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1310, type, sP3: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1311, type, sP4: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1312, type, sP5: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1313, type, sP6: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1314, type, sP7: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1315, type, sP8: nat > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1316, type, sP9: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1317, type, sP10: vEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1318, type, sP11: nat > vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1319, type, sP12: nat > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1320, type, sP13: nat > $o).
% 20.23/7.37  thf(func_def_1321, type, sP14: list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1322, type, sP15: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1323, type, sP16: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 20.23/7.37  thf(func_def_1324, type, sP17: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 20.23/7.37  thf(func_def_1325, type, sP18: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 20.23/7.37  thf(func_def_1326, type, sP19: nat > nat > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1327, type, sP20: nat > vEBT_VEBT > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1328, type, sP21: nat > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1329, type, sP22: list_VEBT_VEBT > nat > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1330, type, sP23: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1331, type, sP24: nat > nat > list_VEBT_VEBT > nat > nat > $o).
% 20.23/7.37  thf(func_def_1332, type, sP25: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1333, type, sP26: list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1334, type, sP27: nat > list_VEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1335, type, sP28: nat > list_VEBT_VEBT > nat > nat > nat > $o).
% 20.23/7.37  thf(func_def_1336, type, sP29: vEBT_VEBT > list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1337, type, sP30: list_VEBT_VEBT > nat > $o).
% 20.23/7.37  thf(func_def_1338, type, sK31: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1339, type, sK32: list_VEBT_VEBT > (vEBT_VEBT > $o) > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1340, type, sK33: nat > (nat > vEBT_VEBT > $o) > nat).
% 20.23/7.37  thf(func_def_1341, type, sK34: nat > (nat > vEBT_VEBT > $o) > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1342, type, sK35: list_VEBT_VEBT > (nat > vEBT_VEBT > $o) > nat > nat).
% 20.23/7.37  thf(func_def_1343, type, sK36: (nat > vEBT_VEBT > $o) > nat > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1344, type, sK37: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1345, type, sK38: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1346, type, sK39: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1347, type, sK40: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1348, type, sK41: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1349, type, sK42: nat > vEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1350, type, sK43: nat > vEBT_VEBT > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1351, type, sK44: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1352, type, sK45: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1353, type, sK46: vEBT_VEBT > nat > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1354, type, sK47: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1355, type, sK48: vEBT_VEBT > nat > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1356, type, sK49: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1357, type, sK50: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1358, type, sK51: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1359, type, sK52: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1360, type, sK53: vEBT_VEBT > nat > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1361, type, sK54: vEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1362, type, sK55: vEBT_VEBT > nat > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1363, type, sK56: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1364, type, sK57: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1365, type, sK58: nat > vEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1366, type, sK59: nat > vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1367, type, sK60: nat > vEBT_VEBT > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1368, type, sK61: nat > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1369, type, sK62: nat > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1370, type, sK63: vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1371, type, sK64: vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1372, type, sK65: nat > list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1373, type, sK66: list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1374, type, sK67: list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1375, type, sK68: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1376, type, sK69: nat).
% 20.23/7.37  thf(func_def_1377, type, sK70: (vEBT_VEBT > $o) > list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1378, type, sK71: (vEBT_VEBT > $o) > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1379, type, sK72: vEBT_VEBT > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1380, type, sK73: list_VEBT_VEBT > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1381, type, sK74: nat > list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1382, type, sK75: nat > vEBT_VEBT > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1383, type, sK76: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 20.23/7.37  thf(func_def_1384, type, sK77: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 20.23/7.37  thf(func_def_1385, type, sK78: list_VEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1386, type, sK79: list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1387, type, sK80: list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1388, type, sK81: (list_VEBT_VEBT > $o) > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1389, type, sK82: vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1390, type, sK83: vEBT_VEBT > $o).
% 20.23/7.37  thf(func_def_1391, type, sK84: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1392, type, sK85: vEBT_VEBT > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1393, type, sK86: vEBT_VEBT > option4927543243414619207at_nat).
% 20.23/7.37  thf(func_def_1394, type, sK87: vEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1395, type, sK88: vEBT_VEBT > list_VEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1396, type, sK89: nat > list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1397, type, sK90: list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1398, type, sK91: list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1399, type, sK92: list_VEBT_VEBT > nat > nat).
% 20.23/7.37  thf(func_def_1400, type, sK93: nat > nat > list_VEBT_VEBT > nat > nat > nat).
% 20.23/7.37  thf(func_def_1401, type, sK94: list_VEBT_VEBT > nat > nat > nat > nat > nat).
% 20.23/7.37  thf(func_def_1402, type, sK95: (vEBT_VEBT > $o) > list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1403, type, sK96: nat > nat).
% 20.23/7.37  thf(func_def_1404, type, sK97: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1405, type, sK98: list_VEBT_VEBT > nat > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1406, type, sK99: vEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1407, type, sK100: list_VEBT_VEBT > vEBT_VEBT).
% 20.23/7.37  thf(func_def_1408, type, sK101: list_VEBT_VEBT > nat).
% 20.23/7.37  thf(func_def_1409, type, sK102: nat > list_VEBT_VEBT).
% 20.23/7.37  thf(func_def_1411, type, ph104: !>[X0: $tType]:(X0)).
% 20.23/7.37  thf(f24734,plain,(
% 20.23/7.37    $false),
% 20.23/7.37    inference(avatar_sat_refutation,[status(thm)],[f24679,f24683,f24701,f24718,f24730,f24733])).
% 20.23/7.37  thf(f24733,plain,(
% 20.23/7.37    ~spl103_1 | ~spl103_8),
% 20.23/7.37    inference(avatar_contradiction_clause,[status(thm)],[f24732])).
% 20.23/7.37  thf(f24732,plain,(
% 20.23/7.37    $false | (~spl103_1 | ~spl103_8)),
% 20.23/7.37    inference(trivial_inequality_removal,[status(thm)],[f24731])).
% 20.23/7.37  thf(f24731,plain,(
% 20.23/7.37    ($false = $true) | (~spl103_1 | ~spl103_8)),
% 20.23/7.37    inference(forward_demodulation,[status(thm)],[f24700,f24675])).
% 20.23/7.37  thf(f24675,plain,(
% 20.23/7.37    ( ! [X0 : nat] : (($false = (vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0))) ) | ~spl103_1),
% 20.23/7.37    inference(avatar_component_clause,[status(thm)],[f24674])).
% 20.23/7.37  thf(f24674,definition,(
% 20.23/7.37    spl103_1 <=> ! [X0 : nat] : ($false = (vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0))),
% 20.23/7.37    introduced(definition,[new_symbols(naming,[spl103_1])],[avatar_definition])).
% 20.23/7.37  thf(f24700,plain,(
% 20.23/7.37    ((vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true) | ~spl103_8),
% 20.23/7.37    inference(avatar_component_clause,[status(thm)],[f24699])).
% 20.23/7.37  thf(f24699,definition,(
% 20.23/7.37    spl103_8 <=> ((vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    introduced(definition,[new_symbols(naming,[spl103_8])],[avatar_definition])).
% 20.23/7.37  thf(f24730,plain,(
% 20.23/7.37    ~spl103_3 | ~spl103_7),
% 20.23/7.37    inference(avatar_contradiction_clause,[status(thm)],[f24729])).
% 20.23/7.37  thf(f24729,plain,(
% 20.23/7.37    $false | (~spl103_3 | ~spl103_7)),
% 20.23/7.37    inference(trivial_inequality_removal,[status(thm)],[f24728])).
% 20.23/7.37  thf(f24728,plain,(
% 20.23/7.37    ($false = $true) | (~spl103_3 | ~spl103_7)),
% 20.23/7.37    inference(backward_demodulation,[status(thm)],[f24697,f24682])).
% 20.23/7.37  thf(f24682,plain,(
% 20.23/7.37    ( ! [X0 : nat] : (($false = (vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0))) ) | ~spl103_3),
% 20.23/7.37    inference(avatar_component_clause,[status(thm)],[f24681])).
% 20.23/7.37  thf(f24681,definition,(
% 20.23/7.37    spl103_3 <=> ! [X0 : nat] : ($false = (vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0))),
% 20.23/7.37    introduced(definition,[new_symbols(naming,[spl103_3])],[avatar_definition])).
% 20.23/7.37  thf(f24697,plain,(
% 20.23/7.37    ((vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true) | ~spl103_7),
% 20.23/7.37    inference(avatar_component_clause,[status(thm)],[f24696])).
% 20.23/7.37  thf(f24696,definition,(
% 20.23/7.37    spl103_7 <=> ((vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    introduced(definition,[new_symbols(naming,[spl103_7])],[avatar_definition])).
% 20.23/7.37  thf(f24718,plain,(
% 20.23/7.37    ~spl103_2),
% 20.23/7.37    inference(avatar_contradiction_clause,[status(thm)],[f24717])).
% 20.23/7.37  thf(f24717,plain,(
% 20.23/7.37    $false | ~spl103_2),
% 20.23/7.37    inference(subsumption_resolution,[status(thm)],[f24678,f24423])).
% 20.23/7.37  thf(f24423,plain,(
% 20.23/7.37    (thesis != $true)),
% 20.23/7.37    inference(cnf_transformation,[status(thm)],[f24024])).
% 20.23/7.37  thf(f24024,plain,(
% 20.23/7.37    (thesis != $true)),
% 20.23/7.37    inference(flattening,[status(thm)],[f21580])).
% 20.23/7.37  thf(f21580,plain,(
% 20.23/7.37    ~(thesis = $true)),
% 20.23/7.37    inference(fool_elimination,[status(thm)],[f21579])).
% 20.23/7.37  thf(f21579,plain,(
% 20.23/7.37    ~thesis),
% 20.23/7.37    inference(rectify,[status(thm)],[f9832])).
% 20.23/7.37  thf(f9832,negated_conjecture,(
% 20.23/7.37    ~thesis),
% 20.23/7.37    inference(negated_conjecture,[status(cth)],[f9831])).
% 20.23/7.37  thf(f9831,conjecture,(
% 20.23/7.37    thesis),
% 20.23/7.37    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',conj_1)).
% 20.23/7.37  thf(f24678,plain,(
% 20.23/7.37    (thesis = $true) | ~spl103_2),
% 20.23/7.37    inference(avatar_component_clause,[status(thm)],[f24677])).
% 20.23/7.37  thf(f24677,definition,(
% 20.23/7.37    spl103_2 <=> (thesis = $true)),
% 20.23/7.37    introduced(definition,[new_symbols(naming,[spl103_2])],[avatar_definition])).
% 20.23/7.37  thf(f24701,plain,(
% 20.23/7.37    spl103_7 | spl103_8),
% 20.23/7.37    inference(avatar_split_clause,[status(thm)],[f24552,f24699,f24696])).
% 20.23/7.37  thf(f24552,plain,(
% 20.23/7.37    ((vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true) | ((vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    inference(binary_proxy_clausification,[status(thm)],[f24551])).
% 20.23/7.37  thf(f24551,plain,(
% 20.23/7.37    (((vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) | (vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69)) = $true)),
% 20.23/7.37    inference(beta_eta_normalization,[status(thm)],[f24458])).
% 20.23/7.37  thf(f24458,plain,(
% 20.23/7.37    (((^[Y0 : vEBT_VEBT]: ((^[Y1 : nat]: ((vEBT_V5719532721284313246member @ Y0 @ Y1) | (vEBT_VEBT_membermima @ Y0 @ Y1))))) @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    inference(definition_unfolding,[status(thm)],[f24351,f24401])).
% 20.23/7.37  thf(f24401,plain,(
% 20.23/7.37    (vEBT_V8194947554948674370ptions = (^[Y0 : vEBT_VEBT]: ((^[Y1 : nat]: ((vEBT_V5719532721284313246member @ Y0 @ Y1) | (vEBT_VEBT_membermima @ Y0 @ Y1))))))),
% 20.23/7.37    inference(cnf_transformation,[status(thm)],[f9895])).
% 20.23/7.37  thf(f9895,plain,(
% 20.23/7.37    (vEBT_V8194947554948674370ptions = (^[Y0 : vEBT_VEBT]: ((^[Y1 : nat]: ((vEBT_V5719532721284313246member @ Y0 @ Y1) | (vEBT_VEBT_membermima @ Y0 @ Y1))))))),
% 20.23/7.37    inference(fool_elimination,[status(thm)],[f9894])).
% 20.23/7.37  thf(f9894,plain,(
% 20.23/7.37    (vEBT_V8194947554948674370ptions = (^[X0 : vEBT_VEBT, X1 : nat] : ((vEBT_VEBT_membermima @ X0 @ X1) | (vEBT_V5719532721284313246member @ X0 @ X1))))),
% 20.23/7.37    inference(rectify,[status(thm)],[f4])).
% 20.23/7.37  thf(f4,axiom,(
% 20.23/7.37    (vEBT_V8194947554948674370ptions = (^[X2 : vEBT_VEBT, X3 : nat] : ((vEBT_VEBT_membermima @ X2 @ X3) | (vEBT_V5719532721284313246member @ X2 @ X3))))),
% 20.23/7.37    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',fact_3_both__member__options__def)).
% 20.23/7.37  thf(f24351,plain,(
% 20.23/7.37    ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    inference(cnf_transformation,[status(thm)],[f24174])).
% 20.23/7.37  thf(f24174,plain,(
% 20.23/7.37    ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    inference(skolemisation,[status(esa),new_symbols(skolem,[sK69])],[f16117,f24173])).
% 20.23/7.37  thf(f24173,plain,(
% 20.23/7.37    ? [X0 : nat] : ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) = $true) => ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ sK69) = $true)),
% 20.23/7.37    introduced(definition,[],[choice_axiom])).
% 20.23/7.37  thf(f16117,plain,(
% 20.23/7.37    ? [X0 : nat] : ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) = $true)),
% 20.23/7.37    inference(fool_elimination,[status(thm)],[f16116])).
% 20.23/7.37  thf(f16116,plain,(
% 20.23/7.37    ? [X0 : nat] : (vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0)),
% 20.23/7.37    inference(rectify,[status(thm)],[f1])).
% 20.23/7.37  thf(f1,axiom,(
% 20.23/7.37    ? [X0 : nat] : (vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0)),
% 20.23/7.37    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062)).
% 20.23/7.37  thf(f24683,plain,(
% 20.23/7.37    spl103_3 | spl103_2),
% 20.23/7.37    inference(avatar_split_clause,[status(thm)],[f24637,f24677,f24681])).
% 20.23/7.37  thf(f24637,plain,(
% 20.23/7.37    ( ! [X0 : nat] : (($false = (vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0)) | (thesis = $true)) )),
% 20.23/7.37    inference(binary_proxy_clausification,[status(thm)],[f24635])).
% 20.23/7.37  thf(f24635,plain,(
% 20.23/7.37    ( ! [X0 : nat] : ((((vEBT_V5719532721284313246member @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) | (vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0)) != $true) | (thesis = $true)) )),
% 20.23/7.37    inference(beta_eta_normalization,[status(thm)],[f24462])).
% 20.23/7.37  thf(f24462,plain,(
% 20.23/7.37    ( ! [X0 : nat] : ((thesis = $true) | (((^[Y0 : vEBT_VEBT]: ((^[Y1 : nat]: ((vEBT_V5719532721284313246member @ Y0 @ Y1) | (vEBT_VEBT_membermima @ Y0 @ Y1))))) @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) != $true)) )),
% 20.23/7.37    inference(definition_unfolding,[status(thm)],[f24370,f24401])).
% 20.23/7.37  thf(f24370,plain,(
% 20.23/7.37    ( ! [X0 : nat] : ((thesis = $true) | ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) != $true)) )),
% 20.23/7.37    inference(cnf_transformation,[status(thm)],[f24047])).
% 20.23/7.37  thf(f24047,plain,(
% 20.23/7.37    ! [X0 : nat] : ((thesis = $true) | ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) != $true))),
% 20.23/7.37    inference(ennf_transformation,[status(thm)],[f15842])).
% 20.23/7.37  thf(f15842,plain,(
% 20.23/7.37    ! [X0 : nat] : (((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) = $true) => (thesis = $true))),
% 20.23/7.37    inference(fool_elimination,[status(thm)],[f15841])).
% 20.23/7.37  thf(f15841,plain,(
% 20.23/7.37    ! [X0 : nat] : ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0) => thesis)),
% 20.23/7.37    inference(rectify,[status(thm)],[f9830])).
% 20.23/7.37  thf(f9830,axiom,(
% 20.23/7.37    ! [X4 : nat] : ((vEBT_V8194947554948674370ptions @ (nth_VEBT_VEBT @ treeList @ maxs) @ X4) => thesis)),
% 20.23/7.37    file('/export/starexec/sandbox/tmp/tmp.1f8NBhQv2G/DTF2THF_22391.p',conj_0)).
% 20.23/7.37  thf(f24679,plain,(
% 20.23/7.37    spl103_1 | spl103_2),
% 20.23/7.37    inference(avatar_split_clause,[status(thm)],[f24636,f24677,f24674])).
% 20.23/7.37  thf(f24636,plain,(
% 20.23/7.37    ( ! [X0 : nat] : ((thesis = $true) | ($false = (vEBT_VEBT_membermima @ (nth_VEBT_VEBT @ treeList @ maxs) @ X0))) )),
% 20.23/7.37    inference(binary_proxy_clausification,[status(thm)],[f24635])).
% 20.23/7.37  % SZS output end Proof for DTF2THF_22391
% 20.23/7.37  % (22661)------------------------------
% 20.23/7.37  % (22661)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 20.23/7.37  % (22661)Termination reason: Refutation
% 20.23/7.37  
% 20.23/7.37  % (22661)Memory used [KB]: 32366
% 20.23/7.37  % (22661)Time elapsed: 0.527 s
% 20.23/7.37  % (22661)Instructions burned: 1624 (million)
% 20.23/7.37  % (22661)------------------------------
% 20.23/7.37  % (22661)------------------------------
% 20.23/7.37  % (22498)Success in time 2.123 s
%------------------------------------------------------------------------------